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<h2>HL Paper 2</h2><div class="specification">
<p>The function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> is defined by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = \frac{{2\,{\text{ln}}\,x + 1}}{{x - 3}}">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mrow>
<mi>x</mi>
<mo>−<!-- − --></mo>
<mn>3</mn>
</mrow>
</mfrac>
</math></span>, 0 < <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> < 3.</p>
</div>
<div class="specification">
<p>Draw a set of axes showing <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span> values between −3 and 3. On these axes</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, or otherwise, find the coordinates of the point of inflexion on the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f\left( x \right)"> <mi>y</mi> <mo>=</mo> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>sketch the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f\left( x \right)"> <mi>y</mi> <mo>=</mo> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math></span>, showing clearly any axis intercepts and giving the equations of any asymptotes.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>sketch the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = {f^{ - 1}}\left( x \right)"> <mi>y</mi> <mo>=</mo> <mrow> <msup> <mi>f</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math></span>, showing clearly any axis intercepts and giving the equations of any asymptotes.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, or otherwise, solve the inequality <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) > {f^{ - 1}}\left( x \right)"> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>></mo> <mrow> <msup> <mi>f</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>finding turning point of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f'\left( x \right)"> <mi>y</mi> <mo>=</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math></span> or finding root of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f''\left( x \right)"> <mi>y</mi> <mo>=</mo> <msup> <mi>f</mi> <mo>″</mo> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math></span> <em><strong> (M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 0.899"> <mi>x</mi> <mo>=</mo> <mn>0.899</mn> </math></span> <em><strong> A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f\left( {0.899048 \ldots } \right) = - 0.375"> <mi>y</mi> <mo>=</mo> <mi>f</mi> <mrow> <mo>(</mo> <mrow> <mn>0.899048</mn> <mo>…</mo> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mo>−</mo> <mn>0.375</mn> </math></span> <em><strong>(M1)A1</strong></em></p>
<p>(0.899, −0.375)</p>
<p><strong>Note:</strong> Do not accept <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 0.9"> <mi>x</mi> <mo>=</mo> <mn>0.9</mn> </math></span>. Accept <em>y</em>-coordinates rounding to −0.37 or −0.375 but not −0.38.<br> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAcsAAAGLCAYAAACoWrkrAAAgAElEQVR4AeydCVhU1fvHv+GwyioiEiERESGigIqISIgIrogSGn8zM/NnRuaSmZkpmZmamWZmRua+h6gZERkRIW5oboiAiIiIiIgIItIE/+dcZNhmYIaZgbkz730emHvP8p73fM6dee8595z3PFVdXV0NOogAESACRIAIEAGJBLQkxlAEESACGkkgOzsLMTExGll3qjQRkESAjKUkMhROBDSQwNatW3EkNg4uLs4YMXSYBhKgKhMB8QSeomFY8WAolAhoGoGiomJ07doFZY8eQVcgwKw5c2Bna4uZM2dqGgqqLxFoQoB6lk2QUAAR4BeBRw8f4tKlyyKlU86miM5lOYmM3ABfX1/OULJ8vd16ITJyoywiKC0RUFsCArWtGVWMCGgAgaqqKsTFxeLtt99BYnISNm5Yj4MHD+Pi5Ssioxd3OA6F/xZCr0IAoQ4grAT0IICZVUf4+Q0VUbpz5y6ec3AUXVtaWSE3J1d0TSdEQJMJkLHU5NanuvOegJaWFkaPCcHfJ09j0qTJOBR9CCtXrm5QL8POBsADMwiNBIBQCIERINQSQKBj0CBdVWUljIwNRWHCKpauSnRNJ0RAkwmQsdTk1qe6qw2Bof7+KMjPg7m5WZM6eXl5NwkTF2BgaoqC/BxRVHFuETqbmoqu6YQIaDIBMpaa3PpUd7UhkJObgz279mDr5q1gvc36x7KlS3Hnzj0ItAAhtCCoroLwqSo8a2uLWbNni5K+HByEYYNHA5tqgtLzUjF6TKgonk6IgCYTIGOpya1Pdec9AbYeUqAlgJ+vL1xcnBAdtR/mlpbw9fEV1W3BwoWi8+ZO3Pt4oJeXK86mnIKzqzsS4pOwc/vW5rJQHBHQGAINH0E1ptpUUSKgHgSOHo2Hoakx7OzssXzlaty796CBoZS1lod++gnHjp3Ed6vXYevmTbCxsZVVBKUnAmpJgNZZqmWzUqWIABEgAkRAkQSoZ6lImiSLCBABIkAE1JIAGUu1bFaqFBEgAkSACCiSABlLRdIkWUSACBABIqCWBMhYqmWzUqWIABEgAkRAkQTIWCqSJskiAkSACBABtSRAxlItm5UqpWkEHguFyLySiejoKFHVN2/eBBZOBxEgAvITIGMpP0OSQATanUDlw0fIvJqJceNeAduFhB2xcUdx9Ldf2l03UoAIqAMBMpbq0IpUB40nYGRihOEjh8PU1JhzbceADPXzg3/gCI1nQwCIgCIIkLs7RVAkGURARQhYWViioDAPOnoCmFpYirbpUhH1SA0iwFsC1LPkbdOR4kSgKQE7ZwfkFRTg4MGDGBMc1DQBhRABJRE4mpCA0DEh6NH9BezcvlNJpbSfWDKW7ceeSiYCCifgYOuIyA0/4NX/e1XhskkgEWiOQGdTY+yPjsLvf/yFTZs3NpeUl3FkLHnZbKQ0ERBPoEvnTli/fg3YO8zag82IjYiIEE38qQ2nTyJQn0Bubg7WrV3HzaDOyspEVFR0/egWz11d3bk0WgIBwsNntpiebwnonSXfWoz0JQJiCDCDePJEMoYOHcrtQFI/yWsTJmDfvn1IPnkccb/+hujowzAzMYCvn3/9ZHSu4QTYDjPpman45quVMDa1hJ5enXk4e+EUfvxuB3QEQBXbF7UKEGhp4YMPP4KVpYWIHNsybuGihZgZXrdPKrs3P5g1Cyu+/BK6urqitHw7oV1H+NZipC8RaERgzbov8Xd8EmbOmAkfv7p9LFmy6dPexr59u3Hv/gN0MjWG/9AA2NnZopNhJ8xbML+RJLrUdAInTqVgwby5iE9IaDWKy5cvoV+/frhbXMJNMGPGUk9bG/fv34eJiUmr5bZ3xrpHh/bWhMonAkSgVQSmT58J9qcraPh1XrlyObZs24STx0+id/9++PnQzxgTGor4ykq8/+EHrSqLMqk3Aa0qIW7eut2kkufOncNPP+1pEj5zxnuwqNezZAm6d++Bfn37ourxY6DRPdlEQCsCjsbFwdHJGTY21q3I3fosDb9drZdDOYkAEWgnAo2NJFNj3YYN+Oijj/Hnn3+ip6srIAQ3PJuYEA8PLy+cOX1arLZJSQlIS81G37694Ope8w5KbEIKVCsC2dlZOJF8AmbmFnBz64Xt27fDy9sL9nb2XD1dXV3B/iQdrPd4cP9e+PoHID8/D8ODgqHfsSOXXFtLCwEBftywraT80ob/GheHsSGjsX/LbjKW0kKjdESACEgmkJ+TjYOHDsHb25tLJBBUQYgqODo5YeWyZfhq1VpUVVVBS6tujl9U1EEIq/6FS28XlJWXSxZOMWpHQE/PgLsXhg4NQLduT+Pe3XsiQylNZdkDW2VVFbZv3QxPVx/MmTVLlI3dY7/99ofourUnbGh47OjR2PbjFowMCW6tmFbno3eWrUZHGYkAfwhoa+vixrXrsLKxQlJiIl5/6w1cvXxVVIHHjx+jd283XLp0WRRGJ0RAlQiw3uufR+PBDHp7HHWPle1ROpVJBIhA2xAQsJHYKq4seycHlBSUiMr9OTYWo4KDUQUtxB09KgqnEyKgKAJsFEPeg/Ve28tQMt3JWMrbgpSfCPCBgLAKgidfd0M9Qwh0INqRZNTQoXjRwREfzn4PAf60nIQPzcknHR9VVqJDhw4oKal7QJNW/19iYjBi2DBpkys1HRlLpeIl4URANQgIBAJRz1Kgrw9hBVDJZis+OeLjYzFu8iu1l/RJBBRGoP57cVmExsbE4pXxoXj99Tdkyaa0tGQslYaWBBMB1SEgFFaKlNEXCKAlAB7cu8+FZWdlwdS0M3S19EVp6IQItDeBt999B6tXf4PQ8aHtrQpXvkouHUlKSkJJSRH8fP1F049VghYpQQT4SkAgEA3DsipYdrFEelY6rG2scTgmBuNDVeMHia94SW/JBNi7xl8OHYLek6UkklM2jEk9f16lfv9Vrme5Zs1XSEpKxI8/bMNLgYMb0qMrIkAEWkVAR1izdKQ2s6OrI7LS07nLo7FxGPfKuNoo+iQCCicwPCioidMMcYVkpmeKgmvXaYoC2vlE5XqWI4YNh4OjI4fFzKzOGXQ7c6LiiQCvCVQKtBr0LO1tHPGgpBylJaV4+tmusLS04nX9SHn+E/g5JgZBI0YgIy0DXa26Ys+eHUg4lYydm7ZzmwCMGjUKR+PjG1R0w/r1yMnLaxDGLgICAuDnW+f6MbegAF9EfIJBQwPh6GCPN978H5YsjUCAn/TLUOReZ7l//26kp2dh4cKFnMIbNqzHq//3WoNdDzLT03Hh0iVwbkRQZ58tLS1Fi6Yb1zY3Nw8xsUcwbeq0xlF0TQSIgIwEtLW1ce3adZHXk6XLluHurVtwdHHAuJdfg7m5mYwSKTkRkI4AWx/5+dKl+HDhQom9SzbrNXj0aPzw/Y+YNHmiSLCZkQmKS2tm0a5atQpz584Vxcl6cuHsWXzyyScYO+4V+Pn7N3AAL42sOsslTWoxacaEjMf4MWPA3jMybyH5eXnIzs1GT5OeotRdn34GhgbGouvaEx0DvdrTBp+JiUmYEDYeAUOHk7FsQIYuiEArCTTqWerpCPDw0b+YNnVGAy8+rZRO2YhAswSYkZo9ezZ0JThSFz7+F99++30DQ8kEWllbcr3K+w8e4EWn7k3KiImNRWFBTc+ynjMquLr3hYtznQ1iGXu4uyMmLhY79u+Hvo5OE1ktBchtLNk2LXZOTkhJSeGMpa29A3r2aKhkdlYmTp8500QXaxsbDA1o2g328fFGYlIiPD09ERMbguFDhzbJSwFEgAhIT6DxO0ttdEAlKslQSo+QUspJoLklJKPHjBYr3dTUFBfT05H6zz+YPGVKkzSy2Iac7Cz0798fF8+ehYenZxNZLQXIbSxZAXoCAbf9SnJSMnx9fZqUyRw5c86cm8RIDrCzs0docCgKxIxHS85FMUSACIgj0Pid5bP2zyL6sGyb+4qTS2FEoCUCzJG6g50d54u4ftpDhw9zl6ODguoHNzi3tLbBtsjv8fnyLxqEy3KRn5+P7MwspGVmcu8yd+zag7KKigbvNKWRp5DZsGzBc05uLu4Vy+Z8V5yC9b085N3Ow/9NmiQuGYURASLwhAB7J9SiO7FGs2FfcOiBa7duEkMioHQCrEeZce1ag70smaFk7yjvFzXv1cfB0QEzZ81uMAdGVoUrKsqRmJyIKVMmc3u+Pt2lCwY82WBAFllyT/Bhhe3fvxdHjhzG1q07ZSm7SVo2M2/YiGGwtrGCl5cvgkY23fW9SSYKIAIaTID96Lw7YwZu3bqFA4cOYdTw4WJpaOvr4kZGjSP12gSW5ha4mpcLIz3xcwdq09EnEVAkgazsLDz/3PPYt2sPQsPGSxR9KiWFG7WUdVRSokA5IxQyDHsz56bchpLVw8jEiJsoJGedKDsR0AgCbKcQNlEhJycHl1LS0HegKx49qnNhVx9C43eWLE6gp4WKklIylvVB0bnSCbA9Ms9fPN9kbkttwadOncDcOXOwbNVqePTpUxvc7p9yGUsfbx8Ejw9F+Ix32r0ipAAR0DQCurq6CHgy+c3B/Vm89tprEhE0fmfJEuoYGOD+g7ImO91LFEIRRKAVBNhrAkP9jrhz6ybMLCw4CY0ngdYX6+Hhid8TEiUuM6mfti3P5TKWGyM3wsnJSSp9GbDKh49EY89szcvNmzXvTDp37QoPDw9OzqOHD/HnH3UbhfYbMFC0Bqx+HuNOnURrNJns2J8PQbuDNiejd//+sHzSKJdSU3EjK4sLNzQ2hs+Thaosz57t24En843ZDFw2qYgdbGZvampqTR49PYSMrxsq2Ll1K9t0njv69vVE9+41DhQuX07H6dMnuHA2LX982IQnqYCoqP0oK6vZTNetVy/RZKfc3BzEH40X6fDKhAmiG2T37p24nXsbVR2A3n3d4OPty81cZC+rd+7cjqdYBID/TX8LRkY1O5Ifjj6MK9lX0EHYAS49XRDwZN+3oqJibNoaCW1hB26TptenvCFiGvtLLNLSUlH1FGD/nAOCx9S8bGdD4hsjN3Dl/If/EBY2HjY2tlyZRw4dQco/Kdy5ra0NJk+umaXG2m7FF3Uv4ieEhYkcTMTGxODEqVNcHktLC0yfHs6ds3Zga7Bqj7HBwSI+bLuo5KQkLsrQ0LDBGqtlEUtQiSo8ePAAQSOHw9evZrcMtlfjT/v2cXnYu/RVa9bUisbSiAjcvXuXux46fDjYHzvOXTiHLd//IEr3+erVoqnla9Z+heuZNfdP/wEDMD4sjEvHhpLWffmVKM/Hn3wqYhq5aRNOJZ9AeUUlvLy9ET59Kpcuv6AQc+fUrRNbtmwJbG1rmG6I3ISkhEQunVsvF8ydV5OOtd2779ZtpOvi7Ij5CxZw6R6WluJ/b9U8qGZmpmHVqtUifbZu34nPPl2MvNx8zg+ssKKSc6TOdoB4bcJEVFZW4PbN28jIuAAHBzsu39bt27Fvzy7W50TXrl2wadMmLpy9Dx01ajQEAi0wH7PhM2aKZqjv378fO3bs4MJZG+3du1+kQ9iEMJSXlUMorMLrr7+G0Ccu9RLiE/DFl19y664FAh1ER0eLZuVOmjwZmZk1noVeDpuAOeE19wn7Tr47611OtkALOBR1SPTDO2fOHJw4VfPdCx4+EvOe8MnKysbESTXfQ5Zn+9a9sLWz4WRERCxB3NFY7tzP1w9Ln9yDhfkFGB06RlTOt9/+gB49apYsLImIwE8H9gEVQN+X+mFT5GYuXUlxMQa+1A+o0AP0hPjuu+/h5VWz6fbyiC+w60BNOkd7R+yPrplUVVpRgQEe7pws6AFsDWFAQM39uHrtV9iyPhIs/OmuXREbV7MQn7WDq2sPrkymw8efLUZoaM1vU2TkRixaFMHFmRh1xJWMur1K+7i7Iy8/n4tbtHAhpj9heig6Cm+9XdfRuZ6bK/r9GTEsEGfPXeDyTAsPR8STdfTJyUkICalzjXj5wgVRO0yeNAmxcXFcngnjJmDV2lXc+Y2sbO7+6NLtGc4xxqXLV2BvV3PfT5s2HTv2bINWJTB8bDD27qx5lXevoAgvvPgcl5/dP8eOHYO7uzt33V7/5DKW0hpKVrmKhw9x++5dOJrUeOXZsWtfzY0HYMjgISJjyYzKO/V22d730z6Ym9d0xX86eJD7YjJ5/fr2FhnLykePuB+UDto1xvKH73+Apb8fx/TA4cPYEhnJnTs97yAylszUbNy4AULmUZqt57G0FBnLf86cwebtWyGsAqwsLBoYy81bt+JBRQXYl8/AQEdkLNPTL2H9xo0QVAmho2fYwFju2bOP8zLB8kydMkVkDLKyc0U6sHzjnvwQM31SU9OQmpbG2XJTY2P4+tTUp6ykDMeOn6y18WCGD6gxllnXMnEy+RjYZkwCIx0E4MmyHC0hjh+ryVMlFGLchFdgjppF6Nk3spF4PJmT9/jRI5GxrEIVjp88CbYNHXueCAkN4Tixfw/LH+L27bvAf1UwrLduStihA+7m3al5mOighcdVj+ryPKrE7bw7QActaNXb247NlGMPBdCuaYeK8pqHCpax8mGFqJzO5hUiWezkzr17qKgUolMXQ+gZGIji2Llp587cNTOW9Q8TMzPRg079PB119UV5WPr6U9xNjI1FcSb16mqgV1cOy6NTr6xOxsawtbMFK/+Zp7uKVDDU04OLS93DZX0dHJ62RsmTOJa39jDQ02mQhz2c1B5a+lqiOENDHQQGDsbdO3fR0ciIcz7wxhtvAjpaMDcywdtPfhRZ3QYM6A9t7Q7IzcnF/ftlteLg7OSE4U8eIIwNDUXhLM/w4XVLvJ7t9owoztHJCQEB/twEI8ak/sGm9bOHGXbY2dUYZHZubW3VQF593iFjRqO4uIRzXWJf70HcxtYWU6ZMBVuqxu5hA7M6JwpBI4PQq1evJ+XUPPCyiy6dO9et066qQqdOplwa9m94gD/snnC2tq5jamhsKCqH3fwWljU9IZYnKDgIzk/ayNzCUiRLz8gIixd/KrqufehmASNfDoSDSzcuzszMXJSmo44OFi9eLLp2cnIWnQ8dMgS2zzzNXXfUq/NixjjVlFPzuO7uXjdEOXJkEJwcnbgHI/YAUv/49ttvUVlZycXVPvCyeE9vH0RF7UWVEFwc8+Fae3z66QpUVD7g4qys67w7OTm7YP/evVx6lq9+O8ydPQ9Tpkzh4jpZdqoVhUsXznHna1atgZ+fL56x7SKKi1gSgVlPHoJYZ6b26GRpjuRTKTUubLQEePYJj9r49vhUyASf9lCcymyeAHsKrf8j1HxqilUHAj169MCPP26Bh0fdj2htvfT19XEt4xqsbOp++BbMmwc2QjN//vzaZPRJBBROIHBYIPr37YsPF0WIeq4KL6QNBNY9SrRBYVRE2xEgQ9l2rFWlJMuuXeD0xK9yY52EaOhIncVbWFmiIK+wcVK6JgIKJfDbr78pVF57CSNj2V7kqVwiICeB9EtpmPfBPAwNGo5HpQ/xbvgM0ZyAxqIF3NvqhqEGBoYoelj3bqthLF0RgdYTSD5xAm4uLiq1xVbra1OTUyFOCeRVgvITASIgOwHHHk5Y/fXX8PX1x/Tp0zF6TM3EFHGShGi46whLw96h3r97R1xyCiMCrSbAXJUGDhmCWDZ5EQCbxLd8+XLus9VCVSAjGUsVaARSgQi0loC9vR2cHB1afIJnPUs2FFv/YNtyZec03d6ofho6JwKyEGAbaowYNgKrV3+NMaNHcVnZJL4PP/wQbCImnw8ahuVz65HuREBKAuJ6lg5OjsgvyJVSAiUjAi0TYDOdv/v2O0yYWLd07t8ns99bdMnYsvh2TUHGsl3xU+FEoG0IiOtZdrW2hrC8EmydIFtWQwcRkJcAWzZTf+lMrTwdtm6O5wf/a8DzBiD1iUBbEBDXs2Tr6oxNjXH73r22UIHKUFMCCYkJWBSxSGLt2H32+N//GjhSl5hYhSPIWKpw45BqREBRBMT1LJlsIxMTlBTR8hFFcdY0OcxQjhoxCjbWdU4n1JUBGUt1bVmqFxGoR0Bcz5JFG5uaI5/2jK1Hik6lJcBcJ4aODuEm80ydOlnabLxNR+8sedt0pDgRkJ6ApJ6lvc2zuHWblo9IT5JS1hJgO95cuXYV5i2872YTe0xMjHDj2nWRH9laGXz6pJ4ln1qLdCUCrSTAepbiDht7axQV0DCsODYUJp5ASUndhs0tGcpaCcznN/O1zedD/DeIzzUi3YkAEWhCQCDBWFpZWCEzJ6dJegogAuIIHD0aj66du6C+wRSXrn5Y7dKR+mF8PCdjycdWI52JgIwEhKgUm8PU2BDlD+6LjaNAIlCfAPPMM2rUMHzz7TcyzWxls2HZDiv1d9mpL5cv5/TOki8tRXoSATkICKDD7SXYWATbfik3l7z4NOZC100JnEhOxmeffY4pU2v2Z22aQnJIRETNXpuSU6h+DBlL1W8j0pAIyE2A9Swbu7tjQi0sbVBIS0fk5qsJAubN0+yt3GgYVhPucqqjxhNgPUtxxzN2XVH24AEePXwoLprCNJzA0fh4nLuQKhcFNht27doveX+PkbGU6zagzESAHwQkvbM00jGCnp4ebt25y4+KkJZtRoBN5hkVOAzHjyXKXeasWXPBZsTy+SBjyefWI92JgJQEJL2zZNltrK2Rk5UupSRKpgkEUlJSMGxYIFZ+vYbb/k0T6txSHemdZUuEKJ4IqAEBSe8sWdXMLSyQX1SsBrWkKiiKQJ8+fbBnzy6EhITKLVJdlo6QsZT7ViABRED1CUh6Z8k0N7XpirL7tHxE9VuxbTVUhKFkGrOlI9XV1W2rvBJKo2FYJUAlkURA1QhIemfJ9LTrbI2iwiJVU5n0aWMCzCm6r58fHguFbVwyP4ojY8mPdiItiYBcBJrrWVpYmiMrJ1su+ZSZ3wTYrNdhgYEIGjmS6wnyuzbK0Z6MpXK4klQioGIEJPcWHOztkZku3/IAFassqSMjgVlvv4vlK1dhzpw5MuZsOTnrqVpYmKO4kN8+iOmdZcttTSmIAO8JSHKkzipm2qkL7t2hd5a8b2Q5KnDm/Bno6urKIUFyVvbO8u7de+RIXTIiiiECREBVCLAtuiQdlhaWKPm3VFI0haspgUupdaMJyjKUDJ26vAOlYVg1/SJQtYhAfQLN9Sy7WFtC+KCC9x5W6teXzpsnwJyi9+3jjvj4o80nVECsujhSJ2OpgJuBRBABVScgaYsupjfbxNfE2Ayp6ZmqXg3STwEEkpKSEDhkCOcU3c/PXwESWxbBHKkbGXVsOaEKpyBjqcKNQ6oRAcURkDzBh5Vh+6ItcjLJi4/ieKuuJObeUFmTeVS31vJrRsZSfoYkgQioPIHmhmGZ8t26Po+75JhA5dtREQoy7zwzZ8xQhCipZLB3lhs3buT9MD8ZS6mamxIRAX4TaG6CD6uZkbk+ysvL+F1J0l4iAebr9VB0tMR4ZUawd5ZvvfUWKsrJkboyOZNsIkAEFECgpZ5l506dUVxI/mEVgFrlRDBDyd5RZmW3j+OJ2tmwQskTslWOmTiFqGcpjgqFEQE1I9DcBB9WVVsbG/yTdl7Nak3VKS4p4QzlRx9/rBSHA9IQ1tZSDzOj0k4JsrLY7DwBnrbtAn0Bv2dSSXNTURoioDwCzU/wsbdzQPoFmuCjPP7tI9nMxAR//PUXXHv2bB8FAGhpaamFI3WVNpZRBw9i4zff4b/yCri49YR9d0c87+CAwQN90c3uWXTsSAa03b4BVDCvCLQ0DGtra4eHFSW8qhMpK5kAG/pk7wrZ0Z6GUrKG/It5qpoHe6fk5OQgMTEB9+/fR2JCIpJSkoDyKjg6OcPH2xvWtjZ4+eVxMDQxEt0g/GsK0pgIKI+Avr42MjKuw8bGWmwhpRUVeN7GFgWFBWLjKZA/BJJPnEBoSAhSzp6DlaUFfxRXcU15YSzFMWRj8UcOHkDunTs4djQBKZdS0FFLHyOCgvHCi/YYGhAAB0cncVkpjAhoHAFtfV1cy7gm0VgyIJbWFkhOPAV7ezuN46MuFWaTeQYPHoyly5djxvTpKlEt1st1euEFnDl5EmYW/DXevDWWje8C1iDply4h5vffkHbpMv6K/xtaWlXo79Ufvr4+8BngC8ceZDwbc6NrzSCgr6+PjIyrzRpLb29vzJ49GyEhIZoBRQ1rOXnyZLi6umDmTMXvHtJaXFVVVejQoQPu3L4DCx73dFX6naUsjcPG53u6unJ/LB9nPK9k4PjffyI+PhGfr/wCBro6GDBgIPwD/OHp7QMbKytZiqC0RIDHBJqf4MMq5uTkjOzsLB7XkVTfvHkzQVASAbUxlo35cMazR3f07NEd06aHc8bzdEoKkhMS8MPG7zHrnXfh5OwMTy9vBAcHwcXVld53NoZI12pDoKUJPqyiVlYWqBS2bFTVBoqaVIS9o3RxcoaRiZFK1ojNhl28eDH0DAxUUj9plVKPBTBS1JYZT29PT8ybPx+/Hf0DlzIyMD08HOUPyhA2fjxesLPD9OnTsXP3dpSU0KxAKZBSEh4RaMmDD6uKjo4BKsrIiw+PmhXMUA4LDMSBgwdUWm1ypK7SzdO8cmz9EXs3s/rr1cjMykJ09CF07/4i1n25Hi+88DxCxodg1covUVhY1LwgiiUCPCAgTc/S0soC6Zk0DMuD5uRUZJN5mKEMnz4NkyZN4ovavNVTY3qWLbWQex93zJgxEydSTuBKxlX4efvhn/Nn0bPni/Ds0wdr1qzBFdqVoSWMFK+iBFry4MPUdnNzw5kTp1W0BqRWYwJ2dvZYtmwZli1f2ThKpa7ZBJ+oqL28d6SuNrNhlXV3lJY+RMyRw/g97i/8EnsIttbWCB47FgH+AXD36KOsYkkuEVAogZbWWbLCSh8+xPPdnsX1gnzoP1nQrlAlSJjGEnjqqadw48bNZmdjqzoc6lm20EJsw9LxYWH4YfN3uJ53DYsiluD69RsY82oI1+NctCgCF86da0EKRROB9iUgzTCsQFsbzI2n8N+H7asslS6RADrKu1IAACAASURBVBt6nfa//0mMV+UIaUY3VFl/MpYytI6ulj6GjxyO777/DhkXM7AoYhEK7t7C2LFj4eXliaVLliLzCu02LwNSStpGBKSZ4KOvo8Npk5Od10ZaUTGyEGCTeQYNGgTLrp1lyaYyaYXg97YjNAyrgFuJjclHH4pGbGwMYo4c4dzwjQwYirBJk8ndlAL4kgj5CUjjwYeV4u83BNPD3yLHBPIjV6gEtm68m7UVpk2biiVLlilUNgmTjgAZS+k4SZ2KGc6d27fiYGwMkuIS4OvngwD/oXh1ymRaxyk1RUqoaALSePBhZS5auAgGxoaYP2+eolUgeXISyMnOga2drZxSKHtrCdAwbGvJScjHFuBOnDQZUbv34+q1a/Dx9cX2vbvxnI01pk+bhpjYWDCDSgcRUAQB1uOIjo7C3r27WxAnnbMBM3MTPLh7twVZFN1WBEpLSkVFkaEUoWiXEzKWSsTOPGqEh89AQnw8EhKSuN1R5s2bC1en7li0cCEuX7qgxNJJtCYQeGPCRMQcicVnn69Az56unKcqcfWWZoIPy8cm+BSWklMOcQzbOoxN5nn6madx6tSJti5aoeWxBzovby8U5vN7Rxsylgq9LSQLc3B0wMIFC3HpwiV8Hfkd7t67h8DAEfAN8MWmjZEoKiqWnJliiIAYApevXMGHixYiclMkNyO7vPwBjiXEi0nJtlCXbjSjc+encSP9ulgZFNh2BGp3D3lv9mx4eHi2XcFKKIl5Tzt+7DgqeO5KkYylEm6OlkT6+vji22+/xeXLlzFtynTsPrQfPbu/iKmTJuFoUoLE3kFLcileswh0f/FF9HB2FlW6l5sbbGzqttdibhuLCwu5P6FQuq+6p0cfXMq6JJJJJ+1DIDMzEzNnhiNiyZL2UUAJpfJ96QhN8FHCTdEakczl3v69+7Et8nsYGhkj6OWxmDptOs2mbQ1MDczDhrrGjBmDmJ9/FtU+MHAw4uLiAQFbPFkl9aJwaytLXL6SARMTE5EsOiEC8hBgvmFnzp4N5maUrwcZSxVsucNHDmNv1F4kHD4Kv6H+CBkfiuCgYBXUlFRSFQJLl0Rg5sz3JO48IY0Hn9q62DvaI/bIUTg41PVSa+PoU3kE2NArm/zn4eGhvEJIcqsJSDc202rxlLE1BIJGBmHn5p24fO0qXHo5Y9GHC/F89xewatly5Ofnt0Yk5VFjAgcOHsT48WESDWVN1aXfjc9M3wi5edlqTEz1qsYMJXM4EBt7RPWUI404AmQsVfhGMDExwrx5C3D6/Dl89/V3uJieij7urggNDkEcLUFR4ZZrO9XWb9iA0uJiFBYVIP5oHPbu3iuhcOmWjrDMDk4uyEpPlyCHghVNgG24PXjwYLDJPIsWqc87yvqckpMT8fjx4/pBvDuX/nGTd1VTH4XZbDJ/fz/uL7+wELs2bcHc+fMgmD8PL7/6f5g8YRKsrKzUp8JUE6kIsCUFzMk/m7wjELDZrgK8/957EvJK/1V36eWE3DxyeScBpMKD2e4hu3Ztw4gRoxUuWxUEsjtzwICXpH5nrgo6i9OB3lmKo8KTsJgjMdi8eROOJx3HsOGBmDptGjw8+T3NnCfoeaemLO8sN2zYgAvnLmHDxvW8qyefFGbvJ5kTE0042K4jt27cgpUNfx/qNaOl1PRuZE7d90dF4diJ47Czd0BIaAg8vbywceNGWn6ipm3eFtXqZGqM27dutkVRGlvGuQupCAwM5P0ej7I0IN8dqZOxlKW1VTQtc4O1YOEC5ObmITx8Ovbs3IVnHWyxaMFC5ObmqKjWpFbbEpB+GNbO3hFpeVltq54GlXb58iWMCPRH3359od+xo0bUvLq6mtd7WbJGomFYNb1Vz6acwg8//oiD+w9jgHc/TAsPh7+/v5rWlqrVEgFZhmEfPnyI57o9ixsF+eT8vyWwrYhn30MvLw/aPaQV7NozCxnL9qTfBmUXFhVjY+RG7NmyDcamxpg8ZTJeGfd/LSwzaAPFqIg2JSCLsWSKOTjY4/CRWDg5OrSpnppQGJsVqqurqwlVVas60jCsWjVn08pYmJth4fz5OHP5DGa/9z727P4J7u49sGDhQuTk5jbNQCFqSkD6YVgGwMzMHPm5tNZSUTfDpdRUkShNM5Scd6mQEHKkLroD6ESlCehq6SM0NAR/xP+OrTt3Izc3F54e7pg0aSJOnTql0rqTcoogIP06S1aanb0dcnLoYUoR5JmhDPT3x87dOxUhjncy2NK3gwcOkCN13rUcKQwvTy9s37oVp06dg6ODE8aMHoWAgADs3b+X9tqk+4Mj4OzoiLyCPLof5LwfLl26wBnKCZMnY0LYBDml8Ts73x2p0zAsv+8/ubS3sbGumUWbl49Ro0bh808/g2uPHli/fp1GTWmXCyJvMss2DGtjY4vMtGyNWQeorGbU1dbF7Pfew8ply5RVBC/kzn5vBnQM9XihqyQlaYKPJDIaGM7eLfweG4d1a9Yh/WoaXnvtdUyfPo28A6nBvSDrBJ9ffonBFytWIiExQQ1qT1UgAvIToJ6l/AzVRgJ7tzBy5HD8dvRXHNj3E3JysjhftFOn/g8Xzp5Vm3pSRVomYG3dDXfu3G45IaVoQoCto1y/fkOTcArgNwEylvxuP6Vp7+7RB1u3bkdS8imYmXVC0NixCA0JQXxCAr3HUhp1ZQqWbRjWzs4WRcVF5AlKxiY5d+Ect3tITk6mjDnVNzlz65eZmc37e4mMpfreowqpGfvRXLlyOc6c+QdeXl6YOnkS/Hz9sP/gAd7f/AoBxBshss2GZTvemBqb4uKFc7ypYXsrWlpSihGBwzD25ZexcuXq9lZHZcpn/m9feOE53MkvUBmdWqMIGcvWUNPAPObmZtxEhcysbEyaPAGfzl+IAX3cOT+0bFcBOtSPwPMvvICsLHJ7J23LGpkY4dChQ9hAQ7DSIuNVOjKWvGqu9leWPSVOnjwVF65cxnsffoyd27fD7gVbrFv5JdiTNR2qSkC2YVhWi6e7dcP9e/dVtUIqqVcfDw+V1IuUkp8AGUv5GWqshLDxoUhMSsLW77Yi/ngievTsjoWLFiE/P19jmahuxWUbhmX1sLWyRmEetWVzbXrh0iVYW1shMz2tuWQaH3f12nV0sbLkNQcylrxuPtVQ3tfPF9HRhxC9PxpZOdno794bM2a9i6xs2vFENVqodVrY2tngn9SLrcusAbnYrNchgwdhfGgY7B2dNKDGra+ivZ0t753yk7FsfftTzkYE2Aza3Vu340jCn6iurICvlwfemvImPXU34tQ+l7IPwzo6OiMtjXpMktpr308/Ydy4MKxesxr0QyqJkvqEUxurT1uqTE16ODrim2+/55adGJt1ho+vHyZMmIBz52itZvs1kuzDsJaWVnj4sKT9VFbxkiMWRWDduq9VXMv2V48tHXln6tvIzeO3r2Eylu1/L6mtBmxT6pWrluPchQtwdHDAkMBAhISGIik5WW3rrE4V62ptiYryShSXkMGsbVe2jpJtsk6H9ATYpMD1P2xAfh4tHZGeGqXUSAKWFhZYFBGBa1evw62XC8aHhWLM8FE4Gk+u1NruhpB9GJZ5dLKzt0dyErUTa6fLZ9MROGQINm2ObLtmU6uSZB/dUKXqU89SlVpDzXUxMuqIhQsX4eqFK+jt2x/hU6dgWGAgEuLJK5Dym751P1ROTi64eZPc3rHJPIOGeWNC2ESw4Vc6ZCMQPHYsLMwtZMukYqnJWKpYg2iCOvomRlg4bwFSzp7D8CFDMTl8MkYNG4aY2FhypadiN4C1pRnKyspUTKu2V8fG2hYffPAhN5mn7Uvnf4nRUVHcKAWfa0LGks+tx3PdmceTGXNn43LKJc737OyZ4ZwrvZ9jyGgqvmllH4ZlOlh1tcGly5cVrw7PJLJ7dc6cOTzTmtRVJAEyloqkSbJaRUC/Y0dMmzYNF1LTETZxPN4Pfxv+/n6IiYmhnmariIrL1LphWJc+bhr7zvLsuYsICxsvDiaFyUig9OFD3vuSJmMpY6NTcuURYBNKpk2djstZVxEyfjzCw2ciwNcf0YcPK69QktwsAScHR40chmXvKEcNC4CpsXGzfChSOgLGhoY4fzZFusQqmoqMpYo2jCarxaaah0+bjiuZaQiZMAbz358HH19v7p2mJnORr+6tG4bVMzGCUFiFR48eyVc8z3K/+n//h7EvB2PDRpr5qrima909qLjy5ZNExlI+fpRbiQRYT3P6tBk4d/YMwkLDMHfuLAzxG0KzZ1vFvHXDsEZ6eujc2QrHjx1vVal8zfTr739g3TrawFmx7de6e1CxOrReGhnL1rOjnG1EgL3TnB4ejjMnz2DsmCBMnToFo0YNQ1zc0TbSQLOL8fLsh/MX/1F7CLn1NgBga4PpUByBv/76A06OzooT2A6SyFi2A3QqsnUEOKM5YwbOnruAgIDheOft/2HM6FFITiKPQC0Tbf0QmLWdDUpKHrZcBI9TZGVnYdDAATh6NI7HtVBd1X18/MBmFPP5IGPJ59bTUN2Zc4OZM2fizJnz8Orng5AJoQgLDUXy2RMaSkSaard+CMxARw/37t2RphBepsnKyuQ88wwa+BL8/QN4WQdSWvkEyFgqnzGVoCQC7En1/QXv49rlDDg6OSMkcBQmTZgI5r+TDsURcHV1Rfzv8YoTqGKSLmdmY8jgwYjcvFnFNFMfdSIiIni/z+1T1dXV1erTJFQTTSZQWlKKT1d8hn3bt2Gw/1AsXPQR7OzsNRmJqO76+vrIyLgKGxtrUZi0J6UVFXjBxgYZublgE37oIAKyEnjqqadw8uRxeHh4yppVZdJTz1JlmoIUkZcA62muXLYcx06dgYlZJwz0HIg5M2chJ5ffWwPJy6Umf+uHYTkDqaWFimL12X0kMz0dR4+qb29ZMfeMoqW0/r25ojVpjTwylq2hRnlUmoC1lRVWr16Fv5OPobS8FN4enpi3YAGKiopVWm9VVs7S0hL/XDyvyipKrduVzHT4B/gjLi5W6jyUUD4Cfd3coGegI5+Qds5NxrKdG4CKVx4BO3s7REZuwq9//Ib8nCz07ueG5UuW4dFD9Z7ZKZ6ofE/1Pv7+OHue/8tH2APTyGHDEBQ0EitXrhSPikIVTuDU2bPo2aOnwuW2pUAylm1Jm8pqFwI9uvfA9p17sW/bLvx9/BhcXFywYcMG3vuqlA1m64dhWTn2tta4X1gkW5EqmNrc3AwrPvucHA6oYNuoukpkLFW9hUg/hRHw8PLCL7/+gm+++xbbt+9E/z7u2Lt3r8Lkq7MgAwNjpGdlqkUVmd9hOoiArATIWMpKjNLznsDQgKFITk7CBx99hC9WrIC3t6cGeAOSbxi2b9++uJjCz3eWbDKPj7cX75cu8PmLp62tjVOn+L0Omowln+9A0l0uAuNDx+PY8eOYOGEyZs16G8NGj8DZlFNyyVTdzPINw7K1lhX/VaCwoFB1qyhGM2Yoff394ejgCDZJiY72ISAUsvtPvge29tG8rlQylnUs6EwDCejq6mLa9Gk4dvw0Xuo3AGNHj8PkiRNpuYmYe8HY0Bg5eTliYlQ3aNnKVRju74+NmzeD7WZDBxFoLQG6e1pLjvKpFQEzExPMX7AAJ88eh5G5Obz69MGcObNQojZrC+V/qndwcER6Whqv2n3zpsgaQ8krrdVP2R1btsDWzpbXFSNjyevmI+UVTcDS0gpfr1mDhMREFBUV4cXuL2Dl8uWKLqYd5Mk3DMsU7tu3Ny5eVH1jmZub02CmM/3ItcPt1qjICZMmge87udB91KhR6ZIIMAIOjo7YunU7oqIP4Y/4P9Dd8QXs3L1do+FYW1sjL1+1vSGxd5Te3t5Yt26dRrcVVV7xBMhYKp4pSVQjAl6envgt7nes/HI1vvz8Kwzy9uE2n+ZfFeUfhrW3t8fF8xdVtuq1k3kC/Pwxd/ZsldVTExVbvnw572cjk7HUxDuX6iwzgZEjR+L4uWN4OSwMk96ZgrCQUDC3ae19MI806776Ct6eHi2oIv8wbK9ebsjLy1NZD0gCHQEmTpjAvaNsAQZFtzGBDz/8EGx4nM8HGUs+tx7p3qYEdLX0ER4+HWf/ToG9oz0Ge/tixqx329XnrKGJEV4aNAh37txVOgsTMxNYmJsjNV0131uyHWZYD4Z+1JR+K7SyAPlHN1pZsEKy0X2lEIwkRJMIMJdpS5ctR0JiAspKitG7txtWr/2ywaSStuKhKxDAzNxCiuIU80Nl/7wjstLbv0ddW2HWu49YElF7SZ8qSsDU1BBairkF262GPFe/3bhRwUSAmwS0efN2JCUmYsGiRdizeSc+XBKBMUFBKkFn+/atSE1Lh45AAKGwQiE6OfdwRnq6ari9y8zK4pyiu7m5KaRuJER5BIqLS5UnvI0kU8+yjUBTMepLwNvHB4kJCZj94Qf4+P35GDVsGFJSUtq9whUVFWCeU8q5T8Wo4+zsiPMX23+Sz2OhEKMCAjGg/wDs3x+lmMqRFCLQDIGnqqurq5uJpygiQARkIFBa+hDr1q7Fhg3rERQ8BosiFit9fVlubh4GvzQQGdeuSdS0o7Y2rly7Dhsba4lppIlgPnXffvsdnDt3TprkSk1zKTUV3Z2d6R2lUimT8FoC1LOsJUGfREABBIyMOmLBwgVIPnEK5eWl8PLog3Wr1+KRsH330KwUKOar7ujojPy8/HabEVtaUTec3IMMpQLu2LYRwb4XZ8+ebZvClFSKYr5BSlKOxBIBvhJgPbjNm7di+9adOHg4GoM8ByMmJkbh1WHDkSdSTiG/sBD5hfmS5QurJMfJEMMmN3WyscTpf9p+I+jMzCz0cHTAqeRkGTSmpKpAoKysHEIF3YPtVR8ylu1FnsrVCAJePt74IyEBM94Lx7Rp0xA6Zgyys7IVVnc2G5bJLC0thaW55F01BAqcyufq4NTmPmLZrNdhw4bAw8MDbF9SOvhFwMHODvk8c8LfmDAZy8ZE6JoIKIHAhLCJuJqdDTsHe/Tx6oP5Sxbh8ePHCi2puV01uB2SFFSak5Mj8vPyFCRNOjE/Rx8Bm/VKk3mk46VqqWbMnAkHxx6qppZM+tAEH5lwUWIiID8BtnPHgtnzcSHnElYuW4kxY0LkF9qCBH19bWRkyD/BhxWzd/9+bFi/HgkJCS2UStFEQH0IUM9SfdqSasITAo5OToiKPYQvVnyJjxYsxqjhI8BmdirzUGTPsmdPV+RkZSlTXU42c492+bLqOEBQeoWpAJUmQMZSpZuHlFNnAsFBwTh56iR69+mN4QFDMH/+fJSWKGfxtiLfWT5nb4fyygqluvnLz8/HoEFDsGPbZnW+BTSmbpsiI5GdrfwHLGUCJWOpTLokmwi0QIBNqY9YsgR/JP6N9Ow09HupP6L270dVlWJmr9YWr8ieJZtU5OLsjLhYxc/uZfoWFBZi0KBB3P6ZS9ViL9HaVtDcz88+/xwXzil39ETZdMlYKpswyScCUhBwsLdH9N5DWL7kU3w8/yOEjBmNzCzFuZVTZM+SVadnH3fczrstRc1kT8I2CX7zzTewe/ducjggOz7KoSQCZCyVBJbEEoHWEAgKGoN/0i/DwdEZbC/NpUuWov5C/NbIZHkU2bNk8mytbTm/s63Vp6V8c+fOaykJxfOIgAJXLrVbrclYtht6KpgIiCfAhjlXrlyO3//4A38l/gXnF57D3Dlz5Hqfqeie5cCXBuLXo4fEV6AVoWwyz4SwsHbzDNQKlSmLDASYK8bRY1RjgwEZ1G6QlIxlAxx0QQRUh4BrT1f8fvR37NmzB1k52ejfrx+ijxxplYKK7lk6ODhCWF6lkF4vM5TsHaWBoSG0dHVbVT/KRASUTYCMpbIJk3zeEGCu44qKilVOXy8vH0RHRePjTxdj/syZmDB+PPILCrhe2KOH0vmcVXTP0qhjR1hYWeB4UpLcvN55510MGvgSIiMjwXrVdBABVSRATglUsVVIpzYnwGZg7tm8DW/Nmcn9YO/cvhNZWVkoK7+PkUHB8PH2aVGnqP1ROHP+NKoqhRjg549RQ4dKzMO29Dp4+DAcHOzx6v+9BiMTI7Ae1ldfrYONjSVKCh8jYtlCUf5hIwKRl5uLoqJS3LpzG6aGhlizZg3Ky8sw8KWX0KN7895RtLW1cU0Bu46IFAIwd848GHc1xsK5C9Cc96D6ecSds2UillZWNJlHHBw1CXN3dcXy5csQMHQ4f2vEtuiigwhoMoGKf/+tDg4eKUJw8uTJ6pcGDeKuKyoqqrt07iyKk3SScfVqda9ezqLorl27Vt+9d090Xf9k14491QHDA+oHced9+/atTktL5c5nz5xdveOnHdz5F199Uf3N199U37pxi/s7eex09dChAdXOzi7VwcFjq916OVffvnOnibz6AXp6guobN27WD5L7/IsvVlTPeHN6q+SUPXhQ/V+rclImPhKws7OrPnjgEB9VF+lMw7D8fc4hzRVEYO2qtfDz9RdJ27RpE7yeOOvW1dWFr68vt0elKIGYk41r18Hfv+6peUxICDZu3NgkJdsU+u133sS2LTsaxCXEJyA3OwcvvtidC3fr3QtrV6zlzqdNmYbwGeGwsrHi/uLiY7FixRc4c+4snF2ckZWVi9dfe7XZCUCKfmfJFOvj4YG4+KMN6iHNBetB93LrjU0bN0iTnNIQAZUgQMZSJZqBlGhPAuvXr0Hf/v1EKhw+cBB29TZJtrK0RE5+M9tfAUg8kYiuXTuLZHTp3Bnnz18UXdeefPHFCnh7+2Ljxg2YM2sOLjzZRPnchXOwt7evTQY3t744ffo02HtUIyMjUTg7+evPo+jZsyc3XLx0yRL8fSoZsbFxeLH7i1i5bHkDo8mGOOe+O4stHsHSpREoKGi+Hg0KauHC18cXD8rKUFhQ2ELKuujayTxubr0wZeq0ugg6U2sCszlH6o68riMZS143HykvL4Hc/HzcuHkTTo5OIlEPysvQqXMn0bWWjg6qKitF1+JO7t8vg5GRiShKh1vbWLdRcW3EgQOH0bevGz6YvwC9e/fFgIEDOeNWVlGOjiZ1RlFXV5/LUtFoAk9efj569/aoFcd99nRy5j6nT52C0+dPo1cPFyxYuBA5OTkYPHgQvly3lltn+f33P2Dw4CEN8sp7YWTVCRdTz0stJjc3F/0HvIQdzOGAFv38SA2O5wnZriPdu5Ox5HkzkvqaTCA7M42rvolJnaEzMNBDRUWdcaysqIRWC7M0DY0NIRTWywPAQM+wCVqWJihoHNcrnDAxDKysLVt+hJ5AgKp/haL0RcUF3Lmenp4ojJ2wod2gsWMbhLGLrl27Qggt7N8bhZ+ifkJhfi6Yw/O0tIaOyFNTUxEdHdUkf2sDQgKCkJSYKHV2Ly9vbN+6mWa9Sk2MEqoKAXq0U5WWID3ahYCBoTFXbv0lGGz2a3pmndNnNivWxsqqWf36enghN6duj8fM9Ew4OtQNq9ZmNjQ0QNmDu7WX8PL2gUBHD67u7sjIuiIK/+fMGbj1dgN7Z1r/OHn8OOfZp34YO2dG2NzMjAt29+iDyE1bsX27eCfkQgW+wLS1t23AqrFe7DonNxeHDh8WF0VhGkJg986dvHekDtFUHzohAhpIgM2E1dPTqc5IyxDV/tjx49WDBw/mrsvLyqq7dO1SzdKx4+PFi6tv3bolSlt7cvHSpepevdxqL6u7dOlcffduzWzYbTu2VaemXuTi3n///eoZM2Zz5//991+1vZ1tNftkB8t/9dpV7vy92e9Vb9u1izuv/Xfj1q3qj99/v/ZS9Ml0A1B948Z1URg74WbydunKxbH42r+xr7xc/cP33ytkNur5f/6ptn3WVqIsppOTk2N1YGBgA93oQrMI0GxYDXkqomqqLwG2CD44OBgnTp8SVZL5ZB01ahQWLFiAxZ9+ikPR0dywIZuM88WKz/Huu++K0tae9HB2xuyZszFnwTwsmD8PO3fvhbl5TU/vyy++wB9//sUlXblyJYpKCzBv4XwsjViCLdt2iN7d7dq1DcuWLMOy5ctgaGiIiWFhteK5z62RkfAf2dRl2O28PNjbO8DGxrZBetYr/fPP3zE2qCYP+zx5/DR8PLy5DZwtrS0wITQMmzdHorikpEFeaS96urpCUAWkpzd1+s62G2PbbDk5OSE2NlZakZSOCKgmAc16vqHaEoGmBNIyrlS/Mm5c0wgxIQ8elFUvXvyRmJiaINZLrO0p1iZqfM3CxYXVppf0KSnPJ4s/qf7l4EFJ2bhwHUHTdZasp/rN12uqR44cWd21S5fqwYMGVX/7zTfV58+fb1ZW48jAIUOqt23b0jiYu/7l119FvXKxCShQIwioQ8+SPPio5jMMadXGBPbv3w99gTZGjgmWWDLz8hMTcwQvB7/MedyRmLANIzLT03Ho50NoaZcObX1d3Mi4zq3TFKdefkEhjh6N5XqAyYlJYBOevH18MGTwIHh4+cDK0kJcNi5s1Vdf4XpWFr755huJaSiCCPCdABlLvrcg6a8wAsxhgLWVDaysLRUmU5mC2NZd51NS4O3t3WIxzFhey7gGm3rrRyVlYms7jyUm4cSJE4g9chiZ2Zno6dwTLu69EeDnh959+4mGmJmM2F9isOTzpUhOSkZeXh7CwsZj89atsLdrOsFJUpkUTgRUnQAZS1VvIdKPCCiAgLa2Lm5ck9yzbK4I9j4zLiYGqWmpOHjwIAoKCtDDsTt6e/aHm5szvLz80MfDFX/8/ifGjXsZzz77LGJiYkTvYpuTTXFEgC8EyFjypaVITyIgBwF9fX1kZFyVqmfZUjGlpQ9x8PAB3L1zB1F7DyEzNw26Vdp4VPkYlQIhfon6Dd7eni2JoXgNIuDBNjKPWMRrR+q0H44G3bBUVc0lIESdwwR5KRgZdcTECRM5MbNnvwdhVRXmzp2Li2fPw8XBEZPefB2VJcUwsrREsP9Q2NjbwsWlF/p6epIzAnnh8zT//Tt38OhRndMNPlaDjCUfW410JgKyEhAKIFDCpawzgAAAIABJREFUJljM16uBgTECAvxx6tQJrIn8FmsAbl/QM2dO4+zZFJxITsGaNatR9qAM1lbWcHJzgYdLHzzr0A2uPfvAWor3qLJWl9KrFgF+m8kaljQMq1r3FGlDBJRCQJHDsLUKMiftAwe+hHHjxmHme7Ph+mJ3ZFy9KnGmcEFREVL/OY+09HRcPHceZ9MvID8zBx2N9PHcc8/DrrsDXO27w9nNBbY2dmREa0Grwee6tWsxeMhQXvuHJWOpBjciVYEItERAW78DrmXcUMg7S1YWM5SDBg2Ck5MDoqIOcZN5fH39MGNGOEJCQlpSRxTPZt6mXbrEuczLvXkdJ0+cQGZaOgpLiiDQFcD1BVc8Y/cMrK2t4dnHk3O+0MnaFEY6dU7nRcLohAgokQANwyoRLokmAipDQMHDsDo6enh57FgsXbZMVEVPD0+kXrwok7FkHpRcXV25P5GgJydnU87h4sV/UFZehszMTMyZOQcF5QVAmRBWVtbo1b83rMwtYWpsgj59+qKXmxsMjDpy70VpR5PGNOlaXgLUs5SXIOUnAjwgoIxh2MbVTkg4infmzMKls5caRyn0ms3GPX36OC5ePI+qx1W4nHkVx5L+RPH9YlRWVcFUxxDunh5wc3ODQLcDLDp1Qi+XvnB27UETjBTaEtILOxQdBfc+nnKNbDB3k9k3bmD0E/eN0peumJRkLBXDkaQQAZUmoIhhWLZx9MbNm7Bo/kKxdWU7t3Tr9iyuZGQ0cFogNrESAtmQbuXjx0hNvYiLFy+iqLAA9+88QGZ2Ns5eOIWK8nLoCrRhaNgZji/awbGXC6zMLWBsaAwbO1vY29jB1t5OCZqRyOeeew5fffkFRo+Rfoi+MbVHQiE+mjcPq1evbhzVJtc0DNsmmKkQItC+BHSE8n3Vmau/QYOHoJeLC5hRYsOnjQ/9jh3h4++HPXt2IDx8RuNopV8zndgfGw5mf42Ph6WluHv3PnJys5GZlYWi/HxkXs5G4d08ZGZmI68gl8tibmKOLl06w9zWCs42jrCwtkDnTp1gYWWFLp27wt7OFqyudMhKQFtiBnZP7dm+FWmpaQgYORJ+vr5N0uqLueeaJFJiQNM7XomFkWgiQATah0ClnN/0kNGjOUO5Zft2sYaytlb9PT1x9sRZILw2RHU+OxoZgf3Z2tnAx8eniWJVbMlLQSFyC3ORn3Mbd27nI//uXaRcuIB7ubeRd6cQxaWFKLv7AB0NO6KrRRdYWluiU5euMDE3Qbeuz3A+dG1sbWFs3AnmZuYwtTSEvoAMaxPYjQK+WL4cfj6+cHRyQkWFkFuGdPr0SVGqgIAgODi0b6+fhmFFzUEnREB9CcjiG1YchcuXL+HFF7u36MKOrbt07eOOW3kFzRpVcWXwJYwNN1+/kYPc3NsoLsrHg7JyFBfdw42becjMSEfRnQIUP3yIyopyVJZXwsTUBHa2tjC17ARj084w79gRBqamsDA3g72dHZ5+5lnOcb2hng4ERkYw0tPjCwqF6MnumdDQ8Zwv4pYEzpkzh4ZhW4JE8USACLSegE4rVoXXH27t3r2HVIWzPTWtLK0Q9+uv3J6gUmXiWSI2BOvk1J37a0n1ktKHyM7OQnp6Oh48eIAqoRAVFWUoKSpFYmISNkVuRkFhPsDet+pUQascgJYWZ1w7d+0Cm+ftYKSnDz2BARduaKgHOzt7WFtbwsbaDsYW5hBoaYkeYsQNj7ekY3vGr1uzDtGHo2FiZMRxYnWTdDAfxT79+0t8DSApn6LCqWepKJIkhwioMAFZe5bsHeWQQYOwavUazjuPLFX7YvkK5N8pkKkHwGY6WtvYiiYG7d69E+bmllKVnZuXi4MHDuLNN94QvUs8d+EcTp88idcmTxH1cNmPbVrq+QZVsXdwgqWF5O3HahPv3L4ddrZ28PKp2eEl+UQyevfxEMmuTSfvZ1VVFTdB6uaNG8jJzUF58QMIq4DHlZUoEz5EQW4+crKzcDP3NkrKiqBVpQUhqiCo0kKVADDq2BFPP9MNNjY23PKaTp1MIRBooQM6QEub2VsBuj3zDCwsrWBuaYmnO3eR6ERC3rpIkz87Nxeho0bj408+gZ+fH5grRVU9yFiqasuQXkRAgQRkMZZs1iubzOPi7IxtO3fKbBDOppzldh9JzciQKm98fALKS+5ze4my3uy4MSH4ePHHuHfvLg4fPoxvvvlWLAk2HPq/t9+Cp7sHXn/zDXR8Mulm66bNyMnNxbTw6Zg4IYxzmsB+hOcvmI/4uDiYmdcYx8wrmdi1exc8vZpOBqotkNNn9GisWLMaZ86lICs1E4siIrhoNiQ4e/ZMsN50ex2PKish/PdfCIXluHe3DAUFecgvLMK927dQUlqK8op/ISxnn1Uor6xASXERbhfcRfGtPBSUFKFCWMm5QdTW0oZAF9DpYAiBthZ0dHRg2qkTLC0tYWBqCDMTE3Tp1IUzvLq6umDrbHUEAu5PYKAHEwMj6OgLYGFlA/Z63NS4E3T0dEQPP1OmvYmh/kO44db6rNjDgaWlBW7mq/6wvZyv/etXm86JABFoDwJFRcXg3hW6ukosXpZh2KTkE1JN5pFUmHsfd+jq6+J4chJ8fZrOaqyfj3kCWhe5FtG7o7ngA7t3w8DYAH369OGuZ82ahbOnUuDuUXNdm5cZseDgILzyyquYPGVybTDnk3bu/LmiH18fH1989umnWL5yOXy8fLF82XJR2hEjhjVrKFnCHzZuhIOjE150cOT+nv/gWYybEMadM7mBw0bg9/ijUj0UiApW4Im+jg7A/tARZiYWsJdx6QvjKHz0COXlFahgvdfycjx8WMINGVc8KENhURHKK8pRUVKBUmEFHty/j7KyClQ+qkBJeQmElUKUlz5AScVDVFVW4V7hXTyu+heVleVghlBYCc7A3n/wAO49ezWpeVTUXgT4D203fk0UaiaAjGUzcCiKCKg6gVOnTmHr1k0YHxqGsJAQ7IyKEusuXZbZsCFjxoD9yXNMmDAJP+3+qUVjuZANv3nWGdR9Bw6iX7/eoqIH9B+IPT/ta2IsmRErKCziDCX7wa99Vxdz5CCsLCxF12wCzZyF8zljOXzkUJHc3Nw8zgiKAiScREdHw6ve5tpOLm44HP0zXpznyA35Ojk54sD+vQgLmyBBgmoHc8ttnswSVrSmrF0q/2UTnYDRw4ZBIGBGveFx5vR5+Pr7NQxU0SstFdWL1CICREAKAv/735tYuDACPr6+cHDrhcgNG6TI1TQJe0eZkpLSNKKVISOHBuDw70fAhkolHezHdNumTQgeN06UJD09DUZGdX5f2Y4kbE1k44MtcO/UqRNWrlwJJ/sXsGzZUi5JTm4eTLt0FSVny0RuX7/JTQoRBQLYuXMrXvu/V+sHiT0/duxvWNV7p2lqaorbN2+K0jo42GPfvp9E13RSR4AZYiN9E5ibmWBa+DTOo1JdbM3Z5q2b8MrLde3fOF6VrslYqlJrkC5EQAYCOTk5yMnJhZWVFZfLxsICsXFxYiU0NwzLTeYZPASfffaZ2LytCezp7s6tO2TvHCUd9/ILIBQKYfNEf5ZOWFHRpAfCZpA2PphRXLhoIebNm4c/k/7ERx99jMz0NE6evm79xe81g2fMs0/9IyYmBs6uPesHiT2vqKiEsaFhg7jyR3V7g7K1lKdOJDeIp4umBCZOnAQPD48GEey+8/P3b9cJRg0UauGCjGULgCiaCKgqgfS0VFjU6/V06twJWVnpYtWVNAzLfrDY7iE2Xa3AhhwVeUyZMgk7tu+SKLKSTfNsdDzTrRu3tKI2OCc7F7a2TRejMyPLZnSyw9bGFs899yziExK5ma2lJcW12ZGVlYlOpqbcLNHaQFZnDw8v0VBtbbi4z169XJCTlyeKun/vNp61f0Z0zSa5sHd6dMhOYP369Zg9c7bsGdspBxnLdgJPxRIBeQkI2OL1er2usrJyGOg17AW1VAZbNjFj+nQciPm5paQyxwePHcf5aL1w4YLYvKadDbjw0ooKUby/vz9yc/JF1/+cP42hw+veNdZGODraI+NyncN2MyMzDBzQH/4BQ5GWnlmbDDk5eQgaO1Z0zU42rt+AscFBDcIkXfj5BaCinn6ZWTnw8qpZPsLylAvZDig2krJT+BMCbIifvSdmh5eHB06dOAH2vrdxb1OVgZGxVOXWId2IQDMEnJxcUFhc14sqyC+Ek7Oz2BzNDcNODw+XqpclVnAzgWy5QXBoMLZs+UFsKhMjczh2s8OZUydE8TNmzERcfBz3rjMzPR1WXSwx2N+fi2fr8Hbv3s2dr1jxFdatXQ/WN2WL/i2sLNG9R0+wd4gvvxyMuLij3HvKpMQELIlYJJLPTo4fPwbPepN2LqWmomfP7oiNi22Qjl189NFHOHL4ICeL/eC7ODs1cJXHnLX3F+M6r4kgDQ/4cN4HOHz4AEch6pdfYGllhbDxYc1SYe+0VekgY6lKrUG6EAEZCFhZWsDH2wcXzp7lct28dRvs3ZC4o/4wbFFhIUInhqK0pFRcUoWGzZgxAz/tPshtFi1O8FtzZ3KGrTbOyMQIv/z8M37Ysg1JSUk4+PMRkSEfOXwoHBwcuKSjR49CREQEvv7qK84TzsGf63rG32yIRG5eNrZs3Ih169Y1WAfJfoDfeP2NBjOGTYwNERz8Mk4k1xntWn3Mzc0QFR2NHzds4PbW3LGj4bDy338fQ/ibb9Qmp89mCNTOhmUTpmxtW16b+smiRZhSb1lQM6LbJIqcErQJZiqECCiHAHv/tuDThbDrbAUTYzPMmDVTbEG1TgkM9HQ4hwNPP90Vh47UGSKxmRQUOG3aW9DvqIM3Xn0dx06exIABA9CzZ93kGuakfceuXSLvOwoqViYxcVyvUgsBAQFS52M92rWr12HNujVS59HUhEP8BiM4JFim3WiYP+JBg4bA08sD0VHRIpd+khiWPnwoejddUlwMEzMzSUlbF15NBxEgAmpPwECgU33jxs3qV197vTowMLBN63v12vVqHT2DagCivw8+eF+kw92796o/+Oij6op//xWFteXJrVu3qk+ePClTkSzPis9XyJSHEstO4MaN69X29nbVLw0Y2Gzmbdu2VHfr9kz1yePHq7f8uK0aAq3qO7duN5tH1kjqWbbuGYNyEQFeEajtWZoaG7f5VH22pOOFF7s34ZWWcYXzhFMbwYaF2TAsHw7mZo7znsMHZXmuIxs9GTJ4CNioyJ9//ilxBGLd2rVISj6F8BnT0M9TutnOMqGR1bq2Nv29O3dET5X1nzDpvO5pm1gQC7oH6B6ge0DyPfDmm29KNEHnL56vdnFykhgvb0SbTfAxs7BAdXW1RvzNDA/H4sWL1bau7Gns3p07alk/1m7h08PVpm6379yBi0uNT85bN261S72u37jR5AFeIBDgxo3rcumzZs0qtWor9vt4/doNMDbq9lvpP8gP3333XavqlXHlCrp1ewbBwcH4phkPVWynloLi4ma9RjW5EWUIaDNjKYNOlJQIEAEFEGAO1tnwla2ttQKktV6ErY0NvvzyS84IMCnMGKxavbrBLNXWS1evnGy7LTrqCJw7lQpvH294+/hwTjNqfQDXpQDycvO45UN6Bobw9vbC4ZgjSEpMrJ9EIedkLBWCsaGQ/gMGoo9bw10SGqbg99W0N6ZBx6BmQTm/a9JUe9Zu/fsPaBrBw5C83Bw4OTrg559/gUCvqRPrtqwS287q5q1b+P7bb9Dlma54ZdzLchfv7NxLbdqqFoapqTGmvDal9lKjPxMSE/DSEC+MDwnFzu07JbJ4Z9Y7SIiPh5+vL6ZOnoKYw0caOL+XmFHGCJrgIyMwSk4E+Eigo7Y2rly7Dhub9u1lMnZvT3sL/1VVYWPk93xESTq3gkDk5k1wc+kl2npNGhErVy4H8827aFFDpxLS5FVGGjKWyqBKMolAOxFgMwcTEhIwPjS0gQZsNuyNjOuwsqnxp9ogso0v2B6WAz0HYNvuXfBqZuPlNlaLiiMCzRJQuWHY4pIShM8Ix/r167BgwYJmlVflSPa+KHT8eDg6OcHfz19pL53bi8HadetgbmkOExMTfPnVl+2lhtLKTTwRj0C/IWCb0/LlYIaSOUXftu1HsSo3fh/Gts+aO3cO912bP2+O2DzKCGS7pMyYMxPz589rtXjmom7ixIlYvXp1q2WoUkY25Ghvb8t9n0LHN3zQUSU9ZdWFLQeaOGki+vTsyfmBZb+LfD1Uzlh+PO8DjBo1mvP0INDSQuSmSF6yTU0/j/179yI9rWbboBoPIbysShOls7KzYGBgiKKCIrD9/ubPm4/MK3XOq5tk4GFAP7cBqBZUQ8zGGCpbG2Yo7WxscOCng0101BFTkcWffAK33r257xpzBr5u/fom+ZQV8FZ4OISVlYiMbN3328HREdZWVqisrNsuS1m6toVcNiHl8uUMXLt2HelpmVi+fFlbFKv0Mo6fPInvv/0OUT//DLad2eGDNf5hlV6wEgqQ21gyX4v1fyjZuTwOcCO3bEbvJ1Pd2W4DkZGblFBt5Yv08arb/d3dvQc8vX2UX2gbldCpU2dMfeKzsUePnhg4sD8eCR+1UeltU4yuri709Tu2TWEKKuXr1atx4OefwXRvfFQKtCBo4BEV2LBhPfx8a3apHzwkEKuWL2+cTWnXbFbjsmXLEbE0Aq3pbbD8emo0yey92e9x7cZ80QYFjcSDMvXY9isgwJ9zIvDGxDdwKeMy/ANqnOIr7cZSouCanVHlKGDfzu347PMV+PWXX2Bnb493Z72DT5d+1uBF7okTyTh6NAECgRaET55w2TlzphsWVud5nn1p2JOisYU5p5GdjT1Onzwph3btm5U9NKxfsxqP/v2P22evfbVRXOlsN4n6B3u4dxTjoaV+Gr6eC3jUcfFvwa9p42FYbkuvJwaHPfXfun27TZvJ188XAb6BmL9gLiI38vOhWFHA9DvWPZjl5+Vg6bKVihLd7nLY72BpeTFc/7+9c4Grqkr7/+9l0FQiMzWHHF4ickzzlhH5OuYtwSsimikSERHDOGVm3vASOWpEhIqamRmZKZoZ3lIiYnjNYdSUMUQR74Y0OU1/M9OUHOL9f9apQ1zOgcM5Z5+z1t6//fnwOfvsvS6/5/tseFh7P3utezrjdh/3J5jZC8ThYBkVHYPDJSdMSQUiWD4R8wS69qy5TFCvXr0hfhrabpT//Jep6l0aD4cHvg116bTzYpHayl9o3nH77ab/EoUdfgEdsGjpcgTcdTemTZ3qtP5c2ZB4tvXVv/+fqcubb22Gdq3aVXWfviYdqanJVStDVJ1QaEe8p1VeUQEPTw/89rbbakynVeHeNy7qpSieBwUHB+O111+r8c+ppUqWbsOKch4WRqGW6mt1LDn1FXTr3BkxUfmapPtrpVurdguLCtG7b3+I57p62cTfwaaezbAzOxsvzJ6NlBQ1/xFwOFgKh4pni6VlZaZleFp4eeMmj+Y1/Cxm5y85UlzjmPhyW5vb0ataNpxP+3amF5YvX75setB96lQJOnb8eUmeOpUlOxATE4HvvruGisoKbM3MRMAvSwmNCQ/Hte+/w87sXMkU2y7n4MGDeO65Z00VxGK4qYtTTfv79+/HHe19bfpHyPbeXF8yYXYCio8cFlcy3nzrzQYDj+sV1u1RJMKJQNmuXWt07dGjboFaR8Rt2NqbmByg/NJlePs0w9ffXMDtt7epXUTz72Lx6XkL52NGQoJpSS7NO5S4A5Gg9dlnnyE+Ll5ilfZJ2/XxR8j56yd4a9VK+xqQoJZTgmWzpk1xtbwcOTk5iI6uu56ev38AxI8t24iRI1BSXGIKovn79+PJp/5oSzW3l8nbnW9Vg3i28j+9g6yel/1E3/79caiw5mr3BwoKcO2b7zA09OdV7EuOFqNTl5p3FGS3y6xv3bp15l1lPtetWWUKlCKZp+pOTCPVT5gwHoePfI5BPkOwb99niI11zx/pifETsTZ9DVatXIX4ie7R0Eh0Ti8uAmXWth1VgVIsNB0YqJ+JTUTWfMsWN+OBB9Wd8MMpwVL8d5i7cSMWvPCCwxfR4tRUzEuci++vfo/zJaewNH2Vw226o4HoqCi0bN0a93XtitKyc0hMnO8OGZr0KZ5Bh4WG4SZvb/zXn/9jenE4JTVZ2WBpCVJpaSlKS8+h+PARXAkZJt1qGM8+NwPPPDutwTX+zLZZug2bnJyC56Y9b2qj4NABpC1eZi7u8s+UlFQ8EfsERj0y2qbn+2IB65KSE2jWopkpQUgkxqi6iUAppiW8ePEbvLjgL/jpp58QFh6ui2A5bVoCrl27hAfuD8KJf5bixZkzVXUTnDIpQUZGBvr27eO0uR7FA+Evy/6JAP+GV9OWlbx4znf6/HncecfvpPtDKysz6qqfgHhGee1GuU3BpHZL1iYlqKysxLlzpQgI8K9dxeXfY+Ji4flf/4fVb1p+T9TlgtihwwR+/PFHnDx7FhszMjAsJMQ0x6vDjbqpgboPMhohREwc8OGHH6Jjx45OC5Sie3FbSeVAKWwQ2W1dO3VioGzE9cSi1gmIQCneo3zl5VesF2rgTO1sWFHcw8NDikAptCTNX4hPcvJwYO/eBizhaVUIiNeYxN/Bg3v34Uhx3bwVVewQOh0KluJVj3t+/3td3C5QyWnUaiwC58rKTIHy1va3YfEvyVWNJWDpNmxj29C6vMgAnTVrFp557jmtu2L7biAg/jFTeXNIvZjgVsykwY0ESEA7Am18bsawIUOQsz3b7k4sZcPa3ZiGFePi49G8WXMsXaS/KRQ1xCZ90w8PehgB/u6/1e8IKKc8s3REAOuSAAloT0A8szx78qwUq440ZK24DTv60Ufwj88P2/V8tqH2eZ4E7CHg0MjSng5ZhwRIoGECG9dtxEe5zns3V4XbsGYqQb17Y3BYKKY984z5ED9JwO0EGCzd7gIKIIGaBNaty8CExyegiRMnCbc0N2zNXuX6lpaUij379yI7O0cuYVRjFwEx0cyVKz/YVVeWSgyWsniCOkgAwIH9+/H444/hfz/5BIOGDXMqE0vZsE7twImNebf0xsKkJMxNSNDd8nZOxKRMU888+xwWLXpVGb2WhDJYWqLCYyTgJgJBvXqh5ORx9B/k3NUZVLoNa0YfFRmFW9q3wYrlrls6zNw3P51LQMy7rPqmvgWqe4D6SQCoMXq6p4PzM8xVyYatfTEsfnkJXlueBrGGKjcScCcBBkt30mffJABAJPM88OCDDq0Dq1eQPbrdi/GRjyEhYYZeTTSEXfMTExEfr/a8vwyWhrhUaaSsBNauWYcJTz6Gl15+ye4J0W2xTcXbsGa7Zs6Zg5LDJabZwszH+KkWAbF2serLjjFYqnXNUa2OCIh5WddmrMXHuz4yTUyvpWmqZcNWZyEWG39hwYuY/vxUCGbcSMAdBDgpgTuos08ScDEBlSYlsIZmaFgoHri3O+YnLbRWhMclJZCxcRPatm2NECcnrrnSXI4sXUmbfZEAgKysLJc/n1T5Nqz5onlt8WKsXL0SRxWfkNtsj5E+c3JzsDff+pq/KrBgsFTBS9SoGwJr12UgLCwMB128soaq2bDVHR8Q0AET4ydh5lQm+1TnosJ+ZYX6t88ZLFW40qhRFwQyM7fhqSefwJYN7ym9rp87nTFn3lyc+9cZbNq4yZ0y2HcjCXTqGAA/P99G1pKrOJ9ZyuUPqtExgVMnTuDcuXMIGTLE5VZ6NWmC42e/UGIi9YbgiDV0Z06fjr/9fR9at27VUHGeJwGnEGCwdApGNkICchMQCT7nT34BH18fuYXaqC5i7Bi09w9AakqKjTVYjAQcI8DbsI7xY20SqJeAeI9yxvPP11vGVSdVmhu2ISZJKSnYlJGBwqLChoryPAk4hQCDpVMwshESqEtAJPM89ccn8YcB/eqedPERPWTDVkfm7x+AiU9PwcxZs6of5r6kBCIjozFv3jxJ1dkmi8HSNk4sRQKNInD50iU88+ensOX99zWfcMAWYXrIhq1t55TJE3HxnxeQkbG29il+l4yAZzNPyRQ1Xo76FjTeZtYgAc0JtGzVCl+e/wrik5s2BJp7eeHFpGRMmfhnjBwxGmJZL24koBUBJvhoRZbtGpKAGFHKGCD1lA1b+8KKiIhEu/ZtkZaaVvsUv0tC4FDBATRr0QKdO3eRRFHjZTBYNp4Za5CARQJbN2/GoxMm4Or165pOim6x8wYO6i0btrq5ZWWlCAwMxIcf7kJQUFD1U9wnAacR4DNLp6FkQ0YmsH3HDlOg3PLhLukCpdkvesqGNdskPn19/TB18lQkJCZUP8x9EnAqAQZLp+JkY0Yl8HlBgSmZJ3RIiJQI9JYNWxvyjNkJ+P5f32LtmvTap/hdAgKZmVuxe/duCZTYL4G3Ye1nx5okoAwBPaw60hDsrOxsPD/pWewrOAixrBc3eQjERMfCz99X6ddHOLKU53qiEsUIiGeUZ86cUky1fuUOGzIE9/e8DwsWLNCvkYpapodHAAyWil58lO1eAps3bcboCeNx6sw59wqxsXe934Y1Y1iYnIIPNm5AYeEh8yF+SkCg9a2tcPPNN0ugxH4JvA1rPzvWNCiB3JwcDB0+HB9//AkGDuyvBAU9Z8PWdsDChfOx7++fYddHu2qf4ncSsJsAR5Z2o2NFoxJ4aOBAfPyJOoHS7Cc93Aoz21Lf5+TJU/H11xewefPG+orxHAk0igBHlo3CxcIkoCYBPU9KYMkjW7ZtQ8L0GThSckzaV3ks6eYxeQlwZCmvb6hMIgLbt27H0MEPS6SocVL0ODdsfQRGjxqFe+/phBcT59ZXjOdcRCAuNl7pTFiBicHSRRcLu1GXQGbmNowaPQoTHntSXSMMqPylV1KwJj0dx0+dMKD1cplcXnFDLkF2qGGwtAMaqxiHQGVlJebMScD7772PqKhIZQ03SjZsdQd17twRsbGxmD59RvXD3HcDAQ9P9UON+ha4wfHs0jgEPDw8cOz4cYzG90iSAAAaqElEQVQdN1Zaoy9fvoKEGfUHBHEb1tOAN5JemPMCThSXIGvnDmn9ZwRhEaPHYtCgQUqbygQfpd1H8VoROHa0CJ27dNOqeae1e/2HH/Du+vX405/+hPL//MdqMosRZvCxBlWsd7nolaXYd+iAVT7W6vI4CZgJcGRpJsFPEviFwNatmeh+3/04cOCA9EzEmo6Px8SYdN7kaX15WiPehjU7LzIyGre0uRVLUxebD/GTBBpNwPpvV6ObYgUSUJ+AafWQR8fj7bffUW65J/F8Vdw2Nm/i5fyS4hOAhyeuodJ82JCfKcnJGDduDKKio+Dj42NIBu40evv2D9GuXWv06tXbnTIc6vvX3yyHmmFlEtAHgda3tcG776xXMpmneqAU3ujU8V7cd393dO3ayZDPK6tfkWKdy6EhoZg7e071w9x3EYF//OMgPtim9nNjPrN00cXCbkjAUQKXL1/GQw/9T1UzRUXHTPvXb9xAi5tuqveZZfPmTXDy5Bfw9W1fVd9oO19/8w3uv6873n//A/Ture4IR0W/zZs3D1fLy5GanKyifJNm3oZV1nUU7iwCYvUQr5YtERIi51qUZju9W7bErl2fmL/W+azvmWWFATNhawNq17YtJk2firmJc5GXm1f7NL+TQL0EOLKsFw9P6p2AeEY5akw4Nry9HhFREUqae/36dbRo0QI//fRTjWeW1Y3xanITjp89a+iRpeDxY0UF/tA9CJMTpiAqKqo6Iu6TQL0EGCzrxcOTeiZQWlqKu+++G+++/a6ygVL4Z/vWTHz51Vf43R13YEhomMXXI4z86kjtazgrKwtTpkxCQUERvL29ap/mdxKwSIDB0iIWHjQKATHhQOd77tG9uQyWNV0cHhaK7t0fwLz5iTVP8BsJWCHAYGkFDA/rl4B4kV+8n2ikzWirjjTk26NHizA4OBh79u5FgH9AQ8V53kECyclJuHq1HOJ1JlU3vjqiqueo2y4CIpnnzrvuND27sqsBRSsZdbo7a+7q0qUbIiKiMC9xnrUiPO5EAuXlN6D6ZOoMlk68INiU3AQ+ysnB6EcfxRuvvW7xuZ7c6h1XZ5TFn20lNX3WTOTv3oO8/XtsrcJyBibAYGlg5xvN9DdWrsCG9e8hfKy8k6Jr5RMjT3dnjal4leT5ydMwby7XvLTGyFnH+/fvjxBOpO4snGyHBEhAKwJM8LFM1vQqSWAgps+ciXERar46ZNkyHnU2AY4snU2U7UlFIC8vD1cuX5FKE8XIQ0BM5JC4cCFenPui4Z5jy+MFNZQwWKrhJ6q0g8CurCwMHjwYeZ/+rx219VWFt2Gt+3PkiBH47053YWlyivVCPOMQgf3792JPvtrPhhksHboEWFlWAh/l5mJ0eDg+yMxE2MiRssp0mS5mw9aPOmneQixaugRlFy7UX5Bn7SKQnZ2DHTuz7KorSyUGS1k8QR1OJdCze3dsZaCswZTZsDVw1PgSGBiIkSNHIemlv9Q4zi8kYCbAYGkmwU9dERCZjsNGjNCVTY4Yw9uwDdNLnDcXOzK3oqioqOHCLGE4ApzBx3Au16/BuXl52LltG9KWLdOvkXZaxmxY28DNnZuI4uJibN2aaVsFljIMAY4sDeNqfRsqJhwYOngwHnzg1/Ue9W0xrdOCwMzp03HsWCFyc3O1aJ5tKkyAwVJh51H6zwTE4sePjx2LDes3KL16iJb+5G1Y2+h6t/TGjGkzkMiJCmwDZqBSDJYGcrZeTW3etCnOf/UVxo4z3sw8tvqU2bC2kgJi4uJxo/wG0teusb0SS9ZLIDU1FYnz1Z1EXRjHYFmvi3lSZgIXL16qkme0VUSqDG/EDrNhbYMl/ii+sOAvWJT0MicqsA1Zg6WuXr2Ka9euNVhO5gIMljJ7h9qsEhDJPL/73W9x6sQpq2V44lcCvA37Kwtb9sJCQ3GH7514c/lSW4qzjAEIMFgawMl6M1EEytDhQ5GyOA0dOnbQm3ma2CNuw3JrHIH5iYlITktD9TsYjWuBpc0EOJG6mQQ/ScCFBFauWgVUVmLixIku7FXtrvjqiH3+ixoXiXYB7ZGaxKnw7COon1p8z1I/vqQlJGCVgFeTJjh+9gv4+ra3WoYn6hI4UVKC/n37Yv+hQ/Dz9a1bgEcMQ4D3ZgzjarUNzcvJRWFRodpGuFE9s2Htg9+xUyeMGT8OyQtfsq8B1jIRKCgowN79+5WmwWCptPuMIT5v924MHj4U+/YdNIbBGlnJbFj7wM6cMQvbsrZzGjz78Jlq7dy5Ezt27HCgBfdXZbB0vw+ooB4CRYWFGB48GIuXLcPE+Lh6SvJUfQSYDVsfnfrPiVvXsTFxWPDygvoL8my9BCoqK1FZWVlvGZlPMljK7B1qQ7cePbB114eYxGQeh64GZsM6hA9TJk/BofwDyM/Pd6whg9f28FA35DDBx+AXr6zm/1hRAbGKPTfnEGA2rOMcF6elIicrF9k52Y43xhaUI6BumFcONQXbSmBXVhb69e5ja3GWs4EAb8PaAKmBInGxE3G+9CyyshksG0Cly9MMlrp0q7pG7c7Nw6iwMMTGxatrhITKmQ3ruFO8vb0wddpUJCclOd4YW1COAIOlci7Tt+Bnpz2PtDfSEBcXo29D3WAds2Edh/5YTCyufvc917tsJMrlS5ci6ZVXGllLruJ8KCSXPwyv5mBBAZ9VanAV8Dasc6CK5+iTpk7GSwteQnj4GOc0aoBWLl66hKvl5UpbypGl0u7Th/gDe/fCnFDOpB5tfMpsWOdxjY6OgcjqXJexznmNGqClSgZLA3iZJmpGQCTzPNRvALJ27tSsDzZMAs4kIEYYk6dMxZJXF3EJLxvBmiZSDwmxsbScxTiylNMvhlCVv3cvxo8bh9defwMjRowwhM3uMpK3YZ1LPjIyAs2aNcP6dWud27BOWxPBcsiwYUpbx2CptPvUFn9LixZ4bdnrTOZxgRuZDet8yNNnTsfS1CX48ccfnd84W5SOAIOldC4xjiAxO090TJRxDHazpcyGda4DRILPzd43Y8OG9c5tWIetnTpxAkeLi5W2jMFSafepJ37//r1Yu4aJEa72HG/DakN81pw5WLSEo8uG6GZs3Ig3lq1oqJjU5xkspXaPvsTtyd+DAQMG4Pvyq/oyTAFrmA2rjZNCQ0Nx+61tsP69Ddp0oKNWPZuoHW7UVq+jC0nvplz4+hsMHzocy5a9wUnR9e5sg9k3bXYClr7yqtIrahjMZXaZy4nU7cLGSvYQOHOuFAH+fvZUZR0HCXg1aYLjZ7+AWG6Km/MJ9O8/ENHREYiJ4TJyzqcrR4scWcrhB0OoYKB0n5uZDast+8mTn8arSYu07YStu5UAg6Vb8eu7c5HM07p1K1y8eEnfhipiHbNhtXOUyIy9rV0bpKev1q4TtuxWAgyWbsWv38737s1HcPBgzFuYZAqY+rVUDcuYDau9n6ZMnYJFr3J0aYn0unVrkbZ0iaVTyhxjsFTGVWoJ/eD995GUksJkHkncxmxY7R0xJnwMbrnlVojAwK0mgTNnzuH0iTM1Dyr2jauOKOYwVeQuTlumilTqJAGnEZg8eTJeXfQKHo2I5Oo5tah6Kj40U1x+LW/wq1sJ7N6zm88n3eoB653zNqx1Ns48ExEZgRs3KrFr+3ZnNqt8W5wbVnkX0gBnERDJPKHDQ7Hlg/ec1STbsZHAjxUVEO+x1rcxG7Y+Os49N3P6FKQuetW5jSreGoOl4g6kfOcQKCgoMCXzzJo1C3HxE53TKFuxiUD6mrV4MLAHegX1RM+egbhy+YrVesyGtYrGqSeiomPw3cVvkZWd7dR22Zh7CfA2rHv566L3jh06YlnaYsyePVsX9qhihBhRVlaUo7DwKEpLy3DLzc2wZLnljEPehnWtVydPm2JakcS1vcrb24ULF1BaViavQBuUMVjaAIlF6ifg3dIbMbGcuaR+Ss4/e5OnJx6Pia1qOCQkBF5e3lXfRQbikeIi0881VFYd5472BCaMfxwnz51Gfn6+9p0p0MOqVauwZOHLCii1LpHB0jobnqmHQN6ePYiIHFdPCZ5yBQERMM3bkeIS/PHJp8xfMWPGc+jb5w+mH1RUwhP8da+Co/GOt7cXJsbHYekSyyN9jbuXsvmK36h9/amtXspLQv+ixIQDYcOH496O9+rfWIksvP7DDxD/oZt/qkvLzsnBhOhoiFG+ecvM3I5Ll66Yfpo1awY+szSTcc1nbGw89u7PR2FRoWs6lLwX1V8d+fXfUslBU548BMaOGweRzMNnlK71ScVvfoOSI0fqdHrs2FH858f/IDR0eJ1z5gMMlGYSrvsUUz1GRsTgjRVv4o1Vr7uuYwl7mjV3roSqGieJq440jhdLA6Z3KcUfAm7uJyBWoC85cQojR44wLRG1a9cuiDUWa2/NmzfByZNcdaQ2F62/l/2zDEE9AnGoqBA+Pj5ad8f2NSTA27AawtVT02UXLlSZw0BZhcKtOyLDMDIyCpOengg/X1/4+/lh377PLGqq4PNKi1y0Pujb3hdh4aFYuWKl1l2xfY0JcGSpMWA9NL8nPx9DBw/Gp3/7FIE9A/VgkuFs4MjSfS4vPFqIoQ8PxunTp+Hl/eszZfcpcn3PmZs349vvv0dc7K/Z265X4ViPHFk6xk/3tUXW6/ChQzFr5kwGSoW9LUaWzIZ1jwN7dOmBoKAgvPnOO+4RIEGvR4qLUfz55xIosV8Cg6X97AxR8/LXX2PWrOmYm5hoCHv1aqQnKpkN60bnxsbHYfWKFRATSRh1q1D8VV8GS6NeuTbaHT52LGbPZqC0EZe0xfjM0r2uGTliJJo2bYYdWze7V4ibeg8MDEK/fv3c1LtzuuUzS+dw1FUrYlL0q+XlGNR/oK7sMrIxzZs3x8mTp+Hr297IGNxq++r01diy8QN8lPuxW3Wwc/sIcGRpHzfd1hKToj/8cDAOf173fT7dGm0AwypwwwBWym3i49ExOH7qOPbv3S+3UKqzSIDB0iIWYx4sLT2H4cOHYs6cmZg6ZbIxIejUak80ZYKPm30rpiaMjonF6lWr3KzE9d1funwZFy9dcn3HTuyRt2GdCFMPTWVlZ2HYkGF6MIU2VCPQpPlNOHvyLG/DVmPijt2yslIE9gpC4aEi+LRr5w4Jbulz3rx5+O7Cv5Gm8ExGHFm65dKRt1MGSnl9Q2XqE/D19cPAgQOxdlW6+sY00oJKL7VnV2WwbKTD9VZcPKPs2aMbxG0SbvolIF4d4SYHgfi4eLy1/i05xLhQRWW52tcgg6ULLxbZuio8dAiDg4MxfsIEtGrZUjZ51ONEAnx1xIkwHWyqf9/+aPPbO7AxI8PBltSpPmnSZMx8YY46gi0o5TNLC1CMcmj48OHo1+8hzJiRYBSTDWsnXx2Ry/Xpq1Zj3caN2L07Ty5hVGOVAIOlVTT6PyFmE6m+eLD+LTauhU2a/wZnT55ngo8kl4BYm/Tu39+FXR/loUc3rgsriVvqlcHbsPXi0d/Jo8XFVUYxUFah0P2OeHWEmzwEmnt5YVxENN7bsE4eURoqyc7KQsY6tW87M1hqeIHI1nRRYSH69emDjRs3ySaNejQmwEkJNAZsR/NPxkQjPX01rt/Q/4QRBYcOIW/3bjsoyVOFwVIeX2iqRGS9DhgwADNnzURExDhN+2Lj8hHgyFI+n3S5915079od76Yb7zUS+bzRsCIGy4YZ6aJEixbNMDtxLpN5dOHNxhvBkWXjmbmiRmRUFDI26f9OT8eOHdGpW2dXINWsDyb4aIaWDZOAPASYDSuPL2orEZPb7/pwF7r16FH7FL9LRIAjS4mc4WwpBw4VICUl2dnNsj0FCXBkKa/TIiIj8fY778orkMpMBBgsdXohHCooQOiAUJSVfaVTC2lWYwjwmWVjaLm27PhHxmP79s0Qr5PodROvqVVWcgYfvfpXWbvEhRkcHIzo+CgsX75MWTso3HkEOLJ0Hktnt9QzsCf82vthx84dzm5amvYWJSchNiZOGj32COHI0h5qktcR70+KBZxTUlIkV0p5riLAkaWrSNvXT1RMLNas3WBfZSVqecDDU+1wo7Z6JS4S14m8Ul5e1VmHjp2q9rlDAhxZyn0NjBo1CsVHDuDMuTNyC7VTndo3YH82msHSTufLVu1Q4QHc076Dbn/ZZOOtmh6OLOX2WOvWrTCo/xBszdwqt1A71UVGRmDy5El21pajGoOlHH5wSMWBvYcxdPBwjIsKR4B/gENtsbI+CXBkKb9fIyIjkL5Bn1mx/v4B6Natm/xOqEchg2U9cFQ5lbt7OyIjIrA4jck8qvjM1To5snQ18cb312/QIFRcuoHdeWpPC9d4y9WowUkJ1PATVZKAQwS46ohD+FxWWSyX19TTEwuTFrqsT1d0tHv3bly9fBUjwka4ojtN+uDIUhOs2jcq5no9p9NkAO3pGa8HjizV8Plj48djdcYqNcQ2QmV+fj627tjeiBryFWWwlM8nDSoSgXJwcDAydZoM0CAAFmg0AT6zbDQyt1To1rMHfFv5YuvWTLf0z06tE2CwtM5GyjNiPcrhw4ciNiYG06ZNk1IjRclHgCNL+XxiTdGocWOR9+mn1k4rebxdu7Zo376dktrNovnM0kxCkU8xO8+7a9YiLi5WEcWUKQMBPrOUwQu2aSgtPYfAnkE4/+V5iAnwuclBgCNLOfxgswoxOw8Dpc24WPAXAhxZqnMp+Pn5w9fXBzu2bVNHtAGUMlgq4OSiwkKEh4freqJlBdygtEQ+s1TLfVGPRSM7N1ct0TpXy2ApuYOPHTuK4MHB6Nq1E5p7eUmulvJkJcCRpayesaxr5JhHkJ21E1eu6GMlkuSUJMREq/3oiMHS8rUqxdGKyko89eSTiI2Lw/z5SVJoogg1CXBkqZbfAvz94Ofrh9zcbLWEW1FbcaNS+YnUPa3YxsMSEPD08EDm1u3w8fGRQA0lqEyAI0v1vPfooxHIys5GePgY9cTrUDGDpYRO/friRbRr3dqkjIFSQgcpKIkjS/Wc1n/gACxZsRgiA14k9qm8jRwxAt9++53KJoC3YSVzn3hG2bNLF2Tt0O9CsJIhN4QcjizVc3PPnj3QrmVbfJqXo574Woq79eiB/gP71zqq1lcGS4n8JQLlgAEDEB0bg2EjR0qkjFJUJ8CRpZoeHDZqBLJz8tQUrzPVDJYSOfSrL7/C009PQtJCJvNI5BZdSOHIUk03hgwKQe5H6if5iLlhs3epbQeDpUS/Q4NCQpCYmCiRIkrRCwGOLNX05AO9euFq+TUcOHBATQN+US1WHdm8Re35bhks3XwJirleN2/e7GYV7F7vBDiyVNPDIrEneODDOHjwMzUN0JFqBks3OrNETIo+bAj2/f1vblTBro1AgCNLdb3cd+BAbN64SV0DAHh6epp+VDaCE6m7yXsXLlxAjx49ERkxFovTlrlJBbs1CgExIffJk6fh69veKCbrxk7xKlmPLp1x8uRZeHtzFi93OZYjSzeRF+9PrluXzkDpJv5G65YjS3U9Lt657tq1K7ZxYnW3OpHB0o34Q0KGubF3dq0HAl9/8w3i4uMxdOgg7Nq+06pJfGZpFY0SJ/r07o3i4mIltOpVJIOlCz1bVFyMoJ49cfHiJRf2yq70TGDHli2YO3s2Zk6fixGPhOHy5csWzeXI0iIWZQ4OHDQM72/ZoIze2kLT0tIQrfhE6mrPoVTbIxJ/LzpyBKEjhmFg/4Fo1bqVxEopTSUCYlQpNj8/P3Tv1AlNrUyLxpGlSl6tq/WBXoG4fukHlF24AF8F54ouLy9H+Y3yuoYpdIQjSxc5a3naMoQMHIQ1a9dyjkEXMTdSNzt27ERSckqNZdyeefZZBAUFoXevIIg/VtzUJSBeIRk4aBB2bPpAWSNUH5mprl+ZC2d1+mpltFKoWgQ2bd6M997bgOysLJw9+0XVKjXDhg3D/d27m4w5+I/P1TKKausQ6Nq9O0pOn6hzXIUDffr0gX9AgApSrWrkqyNW0Th+Qsz16tveD94tvR1vjC0YnsDlS5dw513/beJQUVFZZ2HgsWPH4L777sfs2bPrsGrevAlOnvyCr47UIaPOgUOHCjFmbBjOnSlVR7SOlPI2rEbOFDPziEnRV69eqVEPbNZoBFq2aoVjR0/j2LGzOHn8bB3zu3e/D506daxzXByo4M1/i1xUOihWISm/Vo7Sc2UqydaNVgZLDVx56sQJDB40CKNHjcJz02Zo0AObNCoBn/bt4OPTFuJTrHM4OyEBRYWFpp8L//4XwsLCLaLxZLC0yEW1g/ff/wBy89SbkLygoABZWVmq4a6hl8GyBg7nfLmtze2YPGkSVq5azT9RzkHKViwQEEkf4mX1T/76V5SWlWHFstfg4WH5V5qvjlgAqOCh4OABKC4uUU55bm4uMtZlKKe7umAm+FSn4aT91q1bYYaF50ZOap7NkEAVgYjIyKr9+nb46kh9dNQ55x/QAZs3qrnwQoU6mC0qtfxvqMWiPEgCJKAqgQpUqiqduqsRCB7wME6dOafkxCaqj8xU11/tMuIuCZCANQKeDJbW0Ch1vLmXF3za+6C4pBh9+/RRRntCQoIyWq0J5cjSGhkeJwEdEWA2rH6c+Ye+fbE3/+/6MUgRSxgsFXEUZZKAIwSYDesIPbnqduvaFYcPF8olygBqGCwN4GSaSALMhtXPNdCpQwecOnVKKYOWr1iO2Ng4pTTXFstgWZsIv5OADgkwG1Y/ThWzNJVdKFUqyeeHKz+g/No1pZ3AYKm0+yieBGwjwJGlbZxUKCWmz/Rt296U5KOCXrNGvjpiJsFPEiABaQlwZCmta+wS5t+pI86cUGdSdTGR+qOjR9llqyyV+OqILJ6gDhLQkED62rdwa9vWGvbApl1JIC4uFv6+HVzZpUN9iWCp+sZVR1T3IPWTAAmQAAloToDPLDVHzA5IgARIgARUJ8BgqboHqZ8ESIAESEBzAgyWmiNmByRAAiRAAqoTYLBU3YPUTwIkQAIkoDkBBkvNEbMDEiABEiAB1QkwWKruQeonARIgARLQnACDpeaI2QEJkAAJkIDqBBgsVfcg9ZMACZAACWhOgMFSc8TsgARIgARIQHUCDJaqe5D6SYAESIAENCfAYKk5YnZAAiRAAiSgOgEGS9U9SP0kQAIkQAKaE/j/9LTmuNlBTasAAAAASUVORK5CYII="></p>
<p>smooth curve over the correct domain which does not cross the <em>y</em>-axis</p>
<p>and is concave down for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> > 1 <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>-intercept at 0.607 <em><strong>A1</strong></em></p>
<p>equations of asymptotes given as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> = 0 and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> = 3 (the latter must be drawn) <em><strong>A1A1</strong></em><br> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img 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"></p>
<p>attempt to reflect graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f"> <mi>f</mi> </math></span> in <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y"> <mi>y</mi> </math></span> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> <em><strong>(M1)</strong></em></p>
<p>smooth curve over the correct domain which does not cross the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>-axis and is concave down for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y"> <mi>y</mi> </math></span> > 1 <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y"> <mi>y</mi> </math></span>-intercept at 0.607 <em><strong>A1</strong></em></p>
<p>equations of asymptotes given as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y"> <mi>y</mi> </math></span> = 0 and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y"> <mi>y</mi> </math></span> = 3 (the latter must be drawn) <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> For <em><strong>FT</strong></em> from (i) to (ii) award max <em><strong>M1A0A1A0</strong></em>.</p>
<p><br><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>solve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = {f^{ - 1}}\left( x \right)"> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <msup> <mi>f</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = x"> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>x</mi> </math></span> to get <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> = 0.372 <em><strong>(M1)</strong></em><em><strong>A1</strong></em></p>
<p>0 < <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> < 0.372 <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Do not award <em><strong>FT</strong> </em>marks.</p>
<p><br><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = - 1 + \ln \left( {\sqrt {{x^2} - 1} } \right)">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo>+</mo>
<mi>ln</mi>
<mo><!-- --></mo>
<mrow>
<mo>(</mo>
<mrow>
<msqrt>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</msqrt>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
</div>
<div class="specification">
<p>The function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> is defined by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = - 1 + \ln \left( {\sqrt {{x^2} - 1} } \right),{\text{ }}x \in D">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo>+</mo>
<mi>ln</mi>
<mo><!-- --></mo>
<mrow>
<mo>(</mo>
<mrow>
<msqrt>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</msqrt>
</mrow>
<mo>)</mo>
</mrow>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>x</mi>
<mo>∈<!-- ∈ --></mo>
<mi>D</mi>
</math></span></p>
</div>
<div class="specification">
<p>The function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g">
<mi>g</mi>
</math></span> is defined by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g(x) = - 1 + \ln \left( {\sqrt {{x^2} - 1} } \right),{\text{ }}x \in \left] {1,{\text{ }}\infty } \right[">
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo>+</mo>
<mi>ln</mi>
<mo><!-- --></mo>
<mrow>
<mo>(</mo>
<mrow>
<msqrt>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</msqrt>
</mrow>
<mo>)</mo>
</mrow>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>x</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mo>]</mo>
<mrow>
<mn>1</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi mathvariant="normal">∞<!-- ∞ --></mi>
</mrow>
<mo>[</mo>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the largest possible domain <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="D">
<mi>D</mi>
</math></span> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> to be a function.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f(x)">
<mi>y</mi>
<mo>=</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span> showing clearly the equations of asymptotes and the coordinates of any intercepts with the axes.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Explain why <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> is an even function.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Explain why the inverse function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f^{ - 1}}">
<mrow>
<msup>
<mi>f</mi>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</math></span> does not exist.</p>
<div class="marks">[1]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the inverse function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{g^{ - 1}}">
<mrow>
<msup>
<mi>g</mi>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</math></span> and state its domain.</p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g'(x)">
<msup>
<mi>g</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, show that there are no solutions to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g'(x) = 0">
<msup>
<mi>g</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0</mn>
</math></span>;</p>
<div class="marks">[2]</div>
<div class="question_part_label">g.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, show that there are no solutions to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="({g^{ - 1}})'(x) = 0">
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mi>g</mi>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<msup>
<mo stretchy="false">)</mo>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0</mn>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">g.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^2} - 1 > 0">
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>1</mn>
<mo>></mo>
<mn>0</mn>
</math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x < - 1">
<mi>x</mi>
<mo><</mo>
<mo>−</mo>
<mn>1</mn>
</math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x > 1">
<mi>x</mi>
<mo>></mo>
<mn>1</mn>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img src="images/Schermafbeelding_2017-08-09_om_15.40.09.png" alt="M17/5/MATHL/HP2/ENG/TZ1/12.b/M"></p>
<p>shape <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 1">
<mi>x</mi>
<mo>=</mo>
<mn>1</mn>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = - 1">
<mi>x</mi>
<mo>=</mo>
<mo>−</mo>
<mn>1</mn>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>-intercepts <strong><em>A1</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> is symmetrical about the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span>-axis <strong><em>R1</em></strong></p>
<p><strong>OR</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f( - x) = f(x)">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mo>−</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span> <strong><em>R1</em></strong></p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> is not one-to-one function <strong><em>R1</em></strong></p>
<p><strong>OR</strong></p>
<p>horizontal line cuts twice <strong><em>R1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Accept any equivalent correct statement.</p>
<p> </p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = - 1 + \ln \left( {\sqrt {{y^2} - 1} } \right)">
<mi>x</mi>
<mo>=</mo>
<mo>−</mo>
<mn>1</mn>
<mo>+</mo>
<mi>ln</mi>
<mo></mo>
<mrow>
<mo>(</mo>
<mrow>
<msqrt>
<mrow>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>1</mn>
</msqrt>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{{\text{e}}^{2x + 2}} = {y^2} - 1">
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>1</mn>
</math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{g^{ - 1}}(x) = \sqrt {{{\text{e}}^{2x + 2}} + 1} ,{\text{ }}x \in \mathbb{R}">
<mrow>
<msup>
<mi>g</mi>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msqrt>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mo>+</mo>
<mn>1</mn>
</msqrt>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>x</mi>
<mo>∈</mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
</math></span> <strong><em>A1A1</em></strong></p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g'(x) = \frac{1}{{\sqrt {{x^2} - 1} }} \times \frac{{2x}}{{2\sqrt {{x^2} - 1} }}">
<msup>
<mi>g</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>1</mn>
</msqrt>
</mrow>
</mfrac>
<mo>×</mo>
<mfrac>
<mrow>
<mn>2</mn>
<mi>x</mi>
</mrow>
<mrow>
<mn>2</mn>
<msqrt>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>1</mn>
</msqrt>
</mrow>
</mfrac>
</math></span> <strong><em>M1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g'(x) = \frac{x}{{{x^2} - 1}}">
<msup>
<mi>g</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mi>x</mi>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mfrac>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g'(x) = \frac{x}{{{x^2} - 1}} = 0 \Rightarrow x = 0">
<msup>
<mi>g</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mi>x</mi>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mfrac>
<mo>=</mo>
<mn>0</mn>
<mo stretchy="false">⇒</mo>
<mi>x</mi>
<mo>=</mo>
<mn>0</mn>
</math></span> <strong><em>M1</em></strong></p>
<p>which is not in the domain of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g">
<mi>g</mi>
</math></span> (hence no solutions to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g'(x) = 0">
<msup>
<mi>g</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0</mn>
</math></span>) <strong><em>R1</em></strong></p>
<p> </p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">g.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="({g^{ - 1}})'(x) = \frac{{{{\text{e}}^{2x + 2}}}}{{\sqrt {{{\text{e}}^{2x + 2}} + 1} }}">
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mi>g</mi>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<msup>
<mo stretchy="false">)</mo>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
</mrow>
</msup>
</mrow>
</mrow>
<mrow>
<msqrt>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mo>+</mo>
<mn>1</mn>
</msqrt>
</mrow>
</mfrac>
</math></span> <strong><em>M1</em></strong></p>
<p>as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{{\text{e}}^{2x + 2}} > 0 \Rightarrow ({g^{ - 1}})'(x) > 0">
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mo>></mo>
<mn>0</mn>
<mo stretchy="false">⇒</mo>
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mi>g</mi>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<msup>
<mo stretchy="false">)</mo>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>></mo>
<mn>0</mn>
</math></span> so no solutions to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="({g^{ - 1}})'(x) = 0">
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mi>g</mi>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<msup>
<mo stretchy="false">)</mo>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0</mn>
</math></span> <strong><em>R1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Accept: equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{{\text{e}}^{2x + 2}} = 0">
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span> has no solutions.</p>
<p> </p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">g.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">g.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">g.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>A scientist conducted a nine-week experiment on two plants, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math>, of the same species. He wanted to determine the effect of using a new plant fertilizer. Plant <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> was given fertilizer regularly, while Plant <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> was not.</p>
<p>The scientist found that the height of Plant <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>,</mo><mo> </mo><msub><mi>h</mi><mi>A</mi></msub><mo> </mo><mtext>cm</mtext></math>, at time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> weeks can be modelled by the function <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>h</mi><mi>A</mi></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mi>sin</mi><mo>(</mo><mn>2</mn><mi>t</mi><mo>+</mo><mn>6</mn><mo>)</mo><mo>+</mo><mn>9</mn><mi>t</mi><mo>+</mo><mn>27</mn></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>≤</mo><mi>t</mi><mo>≤</mo><mn>9</mn></math>.</p>
<p>The scientist found that the height of Plant <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mo>,</mo><mo> </mo><msub><mi>h</mi><mi>B</mi></msub><mo> </mo><mtext>cm</mtext></math>, at time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> weeks can be modelled by the function <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>h</mi><mi>B</mi></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mn>8</mn><mi>t</mi><mo>+</mo><mn>32</mn></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>≤</mo><mi>t</mi><mo>≤</mo><mn>9</mn></math>.</p>
</div>
<div class="specification">
<p>Use the scientist’s models to find the initial height of</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Plant <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Plant <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> correct to three significant figures.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> when <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>h</mi><mi>A</mi></msub><mfenced><mi>t</mi></mfenced><mo>=</mo><msub><mi>h</mi><mi>B</mi></msub><mfenced><mi>t</mi></mfenced></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>For <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>></mo><mn>6</mn></math>, prove that Plant <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> was always taller than Plant <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>For <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>≤</mo><mi>t</mi><mo>≤</mo><mn>9</mn></math>, find the total amount of time when the rate of growth of Plant <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> was greater than the rate of growth of Plant <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>32</mn></math> (cm) <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>h</mi><mi>A</mi></msub><mfenced><mn>0</mn></mfenced><mo>=</mo><mi>sin</mi><mfenced><mn>6</mn></mfenced><mo>+</mo><mn>27</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>26</mn><mo>.</mo><mn>7205</mn><mo>…</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>26</mn><mo>.</mo><mn>7</mn></math> (cm) <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempts to solve <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>h</mi><mi>A</mi></msub><mfenced><mi>t</mi></mfenced><mo>=</mo><msub><mi>h</mi><mi>B</mi></msub><mfenced><mi>t</mi></mfenced></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>4</mn><mo>.</mo><mn>0074</mn><mo>…</mo><mo>,</mo><mn>4</mn><mo>.</mo><mn>7034</mn><mo>…</mo><mo>,</mo><mn>5</mn><mo>.</mo><mn>88332</mn><mo>…</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>4</mn><mo>.</mo><mn>01</mn><mo>,</mo><mn>4</mn><mo>.</mo><mn>70</mn><mo>,</mo><mn>5</mn><mo>.</mo><mn>88</mn></math> (weeks) <em><strong>A2</strong></em></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>h</mi><mi>A</mi></msub><mfenced><mi>t</mi></mfenced><mo>-</mo><msub><mi>h</mi><mi>B</mi></msub><mfenced><mi>t</mi></mfenced><mo>=</mo><mi>sin</mi><mfenced><mrow><mn>2</mn><mi>t</mi><mo>+</mo><mn>6</mn></mrow></mfenced><mo>+</mo><mi>t</mi><mo>-</mo><mn>5</mn></math> <em><strong>A1</strong></em></p>
<p><br><strong>EITHER</strong></p>
<p>for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>></mo><mn>6</mn><mo>,</mo><mo> </mo><mi>t</mi><mo>-</mo><mn>5</mn><mo>></mo><mn>1</mn></math> <em><strong>A1</strong></em></p>
<p>and as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mfenced><mrow><mn>2</mn><mi>t</mi><mo>+</mo><mn>6</mn></mrow></mfenced><mo>≥</mo><mo>-</mo><mn>1</mn><mo>⇒</mo><msub><mi>h</mi><mi>A</mi></msub><mfenced><mi>t</mi></mfenced><mo>-</mo><msub><mi>h</mi><mi>B</mi></msub><mfenced><mi>t</mi></mfenced><mo>></mo><mn>0</mn></math> <em><strong>R1</strong></em></p>
<p><br><strong>OR</strong></p>
<p>the minimum value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mfenced><mrow><mn>2</mn><mi>t</mi><mo>+</mo><mn>6</mn></mrow></mfenced><mo>=</mo><mo>-</mo><mn>1</mn></math> <em><strong>R1</strong></em></p>
<p>so for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>></mo><mn>6</mn><mo>,</mo><mo> </mo><msub><mi>h</mi><mi>A</mi></msub><mfenced><mi>t</mi></mfenced><mo>-</mo><msub><mi>h</mi><mi>B</mi></msub><mfenced><mi>t</mi></mfenced><mo>=</mo><mi>t</mi><mo>-</mo><mn>6</mn><mo>></mo><mn>0</mn></math> <em><strong>A1</strong></em></p>
<p><br><strong>THEN</strong></p>
<p>hence for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>></mo><mn>6</mn></math>, Plant <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> was always taller than Plant <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>recognises that <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>h</mi><mi>A</mi></msub><mo>'</mo><mfenced><mi>t</mi></mfenced></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>h</mi><mi>B</mi></msub><mo>'</mo><mfenced><mi>t</mi></mfenced></math> are required <em><strong>(M1)</strong></em></p>
<p>attempts to solve <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>h</mi><mi>A</mi></msub><mo>'</mo><mfenced><mi>t</mi></mfenced><mo>=</mo><msub><mi>h</mi><mi>B</mi></msub><mo>'</mo><mfenced><mi>t</mi></mfenced></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>1</mn><mo>.</mo><mn>18879</mn><mo>…</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>.</mo><mn>23598</mn><mo>…</mo></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>.</mo><mn>33038</mn><mo>…</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mo>.</mo><mn>37758</mn><mo>…</mo></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn><mo>.</mo><mn>47197</mn><mo>…</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn><mo>.</mo><mn>51917</mn><mo>…</mo></math> <em><strong>(A1)</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award full marks for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mfrac><mrow><mn>4</mn><mi mathvariant="normal">π</mi></mrow><mn>3</mn></mfrac><mo>-</mo><mn>3</mn><mo>,</mo><mo> </mo><mfrac><mrow><mn>5</mn><mi mathvariant="normal">π</mi></mrow><mn>3</mn></mfrac><mo>-</mo><mn>3</mn><mo>,</mo><mo> </mo><mfenced><mrow><mfrac><mrow><mn>7</mn><mi mathvariant="normal">π</mi></mrow><mn>3</mn></mfrac><mo>-</mo><mn>3</mn><mo>,</mo><mo> </mo><mfrac><mrow><mn>8</mn><mi mathvariant="normal">π</mi></mrow><mn>3</mn></mfrac><mo>-</mo><mn>3</mn><mo> </mo><mfrac><mrow><mn>10</mn><mi mathvariant="normal">π</mi></mrow><mn>3</mn></mfrac><mo>-</mo><mn>3</mn><mo>,</mo><mo> </mo><mfrac><mrow><mn>11</mn><mi mathvariant="normal">π</mi></mrow><mn>3</mn></mfrac><mo>-</mo><mn>3</mn></mrow></mfenced></math>.</p>
<p><em>Award</em> subsequent marks for correct use of these exact values.</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.</mo><mn>18879</mn><mo>…</mo><mo><</mo><mi>t</mi><mo><</mo><mn>2</mn><mo>.</mo><mn>23598</mn><mo>…</mo></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>.</mo><mn>33038</mn><mo>…</mo><mo><</mo><mi>t</mi><mo><</mo><mn>5</mn><mo>.</mo><mn>37758</mn><mo>…</mo></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn><mo>.</mo><mn>47197</mn><mo>…</mo><mo><</mo><mi>t</mi><mo><</mo><mn>8</mn><mo>.</mo><mn>51917</mn><mo>…</mo></math> <em><strong>(A1)</strong></em></p>
<p>attempts to calculate the total amount of time <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mfenced><mrow><mn>2</mn><mo>.</mo><mn>2359</mn><mo>…</mo><mo>-</mo><mn>1</mn><mo>.</mo><mn>1887</mn><mo>…</mo></mrow></mfenced><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mn>3</mn><mfenced><mrow><mfenced><mrow><mfrac><mrow><mn>5</mn><mi mathvariant="normal">π</mi></mrow><mn>3</mn></mfrac><mo>-</mo><mn>3</mn></mrow></mfenced><mo>-</mo><mfenced><mrow><mfrac><mrow><mn>4</mn><mi mathvariant="normal">π</mi></mrow><mn>3</mn></mfrac><mo>-</mo><mn>3</mn></mrow></mfenced></mrow></mfenced></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>3</mn><mo>.</mo><mn>14</mn><mo> </mo><mfenced><mrow><mo>=</mo><mi mathvariant="normal">π</mi></mrow></mfenced></math> (weeks) <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>Part (a) In general, very well done, most students scored full marks. Some though had an incorrect answer for part(a)(ii) because they had their GDC in degrees.</p>
<p>Part (b) Well attempted. Some accuracy errors and not all candidates listed all three values.</p>
<p>Part (c) Most students tried a graphical approach (but this would only get them one out of three marks) and only some provided a convincing algebraic justification. Many candidates tried to explain in words without a convincing mathematical justification or used numerical calculations with specific time values. Some arrived at the correct simplified equation for the difference in heights but could not do much with it. Then only a few provided a correct mathematical proof.</p>
<p>Part (d) In general, well attempted by many candidates. The common error was giving the answer as 3.15 due to the pre-mature rounding. Some candidates only provided the values of time when the rates are equal, some intervals rather than the total time.</p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>The voltage <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v">
<mi>v</mi>
</math></span> in a circuit is given by the equation</p>
<p style="text-align: center;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v\left( t \right) = 3\,{\text{sin}}\left( {100\pi t} \right)">
<mi>v</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>3</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>100</mn>
<mi>π<!-- π --></mi>
<mi>t</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t \geqslant 0">
<mi>t</mi>
<mo>⩾<!-- ⩾ --></mo>
<mn>0</mn>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> is measured in seconds.</p>
</div>
<div class="specification">
<p>The current <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="i">
<mi>i</mi>
</math></span> in this circuit is given by the equation</p>
<p style="text-align: center;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="i\left( t \right) = 2\,{\text{sin}}\left( {100\pi \left( {t + 0.003} \right)} \right)">
<mi>i</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>100</mn>
<mi>π<!-- π --></mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>t</mi>
<mo>+</mo>
<mn>0.003</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
</div>
<div class="specification">
<p>The power <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
<mi>p</mi>
</math></span> in this circuit is given by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p\left( t \right) = v\left( t \right) \times i\left( t \right)">
<mi>p</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mi>v</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>×<!-- × --></mo>
<mi>i</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</math></span>.</p>
</div>
<div class="specification">
<p>The average power <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{p_{av}}">
<mrow>
<msub>
<mi>p</mi>
<mrow>
<mi>a</mi>
<mi>v</mi>
</mrow>
</msub>
</mrow>
</math></span> in this circuit from <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t = 0">
<mi>t</mi>
<mo>=</mo>
<mn>0</mn>
</math></span> to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t = T">
<mi>t</mi>
<mo>=</mo>
<mi>T</mi>
</math></span> is given by the equation</p>
<p style="text-align: center;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{p_{av}}\left( T \right) = \frac{1}{T}\int_0^T {p\left( t \right){\text{d}}t} ">
<mrow>
<msub>
<mi>p</mi>
<mrow>
<mi>a</mi>
<mi>v</mi>
</mrow>
</msub>
</mrow>
<mrow>
<mo>(</mo>
<mi>T</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mi>T</mi>
</mfrac>
<msubsup>
<mo>∫<!-- ∫ --></mo>
<mn>0</mn>
<mi>T</mi>
</msubsup>
<mrow>
<mi>p</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="T > 0">
<mi>T</mi>
<mo>></mo>
<mn>0</mn>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the maximum and minimum value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v">
<mi>v</mi>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down two transformations that will transform the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = v\left( t \right)">
<mi>y</mi>
<mo>=</mo>
<mi>v</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</math></span> onto the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = i\left( t \right)">
<mi>y</mi>
<mo>=</mo>
<mi>i</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = p\left( t \right)">
<mi>y</mi>
<mo>=</mo>
<mi>p</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</math></span> for 0 ≤ <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> ≤ 0.02 , showing clearly the coordinates of the first maximum and the first minimum.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the total time in the interval 0 ≤ <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> ≤ 0.02 for which <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p\left( t \right)">
<mi>p</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</math></span> ≥ 3.</p>
<p> </p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{p_{av}}">
<mrow>
<msub>
<mi>p</mi>
<mrow>
<mi>a</mi>
<mi>v</mi>
</mrow>
</msub>
</mrow>
</math></span>(0.007).</p>
<p> </p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>With reference to your graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = p\left( t \right)">
<mi>y</mi>
<mo>=</mo>
<mi>p</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</math></span> explain why <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{p_{av}}\left( T \right)">
<mrow>
<msub>
<mi>p</mi>
<mrow>
<mi>a</mi>
<mi>v</mi>
</mrow>
</msub>
</mrow>
<mrow>
<mo>(</mo>
<mi>T</mi>
<mo>)</mo>
</mrow>
</math></span> > 0 for all <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="T">
<mi>T</mi>
</math></span> > 0.</p>
<p> </p>
<div class="marks">[2]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p\left( t \right)">
<mi>p</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</math></span> can be written as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p\left( t \right) = a\,{\text{sin}}\left( {b\left( {t - c} \right)} \right) + d">
<mi>p</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mi>a</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>b</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>t</mi>
<mo>−</mo>
<mi>c</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mi>d</mi>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b">
<mi>b</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c">
<mi>c</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d">
<mi>d</mi>
</math></span> > 0, use your graph to find the values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b">
<mi>b</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c">
<mi>c</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d">
<mi>d</mi>
</math></span>.</p>
<p> </p>
<div class="marks">[6]</div>
<div class="question_part_label">g.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p>3, −3 <em><strong>A1</strong></em><em><strong>A1</strong></em> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>stretch parallel to the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span>-axis (with <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>-axis invariant), scale factor <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{2}{3}">
<mfrac>
<mn>2</mn>
<mn>3</mn>
</mfrac>
</math></span> <em><strong>A1</strong></em></p>
<p>translation of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} { - 0.003} \\ 0 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>0.003</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> (shift to the left by 0.003) <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Can be done in either order.</p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAXIAAAGjCAYAAAAxRp6DAAAgAElEQVR4Ae2dB9gVxfX/J4KiIiAo2FusgKCCLdj/xt5Fxd5iiUZjjyY/Y4tKEo1GjTUqwWgSJPaaYK9YQFQUEcESCzaqgEi5/+cz+r3su9z3fW/ZvXfLOc9z7+7OTv3OzJnZM2fO/KhQKBSckSFgCBgChkBqEVgktTm3jBsChoAhYAh4BIyRW0MwBAwBQyDlCBgjT3kFWvYNAUPAEDBGbm3AEDAEDIGUI2CMPOUVaNk3BAwBQ8AYubUBQ8AQMARSjoAx8pRXoGXfEDAEDIFEMHJTZbeGaAgYAoZA9QjUhZGLUXOdNGmSe+WVV9z777/vc43b/Pnz3bBhw9y8efOc/FZfJAtpCBgChkC+EKgLI//Rj35UZNBPPvmkO/300915551XdBs7dqzbcccdHe/wa2QIGAKGgCFQPgJty/dau0+YdP/+/d2UKVPctddeW4ywe/fu7txzz3WdOnXyzN2YeREauzEEDAFDoFUE6jIjJxdB5rzSSit5EUvQrU2bNm7jjTduNcPmwRAwBAwBQ6ApAnVj5EoWGXiXLl08I5fb+PHj3RprrKHHosil6GA3hoAhYAgYAs0iUHdGzix8mWWWcdOnT3czZ870C5xDhgxxhx12mJ+18z44U2825/bCEDAEDAFDwCNQd0ZOql27dvWJf/LJJ+6MM85wRx55pFtkkYZkxZqBIWAIGAKpR6Du3BPRSseOHR0y8RtvvNEdffTRbsUVV0w9kFYAQ8AQMAQahUDdGTliE5h5586d3aqrrup69+7dqLJbuoaAIWAIZAKBujNyUEM+fuihh7qTTz45EyBaIQwBQ8AQaCQCdWfk3377rfvTn/7kBg4c6MuthU3b0dnIZmBpGwKGQJoRqNuGoM0228xtsskmHqvf/e53bvHFFy/iJmZedLAbQ8AQMAQMgbIRqBsjnzFjhvvwww/doEGDvHycGbgx8LLryTwaAoaAIdAsAj8q1EmmEWTcwSSNmTdbN/bCEDAEDIGyEKibjLwUw37mmWe8BkuQsZeVa/NkCBgChoAhUESgboy8mOIPdlcwmrXtttu6o446yu/wDL63e0PAEDAEDIHyEaibaEVZYvb90ksvuS222MLbIcf9lFNOcX/+85/lxa6GgCFgCBgCFSBQF0YeFJ18+umnbptttnFcZ82a5dq3b+9n5H//+9/dwQcf7LNeSgxTQZnMqyFgCBgCuUKgLqIVMWaY98477+w+++wzvz0fpFFF3Hrrrb2I5cEHH8wV+FZYQ8AQMASiQCA2Rs4sXD8yyvFuPXv2dOPGjXMPPfSQ69Onj88/W/Xvu+8+t/7667sDDzzQDR48OIpyWRyGgCFgCOQGgdgYuWbhIPnWW2/5mfjs2bPdv/71L7/IGUSYk4Eee+wxx4ETLH5edtllwdd2bwgYAoaAIdACArExctIUM+dot2+++cYhB99rr71KZoeZ+f333+/atWvnGX9JT+ZoCBgChoAhsBACddnZ2a9fP/f222+7tdZaq4m4JZyb9dZbz7388stunXXWCb+yZ0PAEDAEDIFmEIidkWtWDhOH9BzOj9zNrG0YGXs2BAwBQ6BlBGIVrbSctL01BAwBQ8AQiAIBY+RRoGhxGAKGgCHQQASMkTcQfEvaEDAEDIEoEDBGHgWKFochYAgYAg1EwBh5A8G3pA0BQ8AQiAIBY+RRoGhxGAKGgCHQQASMkTcQfEvaEDAEDIEoEDBGHgWKFochYAgYAg1EwBh5A8G3pA0BQ8AQiAIBY+RRoGhxGAKGgCHQQAQi36IfPERC2+4bWD5L2hAwBAyBzCMQCSOHef/zn/907777bhGwn/70p27LLbcsPtuNIWAIGAKGQDwIRMLIx48f74488kg3Z84cn0tm4hwSYWQIGAKGgCEQPwI1MXKJUf74xz+61157zZ8AJLf4sx59CuTdxEHR49pajGozhn1rSNl7Q6A0AlUvdqrzTZgwwY0ZM8bHPn/+/GIqae2UKlexIHZTNwQM+7pBbQllDIGqGbkYNaf+PPfcc/7MzbXXXtuNGDEilRBxgtF1112XyrynPdO/+c1v/IHcaS+H5d8QaBQCVTNyzZ622WYb94c//MHtt99+7ssvv3Rbb721P1yZ9/qFC6ewcg8/y70eV9Im3+edd5474ogjfJK4fffdd+7UU091e+yxh/8de+yx7ttvv10oS/gN/v7973+77bff3u2zzz5u11139V8rvIfkDz8sBrfkB3HVcsst5xZbbDEHxi+++GIxvOIhTs44XXTRRf2vbdu2fq1ioUxW6ED8kydPdquttpp75JFHfLrBKJQ+17PPPtuRLj/lg2uPHj2K+SWswhAfWJ5//vnFdnLhhRe6K664wo0ePTqYjN3nAAHahVHtCNQkI2dWvu222xYPU544caLbd9993cEHH+zef/9916VLFzd16lS39957L5TTmTNnejcYxgcffLDQ+3o5sEALY/n973/vGTpMHbr44ovdJ5984q6//nr//Nvf/tYdffTR7pJLLmk2a6+88or7+c9/7g+S5gxSDpreeeed/VmkHTt29OHuvfde97vf/a5FP3/+85/dyiuv7IYPH+6GDRvmTj/9dHfAAQe4J5980rVp06aYPvn79NNP3aWXXlp023zzzT32RYcqb8455xwfN3XaXP2AHfljRt2hQwe/vkDHfPrpp93yyy/fJNysWbPcr3/9a5+3q666yq2++uo+Z4qbAelnP/uZ/yoCO6P8ILDqqqs2adf5KXl0Ja16Rk4WNJrC0PnReWFUyMoff/xx74Z7t27dFvqps/K+kXTccce5NdZYw+dd+Xj++efdbbfd5o4//ng5uUMPPdTdeeedbtSoUUW38A3Mni8TlQ3NHQ6Tvvbaa73XsWPHul/96lfN+gFPvgQY+IgH2mGHHdwJJ5zgPv/8cwczDBIM8aKLLvJ+8c+PAaBWIk6dr9pSXDDhW2+91X8VkHb//v19HhgMg4M37QEmPXfuXHf33XcXmThxgwlfOksuuaQfrBgUjPKDwBdffOE++uij/BQ4rpIWIqb58+cXdtlll8Idd9zRYsyjR4/mm6owaNCgFv3F+fLRRx8ttG3btnDvvfc2SWbXXXf17lOnTi26z5s3r9C+ffvCueeeW3TjhvLye/HFF32YW2+91T/L06GHHlro3r27d7v00kuLfoJhDzvssKIf4gq+4/nUU08tHH300cW0cHvrrbcKSyyxRGG11VYrnHbaaYUPP/yw+F5pV3ol3sGDB/v0Jk6c6PP68MMPlxUNYfkRbu211y7mBbebbrqp0K5du8K4ceOauPPu+eefL0ybNs2n8fHHHxcWXXTRwgsvvOD9lZWweUo1AhMmTCjwM6oNgZpm5C0NLptssklLrxPx7qGHHvL56NWrV5P8PPPMM15UgLhAtMgii7iuXbu6cePGyanJlTDQiiuu6K/6WkHOjZ79vHnzivLylVZaqRhWXyzywwuF5f7111/3oorLL7+8iWokm6/4SoCuueYa95Of/KSYt2B476GMP8IwM2bGjLy6UqIc/P773//6r4hg+BtvvNHRHvg6QWx02WWXua+//jroxd+DC+K4q6++eqF35mAIGALNI1A1I6fj80NOC4l5ILPl8x4NlqQTMm0YtJgv+UVmjwgDmXZQ7EP5WMT76quvShaLNQH8dOrUqfheYRAtEC+fkcQpeTkew36U5owZM7zse6eddnL/+9//3ODBg4vxcoPo4qabbnLvvfeeu/nmmx1rDiw0M9AojiYByniAgSP6qYYoB/TYY4+5HXfcsRgF7eONN97wA9l9993nVlllFb9wy+BZSky1wgor+DgY+IwMAUOgPASqZuREj2yLRSu0M2655Rb/Qx2RGZU6dnnZaIwvdOCZMTNTFEkXHm2RMCHjhfEHSUwTxsM9zD5IYkiE0z1+gvjIHT9yb9++vTvjjDO8fB2Z+1lnneVghJDS5J7Fz8MPP9zdc889fpaLCqXiCOajtfuXX37Zz8iZ2YdJ8XHVfdgPzwyAzz77rGPwIY/8nnrqKb9mctppp7lf/OIXXo4+dOhQP/M+5ZRTmkRD3NTHlClTmiyUNvFkD4aAIbAQAjVprbDazGagkSNHuiWWWMKhMbHssssulEhSHSZNmuQXOoP5g2kyY2YGDWMR0+SKGzPGUrTuuut6Z/xA+Cc8z2Cz9NJLF2f+uOm94pWfYFgGGBYQUWfs3bu3F1vsueeePv7wHyqK+EVEQ5yVEAwYVcuTTz7Z3XXXXT4o2kYQewTIG9pJUEtx82Ww1VZbucUXX9z75Q/9fAixlIjBBy0cNIAYHEXErUEu6K73djUEDIHSCDSdXpb2U9KVTscP7QY6JfrWMHG5c006IY8VU1VemRWj382sEEavGSgy3enTp7tNN9206KYwXAlDmcVINQig2bHxxhv7mTwDHYQfxcv1ww8/LPpRnEH8GFxg1DA53Amj8PLPFb1vfmEq5TfoZ9q0aT7c/fff79Bx5/fggw96L2jwoE4YJuVBV1QR0X2Xto389+3b1+f5448/9le5w9jRVEH/XERcfBFRxqC73tvVEDAESiNQNSMvFV2Q+ZR6nzQ35LUwclT+gsQmFwj7MRAM5p133vEMRjNiMTDeU25UGBFLIPflnYhwCrPXXnv5d2HZMF818qP0gnHgxqwZZh6koB8Y6QMPPOAH1aC7/ONWyp33qIdivTL4u+GGG7x/sGCzFKT6LRUP6w2sH2y33XZK0l9Z5AQbdvwGwzF4MMAFifj5EmDGHlwQDvqxe0PAEFgYgUgZ+cLRJ9sFRgLzYLYYJBZqkTsPGTLEv8cPeuVoiQT1tHEPMkg20Tz88MN+5k58MENm+MQFIZa58sorm/hBowM/7ColPhg24g0WOHnmx6wYBrfLLrv49PgyYDetZsp8PSB/ZjMWoo3miLiqJZUzyIyDcaGtcthhh7mllloq6FwcDP7xj384NhdBs2fP9mVkg1CY8IMYB3GOkSFgCJSJQG3ai9WHToIe+WOPPeZ1paVHLl1ortOnT/e623vuuWdhxx139Lras2bNKupBjx8/vtClS5fCeeedV3Qj3M0331zYbrvtCrvvvnthwIABhbFjxzYBqTU/Y8aMKfTq1cvrXaPPjt76wIEDC+i0K39fffWV19VHj3z55Zcv4O/ZZ5/16eAH0vWee+7xcf3qV79qko9SDwrDO/TB27RpU0CPXOnipjIrvML07t278MEHHxT9Bt/j56qrrirgB0zAdOjQod5vUI+ccpkeuZDLx9X0yKOp5x8RTZk8P1Jvb731lje0NWjQoEjsg1SbOXZfoi+OKp9mrEDCvaAJ3gfTQdWuZ8+efqYd9BO8xz/PIsUpd6VV6r3cdFU8ChOOS/6C7szembmzAxPNltYoGLaU3+bKHPSrfOIWji+MDTZkUEVkJn/HHXe4Rx991KH5BAXjCcZv99lBALVdCPGbUfUI5Fq0ApNhIHnppZccets8i/FwhZGI8eheUPOM4StpsSic/OMvHCboh3fyI3c9+xc//CkO+de7csMwSLEQXeniYTg9PVNm9O55Vh70TnkLXvVOV8Jwr+egX0RZGNMyMgQMgcoQyDUjBypksehey/BUkMGUYlRsvMFPkCGJMQXDyk+wOuRP8eoaDBeOV36C8XCvMEon7I9nfszIBw4c6NULw3GUelYelUbwOexfeQi6h93Ig9wUl/Iqd8KztoBdGdYn5C8Yb6PvlWddNfArX7jrJze7GgL1QiD3jJzOt8UWW7gBAwb4xU2AFyPhqs6JKIidk4gA0PI488wz/QJkqYoSg9I17Efuugbfh93Cz8pf+Br0R571HhVK9MP79evny+JfVPlHGsF0lIau4XdyDycX9odIBSKfSSThyZUdthiHox1ssMEGXgyk90nMu+UpHwjknpGrmtlwg7w82Cm5h+kgY0YbBE2Sv/zlL17Fju3sbEVHyyRpJEZJ/tl5yy5RSO5Jy+8hhxzidwcnNX/K11//+ldv6ZH9ALQD7arF1k2S8U1afVt+YkAgmjXTymNJgtZKqVxLQ0PX5557zmtSrLHGGgWs84mw6IeGxcknnywnu1aBQFBrpYrgdQvy9NNPe+2f/v37F7CESfuYO3duAcuVaPdg/RLC3ah8BExrpXysWvJpM/LQ4KjZl5zPPfdcx2YbrPaxSUUzdg6jQG8aW+NmT1loZfOKPJzdyxDrKej90064cvAIhtL+7//+z78Pt59sImKlShoCxshL1Ig6I7sVMfqE/Hz33Xcv+hQzZ8cjxrUwy2qUXQSwIcPBHhj5Yn2E+lcbwMwATJx2wqEqcs8uGlayJCJgjLyFWkEVbs0113R/+9vfivJlmDw/OixGw1igQ3bKYqh14hbATNkr6pIfs3GMe2GVkZ27kNqArieeeKJXyeS4Ok5HUlhrDymr9BRn1xh5icqjA3KgA5tTsDUSNHOL92AHRXtFi14yR1siSnNKKQKYZoA5o1mDBctShLVH7LhjKgGTCzB4SNdSYczNEIgSAWPkITRh0nRAOi7WBjlIujnCH1b82HCDKd8XXnihOa/mnkIEqF/UDTH8hSnh5og2w3oJm660K7U5v+ZuCMSBgDHyEKp0XmbWHH7AKTyo7uEWJJ6DbqjPQTB/OnVwxh4MZ/fpQYD65fQlDtxgMNcAX6oE+GW2/stf/tKf0PTmm282aR+lwpibIRAlAsbIS6CJzBsb4cccc0yrjJlOvNtuu3l7IVj4087PEtGaU4oQgHFz6hU7f5F9Q80N0HJnTYVDSTD7ILcUFdmymmIEjJH/UHl0PH4cQHzBBRd4c7XYFQnOvEvVM2FQQ7vqqqv8aTgcXmyUXgTEgFE5RT6+//77e/XCltqB3sHE8Y/NGMWTXiQs52lCwBh5qLaw+YGqGbs81UG56j7k3T/SabGhvdFGG7nbb7+9lBdzSwECYr5csRbJQeJHHnlkMeettQHC0W4Ihw15I0OgXggYIw8gTUflgARo5513bpF5B4L5W8Kia/7kk08mctt+OL/2vDACYtRcL7/8cm/euKWDOoIxKCz2eNBiUTsK+rF7QyAuBIyR/4AsHZEZFR2QszzLNTKlDkw0WO/jk/zxxx+Pq74s3joggAYS56qy9oFqaTlE26EtsEEMZm6MvBzUzE9UCBgj/wFJOuLo0aMdhu45UZ5ZVZBJtwS4/GFFEVvdHF4MEadRehBQfXF2KYS8W26tlUJtAH+cv/rqq6/6o+0UXtfW4rH3hkA1CBgjD6DGyfHojrNTD6q089GZ+/fv721rz507t6o4Atmx2zojIGbMSUp8lXFGaqVEm0HENn/+fN8OCF9pO6o0TfNvCBgjD7SB+++/35uoXWaZZcqejQeC+zDMxtgJyOYgMYagH7tPLgIw3DfeeMOLVU4//fSKvspUKup8tdVWcz169HD/+c9/rA0IGLvGioAx8h/gRe1wxIgRbsstt/QzqGpnUYhX2OGHDXPiMGYea/uNNHLq6plnnvFxHnHEEf5aTTsgDIukzz33XHE2bu0g0qqyyEIIGCP/ARBkmixUlqulEMLRP9KBka1z8ACMHKqGEZSK29zqg8CwYcNc3759/T6CalJUfbNYjhoiuzyNDIG4Ecg1I6fT6ffII4/447vogNXOnhRu11139TM7jlkzSjYCYrxcOe3psccea2KyWHVaSSkII9ss7BKuJo5K0jO/hkCuGXmwg6H/jdoYi50Q74Lvy2kqYgo77bSTn91j/tYoPQg88cQTnpnL9nyl9U9JCUM7QHtpvfXW8xvEvvvuu/SAYDlNJQK5ZuTqeIhUkI8jEqmm84Zrvk+fPo4DB6TGFn5vz8lBQIyXHD377LN+ExA7dGsl4t1ss828adu333671ugsvCHQIgK5Z+TMnkaNGuVn0HS8WojOq4EAEQ2Dw8SJE2uJ0sLGjAD1rzpjXYPFamzn1EKKD/O3ECdNGRkCcSJQW4uNM2d1iFuiEDowIhU6sdxqTX7HHXf0UfC5bpRcBGC61PnkyZM9w0UsFhUhqoOY6RsZAnEikGtGrpkT5y0ye0IcIrdaQd9uu+18FGhBGCUXAc3IGXCxQ4+NnaioZ8+eXlaOGqKRIRAnArlm5HRiduBhqU6zp6hm5MjJWfBikCDOqOKNszHkNW7qhs07nMG67rrrRgYDkwKsIWL24ZNPPrF2EBmyFlEYgVwzcjoaJ8BMnTrVm6AFnKhm5MhZOQKOAyrY6QkZMw83v8Y/q77ZifuTn/zEZ0huteaOeLTNn3UYI0MgLgRyzcg1EwPcjTfe2GMcFbMlnr322svHqcWuqBhEXI0hr/Fif37s2LFeaylKDGgD7BTm8O7XXnstyqgtLkOgCQK5ZuQwVg7LXWGFFdyPf/xjPxuPktmy4Nm5c2cvXgH1qAaJJjVoDzUhQJ2g7498HDPEUdY/ccHEWX956aWXfD6jjL+mglvgTCGQa0Y+YcIExw9VwTgI8QozMlvsigPd6OIcPHiw69atm2NxMkrSwL3pppu6F1980UcttyjTsbgMgVwz8tdff923gCg2gISbEjMvfix6Ih9lUdUoeQiwhvHOO+8U5eNR55A2sMEGG7ivv/7a216xGXnUCFt8IGCM3DnXq1evyFsDMy9+GGDCsiK7+6wTRw5zTRFSPyx2Q9q8U1OEocDUN2nAyCEMaPFsZAhEjUCuGbk+d9maHxdpp2DQpGlcaVm85SMghiorlRyeHTUpDUQ2WMXUoBF1OhafIRA5I1fjTQO0aJMg+ujatWvk2dXsm8VOjCeh3maUDASCbRQ9/6WXXtrPyIPuUeSUNsCPcz+Rk2MqWe0iivgtDkNACETCyOkA+hFx1B1CmY3qSv7effddvy177733jiraheJRR2bBk01HRslCALk1Iq8BAwb4Q5Ojzl2wT7AOw4w86X0jagwsvvogEAkjD84yzjjjDHf00UcnusGSXxn8Z8NGMP9xwI6MlN1906dPjyN6i7NCBFTfI0eO9CFlX0XuFUbXrHfiU5ysw6Cvbjbqm4XLXtSAQCSMXOmjBXLDDTf4RzVgvUvKVbOkt956y8ste/fuHXvWtJjKphCbkcUOd9kJIFpDzxt9/zjqJRgn4jVo9OjRZefPPBoC5SIQGSOfNm2aGzhwoFt99dV92sFGXG5m6umP2RiaClg9jDuvSie42BV3mvXEMq1psQCN6eL27dv7mXPUdaLJDFfWYpCV2w7PtLaWZOc7EkYOE99nn33cr3/9a29BMNlF/t6eihY66WTqcHHlG40FxCva3Rc1w4gr31mOlzpgATp42Hac7WCJJZZw3bt3dxLnZBlbK1v9EaiZkdMhzjvvPHfRRRe5DTfcsFgCdYpSTKuUWzFgzDfkC1klB+MyU65HXkiDmZ/UHWMuokXfCgLUB2skTEB0RqvaaytBa3rNgqdm5PVodzVltg6BDYPoQK6JkVMRw4cP97Pwcg9lIEy409S7QiWnRG4Zzkt00C6IiTSQxTN4MIhA9Uh3QQ6Sf0cbqFc7AHsxVDZs1YMoW48ePbxxLqVXr/IqvaRdrQ9EVyNtq42KRshZl3/5y1/c9ddf7xlTqYaJCdfllluu2WQ++OADx8HH9aRHHnnEyythqvU6wWfu3Lm+iDfddJP/nK9neZOcFiqA33zzjWvbtuqmWFXxHn30Ub9/APXDep2piWEu+sygQYPcaqutVlW+sxZoxowZbv31189asepenh8VSnHfMrNx4YUXOioiuL35nHPO8Qb6TzzxRLf99tt7zRAabpg+++wzd8kll/hGfeSRR4Zfx/ZMcY866ih/rBeaKyp+nLMD0pg9e7br2LGjt1F+1113xVa+tEWMnBqtng4dOtQ165gtZmH+3//+d13SpQ0wacHK5vnnn+8uuOAC3/bibHd1KViNiaCWC62xxho1xpTv4FVPg2iYiy22mIMZ0kBFnH2IgaihQ4f6lXoaLkw93GAJByOvN5EPjFhJnh/OV1z5Qc2NmQczwe+++85ry9Qr7bjKlNZ4Z82a5d544w138MEH160I1DUDR6dOnRzH/8HMjQyBqBCompHTMNFSgaEHGRIzHWZYt956a1HcEnwf9h9VQSqJh0MEOIKrXkT5KTdycmSz48eP9xoM9Urf0mmKAAudiDjQJKp3e2RdBjVU0kf11cgQiAKBmhY7yUCQSYczVKqTtOQ/HD6OZ5go1gjXWmutOKJvMU4Wu6CgSKfFAPYyUgRoj5AWOmHk9W6P66yzjmO9hHZoZAhEhUDNjLyljNS7k7SUF72TDXKJVuRej6uOkzNd4nqgXToNmDkzYhjqMsss42fkpX1G66pBRO0O0Y6RIRAVAlWLVprLgM6nbO59o92Rj7PouOaaa9YtK+rE7O6DsIJnVH8EmFhQF6jMotcP1XuyIZMQiHcw1mVkCESBQOQzcjqGflFkMIo46LxipsyE1JmiiLucOIQH5lJhIMzI681Aysln1v3QBlCHHTNmTJGR16vMagNqezBytcl65cHSyS4CkTPyJENFx+FYLz6rG0VsCUd3GvVLo/ojwL4B2oEMmdU7B5wNikiHwcTIEIgKgVwxchY5UZXE5kWjZsRbb721rzvryFE14criQfUPagQjZwDhx2D+0Ucfec2VynJvvg2B0gjkhpHDuNl8wsYcOjEdqhHEJinUzpDVq2M3Ki+NKH8j0wTne++915/YhJir3iTxyg477ODb4bhx4+qdBUsvowjkgpHTgejEjz/+uK9GySnrXafkYckll/SMZMSIET75Rn0Z1LvsjU4P7PkK4mAHjl2DGoW9zgeVzZ9GY2Pppx+BXDByOjGdlhk5W8FXWGGFhtScGAf65JqNkTej+BEAexYYoUbb9qD+MW383nvvxV9wSyEXCOSCkVOTbMJANZKNQI1inhpQ2N0XlJGLweeixTWwkDKOxRdZozFnwV2DeQMhsaQzgkAuGDmdltnPzJkz6656GGwn5ANmzqYQLP7RkeUW9Gf38SCA/jhUL9O1LS8983MAACAASURBVJWCWbkGlpb82TtDoBwEcsHIYZ7aSdfoz2oqRTJ68qRZejmVZX6qQwCMIfT32ZaP+l8jicEbRo6pBpHyqGe7GgKVIJALRg4gmv00Qu0sXCGY7MQKngYX68RhhKJ/xlwq+vv77rtv9JFXEWPPnj39FyL5svqvAkAL0gSBXDByZkATJkzwBW+EsSwhToclLxAy0nfffVev7BoTAsJcA/l2220XU0rlR0ueMO8MmZy8fNzMZ/MI5IKRU3x2dGITfJVVVmkejZjfiImTzLrrruvzFHOSuY8ezGGcfP1Q/zJc1khgyNPaa6/ts8CMPNguGpkvSzu9COSCkdORkUdutdVW/jAMOk6jOg95gRDxMLiw29QoHgTAWnijtw/mqP01msgT+wk47s1MGje6NrKRfi4Y+YcffujlkdqIkYSqY8GTXabothvFg0BwsGZGzmy8kYN4uJTIySXyCb+zZ0OgEgRywcgli8aMrGZolYAUpV8xEkQrUNLN/kZZ9kbEBd4MmMii0RRJAmmAQbyiTUFyS0L+LA/pQyAXjFwHvG6++eYNE6mEmwayemS2zz77bPiVPUeIAAO3Nl9hLK3RA7mKBuNeddVV3ccff+wHmqTkS/mza7oQyAUjRw6JPLJz586JqZ22bdv6GSKMfN68eYnJVxYzoq8enc7TaKZJ+vwQrVD32qiUReytTPVBIPOMnA6DHDJJszFVLZ/606dPL35ey92u0SHAzJfj/RjI2QjEc6PFGEpfdvHtqyy6+s5rTJll5Jr10Gn4tJYNctyTQuwyhMwKXnw1Qn1jMljH7KldxJdi+TGvvvrqXnvFGHn5mJnP0ghklpFr5jVr1iz36aefFs/o1GyoNBz1dZUusRa86pt6PlKjvrF6qNlvUupf+eDsWIl+8lEjVso4EMgsI9fMWzs66TByiwPIauLUAdDGyKtBr/Uw1Pf48eO9+AobO2KeurYeQ3w+1BaZlU+ZMsWfJRpfahZz1hHILCOns9JZpLGAul8SOnCwQZEnNqhoUwj5VQcP+rP76hCgvh966CEfWGKs6mKKPhR54yc11LFjx0afiMWYGwQyy8hVg+yeZBcdMx8oKYySfHDkGyqRwU0hSRtshGNarw8//LBX82SNJImkfGnCkcQ8Wp6Sj0BmGbkYdlIXOtU0ttlmGzdt2jSvTyw3u9aGAHXP77vvvvN6+ohV2rRpU1ukMYUWI9emtZiSsWgzjkBmGbk+XZGRY2mOji23JNSp8iIjTshyjaJF4IMPPvCmGcQso409mtjYFARZ/UeDZ15jySwj14yczowOMSS3pFQ2+dFBwCx4mlglmpoRjhJXoLEit2hSiC6WFVdc0Z8jiz0gI0OgWgQyy8jpuJ999pmbOHGi30EHQLgliZmTn65du7qVV165iZy82sq0cN8joDrWwR06kSmJ+NAG+vXr51jsJN/6JTGvlqfkIpBZRg7ksiwozQDc6DhJIx37JQaUtPylNT/s6ITYmp9UbMnXFlts4UVAH330UVqhtnw3GIHMMnI6iDZaIIdMIgMXc+HoN9Mlj64nqK6Z5bZv396L1uQWXSrRxSQbMBKvJDmv0ZXaYooSgcwycjoDn9boaSOHTBqJiZMvNgb973//8waUgu5Jy3Na8gOG/BgcEVvoOan5l467Nq9ZG0hqTSU3X5ll5HQGRCtJ3AgUbg6IVubMmeMZj83GwuhU94w6H6cv7bDDDtVFUMdQfDFi0EsqiNYG6gh+RpLKLCNH3jh16lTPyJM4w6GzqsNKPY7NS0nMaxrbuhY6d9ppJ599YZ3EslDn2N0RI7c2kMRaSnaeMsvIteU56TNyGAy7ThEBSV0u2U0mHbmDkXfs2NGf00mOk84caadi5OlA2HKZJAQyy8i1cCQd8iSBHs4LzJx8IiNN8swxnO+kPoMh8nFpK6UBUwZz9jxAachvUus+r/nKHCPXzEvHu6ERknSi47LgqTwnPb9pyB+zW1mXJL9JZ44wcg4ZQRyoNpwGnC2PyUAgc4wcWOkI+kzVrCwZcDefi/XWW6+4KaR5X/amHATmzp3rxVRsBEo6A6c85FHtVCLBcsppfgwBIZApRh6cyfBp3alTJ696GHRXwZN0JX90ZA7i/eabb5KUtVTmhbpHYwVjWWkg6l+HjBgjT0ONJS+PmWLkmn1xRd6sjRZyTx783+eI/MnMLsaTkj7wJBVH5UsGqDTLlXsSr9Q9P9QPMbfMfoKkt9ck4pj3PGWKkasyJ0+e7OWNYuRyT+oVxo2FRohFWjqyMfPqa4tZLWZrZVmw+pjqF5I6ZzDXIFS/lC2lLCCQSUY+btw4Xzcc2pAGUidu166dl+3DxG1WVn3NcVAHs3HwTBMhXjFGnqYaS05eq2bkMBvNGidNmuTuvfded9ddd7mvv/7al07v6l1UGCAba6DNNtus3slXnR4zSLQsmE1ShkbhV3UBEhBQmHF0Xs+ePVM3GMLIke+rHAmA1LKQEgSqZuSaMT777LNun332cf/4xz/cqaee6lD3e/HFFxtWfDqBFjr5VFU+G5ahChKmI2uxK035rqCIsXkV8+PKZiDMHqSNqP9PP/3UW0JMW94tv41FoGpGTrZhNg8++KB76qmn3J133ulGjhzp2rZt6y6//PKGlort+WLi6uANzVAZiYMlcnJk5OQ5Lfkuo2h19SJGqDWHuiZeY2KS6Zt4pUYgcxi8JkaOiteJJ55YFAVwSAJGimbPnt0wKGGI7JBbZZVVGpaHahKGcTP4iBERhzHzypGUBUFmt2nDj/qH0FwxMgQqQaBqRk4nYTGJWUSww8yaNcudffbZleQhUr/khcVONtiIgvmTW5Kuyh/MZ968eW7EiBFJyl4q8iJRFAudUBpFK3xFsFaixfpUAG+ZTAQCVTNy5Z4OpN8tt9zill12Wbfllls2TDY9c+ZMP6uFKYrUyfWctKvyJ71nnWwk96TlN6n5AS8WOjk6D4NZacNvscUW8xMj1niMDIFKEGhbieeg32AnYRZ01VVXuaFDhzp0uLGtfdttt3nv7FQ855xzgkH9PZouEOIZwgTjW8hzBQ50ZGiFFVZwU6ZMqSBk470uvfTS/iuHxWLZ3IgKl8aXrmkO+AqhbIjhpk2b5thWX0tZFR/1zyBO3dcSX9Pc1ueJMjArx7wE+VeZ6pN6/VOhfDNmzPCnONU/9WylWDUjD8KAPe0bbrjBXXzxxW7bbbd1t99+u7vwwgt9o0TUct111wW9N7lnW/rw4cMjabQ0jFdffdXHDyNspPZMk0JW8MAANHr06FTmvYJieq/UF0ycwZ5F8igYF2KJjTfeuIhfmpg55V9iiSW8nRjabhR4VFon9fYPf9hoo43qnWzm0ouEkYMKHQaxyk033eRFK8wqUEXs0qWLZ0xh5Ph83Hvvvb3u9M477xzZ7EmmYPfbbz9v4zucblKf6bQQmD355JNu6623zvRMReWFYfXq1ct16NChJsZFfHzdTZw40be/XXfdNalV3Wy+KMNrr73mnnjiCac+gVuaBqNmC1fiBWWT6d4Sr82pAgSqZuRUQpD0zBmJLIIuv/zyvgEy02JzRnNEI42yoTJAsADLQQ1pImGAWABGzswyLSYGqsFZ5Q2GLeUWfN/aPV92iPVYa1B7rDXO1tKM+v1aa63l9cjZ1BZcsI86nSTEl7a6SQJmzeWh5sXOcMRvvvmm31GpA2XD7+N+hgEiZ1RHjju9qONfZ511fJQMSGktQ9SYlBufRGna1ZlGRiEb6q+88kq5xTZ/hoCrmpHTSVCT45P4yiuv9LNq5N3olQ8ePDjSWXYl9YQeNjPyNHZiyildYhnPqqTsefZLfb/88stu0UUX9RimdRDUpiB2p1KmtLbjPLfFRpS9akZOR+Ez8LjjjvOLlQcccIDXWkFzhWPLGtWRkLmFddsbAWy1aWojk20KqRxBdhaLEaaVAbKpDrEga0yN6kOVI28hGo1A1TJyOgoHN/zpT3/yswYaHW6NbHxfffWVV9tigElrR5ZoxTaFVNY1vvjiC78jcvfdd68sYAJ9036ff/55nzP1qwRm07KUIARqmpEHmaXuueq+3uWUjQptrKl3+lGkhy45s3IYeSMHxSjKUs84ZGwszQuE6jfIydln8fnnnzesL9Wz7iyt2hGompGr0ekazkpz7mF/UTyL4emcTsmZo4i73nFQlk022cSrZbFd36g8BMaMGeM9sqchraR2rF3JmpiktTyW7/ohUDUjr18Wy0/pk08+8fLF5ZZbrvxACfSJnRDU6JiRGZWHgIxlsT6TZmICpHUS07FOc03WN++ZYeTMZtD0oBNoZlNfKKNJjY4s/XG0gNJclmgQKS8W6h5Ko/naYAmpby3YooFl9R9Ex+6bQyAzjBwGyAxGerjNFTjp7nRcRCuQ5L7cW4cuXXPCBaxQhU2zWI02zE9tWKLC0iU3V0NgAQKpZ+R0ZBo/VzqzOoE6+IKipueOrwqs90lcQM4po1FpBFT37CrOAqG1Ar3//vtW71mo0DqUIfWMXEwcrIKbgdLM+Mg7sl7TJW+9B4AVawmYL15//fUzwfgwnLXSSis5TroyMgTKQSD1jFyFZPaKSVQ6QBaIWbkYeZq/LuKsC+Eihte3b9/MiKD69Onj619ljBNHizv9CKSekauhjxo1ytcGohW5pbV6yD9WEO2AgdZrEKwwMAVtvvnmmZiRUxbWSbDmqEGqdSTMR54RSD0j59Oan07VkXwx7cwc7Qs6MbrklM9oYQSEC4uCbKRi8Et7vVNKyiCjc6aCuHC9m8vCCKSekatIzMiXWmopt8wyy3gndXK9T8tV+WZTCEzcZmSt1xy7YDFtkAUmTmlpA1tssYUvuG0Kar3+zYer3vphksCjA3PEF/q3YoRJyl+leaEMnDsJoR+dFQZVKQ7l+AcrMGI2niXiQBbUKW0gz1KtxleWTMzIOSoMzQVtpIgPrvhjFtPW7j4WPLMwOMWBnLDSRjBwyhJWaC4FGbnKGweWFme6EcgEI5elwOCBEmlt9DAi8o6ICFGRyUib72BgNX36dH+8m04FSmu9lyol7RltLMrEL0uDVKnymlv1CGSCkYvZZeHzWoyIToucXB25+irOdkjtfpWhqSwxO8rEpiCR2oae7WoICIFMMHItCMkOOZ05rR1aeafT8mmt3X3WidVkF1zBRCqaWdFWonSUi3bAxATRCvsj0tqeF9SW3cWJQCYYOVYPoRVXXDFOrOoWt5g2C54ynGUduTT8MDqOd1PdZwEnlUHrJGrfpREwV0MgA1orMD0tCGVlVycNk87M4q114ua7KRhR99LwwacGweZDpeeNykUZNUtPT+4tp/VEIPUzcjozMnJUtTQrqyeAcaSlGRmiFT6r33777TiSSX2cMDdET1kQqZWqDMn9TbxWCh1zCyKQekZOZ0ZGjm0KMcBgAdN8L5OsnKiepZlmVHVCfaOxJIuXUcXb6Hioa37a4Eb75jlr7bvROGcp/dQzctTPpk2b5i3fZY3ZSS+ezU7WiUt3O/Ts+XLJElHXqm/pkus5S+W0skSHQOoZOaZrIZ0+Hx00jY+pW7dufiFP51E2PkfJygGyYwxLSZacrNzVlhtNShAX2jpJbVjmIXSqGTmNXaqHWIvL2qyF8sCkTLRSuiu+8sor/oVkyaV9pdsV+b/2Eoi5p7tElvs4EEglI6dBq1FrM1Dv3r2LbnEA1Yg4YeR0ZHSlOTgBUrkbkZ8kpBksvxh5FjaCBbGljJqUIP/ny0P1H/Rn94aAEEglI6eRq6Gzoo8seckll1SZMnOlQ0uXmAEryMQyU8gqCgIO/EaMGOEXBLt27VpFLMkNQttWXTNIcc9Xmdp8cnNuOWsUAqlk5IClhs6GmRVWWKFR+MWerhi5yUm/hzrIzN58883MnAjVXENS/bPgrTbfnF9zzy8CqWXkqjIYHAtCdPBgJ9f7tF/VkbWom/byRJV/RA1YvMyaxgr4BBm2NJdGjx6dyfYdVXvIezypZ+SIHHS8W7ADZKFiGZi0W1VrAVkoVy1lUB2zAAhJ176WOJMWVpMSrpx8xA8xksqetPxafhqPQGoZOY2cXY+IVrLYmdU0ZAzKGLkQ+f4qPJAh0xayTExUXn/99cyXM8t1GHfZUsvIAWbixIkeH2atmsXEDVi945f8H9FK1hlWOdgKA60Z6IulnLBp9YMKKhvfvvjii7QWwfIdMwKpZOR8YvJjVx9EQ5dbzHjVNXrKhDYOx35p0KprBhKcGCp5UBY3A4VhVxlV5vB7ezYEUsnINftG9RDiJJUskmaf2JGhrDD2vJMwEFOTel6WcZHokCPtjAyBUgikkpFTEDo0DRvDQp07d86saAVmzmYnziWdPHmyiVd+aMXs6MUOObNVDXilGngW3GR+Qgu8GsyyUDYrQzQIpJaRU3xmqcHFrqw1cMrDb4MNNvC1zQJf1spYbTNmkbtHjx6uTZs21UaRmnBSsWSHr9V/aqqtrhlNNSNnATArNshL1bpmmrIl8tlnn2V+9lkKh1Ju1P16662XCzzEyBm81CaMoZdqFfl1SzUjZ7EzqLWgRp6V6lRn7d69uy8SjFxuWSljNeVg4Xf+/PmZ3T8QxqRTp06OH5o6qv+stfVwme25MgRSychpzDRkRA3Ss66s2OnyzYYQ1gGkO52u3EefWy36sQicF8K6pyx9UmYx9LyU38rZMgKpZOQUCabGNm2pZrVczHS+1ayLTotmjtQt01maaHINJtJYQUYujKKJPZmxUMYNN9zQL3h/+eWXPpN5KHcyayOZuYqMkcNsgr84i0sj1mELWWbkYEhZ+S233HIO0YrRgo1gEjllnanRr7TgzWCe9fJaG68cgUgYOQ0tTKXcwn5qeX7nnXd8cBmVqiWuJIfV4MgOT2Pk34sUZCgtyfUWdd40aLHIG3ffijrvFl/8CETCyG+//XZ30EEHuQMPPNDdfPPNbs6cObHPGrA9AWV9Rk4ZmYGhuWAy8u87BIw8uMj9vWu2/3v16uVVLTls2sgQCCNQNSPXTPHMM890AwcOdPPmzfMnmh933HFu77339s/hxKJ6Ju1Ro0Z5Jt6uXbuook1kPPqM5suDTUEcNJ1nAg9k5DLvmhcstPmJsoOBzcrzUvPllbNqRk5jQg3su+++cxj4Hzp0qDe1ecYZZ7hHHnnEvfzyy+XloApfDBpvv/12Lnb1CR4xLpklkHverjAw9KmFR57Kj4YWmivGxPNU6+WVtWpGTvSvvfaaO++885rsrjvmmGN8ynHOHDlQAPFN1uXjwSpksRMyC3jOHyjRrVu3IDyZv2fihBhRmks8GxkCQqAmRr7rrru6ZZddVnEV5eKLLLKItw/CzCE8e2jOrRhJCzeKSyZMadhyayFYJl5pLYDFrixQuB2En5srI+Il1E6XX3755rxk1p1dzHyNGBkCYQRqYuThyHh+5pln/KJnsKMFmS0zieBsIviuVHxhN/xr0U/MLewni88dO3b0BsI0iGWhjMF2QHnCz6XKKEYWbF+l/GXFTf2DK6KVr7/+2q+VyD0r5bRy1IZA29qCLwhNw5o1a5a75ZZb3P333+9f0DGnTJniFz8X+Pz+jlkVRMOUKmHYT/hZjReRDrT44ouXHTYcVxqfYV6sR0iHPo1lCOaZ+sSiI8agqEueW2Pmw4cP91GwRT8rOAQxaeleBsKYLGXFdDP9P28aSC3VcbXvImHk6oAXXnihu+KKK1xQfknHDD4ro5Kh04HZgl4JffXVV9475j0rDVtJOknzi11qdvZlqcyIivja4ACNcmjq1KneG3Xfvn37coJkxs/666/vy8I6SVbMEzD5M6odgUgYOcz6uuuu85YI+/Xr53OFGwyeTjpkyJAmMy3c0Tr5z3/+48UFOs6steIQDpo0aZK/9uzZ05UbtrW4k/6esjNzQe0yK2WmTGjhMNBjV7612Th1xCI3qnh5sLETbpOaELFOkJU28O2334aLac9VIFCzjJzOOGjQIK9NcdJJJ/ks0CERqYha6qBizvLb0lXxICdlJi9NjpbCZOUdZae8srVBuSrBLmk4kHfVJ3kL3reUV2aj4JDmsrdUvpbeIVpBvKZ1gpb8puFdHuswrnqpekauSrj11lvdfffd5w499FB39913+w7G5y87L6+55pqS+Q522uB9Sc8BR6XJpggMJqEdkydicRdRBLr7zErTTJXWO/6pf/Yu5FmminhNJwWluf7JeyVtIO1ljTv/VTNyMva3v/3NsZOThacHHnigSV5vu+02X1F0Pn5RVRqfYnTmPfbYI9J4m2Q+gQ9gqE0waK7QoaPCNIHFXShLlB/C3kyevsTCQHDICGJJI0MgiEBNU9qjjjrKb8Wnk8HM+XHP77DDDvPpwGyiYjjEw2YgiFX7qOINApLke8lIGcjyVHaVlSuiFXCQW5LrK+q80a9MlzxqVLMRX9WMPNyReA67RQ0RDVkWANddd10fPW55IelOIyen3HkqO22L8qKu1qVLl1yVXe0bDBCvMZjZIqFQsSsIVM3IGwWfTknZaKONfBbiHjwaVc5S6Uo2rBNySvnJqpsGLb7IsqKxUWldgQGHjXPV4RqVxmH+s4lAqhg5TBv5MBoryIjVubNZNU1LRdlZ4NSnNc95GcRUz8zGMZiWZxn5mmuu6RuGGc9q2j/y/lTTYme9waNDs2KvRb96p9/I9MTMmI0iXtJzHpi5yqj1ka5duzayKhqattq+zFTkqR00FPiEJ566GTlMLE82VtR+xMyYjbLYCclNfrJ8hWEhG4a06Jvl8jZXtiWWWMIxkMkKIv7y1A6awyXv7qli5HRmmBjihbwSjFwMTbOxrGNBOWFWMuvQqVOnrBe5ZPnEsLWfQM8lPZtjrhBIFSOnZtjSrRl5Hhsyg5hOiclLS6WeYeba1Ro0nZwXDFROsGB9yGTkQsSuIJAqRo7KFQazsmL5rZImqFkpZ3diMVCz00riSKtflZ3FToxrYSwrj4O46o8Fz6DWCvgY5RuBVDFyHaogfeo8VZ1mpehQQ6gg5oWZqZwMXjBxGFeemRdqqMGvMuGTp/5gZW2KQKoYuRb58qhHLMal9QFseOeFxLgxxJZnjRW1AYkW33jjjbw0AStnKwikipHLWJBUsFopW6Zea9YVZOTq2JkqaDOFofzIyPMsHwca6hwZOcQhI0aGAAikipFjvhMZaR7Vz8S09TXy1ltv5aYFw8QpPzNyDtXgWQNbbkD4Qc2QcmtT0CuvvJKn4ltZW0AgVYycTRCrrLJKruWj7O5kIMuTBTwNYjByDirhWW4ttO3MvVK5Gcz4jRw5MnNltAJVh0CqGDmbgVjoyeNsLFi9iBeCWgvBd1m9p85nzJjhGXle6z9YbiY0DOZBt6zWvZWrdQQSz8g1C6Eoed4MRIfVj8GM7eqzZ89uvYYz4oN2EJyR55GBBb9CaAOoY06fPj2XXycZadaRFSPxjFwlpREzC2XFPtig9T5PV6lfSh0zL2UXI6e8eW4DDGLaS/Huu+/arDwvHaCFciaekWvmxWYgZqFasW+hTJl/pQVP2WbPeoFpA9T/zJkzvS1ytYmslztcvmC51Q+CNlfC/u05PwgknpFr5oX5WogZebBB56eqFpRUjDxPM3KVtXPnzrmfjdMSpLnCfgL1kQUtxO7yhkDiGTkVAuPWyeHoUee94Uq0IuNZWW+01LdmnnneEBSs5zwO5sHy231TBFLByOnIEiNg/S/vM3Jt05cRqaZVms0nFvYgtugbff9lCg4McHnvD9YenEv8wRJqpDTYxRZbzJuwhbHLPY+VqJ2t+krJOgbUtWbkqF7mue5V14gYO3To4K2Bys2u+UUgNTNy2wy0oJHqs1pfKQveZPMuKFpZZpllslnIKkrVq1cvf2JWFUEtSMYQSDwj1+wbjRUtdOZ9RsbBCu3atSva5wYjflkl6lvHvOX1UIlw3YIJjByTxmyUMso3Aoln5FQPTAoxAjPRLDOsSpriOuus408KygsemO212XjTFrLuuut6B77MaAd5aQtNUbAnEEgFIyejqJ9JNpz3qqPDrr322h6TPGBBeWHkprGyoLbBpG/fvt5h3LhxC17YXS4RSDwj5xOSjSDMyI2RL2ijMPI5c+Z48UrWRU0wLRY782j1ckGNN72jzjfccEPvKLs7WW8HTRGwpyACiWfkdGKZbNVuNtzyTHRY5KMQAxx4ZLkTS1/eZuRNWz2WINlXoUNG8t4vmqKTr6fEM3IY1NixY32taCNMlplWuc1PXyecYQpluRNLh5w9BFkuZ7l1r/oGC4xnaddrJeHNb7YQSDwjB+533nnHo67TcfLemSk/ohVIx99leXCbNGmSLysaK1kupy9kmX/gwI8+kZf9BGVCk0tviWfkMC1EK4svvrjXWsllLYUKTQfm64TTkqRLnuXBTaqHyMizXM5QNbf6CBZrrbWWY4+FUb4RSDwjp3pGjRrlPyHzXVULSk8H5oc6poyJZXmmqq8OM8+woA3ojnUjFoLzZJteZbfrAgQSz8jnzp3rVc/YDATpk3JBEfJ5Bw4wcphcFpm4BitqV+sAfIXgbvQ9AtQ7/QJM9GVm2OQTgcQzcsn/ONrK6HsE6MB0XjbIIHYIMr2sYKTBibJpsRMTtnLPSjmrLYfqXOtGtuBZLZLZCJd4Ri4dWc3IswF7baWgE0N8ViNagbllmcFJfCStpdrQy1ZoZOTQ+PHjs1UwK01FCCSekbOjD5IOeUWly7BnGDeiFWZiYuxZLS5ig0UXXdQhIzf6HgEN3Jg0XmqppWzBM+cNIzWMXHrTOa+vYvFh3nxWz5o1yxtOyhozD5YHRo66Jcwr6F4EI4c3woEr2EyYMCGHKFiRhUDiGbk+q9n4YPQ9ApqNST4Ko5NbVjBSebhygAZrJDAtuWelnFGUQ19mUcRlcaQTgdQwcpORL2hgmo3ppCAteC7wkZ07TLRia0dnVKrs2SlhdSVhQNOPvsFakg1y1WGZhVCJZ+Q0UBgWo+GbnQAAIABJREFUGgtGTRFYY401vAOaPVnsxJRJanUs6olxNUXBnn784x97FV0b5PLbFiJl5HE0JGR/ffr0yW8NNVNymNrSSy/td3cifooD+2aSrquz1E+7d++e2TLWCigD+rfffmtb9WsFMsXhI2PkYiRcdV8tLgqPjY1vvvnG9e7du9qoMhkuOPtm7SCLn9VqRzqrs0ePHr4u1TYyWbFVFkoaXVLVrTIaC5ZiBCJj5GAwfPhw/4kXZDS1YBOcjUUVZy35SWJYFjwx85o1Bkd982Mz0CKLLOIXO+WWxHpoZJ6k0aVBr5F5sbQbg0AkjPyvf/2r22qrrVy/fv0ikdWJaUtjxT6rm28c6FajSy7MmveZzjfIyNkIpPJlbcCKolYwJoaevSY+UcRpcaQLgUgY+bHHHusuuOCCYsnV6YoOFd7os1qfihyiUGucFWYhNd5Ry8uijFxtgEEK8ZGerR0s3DTBhFm56ZIvjE1eXCJh5IDFwlvURMNkNsZJKDYTK40uopUs7u4Uw2YwD+4hsHZQuh2wKUi7oEv7MNcsIxAZI1fHqwascOckLn58KtpGkJYRhZHPmzevaCGwZd/JfRtuA8opOvKUUe1LV7236/cIgJHJyPPbGtrGVXR1zGnTpjlEL2GaOnWqd5oyZUqzdiIwmM9s3Aznh9Fb+Pm1115z66yzzsIvEu4yffp0P2BzcEgpQkberl079/7775d6bW4/INC+fXvPyNOGEwv1dqh27c04shl5OCuaOc2fP99rVVBhwd/kyZPDQRZ65rNaO/oWemkOHgGZ90W8kjWCyTMRYLZp1DICqCAyKcKcgVH+EIhtRg6UMHN2ZD711FMLIcvxbeuvv76XrdMIxfjl8bvvvvO7+tBY0Q5GvbPrAgSwswGxISSNODHjZos5FvyCbYAvurffftuXjcVulQ33oL8FSOT3Dkz69u3rAeCgkU022cTfG075aROxzcgrgTDc4GiYyMeR/TIj59moNAKIHVgQloZPaV/Jdw23AZ5VJjFxShH2l/yS1SeH2hT05ptv1idBSyVRCCSCkZdCRDqx+qw2Zl4Kpe8ZG4xcNklK+0qna/DQ5XSWoD65ZnCTUTkYOc824NUH+6SkEhkjDzLa4H21BdUKfFD1rNq4shwOrGHkYno8R4F/EjBjcFpyySVjUW1NQvmizAO7X2Hm77zzTlHnPsr4La5kIxAJI4dxPP/8876k+rSrlZnwWU0n1qkwNsNoviHJjGmtmDefQmPeoK1E2bJWrqjRFD6IV1hXmDNnTtRJWHwJRyASRj5s2DCvWTBkyBDPeF999dWaiz1u3DiHeU77TGwZSvChA0ue3LLvdL2lTLQBo/IQYC0BJi6xZHmhzFcWEIhEa2XHHXeMFAuYk7ZmRxpxBiNjNoZoBSuRGJhadtllM1NKRCsbbrihyXtbqVH6C+1Aqqg2ALYCWAZfRzIjjxoXGiUyci10Rh1/luKjE8PIIUwa6DM7C2VkMJd6ZRbKE1cZqHPagfoLfSdL7SAu3LIUbyIZOY0S+Sj2I4xaRoAOKzOmI0eO9J6z0Ik54o3NLbYhrOX611vqXGIos7kiVPJzTSQj55OacxqlG5uf6qi8pAx6wkkbaCqPJXkhZJZBanXJy2GyckQ70KD33nvvJStzlpvYEUgkI1cnVsOMHYUUJ8BMjF2RnTp1cqNGjUpxSb7POuXhpzbAWZ08G7WMABjxZYYaos3IW8Yqi28TycjVEE2HvLwmx2wMrN59910fgOc0EsxIeZftGGTkcktjmeqRZ+GD4THEK1nUYKoHjmlOIzGMnE6sH+pTnHhiC13lNS1wQwTBpiBEUjynlZR3DsvA8iV7CYxaR0DMHNtEGgRbD2U+soJAYhh5EFBMcfKZqMYZfGf3TREQRpIlo7Egt6Y+k/8UzDeiFURruIm5J78Ejc9hnz59vAE10yVvfF3UMweJYuTqyKjRmcZKec1ATE4LngyCcisvhuT4Ur5pB5RDg1Nycpj8nMDIIa0xJD/HlsMoEEgMI9fMiyu7OoMW76IoaNbjWG211XwRsyAfhaHDiDgoQ8w96/UXVfk23XRTH9X48eOjitLiSQECkezsjLqcyEelGx113FmNTwvDfFLryyaNZVXeKYfWSOSWxvLUO8/YJmJdweTk9Ua+seklhpEz86LDMhPjkAQYk3Xg1huHMBIjZxBMK6ksHDrCyVLacp7W8jQq3+Amza9G5cHSrS8CiROtjBgxwiOAjNw+q8tvDEHRSppxI+9vvPGGLzhyf5h7mstTfg1G4xO8WCQ2GXk0eKYllsQwcnXWF154wWOn7cZpAbLR+VxiiSUcm2dYKE4r0QZgRC+99JIvgm0Gqq4mWV9inckoPwgkhpHrs5pNLRxf1rVrVxOtVNgO1113XX9SkLCsMHjDvSvfHI6AnLdLly4+T3JveAZTkgG0fbJ4YlRK4G9INhPDyFX6119/vSgb1Sxd7+zaMgIwctYXJk2a1LLHBL+lzjmcBDmv6l/XBGc7MVkDK9ZLaAdffPGFz5fhl5jqiS0jiWHkNLYpU6Z487WonRlVjoB0iNMsXqENoHERtLNjM/Ly2wJYaQ9G1swal49C/nwmhpHTAKX7ajrk1TXEXr16+YBpXuhSG9CuTmPilbcFbQ4TlpXHYCHShkBiGDnAqeGxyGVUGQJ80cDIWV9gV2RaSSZYbbG7+hqULjkzchsIq8cxTSETxcil+ypVujQB2ei8qsNyWlCadcmxFQNpVtloXNOaPmsM7PJlgFfbSGtZLN+tI5AoRq7t5TRCa3ytV17YB5ix0MWuSDowv7SRBnOzs1JdzanOmQzZ7s7qMExjqEQxcolW0L6AjJlX1qToxMHNIOCnjl1ZTI3xTV7VBky8VlsdIJqSLnma2kBtpc5v6MQwchrb2LFjvbEsTrwxqg4BRBJihmCatsGQvDMb58Qjo8oRUH3rgAlMHRhlH4HEMHIaICKB3r17p475JKmZIJZChY8DJtSpk5S/lvJCftG42WCDDVKX95bK1Yh3tIPZs2f7WXna2kEj8Ep7molh5CzQzZkzx2tepB3URuRfn8+SLbNoiJvcG5GnStNUG2AwT1O+Ky1nnP6FG4wcGjNmTJzJWdwJQSAxjFxqZ5tssklCoElXNjTr0kYaiVfknobSqA1svvnmNiOvscLUDrBbI+ZeY5QWPMEIJMaMrQ4O3mKLLRIMV3Kzps6qI/LSpktO/jX4WBuovp1p4EYNFXs1o0aNqj4yC5kaBBIzI4eRr7jiikVDSalBMCEZVQdmQxDMPG3b9Mk/i93dunXzbUADU0LgTWU22Ko/fPjwVObdMl0ZAolh5Mh0TX+8ssoL+4YZ8kOXXPrYaWGI5JM8yzyDBqZwGe25fARoB1OnTk21EbXyS5tvnw1n5HRgfnxWs4lBz/mulupKL+xQQWRgFGOvLrb6h0IcZLt6o8NdxyVqt2x0MVtMSUOg4YxcMy8WumwjUPXNAyYOgSfMkIFRbtXHWr+Q5BvRiiz3pSnv9UOpspS04KlF5MpCm+80IdBwRk6H/fLLL73us3bzWSeuvAlp9g12mAHGJvnEiRMrj6hBIbCdjRjABvPoKkCDYtoWvqNDID8xNZyRA7W2Eq+33noeec3S81MN0ZUU7NSB06R6xmwcEiOPDpH8xiQLkupf+UUi+yVvOCOH8UjtDNmuMfHaG502g4wePTo1eMqGusQB9lVWezvQeoOM0dUeo8WQVAQazsjpsGgroPPKOZ3WgWtvKqhxLrroou61116rPbI6xcDnPzZ2dE5nnZLNdDIdOnRw2CaXBlOmC5vzwjWckTMD59NPn9Q2I6+9RbZt29brkr/yyiupGRjZSt69e/faC28xFBGgL7FLlkFSxrNsolSEJ1M3DWfkoMmp6TByY+LRtC06K3iidoYBrTQQMvLgWa3WFmqvNdpB3759vfEsZuU8G66145rEGBLByJGPSp5HY+NnVD0CdFbJydPyWc1OVDYDkXdjNtXXfTikznG1QybCyGTrueGM/Ouvv/bqhyxyiYFbR66tkYGjdkimYTMIVg9RPdRCZ22lt9BBBMTItZisPhb0Y/fpR6DhjPzZZ5/1KAY/q9MPa2NLwEAopihjZI3NUcupqw1I/bRl3/a2EgRoBywiSzPMJkmVoJcevw1n5NKsYKGLRmYNLZrGI11y6WcneSaGvjtki53R1H0wFuqd9ZI0tINgvpN+H+xPwfvW8o1f/VrzW8n7hpuxZTGuc+fO3updJRk3vy0joC8cZuRqaFyTOFC++OKL/ni3jh07tlwoe1sxAtQ3s3I0w7hXW6g4IgtQRCCI4UMPPeSmTZvmDjzwwFb7FuGYuKJVBnGASlTU8Bn59OnTnbbmR1WovMdDg1l88cX9TEyyUTpxEpn4vHnz3Ntvv+3VJZOYv7S3JdoC6yVpWCtJG9afffaZ23vvvd2hhx7qXnjhhbKyf9FFF/kvz6i/PhvOyGlo0lgpCwnz1CoCYog9e/Z0aCtwhF4SibqXeiSiIJ6NokWAtkD/wu7OrFmzoo08p7Gpf62wwgru3nvvdRzisccee7iRI0c2acO0Z/2YtZ9xxhn+LF1m5JqVRwVhwxk5BYHhGEWPwJZbbumZOKp9SWOSyg8HbkM9evSIHgCL0de7Fr6xgigmZNDUhoBw3HXXXd2TTz7pll56abfVVlu5wYMH+4hp3/LDROrRRx91V199td8r8cwzzzh+UVIiGLl2dUZZsDzHpUakI9OSvNCF6iFkg3l8LTa48K0BNL7U8hcz+KJ5ha2oo446yh177LHFmTjMHHMZmB9BjHj22We7rbfe2v+iRKohjDzcmDRjiLJgeY6LxgPGWkxhizZumiEkBRvyow1LdIak5S8pOFWbD+HJAROLLLJIEetq44s6XJgPRB1/PeIDY36cxvTf//7Xi1luvvlmd/755xeT5/1zzz3nrZLqFDTVTdFTrTfbbrstgkn7GQbWBqwNWBuIqA0sssgihfvvv78wf/58//vpT39aOP744wtx0Y/uueeegj5vax0UKgnPiu8ll1ziV3wx7GMUDwI33nijX+Q69dRT40mgxlhPOukkr1XBQpBRfAjQDmbPnu1++ctfxpdIFTHDA7DQeMwxx1QROllBZsyY4S677DL31VdfuR122MHdc8893qoru9exSHr//fe7nXbaKZ4vz6hHCI1ALV1Jc/To0X70HzRoUNRZyH18YA9xPfnkkwurrbaav08SMOTto48+8m3goIMOKuY3SXnMUl5+8YtfFNZYY43EtYOuXbsW9t9//1RDTVv++uuvC5tuuqlvz/379y/Mnj3bY8273//+94UOHToUZsyYEVs5Y5ORo/Y2dOhQvzoreRBX3SdrLM1WbsAY+SNX9FWRQ8+cOTNxhXzjjTd8nhZbbDGf38RlMEMZwvwB7YBZeVKIGSzHPKaZ6Gfwum222cZv9vnDH/7ghgwZ4hc4Va5Bgwa5//f//p974IEH3DfffCPnSK+RM3KYx/Dhw90mm2ziRowY4S688EJ3wgknFBl4FhY4Iq2BmCLTgMlKOqSNQTElV1W0dgRZVbBVFYgFT2ySJ+m0oCS2yUrBZTBiUxCncd10003urLPO8gvLwXiWXXZZb6p7o4028nZvgu+iuo98iz6H/u63336+ULvvvrufCaI9ceedd7r999+/yNCjKoDF0zICUj3DaFLSVPyMkbdcd1G+lTVMmKfMN0QZfzVxpZWRazKKxU7W91DvZcJ65JFHloQBjZW4KfIZ+ZVXXunzvNtuu/nP5SWWWMIdccQRfqRCj1IgxF0wi/97BDiAl+36b731VuIgYRZjVB8EGNBRQUySNUztb6gPAtGloq/dTp06ud/+9rfu2muv9dfoUqg8pshn5MiH+ISgsGLaG2ywgf+ku++++1z//v0rz6WFqBoBOi8zMOyZJI3efPNNt88++7jf//73ZWWN9qROVFYA81REgMGcrfqcxhUXqX7U71urKxj5yiuv7HWuFbalvAX9BO9bChP3u0MOOcTzudbKGnc+IpuRA+zcuXMd28FRtRHQFJBdTdDLL79cZO5xF8ziX4AAs/K4xRjUN8Q1+FuQi6Z3LHJNnjzZb6Ro3769f9noztA0h9l6ok4wTsfmsLgoWH/cq00E0wu2DRZfOWybg9cVVmHkLxhWcQb9BN/X8568KM+61jP9cFqRMXIiRmaE+ARzpMHCoZUAffHFF+H07bkOCNCB62FvRR2snCLps1qLsa2FUdy6tubf3i+MAAueyKXBUL+FfUXrEq4v8QWutEkMT0HB/ITDhN8rjmhzmu7YImPkgCuA+ZwPVsx3333nUcLmgPykG7b05J56YLGZGXDUql7BDsfXGGK1o48+2p188smtasmgekhbwDwDmygGDBjgLr/8cvftt996cIPth3vEMJdeeqkbM2ZMesBPWE5RQeTLTCqIcfVFDEJhcyTYDoJtBVioZxi57CyRLzQ+Dj/8cPfII4945JQ/wvL1xvrbQQcd5I4//nj31FNPJQzdxmYnMkYO2BwQscwyyxQZBhWBOzuboJ/85Cf+ubFFzl/q0laJY8GT+uUrjLUP9GUHDhzoOzGGgeicvNcviDyMnLZCZ2dWfscdd/gdqGg6idHgn7C33HKL+7//+z937rnnFttWMC67Lw8B9hRgiQ+14ChJ9Yt64+mnn+5OO+00v+4BM6cdcHAIJH/co0VFuyFPWAzs06eP22WXXfzCIbs9YdbklTCo+BHP3Xff7VUouaKXfeutt0ZZjHTHFcVWI3Yv6TdgwIDC5ptvXnzG/bLLLiu0b9/e735SerazU0jEf505c2YB2w9XXXVVk3qJImXq9/bbb/f1O2XKlGL8P//5zwvYl1C74BqkzTbbzO+C22KLLQrTpk3z/ubMmVPo1q1b4bjjjvPP+Ff4F154wft/6qmngtHYfZkIgOP48eM9hgMHDiziWmbwVr0RP+0Afj106FDvHzfsi3Ts2LHw8ccfF9PEHT/4feihhwpLLrlk4aKLLiq+f+yxx/y7G2+80cfD9dFHHy3GOXny5EKvXr0KK6+8sg/TauZy4CGSGTkzb/34NGLEZxSFcOdMRnTImbEzwhrVFwE0FrC6xqes6olrVMRsmlk/6liKf9NNN3WPPfZYMU3SCta9tGik5044jO337dvX70EIrqfwTussUeU5j/GgtYLY89VXXy3WUxQ4qF5pB9DGG2/sr9TbZptt5o9Cw+5IkFh0RdT6+eef+70me+65Z5MwPCg+2u+OO+5YDI7tb/gJhzWQhpFzkTDyIJAYhcHAOvJMCKPraKv8+te/Dnqz+zoiQGNnwRP5Mp1Ov6iyAGNAPBKMF7UyCNOe6ug8c4/BNI74gxCvBIkBB3r99df91TpqEJ3a7tu0aeMNlFFfwTqpLdYFoV955RXPnKlDtQW1A+ozWJdMKmgz5IUBXOI/wqHFxCIoYXg+7LDDiokE49hwww1jKUcxsRTdRM7IaSwcSMrsjAWs//znP/48u6TsJktR3USWVToDC55aYCTiYIeoJSEYMouo0j4gLtJDrQzisNkwcSSWCMt3QeKrDYpTTS6YXp7uqRcWF1H7o854jopoB1j9Q9UVHiAK1qfSo+0xqWDxFVl5t27dFjr6DBVmtOBY5ISC7ZV4WFBFFm/0PQKRM3KibdeunT8JAy0GNntgdB3wqYxghVgl1AcBMF9//fX9uY3Y2oiyDqRlwudvsH5Z+IJKnRc6atSoYsEJJ6KNKBwLYUbRIaC60ZF6US98a4F6qaWWatLXxbxVn+QDN0Rr5IVw8IswyT9XtVfC8WPLOyrO2DjRu3D4vD3HwsgFrq6AGrzPG8iNLi+Nn89QaNiwYb4zRJUnZt7IOpFXqqNR1zpUGZFOmJil6/BZiVjkh1kYxIG2RtEhoLqRCCPqHZ7MvGkHzMohpad2oPrEHV12vgjIC7Nx2gDutBvxCWTnMPjguhrvaB/XXXed15AijNH3CMTCyA3c5CEgfV1UwdRZosgln9F0SEx5QsRNB0MODm2//fb+GvxjNshiKMTp7iLCKp5evXrJ2a4RIAC2/LS4jGooz1EQ8agdILbRvhHiVjugPpUecnGIQR7zHTB/vuzEmLknHKqJEtPoHSqoF198sZ+RR5H3rMRhjDwrNdlMOdQB2AaN3PHpp58udphmglTsvPPOO3szngQkPTosOzfZ7CMNBt7hzqc0ctE99tjDMxUxboUlHCaQCau8l8pQS+9K+Te37xFAhg2x2Bg1hrQDCLEJda12gFtQ6wTlB4g6Rj8cQikC/+RJu36DYRDRcXDxlltu6eXwyjszd0jP/iGHf8bIc1TpbLp47733vMpXFMVW5+GYNkQr119/ve+M3P/jH//w6yMSoZAe/tmRR6dkRo5mE6eP87nMuyeeeMLfY/KYTq0wXCUzZQep0vUe7K8iBFiURqskSsuTqg/aASqOiD4gtYOLLrrIq5XKH20A+0v8WD/bdttt/a5O/EM33HCDZ9Znnnmmf6a9HHDAAb7doD7JgTX8LrjgAnfFFVdYe3DORW790CNvf4lBQAyRDGGV8sEHH/TaK5JZRpFRVAhRM2SL9cMPP+w/k/n8DVq6vP32293PfvYzr6lAmgwqaDYxe2PRSgcfMDODGQQJzScGCQjj/Sx0MWs3qg4BdkXedtttvp4wMx0F0c44QIH1D86HZYcuIhLawcEHH1wcmP/+9797lUNO1BFdffXVXmecr7QOHTp4tURUD1FDJN59993Xa8LJv67I0BHl4EeDhN7l7tqoTU+2s7P+yN92221+x9zll19e18TZyffll18W/vjHP/rzQ1dZZZVi+s8//7zf2Vl0sJvYEbjmmmt8Oxg5cmTsaYUToL75ODvqqKP8qwkTJhT4GdWGgIlWcjR0o4IINeJwAWbt2N5gBoX+sFFjEGDmis0jiLWKepMWt7X4Xu/0s5qeMfKs1myJcqFdwqInn7/1+hRVOnz+aiOQNFZKZNGcYkaAekCshb53lHLycrOtA7fZoGYUHQLGyKPDMvExoee7ww47eBk5C0gwWTHauDIP4+AHcSg3FNRk8Q72V1cEqA/t9I27/lUwpaPBQ1+Hem/X2hAwRl4bfqkLzYISKoBR7+wrBwjpD9uMvBy04vUjRh5vKt/HLibOEzNy1GBlg6Ue6echDWPkeajlH8rITEx6u8yMNFOuFwR0YtTNgnZZ6pW2pbMAARgrm3GQkevLbMHb6O/UztgoRJrI6OUWfWr5jNEYeY7qnQ7Mzj4WHmGqwZlSnDCQDlu1sfOCdUzrxHGi3Xrc4C+bK9p803qo2n3AxLGlo8XW2mO0GISAMXIhkZMrTBUdbEyO1oM0WMh0KjrBcqtH+pbGwgiAP1vjIS0+LuwrOhfVt9Labbfd6rI+E10Jkh+TMfLk11FkOWQmxg9GzuEfdLB6zY5feOEFh6XDUrZXIiugRVQWAtQ54i1MCEuTSMy2rAgq9KQ2Rlp8DaJ6iJvcK4zOvJdAwBh5CVCy7sRs7JtvvvE2v+PswMKRDstOPQwnBc3W6r1dG4MA6qhYQawHUyUNbJAj0uG+Hu2uMag2JlVj5I3BvWGp0oFkWRB9cjpVnKROCyM33eE4kS4/bjFRDnuR9pLcyo+lMp/Ez6DBbFxp6VpZTOa7FALGyEuhknE3LXhi0jbuzkT82J5moatfv34ZRzb5xaM+GFz5sTGInbaYkY1zQCdNFrsxnUuaSkvX5KOW/BwaI09+HUWaQ3UemKo26ESaQInIONEF2mKLLWIfOEokb04BBFT/OGljluTkAW+R3pImX38Qh2tDwXx4B/urCQFj5DXBl97AqICx4KkjuuIqCR0WRo7JUju3NS6Uq4sXGTnWD9FgivvLDK0ldhabeK26umotlDHy1hDK4HuYKyZt67HDEwbBbEzqbjYTS06DWmyxxfwpPG+++WasM2TaAOaKOdTCFrvjqX9j5PHgmvhYkVVCcX9WswGEmT8qj2LiuiYepBxkEC0SFjzjnJFT3+iQazDPAax1L6Ix8rpDnowEOfSWTswJPXESB05wuC7ycaNkIQDzxu4Ns2UWpOMiTv7BJASDeZwDRlz5T0O8xsjTUEsx5JEOxYLnM8884ztX1B2M+PgNHjzYH6DLWYtGyUKAmTJrJXw1PfDAA5EyWdU/V+LGpstmm22WLAAylBtj5BmqzEqLwlmJqJ+NGjXKB6XTRUkzZ8509957r1c543g2o+QhwOIjcmvOWIWxR9UGJD7jevfdd/uFzqB4LXlIpDtHxsjTXX9V554Otvnmm/vw//73v30nVuerOtJQQBY5WVBFzS3quENJ2WMVCMC00SRhgxhfZlgnjJo4NJszVzkVyhY6o0Z3QXzGyBdgkbs7tAhQC7zrrrtiEa+w4Qji8z2qmV7uKinGAjO48kODae7cucUvsyiTZDfnjBkzigud1g6iRHdBXMbIF2CRyzuMWGHKlA4XNT3xxBM+Sk5tN0oeAjBVfrJRr41bUeaUmT7EGglp2ZdZlOguiMsY+QIscnnH0W8Qn79REotbdGI6MCfCGCUPATFVaRSJ6UaZUw0OtAOlF2X8Ftf3CBgjz2lL0OxIRv6ff/75SJFgAZXFzkMOOcTHa504UngjiUxtYLXVVnP82OEZdT0RZ6dOnRy7SI3iQ8AYeXzYJj5mOjLW6JCVw8h5joKIh5lYmzZt3N577x1FlBZHDAgEmXb//v3dp59+6v73v/9F1g4mT57s3nvvvWIbiKp9xQBF6qM0Rp76Kqy+AHTkRRZZxJ100knus88+i2yXJ/EiqkFvmMMLIOvE1ddT3CGpr2OOOcYn8/jjj0cyK6e+WSPhevTRR1v9x1yJkTByKosfOsmXXHKJ3ykWc74t+hoRCM7Gfvazn7kll1zS3X///T5W6rIWYifnU0895XbffXfPFEgrmF4tcVvYeBBAPZDfsGHDIkmA+iYu1kdYTLU2EAmszUYSCSOnkoYGzakvAAARiUlEQVQMGeLOO+88d+655/rtvrUyg2ZzbC8iR4DNOix63nfffcWZUy31x6wOneSdd9458rxahPEhsMsuu/gvKXZ61kq0Hxg5h20bxY9AJIycSjvwwAPdr3/96/hzbCnEggBqiBg2+vzzz2uePcHIu3Tp4vWTY8msRRo5AkzG0F75+uuvvV2UWhNA3v7+++/7PQS1xmXhW0cgEkZOI4AwixkkGHw5M7ty/ATjtftoEQB/qSE++uijPvJq64RwiFW22WabFjOJv2Aa4ecWA9vLWBDYaqut/CD+9NNP+7oJ1k+5CSoMbQDabrvtyg1q/mpAIBJGXip9MXfeqXJ1H34uFd7c6osA2its1R46dKhPOFh/leSEzUVYuttnn32a1HupOMJphJ9LhTG3+BBgly/qqNjHqYXo3w8++KA/SGTNNdesJSoLWyYCbcv0V7U3VJAY6cOkk2n4lItjV2E4PXtuHQHEK9dcc4174YUX3NJLL13VAuWgQYO8/Y611lrLn5reEnOmw0tFDTscPLfkv/USmI9aEcD+zlVXXeUwr1BpG9AEDfsqMPL999+/1b5N/19ppZVqzXbuwzfLyFvrVKq01joe/rC1ECZ2/kGov6Fv3Fo84fD2HB0Cqss99tjD/fnPf3Y33HCDX7Supk5QOcPG9bLLLtsiY1aa1H/btm2tDURXnTXFxOLkFVdc4W6//XZ36qmnVtwvqVeMpX3zzTcO0wwt9W38Uv9GtSPQLCNHxnX99de32BlXXXVVd9lll7VY2Sx6ffDBB01ySgVizH799df3i2J2lmMTeOr+QH3AtNdee23/OcynNQyduiuXiIN6ZkfnwIEDHTNyqLnBAP8QJ7ivvvrqbqmllmrWr/dof7EioPpAFEK/xiLm5Zdf7tVSy01Ycfz1r3/19bnvvvv6M0FbagNYXzSqHYFmGTm2qltbqFDFcQ1XVik3ZTfoV3HonV0bi8Cxxx7rzjrrLHfbbbe5U045ZaF6bSl3aC2hunbwwQf7cK3VbbAdBO9bSsPexYOA8Od66KGHuksvvdSrox500EFlJ0hY1E6xbc4aCQc749YSLyg7cvPYIgLNfteoYlsKjR/9wp22nPDEXa6/lvJh72pDQHXAlV142Ma47rrrfAekXsN1G0xN78eMGeP3EqD9Ipmn4g36131L7+THro1B4KijjvIikRtvvNFnQHXcXG70nitiOVQPjzzyyGLfbq6um3NvLh1zbx6BZhl580Gaf6ONBNg2plKN0odA586dvVhl3Lhx/vOaEjTX4YJ1TAeGmNEH3dOHgOUY8cpxxx3nUEN8/fXXywaEhWs2BCKi44veqH4IRMLI6bh09pdeesnnHANMzXX++hXNUqoWgSOOOMJbw2NWTj02x5h5x48B/F//+pfXcthzzz2t7qsFPkHhLrroIq99xJmrrZH6+p133ukP2h4wYIAtYrYGWsTvI2Hk5InRmw1BVCY2GzBfaZReBE488URvT1zneZYqCQyeH4frcgr7xRdf7Dt/Kb/mli4ElllmGb/WgfbK1KlTy8r8Lbfc4uufiYBRnREoREDz588vhH+tRTt69GhkL4VBgwa15tXe1xEB1eOsWbMKq666aqFfv36FefPmNZuDuXPnFjbaaKPCxhtvXGwDzXou8eL5558vTJs2rcQbc2oUAmoD48aNKyy66KKFs846y9dtc/nB/z//+U/fn6+44oqK2sGECRMK/IxqQyCyGTnjjz619alV5zHJkosAAdUhG3RQQWRzEFdIIhZdcbvjjju83vDvfvc7E6lEgH8SolAbQIUUE8fU/4gRI4r1r7agdjBt2jSv6cTGv9NOO63IB5JQlrzkIRJGrorPC2h5KSeHQhx22GHunHPOcegGQ+q8XDmE4PTTT3e77rpr0cqdDeLZah0XXnih1/PXLk3qXW2AkrI+gpYLOzS17yT4PltoJLc0kTDy5BbPclYLAjBltmuzVRstBu5FvPv5z3/ut9j/8Y9/tNm4gMnYlY1aN910k9/sxY7d8JoJm7/uueced8YZZ7gePXpYO2hQ/Te7IahB+bFkE4YA6oh33XWXP3uT2TcH9G600UZ+gZMF7fPPP9/OY0xYnUWdHVQJ//SnP7lf/epXXq2QDUOc/IQ5BtoDewd+85vfFJO1r7IiFHW7sRl53aBOb0LIPjl78corr/Q2prliGAnVNBi5fUqnt27LyTn1i+ybvQVopDzyyCNebs7h2mzjf/jhhx1rKkaNQ+BHrJU2Ivm33nrL21rBWh67wIySj0CwqTDrCj9XUwIWUzGf26FDh2qCW5iYEVAdh+u7uWQrnY1z+AS0xhprNBeluZeBgM3IywAp617UWbFYd+211zZh0LzTe3CgowY7dfBZfoP+g/eED/oJvyv1Pug//F7Puiq+8JX3cZHyR/xKt7m0wn7lP+gejEfu4StiDvlTHM2lWau76pd4dN/Stdb0LHx1CBgjrw63zIViQw9nrvLpTEeFMIDEJzW7NTFxy4Lnt99+22rZkan/9Kc/9YaTdtttN2+XPMxw8INs9eyzz3b9+/f3foIRY6+eTUkc4IwVPWyls+FEecMvcWIieYUVVvAbUdiQhjU9bGHzLug3GHdU9wx8WAkkf6wfkF64nEpL7s8++6w/PQmDUt26dfPhwmaeg/Gw7R3rkH/5y1+K5QGvk08+2aH2Z2QIeARqU0OvPrRtCKoeu6hDzp49u7DDDjsUPv300+JmDjZ5nHjiiYVDDjmk6HbKKacUDj/8cJ+8No0oL3oeNmxYYcUVVyxMmjTJh7vjjjv8xqKvvvqqGC7ohw1BN95440J+TjrppALpKd4vv/yysMoqqxSeffbZohsRXnbZZYUrr7yyMHToUP+76667CmxSIlzc9O677xbGjx9fWHnllQsnn3xyk3yVSvv9998v9O7du3DVVVf5vA4YMKDQtm3bwoYbbliYMWNGkyDK/xFHHFFo06ZN4eqrry6+592LL75Y6NmzZ2HixIlF9zTe2IagaGqNGURDyBh5Q2AvJioGyXW//fYrnHDCCUXmh9uYMWP8rj4YrfyOGjXKM56RI0cW/SpC/EB9+vQpnHbaaXL2/tZbb73Cb37zm2I88kMY7ezs3r170Q+Be/ToUbjpppuKYXBjsIFpKz8MFjDDICkfQbe47zfddNPCL3/5yxaTIV8w4zDj3XPPPT2mDG4ilW/IkCGFSy+91L8PMnL84efUU08tDrRyUxxpuRojj6amTLSS8y+z//73v/6Mxp133llfaP6KbjCbPTj8QyIK7tu1a+fuvvvuJn4F4fDhw90bb7zhFy8lSuDdxhtvXAyDYTX5UTiuffv29frI3BO2e/fu7g9/+IPfdIQbGhMTJkzwx4cpHOIMRDRbbrmlP6IOUUeSiQ1WqO0FiTMyIfS1ReDN2aeIiDArXIrwc/zxx3vbRmADZqqnUv7NLdsIGCPPdv22WDo6/kMPPeT9wKQh3PhhX7xjx47+JybBsV0c0AvjkL9gAugUQ9gj5z3EFeYFE0ZlMejHe/jhDz/jx4/3gwdOmENl5+jWW2/tGCA4Nuzmm292K664og+BXBlNB+T3I0eO9LL8Pn36uM8++ywYbWLuwWHllVdeSIbO4MOGqw022KCY11mzZnm5+wUXXFB0K3WDcTpOcUL9DwoOnqX8m1t2ETBGnt26bbVkdPyXX37ZmxyFQYr54v755583YeJEhjuLiRzPxj2/IEmVjAEgSIRhds/CHX4IV8oPjH7SpEk+KIwNswBffPGFX+hkdr7NNtsUo23fvr1fnGWxEYaPHXQGCzarEE84b8WAJW5KlSXsrRw/4TDhZ/ANYsz7xx9/3O+YlR426XBmJvr55ajksdAb3m0ZTtees4+AMfLs13GLJYSxoj2hU+zlGcaLFggUZGIcGiJmpCt+uIeBQsFzGAkrdw7a5R6/QT+EkTt+FO9HH33kOD6Or4DDDz/cYSM7SMQNYXIVOx9otaAVwgnwiiPov7V7xRf215x72F+5z8RH/vga4iuCo/Eg3BlY0Qw64IADWo0O/3zJsCfDKN8INGSLftQdI99VWF3pYSTUAzNgZn7cy40YmaGHZ3q8Z1YdnL0rdcKvu+66/hE/ik9hULdjuz9+eIefIPGMH8QMEId6T5kyxV+RE7NpDAuLG264oRenBMPqHtVJZvHsQt1iiy3KZubkkcPGdcqR4gtfOZQYuzLVEGWGSAviiwfLkbfeeqvHA7c5c+b4c1I5KxXZPyRb4NQFXx+oY2r2TlwMiMHB1Qeyv9wh0BBGHkZZnT7sbs/xISDMEXFMnz59oYR69uzpD9FldgiDxT/yXH7IZsOMiQh0vNfEiROL8eGPZ47/gvHgh2vQD555XmeddfxZkYThdCKMNUEMHMOGDfOLpjAz5OKQmGIwL4gall12Wf8ed/nxDi38kS/lvwVvkbwiT7/97W/99nYWj3kmrwxmmIvl6yNMHIbN7+OPP/YLzuFyBTEIh7Xn7CNQd9GKGhybHDC6s9NOO3mU5Z59yJNTQjCHSTIrR7QRrANM0/L82muvFTOMJgXErBAKMxM0T2DyLD6K8EM4hUGUEPTDgLHkkkt6P2weUh6QjYuUDiezsxAI4YZf+ceNDUwsdqLFklRiYDzhhBPcXnvt5bEnn5SBmTciLsrAj9k5Pxg375Gb4y6tF5WdQZg6TCstv/zyjp9RbQjUnZEru3Te7bbbrliJ6qx6b9f4EQBzmB7ycOTREG78YLCIKrCFI2bJDJnFRGyjNEeIHhALwGAIh7iCXZqICxR/0A9fBIhD5EftAMb/t7/9rajFwkCDtcX99tvPx8M94hdpqSB6YPcpByFIPKO4mstrre7ChXjAsDWijNj1RqbPgMTXBVjx9cGuWmEfjkfl4BpME38w+n79+vkg8hcOn+RnxEQSFSU5n4nPWzTq6BZL2hDQppPHHnvMbzi59957/SaTYDmmT59eOProowtsWtlpp538Rh+OgBOxuaVLly6F8847zzspzptvvrmw3XbbFXbffffCgQceWBg7dmxxE08pP2zqwQ+k91OnTi0ceeSRhW222aaw//77+7huuOGGYh4fffTRwo9//GO/aWnttdf2u1DZZRkk4oLYHbn00ksXHn/8cf8s96DfSu+Jg92mt956a6Fz585+l+Vtt93md8cqrr///e8eW/DB//bbb++f2c0Z/LFzc/DgwQrW5ArG+GU3aJCI76OPPvLlD+Ib9GP3+UGgYdYPEz/CZTyDmtkxizvwwAO99UFm3MFZHX40Cwxeg9Agu2b2rs97xSs/wXDhd/jRe/nXNeje3L38hq/4h5QeOvHMXIcOHerl7nofDlfps+IPh1P82EJhwxUzZmT3kN6FwxBXS+9KhcU++CeffOI4IFnUXBx6b9dsImCMPJv1WlGp+MzfbLPNvNoeIi+YgRiLrs1F2Nz7Uu64QWI2zfnRe/zKT/gafBe+5zlIhEUU8/TTT7szzzzTvwqmEfRby73yGIyjlFvwfTn3wTi4hzhajQVk1BW5yj2OcpWTR/PTWAQaJiNvbLEt9SACyCiR01588cVFZzEEXYsvQjfNvS/ljlvQPXivaMNueg5f8S+38L3iEnNjJ+qdd97pNw3pXRzXYH4Ufyk3vSv3WioO1hmGDBniOCBZjL6Uv3LTMH/pRsBm5Omuv0hyL4aHDRTEEAMGDPDxpp0xqFxvv/22P09SYKW5XJQJUwUoCqy55pqZqCfVi12rR8AYefXYZSqkZnWZKlRANKNyibmnmZmrLHY1BISAMXIhYVdDwBAwBFKKgMnIU1pxlm1DwBAwBISAMXIhYVdDwBAwBFKKgDHylFacZdsQMAQMASFgjFxI2NUQMAQMgZQiYIw8pRVn2TYEDAFDQAgYIxcSdjUEDAFDIKUIGCNPacVZtg0BQ8AQEALGyIWEXQ0BQ8AQSCkCxshTWnGWbUPAEDAEhIAxciFhV0PAEDAEUoqAMfKUVpxl2xAwBAwBIWCMXEjY1RAwBAyBlCLw/wEQynAESI2kjgAAAABJRU5ErkJggg=="></p>
<p>correct shape over correct domain with correct endpoints <em><strong>A1</strong></em><br>first maximum at (0.0035, 4.76) <em><strong>A1</strong></em><br>first minimum at (0.0085, −1.24) <em><strong>A1</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
<mi>p</mi>
</math></span> ≥ 3 between <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> = 0.0016762 and 0.0053238 and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> = 0.011676 and 0.015324 <em><strong>(M1)(A1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1A1</strong></em> for either interval.</p>
<p>= 0.00730 <em><strong>A1</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{p_{av}} = \frac{1}{{0.007}}\int_0^{0.007} {6\,{\text{sin}}\left( {100\pi t} \right)} {\text{sin}}\left( {100\pi \left( {t + 0.003} \right)} \right){\text{d}}t">
<mrow>
<msub>
<mi>p</mi>
<mrow>
<mi>a</mi>
<mi>v</mi>
</mrow>
</msub>
</mrow>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mn>0.007</mn>
</mrow>
</mfrac>
<msubsup>
<mo>∫</mo>
<mn>0</mn>
<mrow>
<mn>0.007</mn>
</mrow>
</msubsup>
<mrow>
<mn>6</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>100</mn>
<mi>π</mi>
<mi>t</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mtext>sin</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>100</mn>
<mi>π</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>t</mi>
<mo>+</mo>
<mn>0.003</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</math></span> <em><strong>(M1)</strong></em></p>
<p>= 2.87 <em><strong>A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>in each cycle the area under the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> axis is smaller than area above the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> axis <em><strong>R1</strong></em></p>
<p>the curve begins with the positive part of the cycle <em><strong>R1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a = \frac{{4.76 - \left( { - 1.24} \right)}}{2}">
<mi>a</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mn>4.76</mn>
<mo>−</mo>
<mrow>
<mo>(</mo>
<mrow>
<mo>−</mo>
<mn>1.24</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</mfrac>
</math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a = 3.00">
<mi>a</mi>
<mo>=</mo>
<mn>3.00</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d = \frac{{4.76 + \left( { - 1.24} \right)}}{2}">
<mi>d</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mn>4.76</mn>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mo>−</mo>
<mn>1.24</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</mfrac>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d = 1.76">
<mi>d</mi>
<mo>=</mo>
<mn>1.76</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b = \frac{{2\pi }}{{0.01}}">
<mi>b</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mn>2</mn>
<mi>π</mi>
</mrow>
<mrow>
<mn>0.01</mn>
</mrow>
</mfrac>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b = 628\left( { = 200\pi } \right)">
<mi>b</mi>
<mo>=</mo>
<mn>628</mn>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mn>200</mn>
<mi>π</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c = 0.0035 - \frac{{0.01}}{4}">
<mi>c</mi>
<mo>=</mo>
<mn>0.0035</mn>
<mo>−</mo>
<mfrac>
<mrow>
<mn>0.01</mn>
</mrow>
<mn>4</mn>
</mfrac>
</math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c = 0.00100">
<mi>c</mi>
<mo>=</mo>
<mn>0.00100</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">g.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">g.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mi>x</mi><mo>-</mo><mn>12</mn></mrow><mrow><mn>2</mn><mi>x</mi><mo>-</mo><mn>15</mn></mrow></mfrac><mo>,</mo><mo> </mo><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi><mo>,</mo><mo> </mo><mi>x</mi><mo>≠</mo><mfrac><mn>15</mn><mn>2</mn></mfrac></math>.</p>
</div>
<div class="specification">
<p>Find the coordinates where the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> crosses the</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-axis.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the equation of the vertical asymptote of the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The oblique asymptote of the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> can be written as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>,</mo><mo> </mo><mi>b</mi><mo>∈</mo><mi mathvariant="normal">ℚ</mi></math>.</p>
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> and the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>30</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mn>30</mn></math>, clearly indicating the points of intersection with each axis and any asymptotes.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Express <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mrow><mi>f</mi><mfenced><mi>x</mi></mfenced></mrow></mfrac></math> in partial fractions.</p>
<div class="marks">[3]</div>
<div class="question_part_label">e.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find the exact value of <math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∫</mo><mn>0</mn><mn>3</mn></munderover><mfrac><mn>1</mn><mrow><mi>f</mi><mfenced><mi>x</mi></mfenced></mrow></mfrac><mo>d</mo><mi>x</mi></math>, expressing your answer as a single logarithm.</p>
<div class="marks">[4]</div>
<div class="question_part_label">e.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><strong>Note:</strong> In part (a), penalise once only, if correct values are given instead of correct coordinates.</p>
<p><br>attempts to solve <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mi>x</mi><mo>-</mo><mn>12</mn><mo>=</mo><mn>0</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>-</mo><mn>3</mn><mo>,</mo><mn>0</mn></mrow></mfenced></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>4</mn><mo>,</mo><mn>0</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>Note:</strong> In part (a), penalise once only, if correct values are given instead of correct coordinates.</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>0</mn><mo>,</mo><mfrac><mn>4</mn><mn>5</mn></mfrac></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mn>15</mn><mn>2</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>A0</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>≠</mo><mfrac><mn>15</mn><mn>2</mn></mfrac></math>.<br> Award <em><strong>A1</strong></em> in part (b), if <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mn>15</mn><mn>2</mn></mfrac></math> is seen on their graph in part (d).<br><br></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi></mrow></mfenced><mfenced><mrow><mn>2</mn><mi>x</mi><mo>-</mo><mn>15</mn></mrow></mfenced><mo>≡</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mi>x</mi><mo>-</mo><mn>12</mn></math></p>
<p>attempts to expand <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi></mrow></mfenced><mfenced><mrow><mn>2</mn><mi>x</mi><mo>-</mo><mn>15</mn></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>a</mi><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>15</mn><mi>a</mi><mi>x</mi><mo>+</mo><mn>2</mn><mi>b</mi><mi>x</mi><mo>-</mo><mn>15</mn><mi>b</mi><mo>≡</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mi>x</mi><mo>-</mo><mn>12</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p>equates coefficients of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>1</mn><mo>=</mo><mo>-</mo><mfrac><mn>15</mn><mn>2</mn></mfrac><mo>+</mo><mn>2</mn><mi>b</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>=</mo><mfrac><mn>13</mn><mn>4</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>y</mi><mo>=</mo><mfrac><mi>x</mi><mn>2</mn></mfrac><mo>+</mo><mfrac><mn>13</mn><mn>4</mn></mfrac></mrow></mfenced></math></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>attempts division on <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mi>x</mi><mo>-</mo><mn>12</mn></mrow><mrow><mn>2</mn><mi>x</mi><mo>-</mo><mn>15</mn></mrow></mfrac></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>x</mi><mn>2</mn></mfrac><mo>+</mo><mfrac><mn>13</mn><mn>4</mn></mfrac><mo>+</mo><mo>…</mo></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>=</mo><mfrac><mn>13</mn><mn>4</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>y</mi><mo>=</mo><mfrac><mi>x</mi><mn>2</mn></mfrac><mo>+</mo><mfrac><mn>13</mn><mn>4</mn></mfrac></mrow></mfenced></math></p>
<p> </p>
<p><strong>METHOD 3</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mi>x</mi><mo>-</mo><mn>12</mn></mrow><mrow><mn>2</mn><mi>x</mi><mo>-</mo><mn>15</mn></mrow></mfrac><mo>≡</mo><mfrac><mi>x</mi><mn>2</mn></mfrac><mo>+</mo><mi>b</mi><mo>+</mo><mfrac><mi>c</mi><mrow><mn>2</mn><mi>x</mi><mo>-</mo><mn>15</mn></mrow></mfrac></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mi>x</mi><mo>-</mo><mn>12</mn><mo>≡</mo><mfrac><mrow><mfenced><mrow><mn>2</mn><mi>x</mi><mo>-</mo><mn>15</mn></mrow></mfenced><mi>x</mi></mrow><mn>2</mn></mfrac><mo>+</mo><mfenced><mrow><mn>2</mn><mi>x</mi><mo>-</mo><mn>15</mn></mrow></mfenced><mi>b</mi><mo>+</mo><mi>c</mi></math></p>
<p>equates coefficients of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> : <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>1</mn><mo>=</mo><mo>-</mo><mfrac><mn>15</mn><mn>2</mn></mfrac><mo>+</mo><mn>2</mn><mi>b</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>=</mo><mfrac><mn>13</mn><mn>4</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>y</mi><mo>=</mo><mfrac><mi>x</mi><mn>2</mn></mfrac><mo>+</mo><mfrac><mn>13</mn><mn>4</mn></mfrac></mrow></mfenced></math></p>
<p> </p>
<p><strong>METHOD 4</strong></p>
<p>attempts division on <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mi>x</mi><mo>-</mo><mn>12</mn></mrow><mrow><mn>2</mn><mi>x</mi><mo>-</mo><mn>15</mn></mrow></mfrac></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mi>x</mi><mo>-</mo><mn>12</mn></mrow><mrow><mn>2</mn><mi>x</mi><mo>-</mo><mn>15</mn></mrow></mfrac><mo>=</mo><mfrac><mi>x</mi><mn>2</mn></mfrac><mo>+</mo><mfrac><mrow><mstyle displaystyle="true"><mfrac><mrow><mn>13</mn><mi>x</mi></mrow><mn>2</mn></mfrac></mstyle><mo>-</mo><mn>12</mn></mrow><mrow><mn>2</mn><mi>x</mi><mo>-</mo><mn>15</mn></mrow></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mstyle displaystyle="true"><mfrac><mrow><mn>13</mn><mi>x</mi></mrow><mn>2</mn></mfrac></mstyle><mo>-</mo><mn>12</mn></mrow><mrow><mn>2</mn><mi>x</mi><mo>-</mo><mn>15</mn></mrow></mfrac><mo>=</mo><mfrac><mn>13</mn><mn>4</mn></mfrac><mo>+</mo><mo>…</mo></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>=</mo><mfrac><mn>13</mn><mn>4</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>y</mi><mo>=</mo><mfrac><mi>x</mi><mn>2</mn></mfrac><mo>+</mo><mfrac><mn>13</mn><mn>4</mn></mfrac></mrow></mfenced></math></p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p> <img 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"></p>
<p>two branches with approximately correct shape (for <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>30</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mn>30</mn></math>) <em><strong>A1</strong></em></p>
<p>their vertical and oblique asymptotes in approximately correct positions with both branches showing correct asymptotic behaviour to these asymptotes <em><strong>A1</strong></em></p>
<p>their axes intercepts in approximately the correct positions <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Points of intersection with the axes and the equations of asymptotes are not required to be labelled.</p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempts to split into partial fractions: <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>2</mn><mi>x</mi><mo>-</mo><mn>15</mn></mrow><mrow><mfenced><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>-</mo><mn>4</mn></mrow></mfenced></mrow></mfrac><mo>≡</mo><mfrac><mi>A</mi><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow></mfrac><mo>+</mo><mfrac><mi>B</mi><mrow><mi>x</mi><mo>-</mo><mn>4</mn></mrow></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>x</mi><mo>-</mo><mn>15</mn><mo>≡</mo><mi>A</mi><mfenced><mrow><mi>x</mi><mo>-</mo><mn>4</mn></mrow></mfenced><mo>+</mo><mi>B</mi><mfenced><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mn>3</mn></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mo>=</mo><mo>-</mo><mn>1</mn></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mfrac><mn>3</mn><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow></mfrac><mo>-</mo><mfrac><mn>1</mn><mrow><mi>x</mi><mo>-</mo><mn>4</mn></mrow></mfrac></mrow></mfenced></math></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">e.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∫</mo><mn>0</mn><mn>3</mn></munderover><mfenced><mrow><mfrac><mn>3</mn><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow></mfrac><mo>-</mo><mfrac><mn>1</mn><mrow><mi>x</mi><mo>-</mo><mn>4</mn></mrow></mfrac></mrow></mfenced><mo>d</mo><mi>x</mi></math></p>
<p>attempts to integrate and obtains two terms involving ‘ln’ <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msubsup><mfenced open="[" close="]"><mrow><mn>3</mn><mo> </mo><mi>ln</mi><mfenced open="|" close="|"><mrow><mi>x</mi><mo>+</mo><mn>3</mn></mrow></mfenced><mo>-</mo><mi>ln</mi><mfenced open="|" close="|"><mrow><mi>x</mi><mo>-</mo><mn>4</mn></mrow></mfenced></mrow></mfenced><mn>0</mn><mn>3</mn></msubsup></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>3</mn><mo> </mo><mi>ln</mi><mo> </mo><mn>6</mn><mo>-</mo><mi>ln</mi><mo> </mo><mn>1</mn><mo>-</mo><mn>3</mn><mo> </mo><mi>ln</mi><mo> </mo><mn>3</mn><mo>+</mo><mi>ln</mi><mo> </mo><mn>4</mn></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>3</mn><mo> </mo><mi>ln</mi><mo> </mo><mn>2</mn><mo>+</mo><mi>ln</mi><mo> </mo><mn>4</mn><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mi>ln</mi><mo> </mo><mn>8</mn><mo>+</mo><mi>ln</mi><mo> </mo><mn>4</mn></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi>ln</mi><mo> </mo><mn>32</mn><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mn>5</mn><mo> </mo><mi>ln</mi><mo> </mo><mn>2</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> The final <em><strong>A1</strong></em> is dependent on the previous two <em><strong>A</strong></em> marks.</p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">e.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>The function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> is defined by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = {\text{sec}}\,x + 2">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mtext>sec</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0 \leqslant x < \frac{\pi }{2}">
<mn>0</mn>
<mo>⩽<!-- ⩽ --></mo>
<mi>x</mi>
<mo><</mo>
<mfrac>
<mi>π<!-- π --></mi>
<mn>2</mn>
</mfrac>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the range of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f"> <mi>f</mi> </math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f'\left( x \right)"><msup><mi>f</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></math></span>, stating its domain.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right)"> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math></span> ≥ 3 <em><strong>A1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = {\text{sec}}\,y + 2"> <mi>x</mi> <mo>=</mo> <mrow> <mtext>sec</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>y</mi> <mo>+</mo> <mn>2</mn> </math></span> <em><strong>(M1)</strong></em></p>
<p><strong>Note:</strong> Exchange of variables can take place at any point.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\,y = \frac{1}{{x - 2}}"> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>y</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mi>x</mi> <mo>−</mo> <mn>2</mn> </mrow> </mfrac> </math></span> <em><strong>(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f'\left( x \right) = {\text{arccos}}\left( {\frac{1}{{x - 2}}} \right)"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mtext>arccos</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>1</mn> <mrow> <mi>x</mi> <mo>−</mo> <mn>2</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> ≥ 3 <em><strong>A1A1</strong></em></p>
<p><strong>Note:</strong> Allow follow through from (a) for last <em><strong>A1</strong></em> mark which is independent of earlier marks in (b).</p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The population, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math>, of a particular species of marsupial on a small remote island can be modelled by the logistic differential equation</p>
<p style="padding-left: 180px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>P</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mi>k</mi><mi>P</mi><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mi>P</mi><mi>N</mi></mfrac></mrow></mfenced></math></p>
<p>where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> is the time measured in years and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>,</mo><mo> </mo><mi>N</mi></math> are positive constants.</p>
<p>The constant <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math> represents the maximum population of this species of marsupial that the island can sustain indefinitely.</p>
</div>
<div class="specification">
<p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>0</mn></msub></math> be the initial population of marsupials.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>In the context of the population model, interpret the meaning of <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>P</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>P</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><msup><mi>k</mi><mn>2</mn></msup><mi>P</mi><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mi>P</mi><mi>N</mi></mfrac></mrow></mfenced><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mrow><mn>2</mn><mi>P</mi></mrow><mi>N</mi></mfrac></mrow></mfenced></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence show that the population of marsupials will increase at its maximum rate when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>=</mo><mfrac><mi>N</mi><mn>2</mn></mfrac></math>. Justify your answer.</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence determine the maximum value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>P</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By solving the logistic differential equation, show that its solution can be expressed in the form</p>
<p style="padding-left:150px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mi>t</mi><mo>=</mo><mi>ln</mi><mfrac><mi>P</mi><msub><mi>P</mi><mn>0</mn></msub></mfrac><mfenced><mfrac><mrow><mi>N</mi><mo>-</mo><msub><mi>P</mi><mn>0</mn></msub></mrow><mrow><mi>N</mi><mo>-</mo><mi>P</mi></mrow></mfrac></mfenced></math>.</p>
<div class="marks">[7]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>After <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn></math> years, the population of marsupials is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><msub><mi>P</mi><mn>0</mn></msub></math>. It is known that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mo>=</mo><mn>4</mn><msub><mi>P</mi><mn>0</mn></msub></math>.</p>
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> for this population model.</p>
<div class="marks">[2]</div>
<div class="question_part_label">f.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>rate of growth (change) of the (marsupial) population (with respect to time) <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[1 mark] </strong></em></p>
<p><strong><br>Note:</strong> Do not accept growth (change) in the (marsupials) population per year.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>attempts implicit differentiation on <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>P</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mi>k</mi><mi>P</mi><mo>-</mo><mfrac><mrow><mi>k</mi><msup><mi>P</mi><mn>2</mn></msup></mrow><mi>N</mi></mfrac></math> be expanding <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mi>P</mi><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mi>P</mi><mi>N</mi></mfrac></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>P</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mi>k</mi><mfrac><mrow><mo>d</mo><mi>P</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>-</mo><mn>2</mn><mfrac><mrow><mi>k</mi><mi>P</mi></mrow><mi>N</mi></mfrac><mfrac><mrow><mo>d</mo><mi>P</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></math> <em><strong>A1A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi>k</mi><mfrac><mrow><mo>d</mo><mi>P</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mrow><mn>2</mn><mi>P</mi></mrow><mi>N</mi></mfrac></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>P</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mi>k</mi><mi>P</mi><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mi>P</mi><mi>N</mi></mfrac></mrow></mfenced></math> and so <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>P</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><msup><mi>k</mi><mn>2</mn></msup><mi>P</mi><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mi>P</mi><mi>N</mi></mfrac></mrow></mfenced><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mrow><mn>2</mn><mi>P</mi></mrow><mi>N</mi></mfrac></mrow></mfenced></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>attempts implicit differentiation (product rule) on <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>P</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mi>k</mi><mi>P</mi><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mi>P</mi><mi>N</mi></mfrac></mrow></mfenced></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>P</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mi>k</mi><mfrac><mrow><mo>d</mo><mi>P</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mi>P</mi><mi>N</mi></mfrac></mrow></mfenced><mo>+</mo><mi>k</mi><mi>P</mi><mfenced><mrow><mo>-</mo><mfenced><mfrac><mn>1</mn><mi>N</mi></mfrac></mfenced><mfrac><mrow><mo>d</mo><mi>P</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p>substitutes <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>P</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mi>k</mi><mi>P</mi><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mi>P</mi><mi>N</mi></mfrac></mrow></mfenced></math> into their <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>P</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>P</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mi>k</mi><mfenced><mrow><mi>k</mi><mi>P</mi><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mi>P</mi><mi>N</mi></mfrac></mrow></mfenced></mrow></mfenced><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mi>P</mi><mi>N</mi></mfrac></mrow></mfenced><mo>+</mo><mi>k</mi><mi>P</mi><mfenced><mrow><mo>-</mo><mfenced><mfrac><mn>1</mn><mi>N</mi></mfrac></mfenced><mi>k</mi><mi>P</mi><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mi>P</mi><mi>N</mi></mfrac></mrow></mfenced></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mi>k</mi><mn>2</mn></msup><mi>P</mi><msup><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mi>P</mi><mi>N</mi></mfrac></mrow></mfenced><mn>2</mn></msup><mo>-</mo><msup><mi>k</mi><mn>2</mn></msup><mi>P</mi><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mi>P</mi><mi>N</mi></mfrac></mrow></mfenced><mfenced><mfrac><mi>P</mi><mi>N</mi></mfrac></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mi>k</mi><mn>2</mn></msup><mi>P</mi><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mi>P</mi><mi>N</mi></mfrac></mrow></mfenced><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mi>P</mi><mi>N</mi></mfrac><mo>-</mo><mfrac><mi>P</mi><mi>N</mi></mfrac></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p>so <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>P</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><msup><mi>k</mi><mn>2</mn></msup><mi>P</mi><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mi>P</mi><mi>N</mi></mfrac></mrow></mfenced><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mrow><mn>2</mn><mi>P</mi></mrow><mi>N</mi></mfrac></mrow></mfenced></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[4 marks] </strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>P</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mn>0</mn><mo>⇒</mo><msup><mi>k</mi><mn>2</mn></msup><mi>P</mi><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mi>P</mi><mi>N</mi></mfrac></mrow></mfenced><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mrow><mn>2</mn><mi>P</mi></mrow><mi>N</mi></mfrac></mrow></mfenced><mo>=</mo><mn>0</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>=</mo><mn>0</mn><mo>,</mo><mfrac><mi>N</mi><mn>2</mn></mfrac><mo>,</mo><mi>N</mi></math> <em><strong>A2</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong> </em>for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>=</mo><mfrac><mi>N</mi><mn>2</mn></mfrac></math> only.</p>
<p>uses the second derivative to show that concavity changes at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>=</mo><mfrac><mi>N</mi><mn>2</mn></mfrac></math> or the first derivative to show a local maximum at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>=</mo><mfrac><mi>N</mi><mn>2</mn></mfrac></math> <em><strong>M1</strong></em><br><br><strong>EITHER</strong></p>
<p>a clearly labelled correct sketch of <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>P</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac></math> versus <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> showing <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>=</mo><mfrac><mi>N</mi><mn>2</mn></mfrac></math> corresponding to a local maximum point for <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>P</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></math> <em><strong>R1</strong></em></p>
<p><img src="data:image/png;base64,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"></p>
<p><br><strong>OR</strong></p>
<p>a correct and clearly labelled sign diagram (table) showing <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>=</mo><mfrac><mi>N</mi><mn>2</mn></mfrac></math> corresponding to a local maximum point for <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>P</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></math> <em><strong>R1</strong></em></p>
<p><br><strong>OR</strong></p>
<p>for example, <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>P</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>3</mn><msup><mi>k</mi><mn>2</mn></msup><mi>N</mi></mrow><mn>32</mn></mfrac><mfenced><mrow><mo>></mo><mn>0</mn></mrow></mfenced></math> with <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>=</mo><mfrac><mi>N</mi><mn>4</mn></mfrac></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>P</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>3</mn><msup><mi>k</mi><mn>2</mn></msup><mi>N</mi></mrow><mn>32</mn></mfrac><mfenced><mrow><mo><</mo><mn>0</mn></mrow></mfenced></math> with <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>=</mo><mfrac><mrow><mn>3</mn><mi>N</mi></mrow><mn>4</mn></mfrac></math> showing <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>=</mo><mfrac><mi>N</mi><mn>2</mn></mfrac></math> corresponds to a local maximum point for <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>P</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></math> <em><strong>R1</strong></em></p>
<p>so the population is increasing at its maximum rate when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>=</mo><mfrac><mi>N</mi><mn>2</mn></mfrac></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[5 marks] </strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>substitutes <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>=</mo><mfrac><mi>N</mi><mn>2</mn></mfrac></math> into <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>P</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>P</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mi>k</mi><mfenced><mfrac><mi>N</mi><mn>2</mn></mfrac></mfenced><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mstyle displaystyle="true"><mfrac><mi>N</mi><mn>2</mn></mfrac></mstyle><mi>N</mi></mfrac></mrow></mfenced></math></p>
<p>the maximum value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>P</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>k</mi><mi>N</mi></mrow><mn>4</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>attempts to separate variables <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><mfrac><mi>N</mi><mrow><mi>P</mi><mfenced><mrow><mi>N</mi><mo>-</mo><mi>P</mi></mrow></mfenced></mrow></mfrac><mo>d</mo><mi>P</mi><mo>=</mo><mo>∫</mo><mi>k</mi><mo> </mo><mo>d</mo><mi>t</mi></math></p>
<p>attempts to write <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>N</mi><mrow><mi>P</mi><mfenced><mrow><mi>N</mi><mo>-</mo><mi>P</mi></mrow></mfenced></mrow></mfrac></math> in partial fractions form <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>N</mi><mrow><mi>P</mi><mfenced><mrow><mi>N</mi><mo>-</mo><mi>P</mi></mrow></mfenced></mrow></mfrac><mo>≡</mo><mfrac><mi>A</mi><mi>P</mi></mfrac><mo>+</mo><mfrac><mi>B</mi><mfenced><mrow><mi>N</mi><mo>-</mo><mi>P</mi></mrow></mfenced></mfrac><mo>⇒</mo><mi>N</mi><mo>≡</mo><mi>A</mi><mfenced><mrow><mi>N</mi><mo>-</mo><mi>P</mi></mrow></mfenced><mo>+</mo><mi>B</mi><mi>P</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo> </mo><mi>B</mi><mo>=</mo><mn>1</mn></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>N</mi><mrow><mi>P</mi><mfenced><mrow><mi>N</mi><mo>-</mo><mi>P</mi></mrow></mfenced></mrow></mfrac><mo>≡</mo><mfrac><mn>1</mn><mi>P</mi></mfrac><mo>+</mo><mfrac><mn>1</mn><mfenced><mrow><mi>N</mi><mo>-</mo><mi>P</mi></mrow></mfenced></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><mfenced><mrow><mfrac><mn>1</mn><mi>P</mi></mfrac><mo>+</mo><mfrac><mn>1</mn><mfenced><mrow><mi>N</mi><mo>-</mo><mi>P</mi></mrow></mfenced></mfrac></mrow></mfenced><mo>d</mo><mi>P</mi><mo>=</mo><mo>∫</mo><mi>k</mi><mo> </mo><mo>d</mo><mi>t</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>ln</mi><mo> </mo><mi>P</mi><mo>-</mo><mi>ln</mi><mfenced><mrow><mi>N</mi><mo>-</mo><mi>P</mi></mrow></mfenced><mo>=</mo><mi>k</mi><mi>t</mi><mfenced><mrow><mo>+</mo><mi>C</mi></mrow></mfenced></math> <em><strong>A1A1</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>A1</strong> </em>for <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mi>ln</mi><mfenced><mrow><mi>N</mi><mo>-</mo><mi>P</mi></mrow></mfenced></math> and <em><strong>A1</strong> </em>for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mi>P</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mi>t</mi><mfenced><mrow><mo>+</mo><mi>C</mi></mrow></mfenced></math>. Absolute value signs are not required.</p>
<p> </p>
<p>attempts to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>0</mn></msub></math> <em><strong>M1</strong></em></p>
<p>when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn><mo>,</mo><mo> </mo><mi>P</mi><mo>=</mo><msub><mi>P</mi><mn>0</mn></msub></math> and so <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mo>=</mo><mi>ln</mi><mo> </mo><msub><mi>P</mi><mn>0</mn></msub><mo>-</mo><mi>ln</mi><mfenced><mrow><mi>N</mi><mo>-</mo><msub><mi>P</mi><mn>0</mn></msub></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mi>t</mi><mo>=</mo><mi>ln</mi><mfenced><mfrac><mi>P</mi><mrow><mi>N</mi><mo>-</mo><mi>P</mi></mrow></mfrac></mfenced><mo>-</mo><mi>ln</mi><mfenced><mfrac><msub><mi>P</mi><mn>0</mn></msub><mrow><mi>N</mi><mo>-</mo><msub><mi>P</mi><mi>o</mi></msub></mrow></mfrac></mfenced><mo> </mo><mfenced><mrow><mo>=</mo><mi>ln</mi><mfenced><mfrac><mstyle displaystyle="true"><mfrac><mi>P</mi><mrow><mi>N</mi><mo>-</mo><mi>P</mi></mrow></mfrac></mstyle><mstyle displaystyle="true"><mfrac><msub><mi>P</mi><mn>0</mn></msub><mrow><mi>N</mi><mo>-</mo><msub><mi>P</mi><mn>0</mn></msub></mrow></mfrac></mstyle></mfrac></mfenced></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p>so <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mi>t</mi><mo>=</mo><mi>ln</mi><mfrac><mi>P</mi><msub><mi>P</mi><mn>0</mn></msub></mfrac><mfenced><mfrac><mrow><mi>N</mi><mo>-</mo><msub><mi>P</mi><mn>0</mn></msub></mrow><mrow><mi>N</mi><mo>-</mo><mi>P</mi></mrow></mfrac></mfenced></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>attempts to separate variables <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><mfrac><mn>1</mn><mrow><mi>P</mi><mfenced><mrow><mn>1</mn><mo>-</mo><mstyle displaystyle="true"><mfrac><mi>P</mi><mi>N</mi></mfrac></mstyle></mrow></mfenced></mrow></mfrac><mo>d</mo><mi>P</mi><mo>=</mo><mo>∫</mo><mi>k</mi><mo> </mo><mo>d</mo><mi>t</mi></math></p>
<p>attempts to write <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mrow><mi>P</mi><mfenced><mrow><mn>1</mn><mo>-</mo><mstyle displaystyle="true"><mfrac><mi>P</mi><mi>N</mi></mfrac></mstyle></mrow></mfenced></mrow></mfrac></math> in partial fractions form <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mrow><mi>P</mi><mfenced><mrow><mn>1</mn><mo>-</mo><mstyle displaystyle="true"><mfrac><mi>P</mi><mi>N</mi></mfrac></mstyle></mrow></mfenced></mrow></mfrac><mo>≡</mo><mfrac><mi>A</mi><mi>P</mi></mfrac><mo>+</mo><mfrac><mi>B</mi><mrow><mn>1</mn><mo>-</mo><mfrac><mi>P</mi><mi>N</mi></mfrac></mrow></mfrac><mo>⇒</mo><mn>1</mn><mo>≡</mo><mi>A</mi><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mi>P</mi><mi>N</mi></mfrac></mrow></mfenced><mo>+</mo><mi>B</mi><mi>P</mi></math> </p>
<p> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo> </mo><mi>B</mi><mo>=</mo><mfrac><mn>1</mn><mi>N</mi></mfrac></math> <em><strong>A1</strong></em></p>
<p><em><strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mrow><mi>P</mi><mfenced><mrow><mn>1</mn><mo>-</mo><mstyle displaystyle="true"><mfrac><mi>P</mi><mi>N</mi></mfrac></mstyle></mrow></mfenced></mrow></mfrac><mo>≡</mo><mfrac><mn>1</mn><mi>P</mi></mfrac><mo>+</mo><mfrac><mn>1</mn><mrow><mi>N</mi><mfenced><mrow><mn>1</mn><mo>-</mo><mstyle displaystyle="true"><mfrac><mi>P</mi><mi>N</mi></mfrac></mstyle></mrow></mfenced></mrow></mfrac></math></strong></em></p>
<p><em><strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><mfrac><mn>1</mn><mi>P</mi></mfrac><mo>+</mo><mfrac><mn>1</mn><mrow><mi>N</mi><mfenced><mrow><mn>1</mn><mo>-</mo><mstyle displaystyle="true"><mfrac><mi>P</mi><mi>N</mi></mfrac></mstyle></mrow></mfenced></mrow></mfrac><mo>d</mo><mi>P</mi><mo>=</mo><mo>∫</mo><mi>k</mi><mo> </mo><mo>d</mo><mi>t</mi></math></strong></em></p>
<p><em><strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>ln</mi><mo> </mo><mi>P</mi><mo>-</mo><mi>ln</mi><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mi>P</mi><mi>N</mi></mfrac></mrow></mfenced><mo>=</mo><mi>k</mi><mi>t</mi><mfenced><mrow><mo>+</mo><mi>C</mi></mrow></mfenced></math> A1A1</strong></em></p>
<p><strong><br>Note:</strong> Award <em><strong>A1</strong> </em>for <em><strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mi>ln</mi><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mi>P</mi><mi>N</mi></mfrac></mrow></mfenced></math></strong></em> and <em><strong>A1</strong> </em>for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mi>P</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mi>t</mi><mfenced><mrow><mo>+</mo><mi>C</mi></mrow></mfenced></math>. Absolute value signs are not required.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mfenced><mfrac><mi>P</mi><mrow><mn>1</mn><mo>-</mo><mstyle displaystyle="true"><mfrac><mi>P</mi><mi>N</mi></mfrac></mstyle></mrow></mfrac></mfenced><mo>=</mo><mi>k</mi><mi>t</mi><mo>+</mo><mi>C</mi><mo>⇒</mo><mi>ln</mi><mfenced><mfrac><mrow><mi>N</mi><mi>P</mi></mrow><mrow><mi>N</mi><mo>-</mo><mi>P</mi></mrow></mfrac></mfenced><mo>=</mo><mi>k</mi><mi>t</mi><mo>+</mo><mi>C</mi></math></p>
<p>attempts to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>0</mn></msub></math> <em><strong>M1</strong></em></p>
<p>when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn><mo>,</mo><mo> </mo><mi>P</mi><mo>=</mo><msub><mi>P</mi><mn>0</mn></msub></math> and so <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mo>=</mo><mi>ln</mi><mfenced><mfrac><mrow><mi>N</mi><msub><mi>P</mi><mn>0</mn></msub></mrow><mrow><mi>N</mi><mo>-</mo><msub><mi>P</mi><mn>0</mn></msub></mrow></mfrac></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mi>t</mi><mo>=</mo><mi>ln</mi><mfenced><mfrac><mrow><mi>N</mi><mi>P</mi></mrow><mrow><mi>N</mi><mo>-</mo><mi>P</mi></mrow></mfrac></mfenced><mo>-</mo><mi>ln</mi><mfenced><mfrac><mrow><mi>N</mi><msub><mi>P</mi><mn>0</mn></msub></mrow><mrow><mi>N</mi><mo>-</mo><msub><mi>P</mi><mn>0</mn></msub></mrow></mfrac></mfenced><mo> </mo><mfenced><mrow><mo>=</mo><mi>ln</mi><mfrac><mstyle displaystyle="true"><mfrac><mi>P</mi><mrow><mi>N</mi><mo>-</mo><mi>P</mi></mrow></mfrac></mstyle><mstyle displaystyle="true"><mfrac><msub><mi>P</mi><mn>0</mn></msub><mrow><mi>N</mi><mo>-</mo><msub><mi>P</mi><mn>0</mn></msub></mrow></mfrac></mstyle></mfrac></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mi>t</mi><mo>=</mo><mi>ln</mi><mfrac><mi>P</mi><msub><mi>P</mi><mn>0</mn></msub></mfrac><mfenced><mfrac><mrow><mi>N</mi><mo>-</mo><msub><mi>P</mi><mn>0</mn></msub></mrow><mrow><mi>N</mi><mo>-</mo><mi>P</mi></mrow></mfrac></mfenced></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>METHOD 3</strong></p>
<p>lets <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi><mo>=</mo><mfrac><mn>1</mn><mi>P</mi></mfrac></math> and forms <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>u</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mo>-</mo><mfrac><mn>1</mn><msup><mi>P</mi><mn>2</mn></msup></mfrac><mfrac><mrow><mo>d</mo><mi>P</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></math> <em><strong>M1</strong></em></p>
<p>multiplies both sides of the differential equation by <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mn>1</mn><msup><mi>P</mi><mn>2</mn></msup></mfrac></math> and makes the above substitutions <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mn>1</mn><msup><mi>P</mi><mn>2</mn></msup></mfrac><mfrac><mrow><mo>d</mo><mi>P</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mi>k</mi><mfenced><mrow><mfrac><mn>1</mn><mi>N</mi></mfrac><mo>-</mo><mfrac><mn>1</mn><mi>P</mi></mfrac></mrow></mfenced><mo>⇒</mo><mfrac><mrow><mo>d</mo><mi>u</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mi>k</mi><mfenced><mrow><mfrac><mn>1</mn><mi>N</mi></mfrac><mo>-</mo><mi>u</mi></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>u</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>+</mo><mi>k</mi><mi>u</mi><mo>=</mo><mfrac><mi>k</mi><mi>N</mi></mfrac></math> (linear first-order DE)<em><strong> A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>IF</mtext><mo>=</mo><msup><mtext>e</mtext><mrow><mo>∫</mo><mi>k</mi><mo> </mo><mo>d</mo><mi>t</mi></mrow></msup><mo>=</mo><msup><mtext>e</mtext><mrow><mi>k</mi><mi>t</mi></mrow></msup><mo>⇒</mo><msup><mtext>e</mtext><mrow><mi>k</mi><mi>t</mi></mrow></msup><mfrac><mrow><mo>d</mo><mi>u</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>+</mo><mi>k</mi><msup><mtext>e</mtext><mrow><mi>k</mi><mi>t</mi></mrow></msup><mi>u</mi><mo>=</mo><mfrac><mi>k</mi><mi>N</mi></mfrac><msup><mtext>e</mtext><mrow><mi>k</mi><mi>t</mi></mrow></msup></math><em><strong> (M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mo>d</mo><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mfenced><mrow><mi>u</mi><msup><mtext>e</mtext><mrow><mi>k</mi><mi>t</mi></mrow></msup></mrow></mfenced><mo>=</mo><mfrac><mi>k</mi><mi>N</mi></mfrac><msup><mtext>e</mtext><mrow><mi>k</mi><mi>t</mi></mrow></msup></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi><msup><mtext>e</mtext><mrow><mi>k</mi><mi>t</mi></mrow></msup><mo>=</mo><mfrac><mn>1</mn><mi>N</mi></mfrac><msup><mtext>e</mtext><mrow><mi>k</mi><mi>t</mi></mrow></msup><mfenced><mrow><mo>+</mo><mi>C</mi></mrow></mfenced><mo> </mo><mfenced><mrow><mfrac><mn>1</mn><mi>P</mi></mfrac><msup><mtext>e</mtext><mrow><mi>k</mi><mi>t</mi></mrow></msup><mo>=</mo><mfrac><mn>1</mn><mi>N</mi></mfrac><msup><mtext>e</mtext><mrow><mi>k</mi><mi>t</mi></mrow></msup><mfenced><mrow><mo>+</mo><mi>C</mi></mrow></mfenced></mrow></mfenced></math><em><strong> A1</strong></em></p>
<p>attempts to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>0</mn></msub></math> <em><strong>M1</strong></em></p>
<p>when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn><mo>,</mo><mo> </mo><mi>P</mi><mo>=</mo><msub><mi>P</mi><mn>0</mn></msub><mo>,</mo><mo> </mo><mi>u</mi><mo>=</mo><mfrac><mn>1</mn><msub><mi>P</mi><mn>0</mn></msub></mfrac></math> and so <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mo>=</mo><mfrac><mn>1</mn><msub><mi>P</mi><mn>0</mn></msub></mfrac><mo>-</mo><mfrac><mn>1</mn><mi>N</mi></mfrac><mfenced><mrow><mo>=</mo><mfrac><mrow><mi>N</mi><mo>-</mo><msub><mi>P</mi><mn>0</mn></msub></mrow><mrow><mi>N</mi><msub><mi>P</mi><mn>0</mn></msub></mrow></mfrac></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>e</mtext><mrow><mi>k</mi><mi>t</mi></mrow></msup><mfenced><mfrac><mrow><mi>N</mi><mo>-</mo><mi>P</mi></mrow><mrow><mi>N</mi><mi>P</mi></mrow></mfrac></mfenced><mo>=</mo><mfrac><mrow><mi>N</mi><mo>-</mo><msub><mi>P</mi><mn>0</mn></msub></mrow><mrow><mi>N</mi><msub><mi>P</mi><mn>0</mn></msub></mrow></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>e</mtext><mrow><mi>k</mi><mi>t</mi></mrow></msup><mtext>=</mtext><mfenced><mfrac><mi>P</mi><mrow><mi>N</mi><mo>-</mo><mi>P</mi></mrow></mfrac></mfenced><mfenced><mfrac><mrow><mi>N</mi><mo>-</mo><msub><mi>P</mi><mn>0</mn></msub></mrow><msub><mi>P</mi><mn>0</mn></msub></mfrac></mfenced></math><em><strong> A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mi>t</mi><mo>=</mo><mi>ln</mi><mfrac><mi>P</mi><msub><mi>P</mi><mn>0</mn></msub></mfrac><mfenced><mfrac><mrow><mi>N</mi><mo>-</mo><msub><mi>P</mi><mn>0</mn></msub></mrow><mrow><mi>N</mi><mo>-</mo><mi>P</mi></mrow></mfrac></mfenced></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[7 marks]</strong></em></p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>substitutes <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>10</mn><mo>,</mo><mo> </mo><mi>P</mi><mo>=</mo><mn>3</mn><msub><mi>P</mi><mn>0</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mo>=</mo><mn>4</mn><msub><mi>P</mi><mn>0</mn></msub></math> into <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mi>t</mi><mo>=</mo><mi>ln</mi><mfrac><mi>P</mi><msub><mi>P</mi><mn>0</mn></msub></mfrac><mfenced><mfrac><mrow><mi>N</mi><mo>-</mo><msub><mi>P</mi><mn>0</mn></msub></mrow><mrow><mi>N</mi><mo>-</mo><mi>P</mi></mrow></mfrac></mfenced></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn><mi>k</mi><mo>=</mo><mi>ln</mi><mo> </mo><mn>3</mn><mfenced><mfrac><mrow><mn>4</mn><msub><mi>P</mi><mn>0</mn></msub><mo>-</mo><msub><mi>P</mi><mn>0</mn></msub></mrow><mrow><mn>4</mn><msub><mi>P</mi><mn>0</mn></msub><mo>-</mo><mn>3</mn><msub><mi>P</mi><mn>0</mn></msub></mrow></mfrac></mfenced><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mi>ln</mi><mo> </mo><mn>9</mn></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>220</mn><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mfrac><mn>1</mn><mn>10</mn></mfrac><mi>ln</mi><mo> </mo><mn>9</mn><mo>,</mo><mo>=</mo><mfrac><mn>1</mn><mn>5</mn></mfrac><mi>ln</mi><mo> </mo><mn>3</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">f.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>An extremely tricky question even for the strong candidates. Many struggled to understand what was expected in parts (b) and (c). As the question was set all with pronumerals instead of numbers many candidates found it challenging, thrown at deep water for parts (b), (c) and (e). It definitely was the question to show their skills for the Level 7 candidates provided that they did not run out of time.</p>
<p>Part (a) Very well answered, mostly correctly referring to the rate of change. Some candidates did not gain this mark because their sentence did not include the reference to the rate of change. Worded explanations continue being problematic to many candidates.</p>
<p>Part (b) Many candidates were confused how to approach this question and did not realise that they<br>needed to differentiate implicitly. Some tried but with errors, some did not fully show what was required.</p>
<p>Part (c) Most candidates started with equating the second derivative to zero. Most gave the answer <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>=</mo><mfrac><mi>N</mi><mn>2</mn></mfrac></math>omitting the other two possibilities. Most stopped here. Only a small number of candidates provided the correct mathematical argument to show it is a local maximum.</p>
<p>Part (d) Well done by those candidates who got that far. Most got the correct answer, sometimes not fully simplified.</p>
<p>Part (e) Most candidates separated the variables, but some were not able to do much more. Some candidates knew to resolve into partial fractions and attempted to do so, mainly successfully. Then they integrated, again, mainly successfully and continued to substitute the initial condition and manipulate the equation accordingly.</p>
<p>Part (f) Algebraic manipulation of the logarithmic expression was too much for some candidates with a common error of 0.33 given as the answer. The strong candidates provided the correct exact or rounded value.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">f.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = 2{\sin ^2}x + 7\sin 2x + \tan x - 9,{\text{ }}0 \leqslant x < \frac{\pi }{2}">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>2</mn>
<mrow>
<msup>
<mi>sin</mi>
<mn>2</mn>
</msup>
</mrow>
<mi>x</mi>
<mo>+</mo>
<mn>7</mn>
<mi>sin</mi>
<mo><!-- --></mo>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mi>tan</mi>
<mo><!-- --></mo>
<mi>x</mi>
<mo>−<!-- − --></mo>
<mn>9</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>0</mn>
<mo>⩽<!-- ⩽ --></mo>
<mi>x</mi>
<mo><</mo>
<mfrac>
<mi>π<!-- π --></mi>
<mn>2</mn>
</mfrac>
</math></span>.</p>
</div>
<div class="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u = \tan x">
<mi>u</mi>
<mo>=</mo>
<mi>tan</mi>
<mo><!-- --></mo>
<mi>x</mi>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine an expression for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f’(x)"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></span> in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch a graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f’(x)"> <mi>y</mi> <mo>=</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></span> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0 \leqslant x < \frac{\pi }{2}"> <mn>0</mn> <mo>⩽</mo> <mi>x</mi> <mo><</mo> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>-coordinate(s) of the point(s) of inflexion of the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f(x)"> <mi>y</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></span>, labelling these clearly on the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f’(x)"> <mi>y</mi> <mo>=</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Express <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sin x"> <mi>sin</mi> <mo></mo> <mi>x</mi> </math></span> in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mu "><mi>u</mi></math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Express <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sin 2x"> <mi>sin</mi> <mo></mo> <mn>2</mn> <mi>x</mi> </math></span> in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u"> <mi>u</mi> </math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = 0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </math></span> can be expressed as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u^3} - 7{u^2} + 15u - 9 = 0"> <mrow> <msup> <mi>u</mi> <mn>3</mn> </msup> </mrow> <mo>−</mo> <mn>7</mn> <mrow> <msup> <mi>u</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>15</mn> <mi>u</mi> <mo>−</mo> <mn>9</mn> <mo>=</mo> <mn>0</mn> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Solve the equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = 0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </math></span>, giving your answers in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\arctan k"> <mi>arctan</mi> <mo></mo> <mi>k</mi> </math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k \in \mathbb{Z}"> <mi>k</mi> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> </math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f’(x) = 4\sin x\cos x + 14\cos 2x + {\sec ^2}x"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>4</mn> <mi>sin</mi> <mo></mo> <mi>x</mi> <mi>cos</mi> <mo></mo> <mi>x</mi> <mo>+</mo> <mn>14</mn> <mi>cos</mi> <mo></mo> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mrow> <msup> <mi>sec</mi> <mn>2</mn> </msup> </mrow> <mi>x</mi> </math></span> (or equivalent) <strong><em>(M1)A1</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img src="images/Schermafbeelding_2018-02-08_om_16.47.49.png" alt="N17/5/MATHL/HP2/ENG/TZ0/11.a.ii/M"> <strong><em>A1A1A1A1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>A1 </em></strong>for correct behaviour at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 0"> <mi>x</mi> <mo>=</mo> <mn>0</mn> </math></span>, <strong><em>A1 </em></strong>for correct domain and correct behaviour for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x \to \frac{\pi }{2}"> <mi>x</mi> <mo stretchy="false">→</mo> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </math></span>, <strong><em>A1 </em></strong>for two clear intersections with <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>-axis and minimum point, <strong><em>A1 </em></strong>for clear maximum point.</p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 0.0736"> <mi>x</mi> <mo>=</mo> <mn>0.0736</mn> </math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 1.13"> <mi>x</mi> <mo>=</mo> <mn>1.13</mn> </math></span> <strong><em>A1</em></strong></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to write <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sin x"> <mi>sin</mi> <mo></mo> <mi>x</mi> </math></span> in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u"> <mi>u</mi> </math></span> only <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sin x = \frac{u}{{\sqrt {1 + {u^2}} }}"> <mi>sin</mi> <mo></mo> <mi>x</mi> <mo>=</mo> <mfrac> <mi>u</mi> <mrow> <msqrt> <mn>1</mn> <mo>+</mo> <mrow> <msup> <mi>u</mi> <mn>2</mn> </msup> </mrow> </msqrt> </mrow> </mfrac> </math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\cos x = \frac{1}{{\sqrt {1 + {u^2}} }}"> <mi>cos</mi> <mo></mo> <mi>x</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <msqrt> <mn>1</mn> <mo>+</mo> <mrow> <msup> <mi>u</mi> <mn>2</mn> </msup> </mrow> </msqrt> </mrow> </mfrac> </math></span> <strong><em>(A1)</em></strong></p>
<p>attempt to use <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sin 2x = 2\sin x\cos x{\text{ }}\left( { = 2\frac{u}{{\sqrt {1 + {u^2}} }}\frac{1}{{\sqrt {1 + {u^2}} }}} \right)"> <mi>sin</mi> <mo></mo> <mn>2</mn> <mi>x</mi> <mo>=</mo> <mn>2</mn> <mi>sin</mi> <mo></mo> <mi>x</mi> <mi>cos</mi> <mo></mo> <mi>x</mi> <mrow> <mtext> </mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mn>2</mn> <mfrac> <mi>u</mi> <mrow> <msqrt> <mn>1</mn> <mo>+</mo> <mrow> <msup> <mi>u</mi> <mn>2</mn> </msup> </mrow> </msqrt> </mrow> </mfrac> <mfrac> <mn>1</mn> <mrow> <msqrt> <mn>1</mn> <mo>+</mo> <mrow> <msup> <mi>u</mi> <mn>2</mn> </msup> </mrow> </msqrt> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sin 2x = \frac{{2u}}{{1 + {u^2}}}"> <mi>sin</mi> <mo></mo> <mn>2</mn> <mi>x</mi> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> <mi>u</mi> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mrow> <msup> <mi>u</mi> <mn>2</mn> </msup> </mrow> </mrow> </mfrac> </math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2{\sin ^2}x + 7\sin 2x + \tan x - 9 = 0"> <mn>2</mn> <mrow> <msup> <mi>sin</mi> <mn>2</mn> </msup> </mrow> <mi>x</mi> <mo>+</mo> <mn>7</mn> <mi>sin</mi> <mo></mo> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mi>tan</mi> <mo></mo> <mi>x</mi> <mo>−</mo> <mn>9</mn> <mo>=</mo> <mn>0</mn> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{2{u^2}}}{{1 + {u^2}}} + \frac{{14u}}{{1 + {u^2}}} + u - 9{\text{ }}( = 0)"> <mfrac> <mrow> <mn>2</mn> <mrow> <msup> <mi>u</mi> <mn>2</mn> </msup> </mrow> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mrow> <msup> <mi>u</mi> <mn>2</mn> </msup> </mrow> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mn>14</mn> <mi>u</mi> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mrow> <msup> <mi>u</mi> <mn>2</mn> </msup> </mrow> </mrow> </mfrac> <mo>+</mo> <mi>u</mi> <mo>−</mo> <mn>9</mn> <mrow> <mtext> </mtext> </mrow> <mo stretchy="false">(</mo> <mo>=</mo> <mn>0</mn> <mo stretchy="false">)</mo> </math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{2{u^2} + 14u + u(1 + {u^2}) - 9(1 + {u^2})}}{{1 + {u^2}}} = 0"> <mfrac> <mrow> <mn>2</mn> <mrow> <msup> <mi>u</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>14</mn> <mi>u</mi> <mo>+</mo> <mi>u</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mrow> <msup> <mi>u</mi> <mn>2</mn> </msup> </mrow> <mo stretchy="false">)</mo> <mo>−</mo> <mn>9</mn> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mrow> <msup> <mi>u</mi> <mn>2</mn> </msup> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mrow> <msup> <mi>u</mi> <mn>2</mn> </msup> </mrow> </mrow> </mfrac> <mo>=</mo> <mn>0</mn> </math></span> (or equivalent) <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u^3} - 7{u^2} + 15u - 9 = 0"> <mrow> <msup> <mi>u</mi> <mn>3</mn> </msup> </mrow> <mo>−</mo> <mn>7</mn> <mrow> <msup> <mi>u</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>15</mn> <mi>u</mi> <mo>−</mo> <mn>9</mn> <mo>=</mo> <mn>0</mn> </math></span> <strong><em>AG</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u = 1"> <mi>u</mi> <mo>=</mo> <mn>1</mn> </math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u = 3"> <mi>u</mi> <mo>=</mo> <mn>3</mn> </math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \arctan (1)"> <mi>x</mi> <mo>=</mo> <mi>arctan</mi> <mo></mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \arctan (3)"> <mi>x</mi> <mo>=</mo> <mi>arctan</mi> <mo></mo> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </math></span> <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Only accept answers given the required form.</p>
<p> </p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the polynomial <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P\left( z \right) \equiv {z^4} - 6{z^3} - 2{z^2} + 58z - 51,\,\,z \in \mathbb{C}">
<mi>P</mi>
<mrow>
<mo>(</mo>
<mi>z</mi>
<mo>)</mo>
</mrow>
<mo>≡<!-- ≡ --></mo>
<mrow>
<msup>
<mi>z</mi>
<mn>4</mn>
</msup>
</mrow>
<mo>−<!-- − --></mo>
<mn>6</mn>
<mrow>
<msup>
<mi>z</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>−<!-- − --></mo>
<mn>2</mn>
<mrow>
<msup>
<mi>z</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>58</mn>
<mi>z</mi>
<mo>−<!-- − --></mo>
<mn>51</mn>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>z</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">C</mi>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = {x^4} - 6{x^3} - 2{x^2} + 58x - 51"> <mi>y</mi> <mo>=</mo> <mrow> <msup> <mi>x</mi> <mn>4</mn> </msup> </mrow> <mo>−</mo> <mn>6</mn> <mrow> <msup> <mi>x</mi> <mn>3</mn> </msup> </mrow> <mo>−</mo> <mn>2</mn> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>58</mn> <mi>x</mi> <mo>−</mo> <mn>51</mn> </math></span>, stating clearly the coordinates of any maximum and minimum points and intersections with axes.</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, or otherwise, state the condition on <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k \in \mathbb{R}"> <mi>k</mi> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> </math></span> such that all roots of the equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P\left( z \right) = k"> <mi>P</mi> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>k</mi> </math></span> are real.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><img 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dx88x9w2tAzcM9d94Bmj0OGnIbNX25uPsXy/r/cDxqMnnnmGdCgNWTAACz9aAkGDxjaCmC1RtQKjpB+cHvNcR8Vzum+ndfkMWfFqApUyZRGFFJT8/9xVlYWqquVRuQfKS8pdE1U0PS0TMyZOwfjxo/DvXfdi2eeegFfffUVystdR2xs3boFixe8j2lXXiMySe7ZE7kDcvH8s8+L30XFhXj5ny/j9jtvF7+7d+2K8887H3977Ml2RNUaUTtIgn7QYlQNOouAPuQ+KpzTfbukqAiZmVmIi4sLCJNoSRyJGhF3/TW1rPr27YvKmiOm0LC6RqTZNFDncs01V7XCb8LE8eK3u1GuW7cG/fr1RUqfPs3pyCT6zjvvCJV/xfKVsMXZkJnZsiM9JycbS5csQX19ffM36kYyAsxec9wdB6fpZs+evRgwIBfCgUdyMUVCdpGoEUkr38OHD2PTpk2m42y321FUVOSVVl1NDUoLS7Bp69eAU94GRjpwj2Rzd8ZmC0kzFjIxcuDplqWkpIRicCMxIR4O2EUcMdp/s2PHdmRmZ2Lv3r2AzS5MdF27dWvB2GbHieNHUVlXg/VrPsaa9RuQEJ8geHdX+OKmAwsXLFiAAbnZuPlP96C4oBANzgYMyh2CR598FBlpGW5WpP6lzazkiltZWQmzPCrbMkx1lOhxll9RcQmOORwiLE1bfsz6zVlHjzmcKC0pNh9Tmx0bt2zE2BFjUXGAtwypnHz1bTLL8NixYzKzk5KXtIGoV69eGDJkiBSmvGXSGRrqOwO2eieqamhdoj2tgqIC1L1eg0E52VKPTaa4U0SPcyAqLy/3KqM3bGQ8qzxSg5zsLCT06Aeb0wHdpkN3Ao8/9jheefqfLl6cTtTVNeAXP++HIQNyQX525Gm3pmcqrYAiM/tU6I4GJCcn47Qhp+GkLifhpxMnUNDkUt8/rT8GDDgN542fiI22DdhWeQTn/PYcjBg8DP1O/rkMMbzkoYsBL61/f/Tu2xea8Ax0LXo7YQMdkC77GTGxf/9+gRlp0oSB2cvsdXV1yM7KEjJ6AcGURwUFBUJGws9VE+T/pckRlVNN5SEcP3oMA4achi4AzEK0rLQYFQcrMHr0aCT1TmRtg1RIDceOYdCQM0S9NKXQABQXF5uVddD5ShuIGhsbTd+FLCy2ccBJti5eafXvkwZyO6ZG313iGSLUmZB7ONdA5HZ75jwHJd5uR3xCEpK7UTOnf8BHKz7EsFEjMOmCc5srWK9eSfjpZ51a4U9mPZtmQ0JCgtiNThpd9+Turm/i4po74d4JvdC1W1c8/sTjmD9/PnZu240XXnwO3bu5omI0E5F8E5eQiPiEhFY8SybRLjvSBpvLj2Gtgeomp4xUxvHx8S0ytkNA7gNHQzw6d9aQ3K2ryNis1Zu9e10R/MeMG4uSomI2+dxo2ex2dBdt0P1E/l+bk9uw6l+GyOPIP88+U/Tq1UO8o9mozItme8c99tjIzNtXXhr3kQwaKTUt+4L27c/H5xs245GHHmrF4qm//g1KSlpHryDtjTqlpKQkYVuvqipv5SJac8S1tytrUMu6kXvB1Al7q/xN+aE74PK6MiX3iMiUvBndmHIwROF9nM1TDPMpcnnNFRTtF5OqQYNzQRMsK16NJ/0s4sSyFNJkEqKrsLh1Rxkq6jZdw0mMjY741ZkXn4mc3R4voCotLcWc2a/iySefbHb1dDsajD7rTBTsK2jlePDjjz/iwosvRJcuXTB58vk4cqSu1d4jCkR7zsSJ6N6tZ6uiIPdtzdEy+LV6KfOHDri8rmRmGll5UZ/pXpPj4IysDjaz7Y0egnB5zVHdzs7JEVYXD/KWuiXrVaRdlhqIaEZOV9H+Iqk46xqEnV9qphGWGSlgDscxlJYUYdq06zFhwkR88803YnF47rx5+Nvf/iY4vuCCCzBo0CCsWLFC/KZZ+PZd23HvvbPE76FDB2PC5PMwb9685vfvL12Be+66Q/z2/E+4b9t5qqBdl+e84imDz3sesZrJ0z4iTo2IQvyAUWt3lZ/5oG78aiOys7OacVU3PAhIWyPiYbdjKm6N6EidbBduHZ06d+6YuOS33KY53amjtLwaF5z7X/jhhx/w6adrW0n0xRdfiN806164aBGmT78Jhw4ewvr16zB/7nzk5rbsQn/7rTdxzbSr8LOf/Qzfffcd/vrnOzBp0uRW+XH/cNrMWlXwIQmjtkAcmN9Ft5bT5eDR+pmZvzg0IhrIv9r0FWbMmGGmKCpvLwhYaiAi+bJzs0FrFjIvGhTI8wkMi85uvrlNc06bDekp3fHt9/+GjU5qtbm6GvJYAhpaORTQjHH9+vXCHfq6P1zXbkE3NTUVa1avQ2FBAf5w443o2WQydcvm+ZdljYgIMp8+6ykjyz2zKZe82TgvDo2ItjA4HA52TzlOHCOVluUGopzMHNQdc3m+yALdqf8EZ+eTZGUXkfnYdScctjj0afJKas2ky4vO8xlpRh2dXknvs7KzPT9pd9+yRuTyhGqXQOIDMl1Z+SJTGecaETeWHBoR7Teja+LEidzixTw9y7XOzKxMlJW6wtHIKl0bOuEk0ojYLo3f1gJX0FM2Ecmc5NChMawRkV8eRW9mvZhbFg1CnGtErgrKd0Irh0aUn5ePHj16gDR6dfEiwNxczBcuKTEJBysOSiVE7tvo3Elqnh1npoOO7ma9aOzjXtBnEtARjlrOXHxMUHqQIQH5DCocGtGevXtEFBEPIdUtEwLhaKKmipbaPxWlxRSuRualgfsYCIfG18gJKXJWcDV2mbhFTl5WHxeEFxsr3LxdB4dGtG7dOuSmtzjdsMIZ48R4axMD2AkpCaipqWu1z0UGWe6gp9ymJBEdQQZQAeRBa0R0MKzplyMMlk7ThWpNgKYtvGtENLRzFJ5LTrM1IgrjRaFvMnObNl3rFFrIct1j60oTQb8sh3TPeNc587TJUtZF69zcGhHnHg3CiSJvc3vq0RoRi3XHBrA7KzC3LBoS+NeI+LR2szWiikMHBH7jxk1wdRsaeY0y7z2T1WFFYT7MzcV8hHr37iWIFBbK29RKyzXcGpGNIo4yXrQmxTuj5hSOookzdyrMtkAKucN78Qpotka0Y9f3IrTPmDGjmmHkd99pJh1zN9y113SAU1JcGlFBgdx4c9waEfcakcbudWV6VWghoDtEBOeWBxa8o6PCOWPuMENotka0Z89upGelM0ulyLkRsNxAlNp0YNuRI/KiK9B6jc68j4h7jUic+eCuFYx/2Ta0MsoUDlJOnRxOLNecm6E0WyPa8vUWZGd2vO+tmRl1Ix0B69VcTQOdGHqg/LA8sDRS0o/Lyy8ic7Kxr6O0bGiNSEBCY4q5ZdnoKA4b35pNaOAE/rWZGpGz/jj27t7TboO2VU9oDRx9879gbi7mC0QUTjnlFNRVyxuIaLbZCbyx5niQaqHi0DT2dRSuDa0tUjLe8S6hgPsYCEYkBSkzNaKSygocq2vAmWee2UostUbUCg5Tf1hyICKN6OCBQ9KAC4vXnDTujWVEp7K6Grux9CpVZCFAGpFaIwquTPL374FTd2Ls2LHBZaC+ChkBSw5EKT1TUFJWGjI4nhlwe8150ua4J2cF6xp2OBAMMw1ax1RrREEVwtYtO9CvX19069om5iGZQtTFgoAlB6I+qSkoKiiQCiC315xU5o1kJk74ZDikzggvVkjD3LLY90kxl5GZa0Tbtm1Denom6Jhuz4si0quLBwHm5sIjVFJSD+nRFayuETnstNjNfGaP2JvfuvHz1BAGKsyTafKy5D7DigHFZhJmrRHRJuBvv92F04ed3kzLfaM2tLqRMP+vJQeinimuI6kLC+UdGW51jcjOu39W1GxLe82Z33ZbUSCNiDsyRisGTP5hlkZUW1WNiopKjBw+vJ0EylmhHSSmPbDkQNS7V28B2A8/yAvzY3WNCGKNgXcab2mvOdOarPeMLdmQPUQ1SyPambdXHIY3ZswYD2pNt2qNqD0mJj2xZP3NzHQFLiwpKZYGm9U1IjLNUeBTdUlCgBlKMjGpfUSBl93XWzahR1IP9OrlCg3WKgfVHlrBYeYP5uZipigteScnJ8Nut6NC4rlEVteI7HRIHXc8tpYis94dr3IpBiHeoKe8RWaWRvTVV9uQlp6GLl3an0LMK2FsU7PkQERFmpGRiaryWmmla3WNCDZN7SOSVltURrIRMGONiAbu7/d/j3HjxrVnVx0D0R4TE59YdiA6OePnKK8qlwad1TUih84fWYEKR8Wak1NFKbKClS8zNKL6+uM4WHwQw4ePbA+dOgaiPSYmPrHsQJSecYo46EoWdpbWiHQddjjZNSJLe80xtyxazlBrRIG19m07vxKOCiNGDPP6ofKa8wqLKQ+Zm4spMnjNtE+vXvjPf/7j9V0wDy2tEVEMI01DHPMZSJb2mmNWUMh9W60RBdayN67biMSERPTt29f7hxbXMr0LHZ6nlh2IUlNSoPYRGa9UdPhfg6Z2khtHLLJSsh8bwiy+GWtEu/fsRu6AXN+OCsprjq2ULTsQJSYmCRBLS+XEnLO0RkRrNWGIvs1Wy2OAkFMcjGfdiYT0NSJdF44K48eNj4HaEfkiWnYgIpdMuoqL5ewlsvwakc6/RhT5zSMEDplbFq0P6c4whMcIAaJAPpWtEdUePYofCn/EiDFneGdDec15x8Wkp8zNxSQpvGSbnu469reoSM5AZGmNKExrRFRslvWaY14jIiw1LQxEvbQ9Mx7J1ohWr14JiiaSmzPQO7vKa847LiY9texARGcS0VVZWSEFOktrRISQU2dfI7K015yUWmc8E3Lf1nXLNuemQxvlybd585fCUaFf334+QVZecz6hkf5CmlH5xx9/xLx585ojAHuGHJFxr8MBOG2ABlRXHxaOCL6jDeug83UodMe6NesQF2cXJ1j6Tt8xrlVVVSgrKxNBJckE4pbHjL/CX0DXUFZWjvnz58DpdB0BbQYttyyEVWVFBQqLi2CPswG6K4AmPSdzjzudrL9UDps2bQScTix+fzHi7HGivEhG16yeqiWZmWT81VBXVYm8rntgi6OyIzwFaVP+ktMH1dHKsjLRHpprFgUlNQFLd5nU1dWhqLgQ1HlqGuFIdV5+2bnpURnS+uv8+fNNpeOmR7hW11ajvKyEqqeoHlQ/g5GReF+/cT2yc7JF/aNgsfTMMy86ZLCytAbzKudDgxPimA0qW2KEuiGdmokGSH0GVB2uRnFRIXSqRIKIi4Qmfje1S83paqOiqrk8Xl0yGHtOckbc1SjpWrVqlaSc/Gfz2Wef+0/U2Ng4cuToxpl33W0obUeJVn2yqrG+vr6jJFLfEa0lS5ZIzdNfZmvXrm2srKz0l0za+3/961+Ncba4xiM15tP86pudjQcOHJDGu5GMFrz7rpFk0tJ89umnjT/++KO0/IxktHjxYiPJpKShuimzj+ndq3fjCy+80CFvKz5e0eF7M14a7dtCof0uc900wqs0XdfhcDAOssZs4dnZGSjcvy9kvjT2/QSaOLo4ZMYDyUB3sh4jQLMyMs0JxScQPoNIa3dy1k0Xg9IalkF5aXZOs3quizuSg8xZPG3rOFRxCEOGnNYhXPpPxvqZDjMJ+GU4aAbMpPQP+GqudNb9Z9gzuQ8KCiScSaRpOE6qMdtFijknPRLMxjEmtEJQbGhlOAXTwQ2lsODwdii0j0hmZ92qoLz8sIk9NnwmHjLRybqWLFkissrNzZGVpconRATC0ERD5DiAzxP79ERBfn4AX/hISuFTThz38dKcxxpzlAMdOuyMM2o3ajr78OembO5fTu2EJCGlXWZn7Q8d16Anb3AwRs9fKmPvv//+e/zyF79ESkqKsQ9UKtMRsPRA1L9HL2HiCnVTK61Pdurc2fTC8CTg1HiP0HYv1nryYPY9meZYxludF0vCjfu0VNJQODUi4ZFhdgUxKf+9e/diygVTTMo98rPlniQZQcTSA1FmVrbAoKSkxAgWPtOQmYzdfZulh24RmaJvcxvnXKa5Fh5Mu7Pxr2Sxn+kAACAASURBVBEJjyrTBPKeMW8Hw2t6FGtg3sUO6Gl9fT327tnrPeJ2QDlFb2LeCYsxnKw9EGW69hKVVoQW5ofMVtwbWlk7lTBF3zZWRSWk4lvKaGbWJvx6m3+af0NHvQvfcfNJEQW3WzUPNYh9RMKlOUSCa9euFVaS3IFZIeYUvZ9zmnCNomTpgSgtLQ02zYbKskqjePhMx60RcXYqFHmb/vFZ/F0wc3nN+SxUM1/wKgxiTwvn5IVm1ZwdmiuyQugFtmXTFiQmJuI3v8wNPbMozUFpRGEoOIo5V1IcmmmO2ObWiNihog16jDNqko9Mc+yjHxOwzOMQyGuOu/yYoBRkXLHmQqeYt28fJk6cCFuXk0LPLEpz4JxAGIXI0hoRgXDyySej8khoGhEpDNwakdEClJfOGc3rz/JgiNKcuPcREUycM2sZseaI361bt2DMmLFRWspy2OYsN6McW34g6t+/P7777jujeHhNR4qC5TUi0KoGb3WwsmmO00xGlTYcGhHnzFrGeUtff70dNTU1yB0U2/uHOMvNa4fq5SFvz+OFAbMfJXePx549u0MiExavuZA4DvxjXXey7yOytGmO2czJrxHxdh0yvOY2bdwo1oyHnzY88AZioS+URhSGwuzbLxOHDlXgeH0IG1Ip/E1na9uUnXZXAMkwFJEiKQEBbo2IAitwdmi0RhSqxp6Xn4dxE8ajW7duEhCP3iyURhSGsktN6ymoUmypYC+ajdkQwkAWLGHG72wOitjMPMtlijXHCGMzKXYsmWPNOXQ9DF5zwbuA0Am2a1auwZmjRzeXUazecE4gjGLM2/MY5UpiukE5Q0RuIW1qpcgK4I2sIBECQ1lRSH3usELKNGeoaAwl4taIwtFxaCHszfr+u29RWVmJoUOHGsLTyomURhSG0qUzR+iqKA1hU2sseM05dTjEYUhhKCRFMmQE2NeIms5XCplxgxm4THPBh2ra/MWXgtJopRGxmlQNFi+zm5RRriSmI3twYkIiig8eCClXq3vNOeya8LwKCaQAP1ZecwEC1kFybo2INj/rIgJ3B0xJfOVy3w4+RMa3336L0WeNRnJyskSuojMrpRGFqdzSszJwuLwqJOpW30dkE6ej8hpclGkupCrZ6mNujUisEQW/ZNOKdyM/QnXfXrFmBcaMGGWElOXTqDWiMBVx/z6p2BfiAXlW14g0nTSi4GecYSpaRbYJAW6NiI4McTLOW1zu28GNfLt27ULFwQqccUZsu227G4vSiNxIMP/t3bs39uzZGxJVq2tEsGnsa0TKNBdSlWz1MbdGRF5otuDGhVZ8G/0RyhrRtm1fif1DFNpHXbxu90bxZpzTGGVJfjo6ACsvL7SByOoakThZTT70HeaoTHMdwhPQS26NSMSDYnT3D2WNaOv27Rg3Xu0fclcopRG5kWD+S2F+6CovLw+astU1IofNJkLtBw2Q+jCsCHBrRAjBlToYoIJdI6Lzh5YsXozTThsUDFlLfqPWiMJUrMOHu2zDoZzUanWNyOZ0QFaofaPFrExzRpHyn45fI+I18QS7RrRz5240NDgwceIk/yDGSAqlEYWpoLOyXIdg0Ya2YC+ra0S0oZX7PCJlmgu2Nrb/jl8jch2O154Tc54Eu0a0e/dOcf7Q+PHjzWEsCnNVGlGYCo3WiOLi41BUVBQ0B1bXiGiNSHcyrj4HXRLqQ28I8GtEvMvLwa4RLVu2DOPGjfMGWcw+UxpRGIs+PSMdxUXFQXNgdY1IbGhl3KBIBaFMc0FXx3YfcmtE3EFPg1kjojXhzz//HG7TfDvQYvSB0ojCWPC9evRGXgh7iayuEdkdOjTdwVpCyjQnD25ujYg76Gkwa0Tf7P5GAHzppZfKA9oCOSmNKIyFmJ6WhvyC/UFzYHWNyLWPKPhYXkEDqz6UggC3RsRrmIPw6NQRWP3csGYDsrNz4F4jlgK0BTJRGlEYCzE1NRV5O/OCDvhndY2I1oe4nRWUaU5eg+DWiORxbiynQNeIqLOdv2g+Ro8eaYxADKVSGlEYC5v2Ejmho7a2NigurK4RaTYNOvOposo0F1RV9PoRt0bklQkTHwa6RrR//34R1meScttuVypKI2oHCd8DWrB0Oh2oqKgIiqjlNSKoOHNBVYwI+cjqGlGga0QbN20QbtsXTb0oQkoocthQGlEYyyInJ0dQLygoDIoLy2tETg3U2DkvZZqTh7bVNaJA9xHNn/cORo0awXqKrLzSNDcnpRGZi2+HudM5JAnx8SguDm4gsrpGFI4QP8o012GVDeil1TWiQNaIKILK7p27Mem8yQFhGCuJlUYU5pLun5EGpRF5KQRdh113sof48cKJehQkAlbXiAJZI1q/br3Yo3bt5VcHiaa1P4tEjUiaoxQ5AeTn57OU4OHyckFLAGrToDl1aE1mJRGevu190+94eyIo9hTxabQwaPZQXV2N//u//zNVzfecpTQ0NOBY3THk798PksfMi07ZjNOBI1WHUVxYiJqmNTSvOAKCH1tbfP09t7mqmVN3inD8dC5wWZnr6PZ9+QXo2bWbEFEqTeKxic9Dhw6J+lFXV2cmlM15U1k6ahqQX5APp4Nk1lzlKHaB6iBXeRHtXOLfmspKETmkpqammQ8zb0hGorVv3z7RLqg90TOz/h49XIuq6qrmtktVio7PotbhNijTPfEwf/F8ZGVmoOLQoeY14WDaEdUXrj7NXVbU13ijKRNXbqckt2wd/ZU2EBGRYD3SOmLQ2zuH0xkUrf79U7Fnz+6mb0l0WqD39xdwOBygKL5UyTkuqnQO3Ynao0c5yOE4gBNOJ+oaHNA0HpoNDU4xa3XU1KG2uSshcd3lEep9C3Q//fST6DS5yo8oN9gcqK0OzkOzhXPjd/XHT+DYsWOIi4sz/lGIKalDo3bBcdU4a8Sg7q+PqTh0AF+s+wJ33HUHautPNLXx4DikCaE/esHl7PsrKkOzaVL/EmmXtJ61W7duGDp0KIt8R44cCYrWoEEDsGXLBgw89VTY7MY3xx08eBADBw5ka+TUAIqLi4OSMdgCqDhUgUG5OaC1NI5r9+7doDWiIaedBqo7Zl40kUhPTwftJeO6SFPgag8kE7UJcsjhlJFm7lwylh0+jIaaY37pPfHEGsQnxOMvf/lLyHWZuw26y9FsTPP3Br+x36z249ZqQ87fHkDHHjKxIDPIzR2AujoHqqqrA8xBGkyG6bLPWshaxDxTIq85nYmm23RruACiLaHJJty2cHDXFZox+zNSE0+vvzob5513XsiDUFt5rfT7Z106RZw40npYmnVG+jVoQLbYSxToAXl6GPbYsHecOtjdtzm95rjt4tyu8LQexjk4uNeDuNo8GZP8dVZr1q3Gjwd/xG133cbFVlTSIVN1pF3+yjbS+A2Jn7T0TLFYnpe3L6B8NGoCJ3gLj73jBH+In4AKIcTE3AO7LQwaCucaGA16nPSMaESvvjwbOdlZOH3w6SHWFmt/zt23GEFT2hqREWLhTtOlSxf06JWEggCDn2qajuOdO4NvGZj30DEqF6qc3J0154ZW7sbnz4wkuy1wayiy+feXn7/yo71D69etw5tvvsU6QPrjOxLfc04gjMofUxoRFUBKSiry8wuM4iPS6boG2wnyLeO7OM0sJFU4OjJO01wkNj6ZtYldQ2ly1ZYpQ0d52YR53Hd39eSzT8Nm13Dhxed3lI16R36ETOuygYDtu2QDySWK0mZkpGPfvsC9Rjp17hxFUgbBKvcUPggWQ/mEu/Fxa5fcEwkx8DXt0wqlXIx+64qs4KOS6joWLVqIP1x3o3JSMABoJE7KYm4gyszKwo8//hDwrMDqseZ08klirg2cpjkD7VNqEn+mJKnEmma57B0M40DU0RrC3HnzRKTtW269RTaslsyPe1JmBETmrscIS+amyc7KRHlpOY47A9uIZ/VYcxp5zTGPRJymOXNrVfhz59aINGbTXEcD+/z583HmmWciNzdXakFEYoctQ0D2CYsBpjuaaBj4PPqSDBo0BA3OBlRWHUHXVGMbKWniRxoR5651bmQdNh12xhkut3zcpjJuetx46g5HRDgFbN++HZs3b8ayZcukQxCJHbYMISNxgI05jSgjI0OU5b4AdheTJ67VNSK7bmM/GI/TNNfRjFpG426bBzc9sWbDFIKKZNWYN7CLoK5tQQYw+/XZ6NW3F373u995easeeUMgEgfYmBuIKIRNUlICvv12l7cy8vnM8mtEupN9IOI0zbFvMPVZk8x5wW2a0x3OgNdZQ5HcdR5R6+6KNqYvWboED9x1H2hrhrqMIaA0ImM4mZoqzm5HcnIKCgoCc+G2ukZENn/LmpNsdlBHxnlZFks3iBrvXjdvXnOvv/66iFJ//qVT3VypvwYQUBqRAZBMT6JpOOWUU1BYGNgBedbXiHR2jYjNNKc74BRRvU2vXc0EwmGao8kE1+U+CoSNXhtCNKufP28+rr72WmQ2mdvbJAn5ZyRqDiELpfYRyYBQTh6nDhwozhcKJDera0TUcXLPlNhMc7zKUCDVSlpaKjuuALLEtEPX4fAbhlSaeO0mScuXL8f//t//4obp18sj0iYn7vbQhrxpPyNRrtZGV9NEj6yMszIyUVpSgoYAArVaXSOiymnVGSApQ66d+Xz1kN00J0I0+djwaYLY5GFJByqG65ozZw5GjhyJMSPGhIuFqKUbie08JgeinNwc1NEBVNXlhitTLGhEhsGQlJDNNCcOquOMFOiK3ScJJkPZkDMGhaLiuujEU9KKuC6X15yL3tc7vsa6detw0/SbuMhbio7SiCKkODOz0sUx0rv3GndYsLpGZGXTnJ00ogiMryWzOfANQW6uNVZTrstrznWY5bv/Woy+ffvh8ssudzOj/gaAgNKIAgDLzKRJib2QEBeHXdu3GCZjdY3IyqY5RzjOWmLeHOx0Mq/xMY98Lq85JyjK9uuvv4oZM6ab7rIdiR224Q6rg4RKI+oAHM5XdDR1YlIiCgqMe87FgkbEWQZEi800p9thczawisftNWezMR+Mx+x95XK/17Bw8TsiyvaMGTNML89I7LBlCB2JAyzzvEYGjHLyGDr0dOTn5xvOzOoakZVNc3A6oGsus47hAlcJO0SA1oc4XbhJI6qtrcXfH3kSl1/9B6SkpHTIn3rpG4FIHGBjdiCiAIkHDh7wXVpt3lhdI7KyaY55C5GoOdxec9yzXPKa02183Yem61i5cgXq6upw723T27RO9TMQBLjrihHe+GqSEW4Y0+Tk5KCkqMQwxVjQiAyDISkhm2mOzIDMazb8pjkbq/MAec1RmB/Oa/Xq1Zg0aQJyBg7mJGs5WkojiqAizcrKwrGGYygoNjYYWV0jsrJpjrzmaEZt5YtmubwzXY3VNLd7925UVlZixl13sRUjL55sYjHXE2NyxaxGlJaWJhD6dvvXhpCyukZkZdNcLHjNcc9yySrHZZojj8DHH38cZ5xxBiZPmGiovcpIxI2pDJ6N5BGJcsXsQESLnXG2OHzzzTdGyk6cR2QoYZQm4jYlEUxspjkK8cOsEXHjSbN3zg6G0zT37bffijOHLr7wwihtXZHFdiRqejE7ENEhdxnZmSguLjZUS6yuEVnZNEfOCnbwrmcYqlQSE9EgxNvBMG1o1XU8/PBf0aNHDwweOlQiYrGbFeeExSjKMTsQEUAjhw9HYWGRIaysvkbE35EZgl1aIqeNN8QPt9ecNKCMZsTUc+wrKMC6Netwxx13tAt8apRVla41ArwTlta0ff1iqk6+yIf3eVZWJoqKjIX5iQWNiLs02ExzTrBGpiYc+U1zvM4YdOAER4f2+COPiBNYb7jhBu7qySIfu1DgPUfKqHyxPRD9JgcHDh4yVOGsrhEp05zRJhOZ6ch5gNPkwrGhtbiwGEuXLsFtM28DmdK5L048OWXjmEAEKk9sD0S/yBSL2Fu2bPWLm9U1ImWa81sFAkrAbZqjfVKcHQzHhtYHH3oQ8fEJuO2229g1zIAKO8oSR+IAG9MDUWpqqqhCOwy4cMeCRsTdnthMcyQYc/RtbtMc9z4ps73myIno/SWLcdef7wLFhuQe2LnbAic9zgmLUblieiAiF27yxvnP//6vX7ysrhFZ2jQXhsgKfiuU5AQ0MHAeFQ6Yu6F11qxZiLPHY+b0mQIp7oFdcvFEVHZKI4qo4nAxM27cOBQV+fecs7pGZG3TnB2u6M18FZB7Bk/lx3lUuJkbWqk90trQfQ/dh+TkZFFo3HgS0UjUHGTU4EiUK6Y1IirUIYOGGHLhjgWNSEYlDyQPNtOc5oCTOfKp1WfwZprmHnjgQdjtcZj+h5bgpuHAMxI1h0Dal6+0kShXzA9E6VnpKMjP9zv7sbpGZGnTHEVWsPrFHDlCmOZs5MQt99q/Px/vL16Mxx57vFkbIgo2sSE55rsrKWBHokYkrSaVlZVh4cKFUoDyl0l5ebk4qdFfOiPvC/ML4dSdePypJ5CVken1k7KKMhBNrpkEVZTDhw+z4UlCU/lxyrh1q+t03PcXL0V8vLmuuUeOHMG+/XmwM7oAl5SUsJdfYVERq5uzGTI++eTTSOqdhMTExFb4NTQ4UF1bjaqqw17bqBkP6TRYrj7Nzb/Mvs2dZ9u/dJRGxF2Nkq5ly5ZJysl/Np999pn/RAZTHDhwoBFA45tvvuHzi1WrVjXW19f7fC/7BdFavHix7Gw7zG/t2rWNhw4d6jCNzJdvvfVWoz0+vrGyslJmtl7z+uqrrxqpnDmvBQsWcJJrpDbBKeNPjY2Ni999V6qMX3zxhWiLb//r7Xb5VlRUNFI75LyWL1/OSU7Qktm3+WJ+wYL3fL0K2/OY13XJhZtmX/v3+Tmt9cRPETeJiHaGNAd5evFIwX0eEY9ULVRIi+Y0uZBjBK0TybwefPBB0IGVV1x1Rbtsw7FG1I4JizzQtMjry5i6gcguwWHDRqC4pKPgpzqOd+4Mcw1IbTBiniKEo6GzOSvYYsNrjtd9W24kh/nz54kI22vXrvVqAqeJBHOTaNMg1U8zEVBlC2D4GWcgL2+fT5ypCdhOHPf53pQXciebhljkWgNzM0MaEYszmx4DXnN0Yirrpl15XUdtbS0eeugRnH/e+Zg40ft5Q+R+rzMPRZwaprtNxOpfpREBoOCnBfv24qhTR1faINHmojGhU+fObZ5a6yft07Bsw4sBrzlu0yM1E1n15dlnn8UPP/yAdevW+2xUDs1O25J9vjfjBffEzAwZoiXP9r1utHAukc/U/mlo0IF9u3b4zNXq7tuyOhWfAPp44QR1MCZfNrf7r8l0PLLn3oDJ3ZBF0FMJ7tu79m7Hk39/Evfffz+ys7M8EGx9q2bMrfGw2i/u+huR+A0YkCv42r1nt0/+rL6hlTpO7hkgrRFpjgafmMt80cB8HhH7mptkxwF/2MvoOI7XH8ftN/0JmVmZYiDqiCY7nh0xo95JR0BNNACQ51zvXr1R1MEheaQRhSMUvfQS95FhOExzwmvOLqNL8yFU02O7ZofmsLZ9jj/WXMeYG3n75ltv4stvvsKnaz8RgU2NfKPSWBMB83uBKMFtxJgRyC/wfUie1TWicJnmOKqHQ3ewR2/mNs2RNsvrrBDaGhE5B/3pzj/hD9ddhwkTvDsoeNYNl9cc7xqRlduEJ7aRcK8GoqZSyMnKwc6dvk1zVl8jCodpLhIagFk8sJuSKPq2xttRB2vKJS+5319xKTKz0vHKK68YKgKX1xzDeqIHN8HK55GFujWIgDLNNQGVQZ5zBfmoPVqLbl27tYPP6hpROExzbPuI2pWm9R6QxqDrfPNKmxYcLdIy7vvz/SK+4xdfbEaXLl0MFYbLa85pKK1KFH0IBFebok9OvxwPyHU5LOz8xrtWZHWNKBxmCLZ9RH5LX34CbtMcd0Mmr7lg6sycOXPw+uxX8cILL2LYsGGGgXcd48EtpWH2VMIQEVAl2wTggAEDxF1+vveNrbGgESlTRIityePzcJjmPMibfksdR6D1ZeXKFbj91ttx7Q3XYsaMGQHxGI59RAExqBKHhIAaiJrgo+OIs7NzkJ/v3WHB6hpROExzIdVc9XErBJw62E9obcWAnx/r16/DVVddg/ETxmHuG3P9pG7/OhwaUTAaX3vO1RMjCKiByAOlM844DXl5eR5PWm6trhGFq9GxbGhtKUa2O27TnM2msXrNBRJZgQah3//+CmTl5GDJkqUBa1JUaOHQiALV+NgqlwUJKWcFj0LNzMjAihUrPJ603MbCPiLuhteyobVrC9AWuXM4HFi5ciW2btmK4pJCFBeX4WDFAZSVlODIkQ7Og9E0JCbEg3wBeib3QVLvBNi1BKSk9ERCQoL4l9w9Gf3T+yM7KwdDhgxCSkoKwOw1JyJvewmH1bb43nnnHdx8880YMuQ0fPzRsqD3C2nMG3bbyqF+m4uAGog88M3MyUJNTQ0KCwuRkZHh8QawukYUDtMc14bWVgVp0g/SKDds2IAPP/wQFKHjm23fNFOiTnjUqBGY2G8C+vTpg8TEJOgn0eHldjjhEIOISEyeb8d1NNQ0wOF0oKGhDlVVtaCD/eiQtvzCfBTlF+FQxaHmvOnGptnQq3cvpKWnIiM9B6mpKRgwJBcjTh+FnJycpqzlGz9sHXiLEx633HIz5syZi+uuuw4vvfRS0IMQCRCOfUStQFY/TEVADUQe8I4YPEL8+n7/9+0GIqtrROEyzXnAH3W3TocDW7duxeuzZ2PD+g1igKAIHZf+/lKMGjMKN994I9JO+Y3XQLqhCEtlVVVVhYqKCpSUlqK0rARfbf4Shw9XgZxtVixdigbdNcDRIJWelY6f9z0Zp576GwwYNAATx01Er769hOt0sFowuW/rXjQi4u3jjz7GfffdJyZ0/3jpWdw6486gzHGeGIVjH5EnfXVvLgJqIPLANysnSxyS95+8/wCTPF4gNjSiYDul1kgZ/xWt+4hoECA35DfeeKNZe77tjttw3nnn4dRTTxWdLh0x/etfn2ocjABSUjmROY7+uTWe/n3SkJOTLcJV0WBAR80Xl5Zh3749yNuTh6+/3o5FixZi7uwGNDgbkBAfj/SMTGRkpGPQgEE4Y9QZGDpoKJKSkgwNUGSa0x2ufT1Op47a2mosWbIEr776Kvbs3YPx48Zj6fLlYvALQDSfScOxj0hNznwWh/QXaiBqAymZUfbn72/zFLC6RhQu0xzLeUTtSjO4B2Qee/7JZzFn/lw4HA248PyL8fa7b2PE6SPazfht4N186XQ6mvf10EBFJkD6d/rggYDHgaeHq8pRkF8ozIe7d+7Gjh07sGX716h7+mk4dScSExPQJzUVWZlZyMrKws/79UPar9ORmpiKxKREdO3aFVVV1cJcuGXjJuzevRsbN25EXt5eOB06zjzrTHzx+RcYNWpUO0yCQ931VTjWiLgnZqHgE+3fqoGoTQn++te/wrp169o8tb5GpGZ/7Yq8+UF5WRlefvVVPP300yLw7U033YC7774Pqal9mtO0vdG5zkBvIkydppGOs2dyCnoOS3FtJr3J9TGZGKuqq8XhkLt378T//Pvf2Ju3D2vWr8HhssOoO1IntKi2MmpxdqSn9gftwbvqqqtw9dVXuxwn2iZUvxUCfhBQA1EbgOhMlNmzZwsbfHJycvPbWNCIjHRkzYDEwE19fT3efnsBHnhgltA2ZsyYjr8+/Ch6etQLXzA4mU8TJT6CnUzY7HYxgIwbl4Jx48a2EonMbqRt/YQTqCw/grq6OuG5l5SUiBVLV+KqaVe1Sm/WD+WsYBaykZGvtIGI3FWtcGVmuryM9u7di7FjXY2SOmhur7lgO5Vgy4BMcw0NPGcDefKoM1iwgq2b9//5fvzz1X8Kdr/44ovm+uDJv697blMS1RczJhK0P8kmznKKQ9e0lhiMDU4n4hLifIkv/bmdnCM4DlH04Jy7zXuQNvW2U6dOpuYfTObSfDrtdt7IuMEIa+SbM84YIpJVH6tuTk6NnDQizsuMTsUf/+GgyWHBCrRuVh0+jCkXTRGD0OiRo8UmZ/ekxB+G4XpPZcc5eYmz2eDgnLg4aZLUgb+4CcCTRmzF66effoo4saQNRBEnWZAM0SF5ffv1Rf73+c050OZCq86O3EKaNaN25+/tL3nNcWhE3mh7e0ZmqEULF+JXp/4aWzdtxbsL3sWWrVvwm9/8xlvyyHomNrTyNWeqL97ct80CRUVWMAvZyMiXr+ZGhryGuBicPRjbvtnWnJY2dXNrRM3EmW7C5TXHoREZgfDo0aOYdsM1uPLaazFs8DB8//2/ccVVHu5mRjIJZxpxDASfxkD7iDra0CobCm5Tp2z+VX4dI6AGIi/4nDwwE+vXr29l6rC6RkQDUaxedFrogCGnYfmS5XjtlVew6pNVHXrERSJO3BotBVl1xm6VicQqENU8qarkpfiGZg9AZWWlCKvifm11jYi7I3PjGu6/H7z3AU4/fTBsJ45j145dAR9PEG7+3fS514iILqdG5PKac0ur/loNATUQeSnRSRPHi6f79rk2tpK2EAsaEQ1G3JdDl+a4GRDrJOsDDzyIyy6/DJMmTcLO775Ddk52QHnEdmI6d4Kv+3CF+OGjR2UbjvYQq3WKt2SjBOX0zCxQzDDaLU4XHXJmdY0oHKY5clawh8Fboba2Ftdccw2efPLv+PN9d2PZso9CCsgZCdWaHC04L24NTDkrmFu6VH/C6SWoBiIf5ZuTm4utX28Vb2NBIwqHaS4cR4VTnLgp/zUFy5cvxdv/ehvPPPWcjxoQXY9d+334tEvu+qKcFcytj8888xTOPutssZHfXErec1cDkXdckJ2dISIqU4OLFY3I6qaI0pISnH322di7by/Wrv0U10y7xkfpR99jKjve8lNdR/TVEt8cZ+fkYs/e3fjtOb9ttTbu+wu5b1Rt8oHnwKFDUXOkBuXl5YgFjShcpjmW2KA2u4iSffY556CkrASffvppQFESfFSRiHrMvhlZA+uJsMpZwdzqw06k7wAAIABJREFUNvWiC8TkLL8gH2efczby8wvaEfw271s88MADmHbNNDHpoYnPnXfejscee0z8a/dBAA/UQOQDrIvPv1hEI87bnxcTGhFVKu7OjMs0V1NxAFOmTBE28C0bt2Lo0KE+Sj16H7OXH22gtfGZAvdu24r1GzaxFhCvhskqmldiFD3k888/R3lZOX73u9+CBh7Pa3DuYNx1111YuXol3lv0Hl544UWkp2WIQMBxcaGFe5JWk0pKSsUJlZ6Mm3FPM/ddu3aZkXWrPDVoIrjjiqUrkNa3n+ikuTpqagD79+ez4OkW+t///jeOHz+B7t1b4om530n/qwH78vaJbDd+sQnd4s2jWXesDrfNvA3HjtXhuWdeQGlJMUpKiqWL1DZDKr+NGzZCZwpLs2fPHlRX16Jnz5ZAvW15kvlb153Yk7cHKRsoArn5jhILP1iKzZu/xHmTJ0BnCChLG61pf1nPnhsFbBTpjujS/9RpmnIv+rY9oPMGSQPUTXBAIQnINN1HlJv3GvHE40/gzj/djgsmX4APFr2HYaNcB4ZSajoD65JLLsGzzz6L1954TRz34T2XAJ82SrrOPPPMRgDqn8JA1QFVB1QdsEgdyMjIaKyvr281SixZtqQxOzun1bNQf0jTiG666SbMnTsXNFmh44npkC26pN/rwNc7vsawYSPEhjqnjVyAASfNynQIzcWtUnu6mAZzT7bPxQsW48U3XsTYM0ZD63oS7NDg0OCiqetS6bnNK3V1x7Bx01pMnjgFzfJRnCHC0yO4pcz7DRs3iJM6uyV3N1fGpnrxwXvv4W8P/x1fbvtSuE7LlIVwpLU9MsfZ7XF44vHHkDtgALqn9HTJpjvFX9oHIzDX2pepI9g0Ika0hmUrPsK5F15gWj1pWxe2b9+B9PQ0pPbpI0xmstpAR/msXLESF158oVgroigLmuT250n7+eefx/qN6/HRx6sEpo4mnM38u2H9JkyaMA5Ew0zZ3HJSmVLfNmbYKEHTTv1o094+ae1D07B62WpMmDq5Vd2kHpQuIkcmuat/fzV69OiFTz5ZK0xv4mXTf4fKDqIgPx+0DaJbNznWDGkDETFEJzpyXMXFxcjONJ/WpEkTseDtBdB0DRm/OqVdgZgla0ODA//+n++RyYQnyZG2Pw0ZGRno2bOnWWK1yrdPn1TQPqLMrAx07ybXnEQN5KKLLhL0tm7ZhLLycqSlpYljtFsxYeKPPr37IDsj00QKrbOmNpGRmYb+qemtX5j4q9/P+4k6YyKJ5qxzBuQIhxNOTP/n+++QkclXhiSsqxzNpdnz5yk+6yaZk2+cdqMo17WffiLWgJoLAQCZ8DPSMpHUO0kEBb74koubX5PZLthL2kAULAOR/N3ESZMFe3v27sWVJ34CQluPC0BUskDz+pFwrWV4giCcFTwfSLinTXm//e05okFv3rpZdCQ0EFn+0nXojMHf3DN1Llwvv+RyZKaZ20G3lUVjxLMt7XD8XrZyJS674CIMGjQIH6/6uNXE7YUXXkCnTp2Fhnb33Xdh2hXTsHDRQjQ4HLjjjttCZlcNRB1A2K1rV/TL6CdikHXqzHuYFO1dUldgCFDneOWVV4IW7mmf0MDcgYFlEMWphXszY8gdbqioPXBPlhw2a5yxZqSsFr2zEFdeezWGjzwDn679rJ3JLSM9HT8c/BF3zHQNOjNvm4kei3pg5swZRrL3m0YNRB1ARHbZU3MG4ttdOzpIpV6FgoDMWHO33347li9fjuXLV4R9nxD3mECRB7gnLzpjbMJw7HOzgf/E4lDaktFvGxvJl6T1VXPsCCZOnoxVH30MitLR9rpo6tRWj8iMT3uKZF3tKcrK2SL5jDztNLGx9ejx46wS2RhcYlkF8kJMZqw5Wsx+/fXX8eKLL+KCC6Z4ocb7yMlsWiVPX5o4cV10HhHnwXjcgyzhqAlHbS5Ew0tn+vSZWLtqlddBiIMzvprLIY0JNCZMmoy/PvooKg6WoGf37iZQ8J6lI1JOjPPOnt+nhUWFeGfOfMQlxmHo0MGYMGFSu29kbWhds2Y17rnnHsyceRvuuOOOdnTC8YB7IkGzWM51GzIccx0DsX37drz/7vsoLSsVJ+dectXFGDzA/E3JZuzjCUddjAaaaiDyU0q5ublIiI/H+o2fIzub78jocESl9gOF4dcFBQWYcukUfLbqM7HgOW78BBw9Wo8LLnB5shnOyEBC2nT4+99fgTPPOguvvPKSgS94klh9iU/Ix2B/pKjQ995zDw7VVeBozVEUFBbi/r/ez1OIXkxUPIRjj4oyzfkp827duiI9IxMb17t2WPtJLu11NGtE98y6B6NzRzZ73Uy/4Qbcfvvd0sPMl5eVYerUi5DWPw0fLVsmDftozIi0Ic6QO7Rpj0MD27JxI665/nps/nQz5r4yB19/vRVdu3aNxiJSPHeAgBqIOgDH/Wr06JHYvns7S8Nz0dRgC8M5PW55Q/lLG0lXLl2JgacPbs7m9GGn44cfCvHZZ581Pwv1hjrBK664Qmxc/fDDD5GcLHcvUqj8cX9P60PczgNmr0lRGT/02CP4Zts2rFu3AU7dwQqrg3mdj1W4CCOmBiIDBXLKL0/BweKDYiexgeRSkpA7bjReZJajqBq5A3Kb2ScPG7rc8eWaXwAI1mvuoUcewmeff45FixZG5Mmq7MVHQUhp+z/TRaY5Ec3ERHp0dhRtlH/nnXdw5ZWX4ZHHHmNtg3Yn78BnIpQRn3V09nbMsPbN6AuHw4H169czUebrUGQLVHSoSGSZmpranLV75lxbW938jG6C9Zpb9sGHePLvT+Lhhx/FxIntnSBaEQnTD26vOREkU+drzjTQUmgtMy/aqb9q1SpUV1fjX//6b+z8ZjcuuOACM0m2zpvRC7E14dj7xVdzoxjblPiewmFBpmnJHxxcHkn++Aj0vXbc5f9C7r1tL4rw5nkF4zVHGteNN/8REydNxIMPytvH4MmXjHtur7n2aMuQwnceYoOplzL2/UXwb2gic/75F+Lu+/4sjin47vvvgs9MfRmRCHDX34gEwS9TmoYJkyZhxw6+ja0OpkbuV/YAE2RlucKwFJaUNX9ZWloq7tNTQ4tjR+F7yDkhISEBS95fwrpvplkYgzfcXnP8zgp0VoHJKpEH1g5dx4SxY5A7aAjy9uV5vFG3VkBADUQGSpFmf+PGjRUNgMNTiFiinfLReGWmpyExKQn79+5uZr+goFDcDz9zePOzYG7uvvtu7N+/Hx988EG7ECTB5Gelb7idFUSoe8bJkl13iPXEzLT+GDNiDEvRcTp/sAgUwUTUQGSkcDRgzKgxaDjWgE2beE+JNMJeJKXpltwTM2fMwEYPnHbv/gZnn302Bg88PWhWFy1aJCInPPfcCxg2bFjQ+Vj2Q+GswNycTZ4sVdceRkmB6xBDh2ZHcXEh4uMSRSR1jnLkdYfnkChyaTDX3MgFoiPOqL398pe/FOtEq1ev7iipvHdRXDL3338/ykpLsXLlCmzfugWLFn2AefPmecXGiNdcYWEhbr75Zpx//nmYOXOm13wi7iFz+bmcFfi0aOqknSYPRC+/PBu/+NUv8eBDs/D2f7+FFcuX451F8yOuqBVDoSOgIisYwJAsEF26dMGYseOwY4f5x5QTS9HqrEC8k8vtpq1bMX/+fJSUlGHr1i1eUTbiNUem0EsvnYr4+AT8619ve80nIh8yerCR/NymXDJbub0hzcJ/1r33Ir1/GgpLizFy9JkYMGCI6TTNkkXl2zECaiDqGB/x1h1wcfLkSXjwwQcNfBF6kmh1VnBLTgrB9dOmuX96/WvEa44i/O7cuRtffvlllG1a5dNOCFwKemo1UxINdNdMu0bUncOHD2Pbtm1e65FZD7nWg83iP5ryZTYgRBM0Lby6Q9CPHj0aNTU1LOtE3DPcFmkj527Dhg149tln8be//R2jRo2KHMYikBMKeqoW1+UWjNkan1xuozs3NRAZKD+3RkRHods0G1asWGHgK5UkFAQoVND111+P0SNHY9ase0PJKja+ZY6sQKByagyaCO+juiurVmZlmjNQsrQN8zg0dO/WDZPPP49nnSjG2xwdcnfo0EFs3LgxKtcFGD2bRQ3mjqwgTuthjDxAXnPCZdxAe1VJog+BGO/ujBWYw2bDSU0H1U2aMBG7d+809mEIqaLZWSEEscWndGzx4sWL8cZrb8Idpy7UPLm/5w7xw92QuQfacMyYlamTr9Vw118+ySRSsuk6fjpxQuQ4fORwsU5ktht3tDsrBAs/hfC59c7bccnFF2Pa9R07OwRLg+M79jU+k12p22JG5mpO05zbPN6WDzN/W835w0ysQs1bDUQGECSPpE6dO4uUOTk5iIuzY/36NQa+DD4Je0cWPKtSv5x+002w2+Pw2ptvSM3X6pmFw2tOLeZbvVbxyRcOjZdPOkmUyAzxE0gjihP7ia688mps36UCL0qCtzmbOXPmiKMdln30IVJ6pjQ/j8YbdtOVxb3mbHDSbim+qqDrUOcR8cHNWLJ8QplBqdOJFqhGjR2BbZu/NPdslBZyZogTcXnm5+fjlptvwbVXX4sLz7844vgLlCHuNSIKQMp5HpHAg9EcyO6soGmwQ51HFGi9DzZ9jHV3QcKk6TjeZJqjHM44bYw4/G358qVBZuj/s1hyVjjq1HH99dPQ7+f9MHfeXP/gREEKbtMqt9ece28dV1GExXTjVN0jV/kqpI0g7dRgO3G8OeXAU3Nw8sknY/tW846FiCVnhTmvvIBtX32DV19/NSpdtZsrRhhvuBsyKUMORqLhcFaAjVHAMNadSCCtkDZSClqLs4I7+dSpl2LF6pXun9L/cs+opQtgMMN9+wpx331/wbXXXY3zJp9n8KvIT8a9RsR5NpALfd30E1ojv5QVh7IQUAORQSRPNLlvu5OPHjkcP/zwf9if/2/3I/U3QAQcNh1//MN16NGrN157xVpecmKNiHENhdtrjts0F2DVUsmjDAE1EBkssM4ea0T0yfgJE0S4n48+NEkrioGS0Rsc2LdvH96ZPxcndTnJYElERzJxVDijWsQda07sI2Kso7QGpjVtKueqATqN7upiQYCxKrHIYxqRthpRcnIyJk6egG3fmBMROFacFcgkN2HCRNPKLVwZc2tE3F5zNk1jPaqETmjVweuyoKk1IrbmowYig1C31Yjos7NHn4PVq1eiqqrKYC7Gk1ndWYHOKqIAsk899YxxUKIoJbdGxO01Jw7FY9T42N23o6iuWYFVaVOM48ePg07S5Lio4+eiRfJUV1cLenFxca3Ey8rJQEODA0uXLsf48eNavQvlh9PhgOPYMWYZq1BSUiLCF4XCu5Fv6djvzZs3Q4uzi5Nc6+rqjHwWdJojR47AbrejoaEh6DwC/fBYjQOFBQUAU2ddc6QcpaVJbGF3nE5d1BWudni4uhpHm9phoGURbHqqN1zyuXnk6NuOHTvmJhcxf6UNRA6HQ2zw5Ig/RR2KGVqIr1IhmWpra1FfX++RxIa0tCz88he/xLp1azBkyCCPd6HdEj1ng5NVRkeDEzQgmF1+pSVFePivD6N3716oqaxDZU2lOHLaDPdc94I6lVtdZU1ohWLoa2pOTth1GxxoQFX1UfHb0KchJjp+HGg4VgkTlHOfnDXoThw9XAuHRlEPzL0cDQ045nCwtgmSiNq92W3CEzma2JqLqU1MyjxpRsK9tIGIjoceOHAgi0w0Uxk6dCgLLSJy6NAhIVtbjYjeTZkyBQs/WIiFCxdK2wNDA21hUSGrjBUVFcjMzkafnj1NxfV399+PpN5JuPeeWXjgoYcwKHcQuicnm0qT8EzLyED/Pn1MpeOZ+e78PAwdytMeiC61iYzMLKSlpXmyYeo9RcM49XQeGemEVu52X1xczNanuQuKZDQb0/3789zkIuavWiMyUBQdzYgmTpqIioMV0s8oIh8h1kuH6UvBc2bPwbp16zB/7nwkJSWBjgpniXCsAdz7ssQaEWMBkqmMNQgpRd9m9mJjhFORYkaAubdjlk4SOVpU93VNnDgRPXr0wObNX/hKEsRzclTldR3VNR2uUzCDYNfAJ6WlpZj1wCxcefXlmDx5svhCtzMdb80LpQE05Cch9+2OJkyyKZJns50RVzNMt7IxiZb83CbrSOJXDUQGSoMagSv6dvvENAudOHkiaLYv79LBrRERPafW2hlDnjzALbfeIrJ77pmXmrMljch0NQyA3fc8opkX2TeMe1kF6zQmcGpEYt8SkyMGCcgpm+y6oPLzj4AaiPxjJKLPd4LrPCJvySePn4T8gnzk5cmzvbLPAJ00+JkzxV20cBGWL12Ol157Damp/Mc7OJx0zLS1LzI9smpE5FDjNN9JwV1qnLTcNK36NxKxVAORkdqmO3G8gzWbSy6/DImJidiwYYOR3Ayl4VafHXSejQnT+PLyctx+z+04/7zzcdXvf99OdofOoK7Y+MP5MyoLLkyZCVL95NRSXJEV2lUf9SAIBDjLzSh7aiAygJSu2XFSB9pCly5dMH78eMydK888Z8ag0JGopDO4Dh/rKFXg726981boDh0vvdJiknPnQmtEdp1hVu2k7TzmaHtuWdr+NWFMb0ui1W+a5bI4frSiyveDpiu8JUj0GOomH4TNlJRG1AxFdN2QB9RPbYKetpWAwtTs2bMX/0ObGKPxcupwSF4jWr16Nd5/93088cTjSE9Pb4cKXxgjO7vXXDthTX5As1yd0VRGEyXODo17YkbFpXEsYJpcL7xlrzQib6hEwTNyU+3UJuhpW7b/8IfrhHnu05UmBUFtS1D2b5sGTaJ2Ul5ehhtuuAlnnXkWpk+f4ZVbEVSSwTIHOOC0aKfiCSynOZfbNOcpp7oPDQHOCYRRTpVpzgBS5FHmTyMi89y4cePw6py54D8bxoAQfpI4dLn7UP76l4dx7Fgd5s2f54cyz2taY+C8mMmB9hFxag3cpkfOQVbUE12Ho4N1Yc66JJuW0ohkI8qUH3Uq/jQiYuXiiy9G0b69yMuPPvMcmR9lzZRo0+rsObNx//1/QUZGhs9SojUiLjO8GetfPgUTgX14Bz5yp+btYKij5lu14RxkRblqGuzgd3LpqE7Jeierncvih/LhbS0yOWfMi2Z/bY+B8Eb+qquuQnxSD3z43iJvryP8mZyOjGJzTZt2Pc48cyRmzbq3Q5m59hFRLdd13qrOH1mB152aNJQ4vnGow3pk3ksWu7F57PvImXfC4oOJNo95W2cb4tHykwYib8dAtOWfCpgiLcz+79nStIu2NMz6LTa0Sljs/vOf/yxMcnPnzjeL1cDz1e2meAR2xAi36Yrqnsbn/SHMgJzmTnbTXEeFG+XvlEYUpQVIo7WvyAptRbrppptwoOgAdu3a0fZVRP+mkEKhzpQ2bdqE2bNn4+GHH0ZWVlYEyeuA6zybCGJJNiu6Dt3JPK9kHG3ZTXOyyyeC8gu1nZshCnPNNUME8/PUNCecMHaU9dixY8W6yMsvv2o+Y1IphLZGRCa5a66ZhuHDz8Dtt98plbOQM3Oas0cqZL4kZsCpnbjYpkiyqvuQWIRsWSmNiA1q2YTssJ04bihTmm1cfPGFWLFiBaqP0nk00XLZQloxvPWWW3Ho0AEsXLgItHBu9OKJrEA+Ebz2fu4+2jjiRkvGXzreBaJwmOasuqFVaUT+6naEviezlRGvOTf7t91xF47VHcOKpe+7H0X8X00E9Q+uO1u/fh3eXvA2nnzuBWRmZhqWlS2yQhiOgXAy+wGR+zZ3ZAXuDbSGK5akhFbd0Ko0IkkVJBzZGPGac/OVnpaGM4afgddfnxNFe4pssOuBu6tW1x7G9dNuwvDhI3HnDO8bV924tP3L5TVHxxVwB2vh9poT0bAlOJu0LSOfvzWNfeDzyYt6ERACSiMKCK7ISmzEa86T47vuuAtfbduG/CgJ+ePQtKAW9P9w3S2oqDyIhQvfidg1AwfsiGMeihjX8V3VTtd54+kxCxgO05xne7bSvdKIorg0A9GISMzzzj8PPZKS8NTjT0WF1DanI+BVlA/eW4SlS9/HG6+9GZBJjh8Q64f4IWcF7r1SnCe0snvNqcgKrM00uEUBVhYjg1igGhGF/Jk5cyaWr1yO+vr6yBCiAy5ofSGQxl5cWIxb7nQd7zDt+mkd5BwZr/i9ynjlZj8KnY6B4PVXYAZURVbgBFwNRAbRDlQjomxpT1FlZSX+8cbLBqmEMZnuRCCd9XV/vE4w+8abb4TENIvXXBjct7m95ujobk5nBWdMmOZ4PS1DakgBfKzWiAIAK9KSBqoREf9paWm4/LLL8fbrb0aaOF74Me6s8PTTz2LD+g2YN2ceUlNTveRl7BGb15yNP8QPt9ccu7OCOFreWDnLSBWIti6DnpXzUGtEUVy6wWhEJO7Mu2Zif34B3nsnsuPPGXVW2LFrOx584AFhdjzvvPNCKlEurzmAP8QPt9ccRXznPPyPND5Lm+ZCqtmR/bHSiCK7fDrkLhiNiDIcM2wURp45Gv+YE9mRFow4K1D0hMsuuRy5ubl49tlnO8Qrsl7yOyswW66EWZXTWUFoKIz2R+U1J69FKY1IHpbsOQWrEVEYlAdnPYBtm7/Ehg0b2Pk2StCvs4Ku45ZbbkFZWRkWvPc2yBkjmq5A1r+iSS43r+FY7OXs0MJhmlORFdy1y/y/4ai/5ktlAoVgNSJihSJy5+bk4umnnzaBM0lZ+nFWmDt3HhYsWIAXXnwFp+YMlESULxvu84j4JHNR4o6sQBpKJJp4ZOKuIivIRLPjvNRA1DE+zW+D1ogA0WDv/8t9oAPj6F9kXr6dFXZ8uwM33nwjrrz6cky/6Xqp7LN4zQmOeT2gGK1WLulsGnhD7lC0Cj7/bWWak9fsInECoQYig+UbikZEJK664ipkZ+fi+eefF8c6GyTLlsyXs0JVVRWm/NcFyM3OxULJDhdsXnMgrzm+TpMKjdtrjttZAXRsCCOk3OVHeFr1qHBOk6rRDkwNRAaRCkUjEiQ0DX//+6NNWtEag1T5knlzVqAKe9lll8DhaMCyjz+Szgyf15x01v1myO01xx1ZgXPPkl+wzUhg4aPClUZkRoVhyjNUjYjYnDp1qjiv54FZD0RctAVvzgqzZj2AjRs2Yf68+cgKIKo2U5EERIbTtZkYY1bAwB1ZQZgBGe2P4THN8ZpzA6rQISRWGlEI4IX705A1oiYBnnnuOezZuwcL3l4QbpFa02/jrDBnzhw8//yzePTvj4q4ea0TR9kvjX9DKzdC3JEVhHyMoy27aY67ABnpKY2IEWzZpGRoRMTT2DFjcdlll2HWA7NQWloqm80Q8mtxVli9bg1uv/1WXHvDdXjg3lkh5Kk+5UKAP7IC4FCGfa7ilUpHaURS4eTNTJZGRFw/88wzgvlHH/0brxAdUHM7K2zfvh3XXHEVxgwfiTdefs30ox34vOY6EN6EV4xWKxf3zJEVbLSGwuisEB7TnAkVIwKyjESN6GeNjY2NMrCZM3sO4hPjodHJlDqga7pp9xVlFeiR0kMG2wby0FBWVoo+ffoYSOs7CeFCC8o28sXR7Hh7wQKsXLkSjz78KDKzXKeaijTkjWSzobi4OKQ4br458f6msuwg9E42PPXUM3DCgcceewpJ8XFiIKL+htYgXP2OexosnjZlZvxeo1w0Deu++BxL5s7Di2++hjh7vHempDzVcORwJeK7JyLO7nIhIAlIGhferhPSZf0W3mTQUFh0EGnpfQUV14KRCz0pInnJpLz8CLp1j0e8kNFdRl4SSnxUUnIQ/fv3pdohEJWYdbusaBZPHpwpKSlN5WYunsQAWSwoXqQwewo/SJvo18hMaIMJ95oN5eXlSO6ZDBvF8qPao1O/QR1qO0iCflBXV4fp06cH/b0ZH0pbjUvqnYSLplwEqjA0W6LOxqz7TVu2YPzYsaJjFx0knRZJHaVJf9etX4+RI0eiW7duogyEXDYXdIHci4jFIkqyhgumTsW4sWPx+uzZ+O7bXUju2RP0nnKtPXoUa9aswaWXXipWvYVc5BJM2AZBl5j29+2HKz7EnTP/JOTbsXU7+vfv7+JHs4m/ms2FMZVr80p8EPdiyNJ11NTUYDHewfnnX4xkwtVdXyTLR7x+vf1bZKSnoGdKKmxCHqqjJJcZfzWB14cffoiLLrpItAVRrkJFcr2ziXFC7v2mTRuRk5ODlJQ+IDOdv/I2Uif8pSEZf091lC53XTDpb/nhKuzevRPjx49v3QYErmiqq64BuAXv0J5//MnHmPJfU8SRitQu6ZRfd4dJ7vluz0hpz3UdW7duxahRowSebi1QVt9GDibUN7/3wTJXmUXQ/25cQ2apa6euooK4O0rK0Kx7AtNmt4fMs/EMdBHSxi2P+2+gMrYFe968+Rg27HTc/qc/YdE77zTj1UXXYbfbm3+7+QyWrj8+aeb38L0PCzftXTu2Iz0jQ5B08+v+6+ZDxl+ShfYReZalGfK5ZNdFfYmLi2uSyyURzWrFe+l/RbZooef67aJlzr27PcTFudqFGVh65ilksdnY2mGc2LSkNWPqpu9G07OOyrrXESfkc+fn/itouwl7DE6ynrvrjQcJObdNk7xOnaQYweTw1JSLNB3eoTukMhZJmZFmfFyYH+RylZ2dhaeeegqL330Xc+bObZU5zWg5LhqEzj77bBSVFOKTtauQnuEyE3LQpn1EmmfrNo2ond292T2bNU2kthl7aMttX5nxu+2gZAYNzzwdGg2wEu1Tnpn7uLehwceb6H4saTVGKgjSBiKpXEVYZk7STk4cN4WrGdOn4/IrLxdeauQowHnt37cfQ4eejurqWjzxxDPIzMjiJM9IywFXR8ZHktvdmMy3XJMXQpFocdJjma+0qx7hodqOjRh4oAYiA4VsczSgU+fOBlIGkUTT8MabbyE7KwfnXnAuCgsLg8gk8E82bdmEYSOGoUuXk7B16xbk5uQIG3jgOQX/BZnmuIhaPegprSNwX5z3F3xXAAAbyUlEQVRaEffAzo1lrNNTA5GhGqCZYppzk+7etSs+WfsJ4u3xOOecc1BUVORyRHInkPiXZrHk4XjWmWchKzMLO3duQwatCWkuRwmJpPxmxRnix9m8zOyXLSkJODtpYlgsmDetAUgRQGVCng4KBSYE1EBkAGinRsva5pjm3OTpyO11a9ahvv64ODbiSGWN+5W0v+T+Ou2GafjjzX/E1TdMx45dO5CcnNKUv8YeGFSaYP4yIidOZo2B02xF4pPTBydNGmg56bk9R/0Vtcz3DjWwy4Szw7zUQNQhPK6X5KbZ6YT5UGXnZGNj0+F5TzzxOLZu2WKAO2NJVq9ejcGDB2Pl8pV488038c5bb7T+UPhVt37E8YtjQyttvBSu8xwChYtGGAZaTq3Prjtce7IY8bXDug5YjDAaImV+72qIjchORBvKjpu1RtRGdBqM6CRXh9OBs846Gy+/+nJIM8/v8r7DRVMvwrnnnotTTjkF27fvwE033dSGatNPZksE1zEQDthht/i6M5nmrBwROxxecy27hrw3F/VUHgJqIDKApebUTPOa80aeNpM+88QzmD5jOm6/9XaMGTMq4AP1aGPcFb+/AqcNPA1fb92Ot956E2s//RTkMu710lwbgr2+M+kh2xqR0/ozWxpnOQ/GI1Mg6Shcl8XnEVwwRiwdNRAZKBpdg3lecz7o2+PteOWVV/DZZ5/B6dDxu9/9DmPHjsVjjz2G/fv3t/uqtrYWtNN95oyZ+PWvf4XRo0dj46ZNePGVF/H//t//4IYbbhLrCO0+9HjAvY7iQdrcWxvgoEK08kVRRWjDG9NFps44PnLhWb9UzgpMtUnpnoaBpqCnpu149sIFhaChi0KafL1jO9atWQOKxPDwww+Lfwnx8Ujp00eY7aqrq1BTUyfSJyYmYto10/DSS6/grPFn4iRbFy+5e3tEIUt4NtF6o27mM9oKyb7B1EyBvORNY4LOONgSnozjkBeJzX+knBXMx9hNQWm8biT8/JV1DIQfMs2vPT2SyAwyefJk8a+6+hXk5eUjPz8fR45UiPQJCUno3b83Mk/OxK9+9at2oYGaM+3gRnfS0chmBh/tgLjJr8gwZ1ltrwk7bvnIN4LzGIhweM0pZwWTG6ZH9mog8gCjo1tujcgXL927p2DUKPo3yleS4J7byAWYXyPi8JojQKyuEZEVidOLjWKNcprmwuE1pwxGwXUlwXxlccN5MJC0/4ZA4taIODsVIXEY7CxcXnMkn9V35rsjbrevveY8ITwdjC7j4fGaMwc7lWt7BNRA1B6Tdk+oj5Z5MF47Al4eeJrmvLw24RGdyMM7GrF5zZmAVqxnSeZijdEnnjQi08KN+ChMTi9EHyzEzGM1EBksam6NyCBb0pK5TlexbnXg9CiTVigBZORk9vAirzmNkWY4NCIr78sKoGqxJLVuzyMZPm6NSDL7BrJzwt50yJiBxNGVRJzlZ+2qToft8ZpzNVZnBapw1i7B6GpSsrlVZWsQUatrROT6y28ONAi+SuYXAe5jICzvrEBrYGro81vvZCVQA5FBJC2tEdGis90mjlo3CIe0ZFxec9IYjtCM+N23Le6soGlQ7tt8lV0NRAaxtrRGRJsTHQ64FoQNAiIhGafXnAR2IzoLsWbDGC2a3OE5TYHcA62rsNXuFq5KrwYig0hbWiOiYwRsNvZTTLm85hxOwO60dlWn8uP08iL3bSejl2U4NrQa7BpUMgkIWLt1SgDInYWlNSISkjZEuoW12F86CNZh43VNDweEnJ6B5L7NCWlYNrQyegWGo75EEk01EBksDatrRE7daflNnwaLOiqTkfs2Z6w5cb4To5dlONy3Vaw5vqagBiKDWFtdIyLTjlWDnhos4qhORpEVONdsyJma08uSfY1I15WzAmOLUAORQbAtrRHpdBYR/xoRQa+85gxWQD/JaFDgHBjIs5lz4GNfIxLanlWN1X4qUxheq4HIIOiW1oiE11wD+xqR8pozWPkMJOMcFAQ75PLPGGsuLGtEBnBXSeQgoAYigzhaWiMirznNxr5GxOU1Z7CIozoZaUO8g5FGIc3ZMAvHGhGbcIqQ2jpstA5YWiMiEMJwVLhR7FU6/wjQIMRqmmN03XZVT36vR148/ZexlVPwTWmiHEWra0R0OCv3htYorxIxz77V9xHxapixXZ3UQGSw/K2uEZGrqsv8YRAQlSyiEOCOvh0T+4giqoStzYwaiAyWr9U1IpvuYHdWIOiV15zBCugnGXf07VjYR0SxI9TFg4AaiAzibHWNSNM1aOLwMYOASEimvOYkgNiUBXf0bSLLuYbCvo9IHDuh3Lfl1dCOc1IDUcf4NL+1ukYEmwYH4pvl5bhRXnPyUObuqLmDnrLvI6KiUSF+5FVQPzmpgcgPQO7XVteIHBpgxzG3uOpvlCHAHX2bthBxOiuEYx+RCvHD1wik6Z7V1dXYsGEDC+eFhYVstEig4pJibNq0iW2fBpk8KisrWWUsLSzC1uM6unTZx1KGeXvzxJGb/7+964+t6rjS314/P0yJQwnJIsRSL7GQy2b/aJuWqCFFCFUoC2kTNUJNomwSFi3pH21YNW2EKiEiEVVN1UjtIjWhbdoiQiwsoipIiRIUUYIoBUK1SqIKVRayiOME47g2Pxab68tNdeb5mveM7TcPz/3ufcO5kn1/zZsf3zl3zpwzZ84cOHAIzbNmpVpmb28v5K+5uTnVcsozZ9Ovp6cHQ0ND+Buxjed6e2k8Km378MMPaeUJLQdPc8uTMhl928WL+RtwOhNEs2fPxsqVK8u/xdSuReCxypJGRBcv4o7lKzC7qZham8ozHh4exsDAALeNYYQvfuWLuGXuLeVVSe1aPri4EGDFXXdh9pw5qZUjGR85cgQtLS2YP39+quWUZy6Cj8mj8k0sXrwYCxcuLK9GetdxjJ4+Xht7+/vRcPQoFdPz589TyxNiMfq23bt3p8cX15izmuYsgAuDAmaQF/BlsYYhJq9v9nmOSPbrYR6iRTN5RqZPisQmyoiZWByTdPSymE4mto1TQWSBVAExLo+MWKR0l4TNLOKq6kw9toRBvOZ89ZCVyXzmkUlkBWIbxZGaiyiTetyy2Lxp0zqlrQVKMinb0NhokbJ+kwR0MQTQNKICx6RaTn22RlReNuOaLfjEK1A1IjeUVUHkBkd6LgUEdI2I3UivP3Py+ig27aQ8ummObArMY+eZBZ1dlMm2ttjUWTUiG5QwgqhxhlVKTWSPgJrm7LGqlpKtobANZWFMnsGMY4SeGgOZc4nV+DZ5r4IoQWKKcxTMwAzyHNEU1fHmFcs0J5Pq7GAtvpvmzPQQ0SGjKHtmMTk/CLzdoVU1IiYjOSwriCOgscFhjpoVE4FQ9ltiFphBWezORQStLIJmHaXwU8QCTcP85BrViFhc67icAAG8D/HjGDOb7NQ0Z4OSXRrpXJgdjETfZrpv68Z4dnxgk4o9aLGpE3uIYVOn3KURk4DvIX6yAJ1lmpPgReyDbZqTzoXZwZjo20RQ2bH0iE2jF8UcsNg2TgWRJVKqEVkClcdkgRjn/D74nQt3q3D1mnPHv8wBi22tVRBZICUTs6oRWQBVYxI1zdUI2BTJpXOhCqMAiCOeCwhbw5wC6rp/ReUTS7RUEFkAJc5BqhFZAFVjEpZp7nrwmpPOhTnSNdP4Rd5kPj2ygsfu20w+se0SVBDZIBVHiHUdkQ1SuUwToogmugN3LqFwVimz7QRxvx5133ZGOq7mbFltFUQWQMmKggIuWaTUJLUgwDPNhfRIeuw5DRnlBgWehiILWpkOC+q+XcuXNXVa1Yimxie3b+MgRgP8jjWXBfgs05y0TXb4ZB7sOQ0xzTHnbARO5lyDum+7414m3Wxrzf06bWuVw3Q6R5RDothWSRy8iFEAbKvlNJ2EwAl4sQcMnERMvaefU2aYOjPViKbGJ9dv1WvOPXlYpjlxVuB10SWc2KY50fjimDiulB08mMWRNVr33J6fHFUjyg8taq6JakQ1Q1b1ByzTnFlFRO7I2KY5tsYgMqhAlO5sPKFec1W/X5cJiGMal9Xm56UaER9zZyVGEmuOt+bFWb1ryEgc2JjOCkYwEIU73X3b46CnqhHV8GHlLalqRO4pwjLNoVCkzxGxTXOFQkB1VjBec8QFrXT3bcPuTC9E99/XZDnqHNFkyNTBc9WI3BOJZZoD+O7bWZiSqMKP7DVHx9M9u+cmR9WIckOK2iuiGlHtmOXpF2z3bXbbS84KvEkbY9Nnes0RzYBs2rHLU42IjbjD8lQjcgjmaFY005zZj4g7R0TVTsDeLxUQDYW5oFU1Inffn2pE7rCk56QakXvIeaY5/hwRu+OUUS7TWUFEXx47NGdcql5zzqC0yUi95mxQ0v2ILFHKazL+HBEbCREKzMgKkJ6DaJoT0yO1s1KvOSoLU2lLbZnjwlQjcgyo9GNFWRXpPt+JcmTPEbFNcyIUmJEVjBAiztsU4xAxVxR5u8G8zhFN1EPUyTOdI3JPKJppLoN1RGzTXMlZgTmuDBAR41UMQ1ypec4Y7rk9Pznm0aTK5Nz8UOIaaqIa0TWAlpefFMWK5DersyMrFCR+H3kbiLywU73XQzWiOqagakTuiUczzcWyjQfJBjgKE9s0x46sYPYjIm47EYrp0T0LXpc55lEj8nPpcArsJRpRU1NTCjnnJ8u4zPSxa9dO7N9/ADNnNmL16m9i9erVFRW9FA3h979+CWfDfjR/Zg4ee/gRzJw5cyyNjLp2727HR6fPoKlYwGOP/Ream5vH3ssFzTSHEKVtBCqKT/WGbZrLIrJCqgCOy5weWUG95sZRIN1bHWRY4uu7RlTuk/TjZ57Biy/+DnPmzMWRI8ewZs0aPP/88xVIPfyfD0Pmqp/auMmEz9mwYUPFVtWbfvQU3n33ffzwySexePFifOMbayreV2SW9k0Gc0RpN+mq/I2zAvNz5s7XsAW7MHcRJlzuVVDX+4M8akRMzq1r+vk+RyTaUCEoKcgXoxD79+/Hz372UyOIli37Kp559pkx+u3duxev730D6x55xDx7/L8fx+uvv47Dhw+b+/feew+/eG4bNm/ebO7vvns1zpw5g507do7lIRc001wGc0Rs0xw7soLQjznXUDKtsrsrPw1GTLpVfPBT3LApO0VV8v3Ke40oLnUsQ0NDePJ7G8eIIaOnRx99FNHQldHh7t27sWzZMhRmziilCwK0trWOaU0vv/QyWpe0Vpji7lj2Nezas2ssX7mgmeYymCNij+DZH7IIWubIWndorfh0pnXDpJttRdn8a1uv3KXzWiOKY8TFgtlOW+Z55sydW4H/8MUQX/rCl8eeHTp6CLfeeuvYvVwsWtCCo0ePmmdvHXwLN3325or3t8ydi7f3vV3xjHfDnyPita1UkoxymZEVZC0r031bBLt2Vm64SjUiNzhmkovXGpHs7hmGkEWDEx07dv4OW7ZsGXv1wckPcMMNN4zdy0WhWMTAQB+Gh0N0dZ6ECJ7y4zOFIqI4Qn9//9hjmmkugzkitmlORrnUyApiyiVOE9GdFca41L+LPGpEzoygvb292LVrF6TDTrSHtK6lrI8//pjGIVJef38vgqCYepnSgcnoT9oneLo8zp07B2nL+EPoJM8PHz+G21rbTATNxLT0/rt/xezZc/DBhx+gq70LDQ0N5uednZ1ob+9AHI+gMWhEZ1cnojDGH/7wCkSD6u7uRvuudpNXAxrw/rv/Z34n80vf//4TGBy8YO4/e9NN2Lp1K1pbW027k7oZ54mgFFxzes8CDA7+HSc6T6BYTId+SV2lnZdxGXI+1X0KHe0dGIlHkNA0aUca508++QSnTp1KrY0T1fmj7o/Q3t5eQbeJ0rl4FoYhzg2cQ/tAO8q9O13kPVkePT096OjowOXLlydL4vy50FH+pMw4jhAEBef4XrhQ+vacV34aGToTRPPmzcM999xjIvJKpiY2lJh8JEaU4/OhAwewfMWKaTS7tp/u27cPX132NTTPmmVWd8saCudtLKvS8PAwXnvtNdx///1lT6d/efJkF954Y++EGfV0R5g/by7Wrl1bWsEeBDh58iTeemsf3nzzTePYnZhG2tqWYP78+Vi79luleGNBgO2/2Y6WRa1Yu/bb2L7915COY+23Ja/S8cc/H8KNN96IR9etQ1tbG/Z07MG2bdvQsWcPVqxYjuZZzaUQ0jLKTgpKrpOzZJVcJ+dqzwAcP34cLQsX4pZ580dr4/AkUahNMJgSrxdGBxIyj/YtwUcOsWOJi2GK5/0HDqKtbTEWLFhwFUQJVK7O0iQx77zyyiuj/CIEc5X7xPkMDPTjnXfewapVq0qYJv8d9y1mYfBo6KJXX30V9957b1IS5Xzw4EEsX778Slkmnp+sHr7yaDpXUVQaLE4njzR+60wQSeVE5XOa4WQtJk+UIogxc0YjZK2GcETabUxU5+Q8GQy1PpeOqq3tyQl/JsL29ttvR6FYal1fXx+ee+5ZvPDLX101KX33qlU4/t7xiue9vX1YvfrrBqM1a/4D21/cXvH+dHc37rvvPvM93XnnnThx4gSCpiJWfn05ZjfPmbBOTh8annH0NVdU7Gp+SMxyrulXUey4G+FN0fh4ZZacFa6Ul2Cb1rnU4CvljQMghVspi1le0oQ0yySuQU6aY3VOuMYq8XWbKAqAEZ56ngnOZvK5VPLAwAD+5wc/wNatzyKKQpwdDvHXv3Vi+7ZfmAQPPfwQ/vKnv0DSyXH+/Hl0d3dh3ePfM/cPPPAQek51G1ORPJDRs4z01q9fb94n/4zXXHKj5+khIJoZcetuEXzM8qYHjv66HIFkoFT+LOtrFUQ2FCjEuNTYaJOyftPI1s+AmZu64yt3oOOll/Gvn/scbr75n/EvzbPw7//2edy1smQWWbp0KbZs3YLNm7egq6sL39mwAT//+f9iSesi0/6FCxfgt7/9PZ747hPm/VObNmHjxo2VJodRpK7WJ9KBkBqZOp0mTJ0rMRK2VESEUJoj9/GNzWYd0fha+HGfxwFE2lYmLyhXuCQmx0sA0pnszgNIxj02CCDzUy/85gWUJuBL08JyXSwUcdttS8aqumnTJhw7dgw7du7Aj3/yE7S0tIy9k4sHH3wQX166FDt27MD6desrfluesDTDUv4khWszPeP3mIstGNij6izWEeWxw3bxdTAHELb1VUFkgVTUGKIBfmtEImzkw1u0aJH5s4AFohnJ32TH4tZWPP3005O95j2P/R1AJCBK58LsOGXgwiyvtLSAO5jIY4ed0Hs6ZybdbOvJpaxtrXKWLkITLvs+RySOOWTzDo/ME6+P4pXvX0nCK8yOOguNyD+qlVrEpJsthiqILJAqioNuo7iV+nvIKClZO+RjK32fIxL6cSMryPfA/Sa0s3LzZapG5AZHei7mcxvx+zMICtwRbkLEMOZYh6PUne6TFmVzllEuN7ICqLGps9gqPI8dtgvuUo3IBYpZ5BHEiDz3mgtivjuuhPgpxoQN68TsaBYGZsE8pDLNNhA8DUUW7hZ5xY3uJ0UscHRdJIl61GLyKGD9HuY7Im8QBSiMiNecv0ccxPQ5Ilr0bX/JNtay0jYQvM9ZooswD+8HEkQwVSMigu2yqDgAGnzXiKLKuG4u8dO80keAJ4KSthjvluQm9bMJFZZ6KddHAaoR1TGdk0CuddyEKaseFgK6RjRlhfRlTQhIDDHmSFeiXTE7NJ0jqokdpkzM5JMpK1L2kj+QKiu8ni693gbCBO2MEEyyDUQ90el6rSs75I4J/EsMXJaF+3YeO2wX/M0cQNjWVwWRJVK+a0QyDxYGTZZouEvG8ppzV2PNqYQAt+vQOSJ3fJdHAcvlJndY0nPyXSOCRG+Oh6m40rzmqK3KpjD+KFecFbgOC9kg61+pfF6pjqEKouoYmRS+a0RhLBoRNxSOes1ZMp9FMhnlske6Mi/FOrJwVshjh+0Cbzaf2NRZBZENSoDZedYyaV0mKyCEbMesR30iIJ0ms+Nkh/jJwlkhjx22C+5k8oltfbXnsUTKd40oiwWtltBrMgsE2J2mLCOKiKa5LJwVLGCvyyRsXrEBSQWRDUrXgUYkc0R61C8CMsrldjAxCjzLnP+RMYispxoREWzXRfmuEcUSaicDWaRec244VYQQs4PxN1L7FXow8bxSavpX3AGLXXsy6HrsKpa3VL57zZmdUokjXKGves3ljcvt62P2IyL2Hlk4K+Sxw7an0OQp8yhgiaw0OTD18MZ3jSjMYD8i9Zpzx/nszkWCnjJNc1k4K7ijTr5yyqOAVUFkySO+a0QF2c9GIytYckP+kknnwuxgTNBTopelOiu44zn2oMWm5iqIbFAC4LtGJJuqsdcRWUKvySwQkM6F28Fwtw3JIrICF08LIjtKwhyw2FZZBZElUr5rROKJy9mizhJwTVYTAvTOJQBVA6sJDEeJ6Zg6qne1bPIoYFUQVaPa6HvfNSKZfM5iq3D1mrNkwCrJpHOhdpyykIi4J1EWzgpVIK/b11Q+sURJBZElUL5rRLJVOPtQrzl3iEvnwh3pcvcjUmcFd7zC5RO7evN7H7t65S6V9xqRrJMnTj4LgdVrLndsXkOFxNefsM37aI2ycFbIY4ddA4EmTaoa0aTQ5P+F9xpRJNG3w/wTQms4IQL8TjMAZOti0pGFs0IeO2wXcPN5pXqteZxUvS65TuG7RhSq11yu+a9a5aTTpHac2nNUI0lu31P5xBIFZSdLoHzXiIpRpBqRJS/kMZmMcqkjXXFUIJpy1VnBHddR+cSy2iqILIHyXSOSoKdZrCNSrzlLBqySLJNRLtFrLgtnhTx22FXYwOp1JrxSpWYqiKoAlLz2XSMCcZOzBFP1mkuQmP5ZOk1mB8MOepqFswITz+lzgH0OeRSwKogs6Xc9aERsZwX1mrNkPotk0mkyOxhRhiQ+IevIwlmB1TZ2OXkUsERWYsPttjzfNaI4ijMxzbmlkubGQyBGkRytndc2v0tiDlhskXQW1eXSpUs4ceKEbbnTStfX10crSyra39+Pzs5OmulDGOXChQvUNg6cHcCpk6fw9zNnpkUb2x8LDQtBweA6a2az7c9qTicjrU96PkIcBRgcHKz599fyAxlxCv2EZ1gffW9vL5qamky511Lna/nN4IWLNB49f/b/cXZggFae4CH8wqShlJl23ya8GYb5W6bxT59++umn18KE439z+PBhnD59evxjvVcEJkRgaGjIdNLNzekJoQkL1ofOEJBRLG9Jq7NqX/cZLVmyBPKXp8OZIMpTo7QuioAioAgoAvWDgM4R1Q+ttKaKgCKgCHiJgAoiL8mqjVIEFAFFoH4QUEFUP7TSmioCioAi4CUCKoi8JKs2ShFQBBSB+kFABVH90EprqggoAoqAlwioIPKSrNooRUARUATqBwEVRPVDK62pIqAIKAJeIqCCyEuyaqMUAUVAEagfBFQQ1Q+ttKaKgCKgCHiJgAoiL8mqjVIEFAFFoH4QUEFUP7TSmioCioAi4CUCKoi8JKs2ShFQBBSB+kHgH8Bne49B7DpXAAAAAElFTkSuQmCC"></p>
<p>shape <em><strong> A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>-axis intercepts at (−3, 0), (1, 0) and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y"> <mi>y</mi> </math></span>-axis intercept at (0, −51) <em><strong> A1A1</strong></em></p>
<p>minimum points at (−1.62, −118) and (3.72, 19.7) <em><strong> A1A1</strong></em></p>
<p>maximum point at (2.40, 26.9) <em><strong> A1</strong></em></p>
<p><strong>Note:</strong> Coordinates may be seen on the graph or elsewhere.</p>
<p><strong>Note</strong>: Accept −3, 1 and −51 marked on the axes.</p>
<p><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>from graph, 19.7 ≤ <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k"> <mi>k</mi> </math></span> ≤ 26.9 <em><strong> A1A1</strong></em></p>
<p><strong>Note:</strong> Award<em><strong> A1</strong></em> for correct endpoints and <em><strong>A1</strong></em> for correct inequalities.</p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>A curve <em>C</em> is given by the implicit equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x + y - {\text{cos}}\left( {xy} \right) = 0">
<mi>x</mi>
<mo>+</mo>
<mi>y</mi>
<mo>−<!-- − --></mo>
<mrow>
<mtext>cos</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mi>y</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span>.</p>
</div>
<div class="specification">
<p>The curve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="xy = - \frac{\pi }{2}">
<mi>x</mi>
<mi>y</mi>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mfrac>
<mi>π<!-- π --></mi>
<mn>2</mn>
</mfrac>
</math></span> intersects <em>C</em> at P and Q.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}} = - \left( {\frac{{1 + y\,{\text{sin}}\left( {xy} \right)}}{{1 + x\,{\text{sin}}\left( {xy} \right)}}} \right)"> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>y</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mfrac> <mo>=</mo> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <mi>y</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mi>y</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mi>x</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mi>y</mi> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the coordinates of P and Q.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that the gradients of the tangents to <em>C</em> at P and Q are <em>m</em><sub>1</sub> and <em>m</em><sub>2</sub> respectively, show that <em>m</em><sub>1</sub> × <em>m</em><sub>2</sub> = 1.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the coordinates of the three points on <em>C</em>, nearest the origin, where the tangent is parallel to the line <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = - x"> <mi>y</mi> <mo>=</mo> <mo>−</mo> <mi>x</mi> </math></span>.</p>
<div class="marks">[7]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p>attempt at implicit differentiation <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1 + \frac{{{\text{d}}y}}{{{\text{d}}x}} + \left( {y + x\frac{{{\text{d}}y}}{{{\text{d}}x}}} \right){\text{sin}}\left( {xy} \right) = 0"> <mn>1</mn> <mo>+</mo> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>y</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mfrac> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mi>y</mi> <mo>+</mo> <mi>x</mi> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>y</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow> <mtext>sin</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mi>y</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </math></span> <em><strong>A1M1A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong> </em>for first two terms. Award <em><strong>M1</strong> </em>for an attempt at chain rule <em><strong>A1</strong> </em>for last term.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {1 + x\,{\text{sin}}\left( {xy} \right)} \right)\frac{{{\text{d}}y}}{{{\text{d}}x}} = - 1 - y\,{\text{sin}}\left( {xy} \right)"> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mi>x</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mi>y</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>y</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mfrac> <mo>=</mo> <mo>−</mo> <mn>1</mn> <mo>−</mo> <mi>y</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mi>y</mi> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}} = - \left( {\frac{{1 + y\,{\text{sin}}\left( {xy} \right)}}{{1 + x\,{\text{sin}}\left( {xy} \right)}}} \right)"> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>y</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mfrac> <mo>=</mo> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <mi>y</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mi>y</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mi>x</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mi>y</mi> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>AG</strong></em></p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p>when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="xy = - \frac{\pi }{2},\,\,{\text{cos}}\,xy = 0"> <mi>x</mi> <mi>y</mi> <mo>=</mo> <mo>−</mo> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mi>y</mi> <mo>=</mo> <mn>0</mn> </math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow x + y = 0"> <mo stretchy="false">⇒</mo> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo>=</mo> <mn>0</mn> </math></span> <em><strong>(A1)</strong></em></p>
<p><strong>OR</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x - \frac{\pi }{{2x}} - {\text{cos}}\left( {\frac{{ - \pi }}{2}} \right) = 0"> <mi>x</mi> <mo>−</mo> <mfrac> <mi>π</mi> <mrow> <mn>2</mn> <mi>x</mi> </mrow> </mfrac> <mo>−</mo> <mrow> <mtext>cos</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mo>−</mo> <mi>π</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </math></span> or equivalent <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x - \frac{\pi }{{2x}} = 0"> <mi>x</mi> <mo>−</mo> <mfrac> <mi>π</mi> <mrow> <mn>2</mn> <mi>x</mi> </mrow> </mfrac> <mo>=</mo> <mn>0</mn> </math></span> <em><strong>(A1)</strong></em></p>
<p><strong>THEN</strong></p>
<p>therefore <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^2} = \frac{\pi }{2}\left( {x = \pm \sqrt {\frac{\pi }{2}} } \right)\left( {x = \pm 1.25} \right)"> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>=</mo> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>=</mo> <mo>±</mo> <msqrt> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </msqrt> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>=</mo> <mo>±</mo> <mn>1.25</mn> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{P}}\left( {\sqrt {\frac{\pi }{2}} ,\, - \sqrt {\frac{\pi }{2}} } \right),\,\,{\text{Q}}\left( { - \sqrt {\frac{\pi }{2}} ,\,\sqrt {\frac{\pi }{2}} } \right)"> <mrow> <mtext>P</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <msqrt> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </msqrt> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mo>−</mo> <msqrt> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </msqrt> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mrow> <mtext>Q</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <msqrt> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </msqrt> <mo>,</mo> <mspace width="thinmathspace"></mspace> <msqrt> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </msqrt> </mrow> <mo>)</mo> </mrow> </math></span> <strong>or</strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P\left( {1.25,\, - 1.25} \right),\,Q\left( { - 1.25,\,1.25} \right)"> <mi>P</mi> <mrow> <mo>(</mo> <mrow> <mn>1.25</mn> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mo>−</mo> <mn>1.25</mn> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mi>Q</mi> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>1.25</mn> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mn>1.25</mn> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><em>m</em><sub>1 </sub>= <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - \left( {\frac{{1 - \sqrt {\frac{\pi }{2}} \times - 1}}{{1 + \sqrt {\frac{\pi }{2}} \times - 1}}} \right)"> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <msqrt> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </msqrt> <mo>×</mo> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <msqrt> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </msqrt> <mo>×</mo> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>M1A1</strong></em></p>
<p><em>m</em><sub>2 </sub>= <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - \left( {\frac{{1 + \sqrt {\frac{\pi }{2}} \times - 1}}{{1 - \sqrt {\frac{\pi }{2}} \times - 1}}} \right)"> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mn>1</mn> <mo>+</mo> <msqrt> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </msqrt> <mo>×</mo> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <msqrt> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </msqrt> <mo>×</mo> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p><em>m</em><sub>1 </sub><em>m</em><sub>2 </sub>= 1 <em><strong>AG</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1A0A0</strong> </em>if decimal approximations are used.<br><strong>Note:</strong> No <strong>FT</strong> applies.</p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>equate derivative to −1 <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {y - x} \right){\text{sin}}\left( {xy} \right) = 0"> <mrow> <mo>(</mo> <mrow> <mi>y</mi> <mo>−</mo> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mtext>sin</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mi>y</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </math></span> <em><strong>(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = x,\,{\text{sin}}\left( {xy} \right) = 0"> <mi>y</mi> <mo>=</mo> <mi>x</mi> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mi>y</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </math></span> <em><strong>R1</strong></em></p>
<p>in the first case, attempt to solve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2x = {\text{cos}}\left( {{x^2}} \right)"> <mn>2</mn> <mi>x</mi> <mo>=</mo> <mrow> <mtext>cos</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>M1</strong></em></p>
<p>(0.486,0.486) <strong>A1</strong></p>
<p>in the second case, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{sin}}\left( {xy} \right) = 0 \Rightarrow xy = 0"> <mrow> <mtext>sin</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mi>y</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> <mo stretchy="false">⇒</mo> <mi>x</mi> <mi>y</mi> <mo>=</mo> <mn>0</mn> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x + y = 1"> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo>=</mo> <mn>1</mn> </math></span> <em><strong>(M1)</strong></em></p>
<p>(0,1), (1,0) <em><strong> A1</strong></em></p>
<p><em><strong>[7 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>A large tank initially contains pure water. Water containing salt begins to flow into the tank The solution is kept uniform by stirring and leaves the tank through an outlet at its base. Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> grams represent the amount of salt in the tank and let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> minutes represent the time since the salt water began flowing into the tank.</p>
<p>The rate of change of the amount of salt in the tank, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}x}}{{{\text{d}}t}}">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
</math></span>, is described by the differential equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}x}}{{{\text{d}}t}} = 10{{\text{e}}^{- \frac{t}{4}}} - \frac{x}{{t + 1}}">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mn>10</mn>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−<!-- − --></mo>
<mfrac>
<mi>t</mi>
<mn>4</mn>
</mfrac>
</mrow>
</msup>
</mrow>
<mo>−<!-- − --></mo>
<mfrac>
<mi>x</mi>
<mrow>
<mi>t</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</mfrac>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> + 1 is an integrating factor for this differential equation.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, by solving this differential equation, show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x\left( t \right) = \frac{{200 - 40{{\text{e}}^{ - \frac{t}{4}}}\left( {t + 5} \right)}}{{t + 1}}">
<mi>x</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mn>200</mn>
<mo>−</mo>
<mn>40</mn>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mfrac>
<mi>t</mi>
<mn>4</mn>
</mfrac>
</mrow>
</msup>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>t</mi>
<mo>+</mo>
<mn>5</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mi>t</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</mfrac>
</math></span>.</p>
<div class="marks">[8]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> versus <span style="display: inline !important;float: none;background-color: #ffffff;color: #000000;font-family: Verdana,Arial,Helvetica,sans-serif;font-size: 14px;font-style: normal;font-variant: normal;font-weight: 400;letter-spacing: normal;text-align: left;text-decoration: none;text-indent: 0px;white-space: normal;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span></span> for 0 ≤ <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> ≤ 60 and hence find the maximum amount of salt in the tank and the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> at which this occurs.</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> at which the amount of salt in the tank is decreasing most rapidly.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The rate of change of the amount of salt leaving the tank is equal to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{x}{{t + 1}}">
<mfrac>
<mi>x</mi>
<mrow>
<mi>t</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</mfrac>
</math></span>.</p>
<p>Find the amount of salt that left the tank during the first 60 minutes.</p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><em><strong>METHOD 1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="I(t) = {{\text{e}}^{\int {P\left( t \right)} \,{\text{d}}t}}">
<mi>I</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>∫</mo>
<mrow>
<mi>P</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</msup>
</mrow>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{{\text{e}}^{\int {\frac{1}{{t + 1}}} \,{\text{d}}t}}">
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>∫</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<mi>t</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</mfrac>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</msup>
</mrow>
</math></span></p>
<p>= <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{{\text{e}}^{{\text{ln}}\left( {t + 1} \right)}}">
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mrow>
<mtext>ln</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>t</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
</msup>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = t + 1">
<mo>=</mo>
<mi>t</mi>
<mo>+</mo>
<mn>1</mn>
</math></span> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>METHOD 2</strong></em></p>
<p>attempting product rule differentiation on <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{\text{d}}}{{{\text{d}}t}}\left( {x\left( {t + 1} \right)} \right)">
<mfrac>
<mrow>
<mtext>d</mtext>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>t</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{\text{d}}}{{{\text{d}}t}}\left( {x\left( {t + 1} \right)} \right) = \frac{{{\text{d}}x}}{{{\text{d}}t}}\left( {t + 1} \right) + x">
<mfrac>
<mrow>
<mtext>d</mtext>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>t</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mi>t</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mi>x</mi>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left( {t + 1} \right)\left( {\frac{{{\text{d}}x}}{{{\text{d}}t}} + \frac{x}{{t + 1}}} \right)">
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>t</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
<mo>+</mo>
<mfrac>
<mi>x</mi>
<mrow>
<mi>t</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p>so <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t + 1">
<mi>t</mi>
<mo>+</mo>
<mn>1</mn>
</math></span> is an integrating factor for this differential equation <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p> </p>
<p>attempting to multiply through by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {t + 1} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mi>t</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span> and rearrange to give <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {t + 1} \right)\frac{{{\text{d}}x}}{{{\text{d}}t}} + x = 10\left( {t + 1} \right){{\text{e}}^{ - \frac{t}{4}}}">
<mrow>
<mo>(</mo>
<mrow>
<mi>t</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
<mo>+</mo>
<mi>x</mi>
<mo>=</mo>
<mn>10</mn>
<mrow>
<mo>(</mo>
<mrow>
<mi>t</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mfrac>
<mi>t</mi>
<mn>4</mn>
</mfrac>
</mrow>
</msup>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{\text{d}}}{{{\text{d}}t}}\left( {x\left( {t + 1} \right)} \right) = 10\left( {t + 1} \right){{\text{e}}^{ - \frac{t}{4}}}">
<mfrac>
<mrow>
<mtext>d</mtext>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>t</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>10</mn>
<mrow>
<mo>(</mo>
<mrow>
<mi>t</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mfrac>
<mi>t</mi>
<mn>4</mn>
</mfrac>
</mrow>
</msup>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x\left( {t + 1} \right) = \int {10\left( {t + 1} \right){{\text{e}}^{ - \frac{t}{4}}}} {\text{d}}t">
<mi>x</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>t</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mo>∫</mo>
<mrow>
<mn>10</mn>
<mrow>
<mo>(</mo>
<mrow>
<mi>t</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mfrac>
<mi>t</mi>
<mn>4</mn>
</mfrac>
</mrow>
</msup>
</mrow>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</math></span> <em><strong>A1</strong></em></p>
<p>attempting to integrate the RHS by parts <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = - 40\left( {t + 1} \right){{\text{e}}^{ - \frac{t}{4}}} + 40\int {{{\text{e}}^{ - \frac{t}{4}}}} \,{\text{d}}t">
<mo>=</mo>
<mo>−</mo>
<mn>40</mn>
<mrow>
<mo>(</mo>
<mrow>
<mi>t</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mfrac>
<mi>t</mi>
<mn>4</mn>
</mfrac>
</mrow>
</msup>
</mrow>
<mo>+</mo>
<mn>40</mn>
<mo>∫</mo>
<mrow>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mfrac>
<mi>t</mi>
<mn>4</mn>
</mfrac>
</mrow>
</msup>
</mrow>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = - 40\left( {t + 1} \right){{\text{e}}^{ - \frac{t}{4}}} - 160{{\text{e}}^{ - \frac{t}{4}}} + C">
<mo>=</mo>
<mo>−</mo>
<mn>40</mn>
<mrow>
<mo>(</mo>
<mrow>
<mi>t</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mfrac>
<mi>t</mi>
<mn>4</mn>
</mfrac>
</mrow>
</msup>
</mrow>
<mo>−</mo>
<mn>160</mn>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mfrac>
<mi>t</mi>
<mn>4</mn>
</mfrac>
</mrow>
</msup>
</mrow>
<mo>+</mo>
<mi>C</mi>
</math></span> <span style="display: inline !important;float: none;background-color: #ffffff;color: #000000;font-family: Verdana,Arial,Helvetica,sans-serif;font-size: 14px;font-style: normal;font-variant: normal;font-weight: 400;letter-spacing: normal;text-align: left;text-decoration: none;text-indent: 0px;white-space: normal;"> </span><em style="color: #000000;font-family: Verdana,Arial,Helvetica,sans-serif;font-size: 14px;font-style: italic;font-variant: normal;font-weight: 400;letter-spacing: normal;text-align: left;text-decoration: none;text-indent: 0px;white-space: normal;"><strong>A1</strong></em></p>
<p><strong>Note:</strong> Condone the absence of <em>C</em>.</p>
<p> </p>
<p><em><strong>EITHER</strong></em></p>
<p>substituting <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t = 0,\,\,x = 0 \Rightarrow C = 200">
<mi>t</mi>
<mo>=</mo>
<mn>0</mn>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>=</mo>
<mn>0</mn>
<mo stretchy="false">⇒</mo>
<mi>C</mi>
<mo>=</mo>
<mn>200</mn>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \frac{{- 40\left( {t + 1} \right){{\text{e}}^{ - \frac{t}{4}}} - 160{{\text{e}}^{ - \frac{t}{4}}} + 200}}{{t + 1}}">
<mi>x</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mo>−</mo>
<mn>40</mn>
<mrow>
<mo>(</mo>
<mrow>
<mi>t</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mfrac>
<mi>t</mi>
<mn>4</mn>
</mfrac>
</mrow>
</msup>
</mrow>
<mo>−</mo>
<mn>160</mn>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mfrac>
<mi>t</mi>
<mn>4</mn>
</mfrac>
</mrow>
</msup>
</mrow>
<mo>+</mo>
<mn>200</mn>
</mrow>
<mrow>
<mi>t</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</mfrac>
</math></span> <em><strong>A1</strong></em></p>
<p>using <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 40{{\text{e}}^{ - \frac{t}{4}}}">
<mo>−</mo>
<mn>40</mn>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mfrac>
<mi>t</mi>
<mn>4</mn>
</mfrac>
</mrow>
</msup>
</mrow>
</math></span> as the highest common factor of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 40\left( {t + 1} \right){{\text{e}}^{ - \frac{t}{4}}}">
<mo>−</mo>
<mn>40</mn>
<mrow>
<mo>(</mo>
<mrow>
<mi>t</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mfrac>
<mi>t</mi>
<mn>4</mn>
</mfrac>
</mrow>
</msup>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 160{{\text{e}}^{ - \frac{t}{4}}}">
<mo>−</mo>
<mn>160</mn>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mfrac>
<mi>t</mi>
<mn>4</mn>
</mfrac>
</mrow>
</msup>
</mrow>
</math></span> <em><strong>M1</strong></em></p>
<p> </p>
<p><em><strong>OR</strong></em></p>
<p>using <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 40{{\text{e}}^{ - \frac{t}{4}}}">
<mo>−</mo>
<mn>40</mn>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mfrac>
<mi>t</mi>
<mn>4</mn>
</mfrac>
</mrow>
</msup>
</mrow>
</math></span> as the highest common factor of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 40\left( {t + 1} \right){{\text{e}}^{ - \frac{t}{4}}}">
<mo>−</mo>
<mn>40</mn>
<mrow>
<mo>(</mo>
<mrow>
<mi>t</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mfrac>
<mi>t</mi>
<mn>4</mn>
</mfrac>
</mrow>
</msup>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 160{{\text{e}}^{ - \frac{t}{4}}}">
<mo>−</mo>
<mn>160</mn>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mfrac>
<mi>t</mi>
<mn>4</mn>
</mfrac>
</mrow>
</msup>
</mrow>
</math></span> giving</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x\left( {t + 1} \right) = - 40{{\text{e}}^{ - \frac{t}{4}}}\left( {t + 5} \right) + C">
<mi>x</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>t</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mo>−</mo>
<mn>40</mn>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mfrac>
<mi>t</mi>
<mn>4</mn>
</mfrac>
</mrow>
</msup>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>t</mi>
<mo>+</mo>
<mn>5</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mi>C</mi>
</math></span> (or equivalent) <em><strong>M1A1</strong></em></p>
<p>substituting <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t = 0,\,\,x = 0 \Rightarrow C = 200">
<mi>t</mi>
<mo>=</mo>
<mn>0</mn>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>=</mo>
<mn>0</mn>
<mo stretchy="false">⇒</mo>
<mi>C</mi>
<mo>=</mo>
<mn>200</mn>
</math></span> <em><strong>M1</strong></em></p>
<p> </p>
<p><em><strong>THEN</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x\left( t \right) = \frac{{200 - 40{{\text{e}}^{ - \frac{t}{4}}}\left( {t + 5} \right)}}{{t + 1}}">
<mi>x</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mn>200</mn>
<mo>−</mo>
<mn>40</mn>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mfrac>
<mi>t</mi>
<mn>4</mn>
</mfrac>
</mrow>
</msup>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>t</mi>
<mo>+</mo>
<mn>5</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mi>t</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</mfrac>
</math></span> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[8 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p> </p>
<p><img 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"></p>
<p>graph starts at the origin and has a local maximum (coordinates not required) <em><strong>A1</strong></em></p>
<p>sketched for 0 ≤ <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> ≤ 60 <em><strong>A1</strong></em></p>
<p>correct concavity for 0 ≤ <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> ≤ 60 <em><strong>A1</strong></em></p>
<p>maximum amount of salt is 14.6 (grams) at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> = 6.60 (minutes) <em><strong>A1A1 </strong></em></p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>using an appropriate graph or equation (first or second derivative) <strong>M1</strong></p>
<p>amount of salt is decreasing most rapidly at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> = 12.9 (minutes) <em><strong>A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p>attempting to form an integral representing the amount of salt that left the tank <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int\limits_0^{60} {\frac{{x\left( t \right)}}{{t + 1}}{\text{d}}t} ">
<munderover>
<mo>∫</mo>
<mn>0</mn>
<mrow>
<mn>60</mn>
</mrow>
</munderover>
<mrow>
<mfrac>
<mrow>
<mi>x</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mi>t</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</mfrac>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int\limits_0^{60} {\frac{{200 - 40{{\text{e}}^{ - \frac{t}{4}}}\left( {t + 5} \right)}}{{{{\left( {t + 1} \right)}^2}}}{\text{d}}t} ">
<munderover>
<mo>∫</mo>
<mn>0</mn>
<mrow>
<mn>60</mn>
</mrow>
</munderover>
<mrow>
<mfrac>
<mrow>
<mn>200</mn>
<mo>−</mo>
<mn>40</mn>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mfrac>
<mi>t</mi>
<mn>4</mn>
</mfrac>
</mrow>
</msup>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>t</mi>
<mo>+</mo>
<mn>5</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>t</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>OR</strong></p>
<p>attempting to form an integral representing the amount of salt that entered the tank minus the amount of salt in the tank at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> = 60(minutes)</p>
<p>amount of salt that left the tank is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int\limits_0^{60} {10} {{\text{e}}^{ - \frac{t}{4}}}\,{\text{d}}t - x\left( {60} \right)">
<munderover>
<mo>∫</mo>
<mn>0</mn>
<mrow>
<mn>60</mn>
</mrow>
</munderover>
<mrow>
<mn>10</mn>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mfrac>
<mi>t</mi>
<mn>4</mn>
</mfrac>
</mrow>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
<mo>−</mo>
<mi>x</mi>
<mrow>
<mo>(</mo>
<mrow>
<mn>60</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>THEN</strong></p>
<p>= 36.7 (grams) <em><strong>A2</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the differential equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><msup><mi>y</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>></mo><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>></mo><mn>2</mn><mi>x</mi></math>. It is given that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>3</mn></math> when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>1</mn></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use Euler’s method, with a step length of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>1</mn></math>, to find an approximate value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>1</mn><mo>.</mo><mn>5</mn></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use the substitution <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>v</mi><mi>x</mi></math> to show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mfrac><mrow><mo>d</mo><mi>v</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><msup><mi>v</mi><mn>2</mn></msup><mo>-</mo><mi>v</mi><mo>-</mo><mn>2</mn></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By solving the differential equation, show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfrac><mrow><mn>8</mn><mi>x</mi><mo>+</mo><msup><mi>x</mi><mn>4</mn></msup></mrow><mrow><mn>4</mn><mo>-</mo><msup><mi>x</mi><mn>3</mn></msup></mrow></mfrac></math>.</p>
<div class="marks">[10]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the actual value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>1</mn><mo>.</mo><mn>5</mn></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Using the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfrac><mrow><mn>8</mn><mi>x</mi><mo>+</mo><msup><mi>x</mi><mn>4</mn></msup></mrow><mrow><mn>4</mn><mo>-</mo><msup><mi>x</mi><mn>3</mn></msup></mrow></mfrac></math>, suggest a reason why the approximation given by Euler’s method in part (a) is not a good estimate to the actual value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>1</mn><mo>.</mo><mn>5</mn></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.iii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>attempt to use Euler’s method <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>x</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><msub><mi>x</mi><mi>n</mi></msub><mo>+</mo><mn>0</mn><mo>.</mo><mn>1</mn><mo>;</mo><mo> </mo><mo> </mo><msub><mi>y</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><msub><mi>y</mi><mi>n</mi></msub><mo>+</mo><mn>0</mn><mo>.</mo><mn>1</mn><mo>×</mo><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><msup><mi>y</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup></mrow><msup><mi>x</mi><mn>2</mn></msup></mfrac></math></p>
<p>correct intermediate <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-values <em><strong>(A1)(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>.</mo><mn>7</mn><mo>,</mo><mo> </mo><mn>4</mn><mo>.</mo><mn>63140</mn><mo>…</mo><mo>,</mo><mo> </mo><mn>5</mn><mo>.</mo><mn>92098</mn><mo>,</mo><mo> </mo><mn>7</mn><mo>.</mo><mn>79542</mn><mo>…</mo></math></p>
<p> </p>
<p><strong>Note:</strong> <em><strong>A1</strong> </em>for any two correct <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-values seen</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>10</mn><mo>.</mo><mn>6958</mn><mo>…</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>10</mn><mo>.</mo><mn>7</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> For the final <em><strong>A1</strong></em>, the value <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn><mo>.</mo><mn>7</mn></math> must be the last value in a table or a list, or be given as a final answer, not just embedded in a table which has further lines.</p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>v</mi><mi>x</mi><mo>⇒</mo><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mi>v</mi><mo>+</mo><mi>x</mi><mfrac><mrow><mo>d</mo><mi>v</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac></math> <em><strong>(A1)</strong></em></p>
<p>replacing <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> with <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mi>x</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac></math> with <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mo>+</mo><mi>x</mi><mfrac><mrow><mo>d</mo><mi>v</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><msup><mi>y</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>⇒</mo><msup><mi>x</mi><mn>2</mn></msup><mfenced><mrow><mi>v</mi><mo>+</mo><mi>x</mi><mfrac><mrow><mo>d</mo><mi>v</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac></mrow></mfenced><mo>=</mo><msup><mi>v</mi><mn>2</mn></msup><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mo>+</mo><mi>x</mi><mfrac><mrow><mo>d</mo><mi>v</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><msup><mi>v</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn></math> (since <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>></mo><mn>0</mn></math>)</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mfrac><mrow><mo>d</mo><mi>v</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><msup><mi>v</mi><mn>2</mn></msup><mo>-</mo><mi>v</mi><mo>-</mo><mn>2</mn></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to separate variables <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><mfrac><mrow><mo>d</mo><mi>v</mi></mrow><mrow><msup><mi>v</mi><mn>2</mn></msup><mo>-</mo><mi>v</mi><mo>-</mo><mn>2</mn></mrow></mfrac><mo>=</mo><mo>∫</mo><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mi>x</mi></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><mfrac><mrow><mo>d</mo><mi>v</mi></mrow><mrow><mfenced><mrow><mi>v</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mfenced><mrow><mi>v</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mrow></mfrac><mo>=</mo><mo>∫</mo><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mi>x</mi></mfrac></math> <em><strong>(A1)</strong></em></p>
<p>attempt to express in partial fraction form <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mrow><mfenced><mrow><mi>v</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mfenced><mrow><mi>v</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mrow></mfrac><mo>≡</mo><mfrac><mi>A</mi><mrow><mi>v</mi><mo>-</mo><mn>2</mn></mrow></mfrac><mo>+</mo><mfrac><mi>B</mi><mrow><mi>v</mi><mo>+</mo><mn>1</mn></mrow></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mrow><mfenced><mrow><mi>v</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mfenced><mrow><mi>v</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mfenced><mrow><mfrac><mn>1</mn><mrow><mi>v</mi><mo>-</mo><mn>2</mn></mrow></mfrac><mo>-</mo><mfrac><mn>1</mn><mrow><mi>v</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>3</mn></mfrac><mo>∫</mo><mfenced><mrow><mfrac><mn>1</mn><mrow><mi>v</mi><mo>-</mo><mn>2</mn></mrow></mfrac><mo>-</mo><mfrac><mn>1</mn><mrow><mi>v</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mrow></mfenced><mo>d</mo><mi>v</mi><mo>=</mo><mo>∫</mo><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mi>x</mi></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>3</mn></mfrac><mfenced><mrow><mi>ln</mi><mfenced open="|" close="|"><mrow><mi>v</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mo>-</mo><mi>ln</mi><mfenced open="|" close="|"><mrow><mi>v</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mrow></mfenced><mo>=</mo><mi>ln</mi><mfenced open="|" close="|"><mi>x</mi></mfenced><mfenced><mrow><mo>+</mo><mi>c</mi></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p style="text-align:left;"><strong>Note:</strong> Condone absence of modulus signs throughout.</p>
<p style="text-align:left;"><strong><br>EITHER</strong></p>
<p>attempt to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math> using <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo> </mo><mi>y</mi><mo>=</mo><mn>3</mn><mo>,</mo><mo> </mo><mi>v</mi><mo>=</mo><mn>3</mn></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>ln</mi><mfrac><mn>1</mn><mn>4</mn></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>3</mn></mfrac><mfenced><mrow><mi>ln</mi><mfenced open="|" close="|"><mrow><mi>v</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mo>-</mo><mi>ln</mi><mfenced open="|" close="|"><mrow><mi>v</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mrow></mfenced><mo>=</mo><mi>ln</mi><mfenced open="|" close="|"><mi>x</mi></mfenced><mo>+</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>ln</mi><mfrac><mn>1</mn><mn>4</mn></mfrac></math></p>
<p>expressing both sides as a single logarithm <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mfenced open="|" close="|"><mfrac><mrow><mi>v</mi><mo>-</mo><mn>2</mn></mrow><mrow><mi>v</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mfenced><mo>=</mo><mi>ln</mi><mfenced><mfrac><msup><mfenced open="|" close="|"><mi>x</mi></mfenced><mn>3</mn></msup><mn>4</mn></mfrac></mfenced></math></p>
<p><strong><br>OR</strong></p>
<p>expressing both sides as a single logarithm <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mfenced open="|" close="|"><mfrac><mrow><mi>v</mi><mo>-</mo><mn>2</mn></mrow><mrow><mi>v</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mfenced><mo>=</mo><mi>ln</mi><mfenced><mrow><mi>A</mi><msup><mfenced open="|" close="|"><mi>x</mi></mfenced><mn>3</mn></msup></mrow></mfenced></math></p>
<p>attempt to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> using <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo> </mo><mi>y</mi><mo>=</mo><mn>3</mn><mo>,</mo><mo> </mo><mi>v</mi><mo>=</mo><mn>3</mn></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></math></p>
<p><strong><br>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mfrac><mrow><mi>v</mi><mo>-</mo><mn>2</mn></mrow><mrow><mi>v</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mfenced><mo>=</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><msup><mi>x</mi><mn>3</mn></msup></math> (since <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>></mo><mn>0</mn></math>)</p>
<p>substitute <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mo>=</mo><mfrac><mi>y</mi><mi>x</mi></mfrac></math> (seen anywhere) <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mstyle displaystyle="true"><mfrac><mi>y</mi><mi>x</mi></mfrac></mstyle><mo>-</mo><mn>2</mn></mrow><mrow><mstyle displaystyle="true"><mfrac><mi>y</mi><mi>x</mi></mfrac></mstyle><mo>+</mo><mn>1</mn></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><msup><mi>x</mi><mn>3</mn></msup></math> (since <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>></mo><mn>2</mn><mi>x</mi></math>)</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>⇒</mo><mfrac><mrow><mi>y</mi><mo>-</mo><mn>2</mn><mi>x</mi></mrow><mrow><mi>y</mi><mo>+</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><msup><mi>x</mi><mn>3</mn></msup></mrow></mfenced></math></p>
<p>attempt to make <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> the subject <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>-</mo><mfrac><mrow><msup><mi>x</mi><mn>3</mn></msup><mi>y</mi></mrow><mn>4</mn></mfrac><mo>=</mo><mn>2</mn><mi>x</mi><mo>+</mo><mfrac><msup><mi>x</mi><mn>4</mn></msup><mn>4</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfrac><mrow><mn>8</mn><mi>x</mi><mo>+</mo><msup><mi>x</mi><mn>4</mn></msup></mrow><mrow><mn>4</mn><mo>-</mo><msup><mi>x</mi><mn>3</mn></msup></mrow></mfrac></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[10 marks]</strong></em></p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>actual value at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mfenced><mrow><mn>1</mn><mo>.</mo><mn>5</mn></mrow></mfenced><mo>=</mo><mn>27</mn><mo>.</mo><mn>3</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>gradient changes rapidly (during the interval considered) OR</p>
<p>the curve has a vertical asymptote at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mroot><mn>4</mn><mn>3</mn></mroot><mo> </mo><mfenced><mrow><mo>=</mo><mn>1</mn><mo>.</mo><mn>5874</mn><mo>…</mo></mrow></mfenced></math> <em><strong>R1</strong></em></p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">c.iii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>Most candidates showed evidence of an attempt to use Euler's method in part a), although very few explicitly wrote down the formulae, they used in order to calculate successive <em>y</em>-value. In addition, many seemed to take a step-by-step approach rather than using the recursive capabilities of the graphical display calculator.</p>
<p>There were many good attempts at part b), but not all candidates recognised that this would help them to solve part c).</p>
<p>Part c) was done very well by many candidates, although there were a significant number who failed to recognise the need for partial fractions and could not make further progress. A common error was to integrate without a constant of integration, which meant that the initial condition could not be used. The reasoning given for the estimate being poor was often too vague and did not address the specific nature of the function given clearly enough.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.iii.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><msqrt><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></msqrt></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mn>2</mn></math>.</p>
</div>
<div class="specification">
<p>The curve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> is rotated <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>π</mi></math> about the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-axis to form a solid of revolution that is used to model a water container.</p>
</div>
<div class="specification">
<p>At <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn></math>, the container is empty. Water is then added to the container at a constant rate of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>4</mn><mo> </mo><msup><mtext>m</mtext><mn>3</mn></msup><mo> </mo><msup><mtext>s</mtext><mrow><mo>-</mo><mn>1</mn></mrow></msup></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the curve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mfenced><mi>x</mi></mfenced></math>, clearly indicating the coordinates of the endpoints.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the inverse function of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> is given by <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>f</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><msqrt><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></msqrt></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State the domain and range of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>f</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the volume, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mo> </mo><msup><mtext>m</mtext><mn>3</mn></msup></math>, of water in the container when it is filled to a height of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math> metres is given by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mo>=</mo><mi>π</mi><mfenced><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac><msup><mi>h</mi><mn>3</mn></msup><mo>+</mo><mi>h</mi></mrow></mfenced></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, determine the maximum volume of the container.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the time it takes to fill the container to its maximum volume.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the rate of change of the height of the water when the container is filled to half its maximum volume.</p>
<div class="marks">[6]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><img src="data:image/png;base64,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"></p>
<p>correct shape (concave down) within the given domain <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mn>2</mn></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>1</mn><mo>,</mo><mn>0</mn></mrow></mfenced></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>2</mn><mo>,</mo><msqrt><mn>3</mn></msqrt></mrow></mfenced><mfenced><mrow><mo>=</mo><mfenced><mrow><mn>2</mn><mo>,</mo><mn>1</mn><mo>.</mo><mn>73</mn></mrow></mfenced></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> The coordinates of endpoints may be seen on the graph or marked on the axes.</p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>interchanging <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> (seen anywhere) <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><msqrt><msup><mi>y</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></msqrt></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>=</mo><msup><mi>y</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><msqrt><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></msqrt></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>f</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><msqrt><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></msqrt></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><msqrt><mn>3</mn></msqrt></math> OR domain <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="[" close="]"><mrow><mn>0</mn><mo>,</mo><msqrt><mn>3</mn></msqrt></mrow></mfenced><mfenced><mrow><mo>=</mo><mfenced open="[" close="]"><mrow><mn>0</mn><mo>,</mo><mn>1</mn><mo>.</mo><mn>73</mn></mrow></mfenced></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>≤</mo><mi>y</mi><mo>≤</mo><mn>2</mn></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>≤</mo><msup><mi>f</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mi>x</mi></mfenced><mo>≤</mo><mn>2</mn></math> OR range <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="[" close="]"><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to substitute <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><msqrt><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></msqrt></math> into the correct volume formula <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mo>=</mo><mi>π</mi><munderover><mo>∫</mo><mn>0</mn><mi>h</mi></munderover><msup><mfenced><msqrt><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></msqrt></mfenced><mn>2</mn></msup><mo>d</mo><mi>y</mi><mo> </mo><mfenced><mrow><mo>=</mo><mi>π</mi><munderover><mo>∫</mo><mn>0</mn><mi>h</mi></munderover><mfenced><mrow><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfenced><mo>d</mo><mi>y</mi></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi>π</mi><msubsup><mfenced open="[" close="]"><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac><msup><mi>y</mi><mn>3</mn></msup><mo>+</mo><mi>y</mi></mrow></mfenced><mn>0</mn><mi>h</mi></msubsup></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi>π</mi><mfenced><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac><msup><mi>h</mi><mn>3</mn></msup><mo>+</mo><mi>h</mi></mrow></mfenced></math> <em><strong>AG</strong></em></p>
<p><strong><br>Note:</strong> Award marks as appropriate for correct work using a different variable e.g. <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>π</mi><munderover><mo>∫</mo><mn>0</mn><mi>h</mi></munderover><msup><mfenced><msqrt><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></msqrt></mfenced><mn>2</mn></msup><mo>d</mo><mi>x</mi></math></p>
<p><em><strong><br>[3 marks]</strong></em></p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to substitute <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>=</mo><msqrt><mn>3</mn></msqrt><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mn>1</mn><mo>.</mo><mn>732</mn><mo>…</mo></mrow></mfenced></math> into <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mo>=</mo><mn>10</mn><mo>.</mo><mn>8828</mn><mo>…</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mo>=</mo><mn>10</mn><mo>.</mo><mn>9</mn><mo> </mo><mfenced><msup><mtext>m</mtext><mn>3</mn></msup></mfenced><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mn>2</mn><msqrt><mn>3</mn></msqrt><mi mathvariant="normal">π</mi></mrow></mfenced><mo> </mo><mfenced><msup><mtext>m</mtext><mn>3</mn></msup></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>time <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mn>10</mn><mo>.</mo><mn>8828</mn><mo>…</mo></mrow><mrow><mn>0</mn><mo>.</mo><mn>4</mn></mrow></mfrac><mfenced><mrow><mo>=</mo><mfrac><mrow><mn>2</mn><msqrt><mn>3</mn></msqrt><mi mathvariant="normal">π</mi></mrow><mrow><mn>0</mn><mo>.</mo><mn>4</mn></mrow></mfrac></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>27</mn><mo>.</mo><mn>207</mn><mo>…</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>27</mn><mo>.</mo><mn>2</mn><mfenced><mrow><mo>=</mo><mn>5</mn><msqrt><mn>3</mn></msqrt><mi mathvariant="normal">π</mi></mrow></mfenced><mfenced><mi>s</mi></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to find the height of the tank when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mo>=</mo><mn>5</mn><mo>.</mo><mn>4414</mn><mo>…</mo><mo> </mo><mfenced><mrow><mo>=</mo><msqrt><mn>3</mn></msqrt><mi mathvariant="normal">π</mi></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">π</mi><mfenced><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac><msup><mi>h</mi><mn>3</mn></msup><mo>+</mo><mi>h</mi></mrow></mfenced><mo>=</mo><mn>5</mn><mo>.</mo><mn>4414</mn><mo>…</mo><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><msqrt><mn>3</mn></msqrt><mi mathvariant="normal">π</mi></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>=</mo><mn>1</mn><mo>.</mo><mn>1818</mn><mo>…</mo></math> <em><strong>(A1)</strong></em></p>
<p>attempt to use the chain rule or differentiate <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mo>=</mo><mi mathvariant="normal">π</mi><mfenced><mrow><mfrac><mn>1</mn><mn>3</mn></mfrac><msup><mi>h</mi><mn>3</mn></msup><mo>+</mo><mi>h</mi></mrow></mfenced></math> with respect to <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>h</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mo>d</mo><mi>h</mi></mrow><mrow><mo>d</mo><mi>V</mi></mrow></mfrac><mo>×</mo><mfrac><mrow><mo>d</mo><mi>V</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><mrow><mi mathvariant="normal">π</mi><mfenced><mrow><msup><mi>h</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfenced></mrow></mfrac><mo>×</mo><mfrac><mrow><mo>d</mo><mi>V</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>V</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mi mathvariant="normal">π</mi><mfenced><mrow><msup><mi>h</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfenced><mfrac><mrow><mo>d</mo><mi>h</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></math> <em><strong>(A1)</strong></em></p>
<p>attempt to substitute <strong>their </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>V</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mn>0</mn><mo>.</mo><mn>4</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>h</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>0</mn><mo>.</mo><mn>4</mn></mrow><mrow><mi mathvariant="normal">π</mi><mfenced><mrow><mn>1</mn><mo>.</mo><mn>1818</mn><msup><mo>…</mo><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfenced></mrow></mfrac><mo>=</mo><mn>0</mn><mo>.</mo><mn>053124</mn><mo>…</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>0</mn><mo>.</mo><mn>0531</mn><mo> </mo><mfenced><msup><mtext>m s</mtext><mrow><mo>-</mo><mn>1</mn></mrow></msup></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>Part a) was generally well done, with the most common errors being to use an incorrect domain or not to give the coordinates of the endpoints. Some graphs appeared to be straight lines; some candidates drew sketches which were too small which made it more difficult for them to show the curvature.</p>
<p>Most candidates were able to show the steps to find an inverse function in part b), although occasionally a candidate did not explicitly swop the <em>x</em> and <em>y</em> variables before writing down the inverse function, which was given in the question. Many candidates struggled to identify the domain and range of the inverse, despite having a correct graph.</p>
<p>Part c) required a rotation around the <em>y</em>-axis, but a number of candidates attempted to rotate around the <em>x</em>-axis or failed to include limits. In the same vein, many substituted 2 into the formula instead of the square root of 3 when answering the second part. Many subsequently gained follow through marks on part d).</p>
<p>There were a number of good attempts at related rates in part e), with the majority differentiating <em>V</em> with respect to <em>t</em>, using implicit differentiation. However, many did not find the value of <em>h</em> which corresponded to halving the volume, and a number did not differentiate with respect to <em>t</em>, only with respect to <em>h</em>.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="question">
<p>The following diagram shows the curve <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msup><mi>x</mi><mn>2</mn></msup><mn>36</mn></mfrac><mo>+</mo><mfrac><msup><mfenced><mrow><mi>y</mi><mo>-</mo><mn>4</mn></mrow></mfenced><mn>2</mn></msup><mn>16</mn></mfrac><mo>=</mo><mn>1</mn></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>≤</mo><mi>y</mi><mo>≤</mo><mn>4</mn></math>.</p>
<p><img style="display:block;margin-left:auto;margin-right:auto;" src="data:image/png;base64,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"></p>
<p>The curve from point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Q</mtext></math> to point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math> is rotated <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>360</mn><mo>°</mo></math> about the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-axis to form the interior surface of a bowl. The rectangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>OPQR</mtext></math>, of height <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo> </mo><mtext>cm</mtext></math>, is rotated <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>360</mn><mo>°</mo></math> about the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-axis to form a solid base.</p>
<p>The bowl is assumed to have negligible thickness.</p>
<p>Given that the interior volume of the bowl is to be <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>285</mn><mo> </mo><msup><mtext>cm</mtext><mn>3</mn></msup></math>, determine the height of the base.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p>attempts to express <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mo>=</mo><mi mathvariant="normal">π</mi><munderover><mo>∫</mo><mi>h</mi><mn>4</mn></munderover><mn>36</mn><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><msup><mfenced><mrow><mi>y</mi><mo>-</mo><mn>4</mn></mrow></mfenced><mn>2</mn></msup><mn>16</mn></mfrac></mrow></mfenced><mo>d</mo><mi>y</mi></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Correct limits are required.</p>
<p> </p>
<p>Attempts to solve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">π</mi><munderover><mo>∫</mo><mi>h</mi><mn>4</mn></munderover><mn>36</mn><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><msup><mfenced><mrow><mi>y</mi><mo>-</mo><mn>4</mn></mrow></mfenced><mn>2</mn></msup><mn>16</mn></mfrac></mrow></mfenced><mo>d</mo><mi>y</mi><mo>=</mo><mn>285</mn></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math> <em><strong>(M1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1</strong> </em>for attempting to solve <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>36</mn><mi>π</mi><mfenced><mrow><mfrac><msup><mi>h</mi><mn>3</mn></msup><mn>48</mn></mfrac><mo>-</mo><mfrac><msup><mi>h</mi><mn>2</mn></msup><mn>4</mn></mfrac><mo>+</mo><mfrac><mn>8</mn><mn>3</mn></mfrac></mrow></mfenced><mo>=</mo><mn>285</mn></math> or equivalent for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math>.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>7926</mn><mo>…</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>793</mn></math> (cm) <em><strong>A2</strong></em></p>
<p> </p>
<p><em><strong>[5 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p>This question was a struggle for many candidates. To start with, many candidates had difficulty understanding the diagram. Some candidates tried to include the base in their equation. </p>
<p>Because of this confusion, the question was poorly attempted. Some only received one mark for rearranging the equation to make <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup></math> the subject but were unable to set the correct definite integral with correct terminals. Again, many candidates tried to solve by hand instead of using their GDC. The correct answer was not seen that often.</p>
<p>Those candidates who recognised that the volume was around the y -axis and used their GDC to solve, usually achieved full marks for this question.</p>
</div>
<br><hr><br><div class="specification">
<p>The function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> is defined by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = {\text{sec}}\,x + 2">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mtext>sec</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0 \leqslant x < \frac{\pi }{2}">
<mn>0</mn>
<mo>⩽<!-- ⩽ --></mo>
<mi>x</mi>
<mo><</mo>
<mfrac>
<mi>π<!-- π --></mi>
<mn>2</mn>
</mfrac>
</math></span>.</p>
</div>
<div class="question">
<p>Use integration by parts to find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {{{\left( {{\text{ln}}\,x} \right)}^2}} {\text{d}}x">
<mo>∫</mo>
<mrow>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</math></span>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><strong>METHOD 1</strong></p>
<p>write as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {1 \times {{\left( {{\text{ln}}\,x} \right)}^2}} {\text{d}}x">
<mo>∫</mo>
<mrow>
<mn>1</mn>
<mo>×</mo>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = x{\left( {{\text{ln}}\,x} \right)^2} - \int {x \times \frac{{2\left( {{\text{ln}}\,x} \right)}}{x}} {\text{d}}x\left( { = x{{\left( {{\text{ln}}\,x} \right)}^2} - \int {2\,{\text{ln}}\,x} } \right)">
<mo>=</mo>
<mi>x</mi>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mo>∫</mo>
<mrow>
<mi>x</mi>
<mo>×</mo>
<mfrac>
<mrow>
<mn>2</mn>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mi>x</mi>
</mfrac>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mi>x</mi>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mo>∫</mo>
<mrow>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>M1A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = x{\left( {{\text{ln}}\,x} \right)^2} - 2x\,{\text{ln}}\,x + \int 2 \,{\text{d}}x">
<mo>=</mo>
<mi>x</mi>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>2</mn>
<mi>x</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>+</mo>
<mo>∫</mo>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</math></span> <em><strong>(M1)</strong></em><em><strong>(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = x{\left( {{\text{ln}}\,x} \right)^2} - 2x\,{\text{ln}}\,x + 2x + c">
<mo>=</mo>
<mi>x</mi>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>2</mn>
<mi>x</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mi>c</mi>
</math></span> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u = {\text{ln}}\,x">
<mi>u</mi>
<mo>=</mo>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}u}}{{{\text{d}}x}} = \frac{1}{x}">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>u</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mi>x</mi>
</mfrac>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {{u^2}{{\text{e}}^u}} {\text{d}}u">
<mo>∫</mo>
<mrow>
<mrow>
<msup>
<mi>u</mi>
<mn>2</mn>
</msup>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mi>u</mi>
</msup>
</mrow>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>u</mi>
</math></span> <em><strong>A</strong><strong>1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {u^2}{{\text{e}}^u} - \int {2u{{\text{e}}^u}} {\text{d}}u">
<mo>=</mo>
<mrow>
<msup>
<mi>u</mi>
<mn>2</mn>
</msup>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mi>u</mi>
</msup>
</mrow>
<mo>−</mo>
<mo>∫</mo>
<mrow>
<mn>2</mn>
<mi>u</mi>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mi>u</mi>
</msup>
</mrow>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>u</mi>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {u^2}{{\text{e}}^u} - 2u{{\text{e}}^u} + \int {2{{\text{e}}^u}} {\text{d}}u">
<mo>=</mo>
<mrow>
<msup>
<mi>u</mi>
<mn>2</mn>
</msup>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mi>u</mi>
</msup>
</mrow>
<mo>−</mo>
<mn>2</mn>
<mi>u</mi>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mi>u</mi>
</msup>
</mrow>
<mo>+</mo>
<mo>∫</mo>
<mrow>
<mn>2</mn>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mi>u</mi>
</msup>
</mrow>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>u</mi>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {u^2}{{\text{e}}^u} - 2u{{\text{e}}^u} + 2{{\text{e}}^u} + c">
<mo>=</mo>
<mrow>
<msup>
<mi>u</mi>
<mn>2</mn>
</msup>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mi>u</mi>
</msup>
</mrow>
<mo>−</mo>
<mn>2</mn>
<mi>u</mi>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mi>u</mi>
</msup>
</mrow>
<mo>+</mo>
<mn>2</mn>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mi>u</mi>
</msup>
</mrow>
<mo>+</mo>
<mi>c</mi>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = x{\left( {{\text{ln}}\,x} \right)^2} - 2x\,{\text{ln}}\,x + 2x + c">
<mo>=</mo>
<mi>x</mi>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>2</mn>
<mi>x</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mi>c</mi>
</math></span> <em><strong>M1A1</strong></em></p>
<p> </p>
<p><strong>METHOD 3</strong></p>
<p>Setting up <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u = {\text{ln}}\,x">
<mi>u</mi>
<mo>=</mo>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}v}}{{{\text{d}}x}} = {\text{ln}}\,x">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>v</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ln}}\,x\left( {x\,{\text{ln}}\,x - x} \right) - \int {\left( {{\text{ln}}\,x - 1} \right)} {\text{d}}x">
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>−</mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mo>∫</mo>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</math></span> <em><strong>M1A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = x{\left( {{\text{ln}}\,x} \right)^2} - x\,{\text{ln}}\,x - \left( {x\,{\text{ln}}\,x - x} \right) + x + c">
<mo>=</mo>
<mi>x</mi>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mi>x</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>−</mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>−</mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mi>x</mi>
<mo>+</mo>
<mi>c</mi>
</math></span> <em><strong>M1A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = x{\left( {{\text{ln}}\,x} \right)^2} - 2x\,{\text{ln}}\,x + 2x + c">
<mo>=</mo>
<mi>x</mi>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>2</mn>
<mi>x</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mi>c</mi>
</math></span> <em><strong>A1</strong></em></p>
<p> </p>
<p> </p>
<p><em><strong>[6 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>Two airplanes, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math>, have position vectors with respect to an origin <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math> given respectively by</p>
<p style="padding-left: 180px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="bold-italic">r</mi><mtext mathvariant="bold-italic">A</mtext></msub><mo>=</mo><mfenced><mtable><mtr><mtd><mn>19</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mi>t</mi><mfenced><mtable><mtr><mtd><mo>-</mo><mn>6</mn></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr><mtr><mtd><mn>4</mn></mtd></mtr></mtable></mfenced></math></p>
<p style="padding-left: 180px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="bold-italic">r</mi><mi mathvariant="bold-italic">B</mi></msub><mo>=</mo><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>12</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mi>t</mi><mfenced><mtable><mtr><mtd><mn>4</mn></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr></mtable></mfenced></math></p>
<p>where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> represents the time in minutes and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>≤</mo><mi>t</mi><mo>≤</mo><mn>2</mn><mo>.</mo><mn>5</mn></math>.</p>
<p>Entries in each column vector give the displacement east of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math>, the displacement north of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math> and the distance above sea level, all measured in kilometres.</p>
</div>
<div class="specification">
<p>The two airplanes’ lines of flight cross at point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the three-figure bearing on which airplane <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> is travelling.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that airplane <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> travels at a greater speed than airplane <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the acute angle between the two airplanes’ lines of flight. Give your answer in degrees.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the coordinates of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the length of time between the first airplane arriving at <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math> and the second airplane arriving at <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mo>(</mo><mi>t</mi><mo>)</mo></math> represent the distance between airplane <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> and airplane <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>≤</mo><mi>t</mi><mo>≤</mo><mn>2</mn><mo>.</mo><mn>5</mn></math>.</p>
<p>Find the minimum value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mo>(</mo><mi>t</mi><mo>)</mo></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ϕ</mi></math> be the required angle (bearing)</p>
<p><strong><br>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ϕ</mi><mo>=</mo><mn>90</mn><mo>°</mo><mo>-</mo><mtext>arctan</mtext><mfrac><mn>1</mn><mn>2</mn></mfrac><mo> </mo><mfenced><mrow><mo>=</mo><mtext>arctan</mtext><mo> </mo><mn>2</mn></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>M1</strong> </em>for a labelled sketch.</p>
<p><br><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mo> </mo><mi>ϕ</mi><mo>=</mo><mfrac><mrow><mfenced><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr></mtable></mfenced><mo>·</mo><mfenced><mtable><mtr><mtd><mn>4</mn></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable></mfenced></mrow><mrow><msqrt><mn>1</mn></msqrt><mo>×</mo><msqrt><mn>20</mn></msqrt></mrow></mfrac><mo> </mo><mfenced><mrow><mo>=</mo><mn>0</mn><mo>.</mo><mn>4472</mn><mo>…</mo><mo>,</mo><mo>=</mo><mfrac><mn>1</mn><msqrt><mn>5</mn></msqrt></mfrac></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ϕ</mi><mo>=</mo><mtext>arccos</mtext><mfenced><mrow><mn>0</mn><mo>.</mo><mn>4472</mn><mo>…</mo></mrow></mfenced></math></p>
<p><br><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>063</mn><mo>°</mo></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Do not accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>063</mn><mo>.</mo><mn>6</mn><mo>°</mo></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>63</mn><mo>.</mo><mn>4</mn><mo>°</mo></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.</mo><msup><mn>10</mn><mi>c</mi></msup></math>.</p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>let <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><msub><mi mathvariant="bold-italic">b</mi><mi>A</mi></msub></mfenced></math> be the speed of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> and let <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><msub><mi mathvariant="bold-italic">b</mi><mi>B</mi></msub></mfenced></math> be the speed of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math></p>
<p>attempts to find the speed of one of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><msub><mi mathvariant="bold-italic">b</mi><mi>A</mi></msub></mfenced><mo>=</mo><msqrt><msup><mfenced><mrow><mo>-</mo><mn>6</mn></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mn>2</mn><mn>2</mn></msup><mo>+</mo><msup><mn>4</mn><mn>2</mn></msup></msqrt></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><msub><mi mathvariant="bold-italic">b</mi><mi>B</mi></msub></mfenced><mo>=</mo><msqrt><msup><mn>4</mn><mn>2</mn></msup><mo>+</mo><msup><mn>2</mn><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mo>-</mo><mn>2</mn></mrow></mfenced><mn>2</mn></msup></msqrt></math></p>
<p><br><strong>Note:</strong> Award <em><strong>M0</strong> </em>for <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><msub><mi mathvariant="bold-italic">b</mi><mi>A</mi></msub></mfenced><mo>=</mo><msqrt><msup><mn>19</mn><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mo>-</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mn>1</mn><mn>2</mn></msup></msqrt></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><msub><mi mathvariant="bold-italic">b</mi><mi>B</mi></msub></mfenced><mo>=</mo><msqrt><msup><mn>1</mn><mn>2</mn></msup><mo>+</mo><msup><mn>0</mn><mn>2</mn></msup><mo>+</mo><msup><mn>12</mn><mn>2</mn></msup></msqrt></math>.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><msub><mi mathvariant="bold-italic">b</mi><mi>A</mi></msub></mfenced><mo>=</mo><mn>7</mn><mo>.</mo><mn>48</mn><mo>…</mo><mo> </mo><mfenced><mrow><mo>=</mo><msqrt><mn>56</mn></msqrt></mrow></mfenced></math> (km min<sup>-1</sup>) and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><msub><mi mathvariant="bold-italic">b</mi><mi>B</mi></msub></mfenced><mo>=</mo><mn>4</mn><mo>.</mo><mn>89</mn><mo>…</mo><mo> </mo><mfenced><mrow><mo>=</mo><msqrt><mn>24</mn></msqrt></mrow></mfenced></math> (km min<sup>-1</sup>) <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><msub><mi mathvariant="bold-italic">b</mi><mi>A</mi></msub></mfenced><mo>></mo><mfenced open="|" close="|"><msub><mi mathvariant="bold-italic">b</mi><mi>B</mi></msub></mfenced></math> so <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> travels at a greater speed than <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>attempts to use <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>speed</mtext><mo>=</mo><mfrac><mtext>distance</mtext><mtext>time</mtext></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>speed</mtext><mi>A</mi></msub><mo>=</mo><mfrac><mfenced open="|" close="|"><mrow><msub><mi>r</mi><mi>A</mi></msub><mfenced><msub><mi>t</mi><mn>2</mn></msub></mfenced><mo>-</mo><msub><mi>r</mi><mi>A</mi></msub><mfenced><msub><mi>t</mi><mn>1</mn></msub></mfenced></mrow></mfenced><mrow><msub><mi>t</mi><mn>2</mn></msub><mo>-</mo><msub><mi>t</mi><mn>1</mn></msub></mrow></mfrac></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>speed</mtext><mi>B</mi></msub><mo>=</mo><mfrac><mfenced open="|" close="|"><mrow><msub><mi>r</mi><mi>B</mi></msub><mfenced><msub><mi>t</mi><mn>2</mn></msub></mfenced><mo>-</mo><msub><mi>r</mi><mi>B</mi></msub><mfenced><msub><mi>t</mi><mn>1</mn></msub></mfenced></mrow></mfenced><mrow><msub><mi>t</mi><mn>2</mn></msub><mo>-</mo><msub><mi>t</mi><mn>1</mn></msub></mrow></mfrac></math> <em><strong>(M1)</strong></em></p>
<p>for example:</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>speed</mtext><mi>A</mi></msub><mo>=</mo><mfrac><mfenced open="|" close="|"><mrow><msub><mi>r</mi><mi>A</mi></msub><mfenced><mn>1</mn></mfenced><mo>-</mo><msub><mi>r</mi><mi>A</mi></msub><mfenced><mn>0</mn></mfenced></mrow></mfenced><mn>1</mn></mfrac></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>speed</mtext><mi>B</mi></msub><mo>=</mo><mfrac><mfenced open="|" close="|"><mrow><msub><mi>r</mi><mi>B</mi></msub><mfenced><mn>1</mn></mfenced><mo>-</mo><msub><mi>r</mi><mi>B</mi></msub><mfenced><mn>0</mn></mfenced></mrow></mfenced><mn>1</mn></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>speed</mtext><mi>A</mi></msub><mo>=</mo><mfrac><msqrt><msup><mfenced><mrow><mo>-</mo><mn>6</mn></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mn>2</mn><mn>2</mn></msup><mo>+</mo><msup><mn>4</mn><mn>2</mn></msup></msqrt><mn>1</mn></mfrac></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>speed</mtext><mi>B</mi></msub><mo>=</mo><mfrac><msqrt><msup><mn>4</mn><mn>2</mn></msup><mo>+</mo><msup><mn>2</mn><mn>2</mn></msup><mo>+</mo><msup><mn>2</mn><mn>2</mn></msup></msqrt><mn>1</mn></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>speed</mtext><mi>A</mi></msub><mo>=</mo><mn>7</mn><mo>.</mo><mn>48</mn><mo>…</mo><mfenced><mrow><mn>2</mn><msqrt><mn>14</mn></msqrt></mrow></mfenced></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>speed</mtext><mi>B</mi></msub><mo>=</mo><mn>4</mn><mo>.</mo><mn>89</mn><mo>…</mo><mfenced><msqrt><mn>24</mn></msqrt></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>speed</mtext><mi>A</mi></msub><mo>></mo><msub><mtext>speed</mtext><mi>B</mi></msub></math> so <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> travels at a greater speed than <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempts to use the angle between two direction vectors formula <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mo> </mo><mi>θ</mi><mo>=</mo><mfrac><mrow><mfenced><mrow><mo>-</mo><mn>6</mn></mrow></mfenced><mfenced><mn>4</mn></mfenced><mo>+</mo><mfenced><mn>2</mn></mfenced><mfenced><mn>2</mn></mfenced><mo>+</mo><mfenced><mn>4</mn></mfenced><mfenced><mrow><mo>-</mo><mn>2</mn></mrow></mfenced></mrow><mrow><msqrt><msup><mfenced><mrow><mo>-</mo><mn>6</mn></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mn>2</mn><mn>2</mn></msup><mo>+</mo><msup><mn>4</mn><mn>2</mn></msup></msqrt><msqrt><msup><mn>4</mn><mn>2</mn></msup><mo>+</mo><msup><mn>2</mn><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mo>-</mo><mn>2</mn></mrow></mfenced><mn>2</mn></msup></msqrt></mrow></mfrac></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mo> </mo><mi>θ</mi><mo>=</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>7637</mn><mo>…</mo><mo> </mo><mfenced><mrow><mo>=</mo><mo>-</mo><mfrac><mn>7</mn><msqrt><mn>84</mn></msqrt></mfrac></mrow></mfenced></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi><mo>=</mo><mtext>arccos</mtext><mfenced><mrow><mo>-</mo><mn>0</mn><mo>.</mo><mn>7637</mn><mo>…</mo></mrow></mfenced><mo> </mo><mfenced><mrow><mo>=</mo><mn>2</mn><mo>.</mo><mn>4399</mn><mo>…</mo></mrow></mfenced></math></p>
<p>attempts to find the acute angle <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>180</mn><mo>°</mo><mo>-</mo><mi>θ</mi></math> using their value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>40</mn><mo>.</mo><mn>2</mn><mo>°</mo></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>for example, sets <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="bold-italic">r</mi><mi mathvariant="bold-italic">A</mi></msub><mfenced><msub><mi>t</mi><mn>1</mn></msub></mfenced><mo>=</mo><msub><mi mathvariant="bold-italic">r</mi><mi mathvariant="bold-italic">B</mi></msub><mfenced><msub><mi>t</mi><mn>2</mn></msub></mfenced></math> and forms at least two equations <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>19</mn><mo>-</mo><mn>6</mn><msub><mi>t</mi><mn>1</mn></msub><mo>=</mo><mn>1</mn><mo>+</mo><mn>4</mn><msub><mi>t</mi><mn>2</mn></msub></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>1</mn><mo>+</mo><mn>2</mn><msub><mi>t</mi><mn>1</mn></msub><mo>=</mo><mn>2</mn><msub><mi>t</mi><mn>2</mn></msub></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>+</mo><mn>4</mn><msub><mi>t</mi><mn>1</mn></msub><mo>=</mo><mn>12</mn><mo>-</mo><mn>2</mn><msub><mi>t</mi><mn>2</mn></msub></math></p>
<p><br><strong>Note:</strong> Award <em><strong>M0</strong> </em>for equations involving <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> only.</p>
<p><br><strong>EITHER</strong></p>
<p>attempts to solve the system of equations for one of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>t</mi><mn>1</mn></msub></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>t</mi><mn>2</mn></msub></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>t</mi><mn>1</mn></msub><mo>=</mo><mn>2</mn></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>t</mi><mn>2</mn></msub><mo>=</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p><br><strong>OR</strong></p>
<p>attempts to solve the system of equations for <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>t</mi><mn>1</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>t</mi><mn>2</mn></msub></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>t</mi><mn>1</mn></msub><mo>=</mo><mn>2</mn></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>t</mi><mn>2</mn></msub><mo>=</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p><br><strong>THEN</strong></p>
<p>substitutes their <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>t</mi><mn>1</mn></msub></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>t</mi><mn>2</mn></msub></math> value into the corresponding <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="bold-italic">r</mi><mi mathvariant="bold-italic">A</mi></msub></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="bold-italic">r</mi><mi mathvariant="bold-italic">B</mi></msub></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mrow><mn>7</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>9</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mtext>OP</mtext><mo>→</mo></mover><mo>=</mo><mfenced><mtable><mtr><mtd><mn>7</mn></mtd></mtr><mtr><mtd><mn>3</mn></mtd></mtr><mtr><mtd><mn>9</mn></mtd></mtr></mtable></mfenced></math>. Accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn></math> km east of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math> km north of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9</mn></math> km above sea level.</p>
<p> </p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempts to find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>t</mi><mn>1</mn></msub><mo>-</mo><msub><mi>t</mi><mn>2</mn></msub></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>t</mi><mn>1</mn></msub><mo>-</mo><msub><mi>t</mi><mn>2</mn></msub><mo>=</mo><mn>2</mn><mo>-</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>5</mn></math> minutes (<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>30</mn></math> seconds) <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">d.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p>attempts to find <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="bold-italic">r</mi><mi mathvariant="bold-italic">B</mi></msub><mo>-</mo><msub><mi mathvariant="bold-italic">r</mi><mi mathvariant="bold-italic">A</mi></msub></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="bold-italic">r</mi><mi mathvariant="bold-italic">B</mi></msub><mo>-</mo><msub><mi mathvariant="bold-italic">r</mi><mi mathvariant="bold-italic">A</mi></msub><mo>=</mo><mfenced><mtable><mtr><mtd><mo>-</mo><mn>18</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>11</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mi>t</mi><mfenced><mtable><mtr><mtd><mn>10</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>6</mn></mtd></mtr></mtable></mfenced></math></p>
<p>attempts to find their <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mo>(</mo><mi>t</mi><mo>)</mo></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><msqrt><msup><mfenced><mrow><mn>10</mn><mi>t</mi><mo>-</mo><mn>18</mn></mrow></mfenced><mn>2</mn></msup><mo>+</mo><mn>1</mn><mo>+</mo><msup><mfenced><mrow><mn>11</mn><mo>-</mo><mn>6</mn><mi>t</mi></mrow></mfenced><mn>2</mn></msup></msqrt></math> <em><strong>A1</strong></em></p>
<p><strong><br>OR</strong></p>
<p>attempts to find <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="bold-italic">r</mi><mi mathvariant="bold-italic">A</mi></msub><mo>-</mo><msub><mi mathvariant="bold-italic">r</mi><mi mathvariant="bold-italic">B</mi></msub></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="bold-italic">r</mi><mi mathvariant="bold-italic">A</mi></msub><mo>-</mo><msub><mi mathvariant="bold-italic">r</mi><mi mathvariant="bold-italic">B</mi></msub><mo>=</mo><mfenced><mtable><mtr><mtd><mn>18</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>11</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mi>t</mi><mfenced><mtable><mtr><mtd><mo>-</mo><mn>10</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>6</mn></mtd></mtr></mtable></mfenced></math></p>
<p>attempts to find their <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mo>(</mo><mi>t</mi><mo>)</mo></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><msqrt><msup><mfenced><mrow><mn>18</mn><mo>-</mo><mn>10</mn><mi>t</mi></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mo>-</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mo>-</mo><mn>11</mn><mo>+</mo><mn>6</mn><mi>t</mi></mrow></mfenced><mn>2</mn></msup></msqrt></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>M0M0A0</strong></em> for expressions using two different time parameters.</p>
<p><br><strong>THEN</strong></p>
<p>either attempts to find the local minimum point of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mo>(</mo><mi>t</mi><mo>)</mo></math> or attempts to find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mo>'</mo><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mn>0</mn></math> (or equivalent) <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>1</mn><mo>.</mo><mn>8088</mn><mo>…</mo><mo> </mo><mfenced><mrow><mo>=</mo><mfrac><mn>123</mn><mn>68</mn></mfrac></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mn>1</mn><mo>.</mo><mn>01459</mn><mo>…</mo></math></p>
<p>minimum value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mo>(</mo><mi>t</mi><mo>)</mo></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.</mo><mn>01</mn><mo> </mo><mfenced><mrow><mo>=</mo><mfrac><msqrt><mn>1190</mn></msqrt><mn>34</mn></mfrac></mrow></mfenced></math> (km) <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>M0</strong> </em>for attempts at the shortest distance between two lines.</p>
<p> </p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>General comment about this question: many candidates were not exposed to this setting of vectors question and were rather lost.</p>
<p>Part (a) Probably the least answered question on the whole paper. Many candidates left it blank, others tried using 3D vectors. Out of those who calculated the angle correctly, only a small percentage were able to provide the correct true bearing as a 3-digit figure.</p>
<p>Part (b) Well done by many candidates who used the direction vectors to calculate and compare the speeds. A number of candidates tried to use the average rate of change but mostly unsuccessfully.</p>
<p>Part (c) Most candidates used the correct vectors and the formula to obtain the obtuse angle. Then only some read the question properly to give the acute angle in degrees, as requested.</p>
<p>Part (d) Well done by many candidates who used two different parameters. They were able to solve and obtain two values for time, the difference in minutes and the correct point of intersection. A number of candidates only had one parameter, thus scoring no marks in part (d) (i). The frequent error in part (d)(ii) was providing incorrect units.</p>
<p>Part (e) Many correct answers were seen with an efficient way of setting the question and using their GDC to obtain the answer, graphically or numerically. Some gave time only instead of actually giving the minimal distance. A number of candidates tried to find the distance between two skew lines ignoring the fact that the lines intersect.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p>Xavier, the parachutist, jumps out of a plane at a height of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="h">
<mi>h</mi>
</math></span> metres above the ground. After free falling for 10 seconds his parachute opens. His velocity, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v\,{\text{m}}{{\text{s}}^{ - 1}}">
<mi>v</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>m</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>s</mtext>
</mrow>
<mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> seconds after jumping from the plane, can be modelled by the function</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v(t) = \left\{ {\begin{array}{*{20}{l}} {9.8t{\text{,}}}&{0 \leqslant t \leqslant 10} \\ {\frac{{98}}{{\sqrt {1 + {{(t - 10)}^2}} }},}&{t > 10} \end{array}} \right.">
<mi>v</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow>
<mo>{</mo>
<mrow>
<mtable columnalign="left" rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>9.8</mn>
<mi>t</mi>
<mrow>
<mtext>,</mtext>
</mrow>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>0</mn>
<mo>⩽<!-- ⩽ --></mo>
<mi>t</mi>
<mo>⩽<!-- ⩽ --></mo>
<mn>10</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mfrac>
<mrow>
<mn>98</mn>
</mrow>
<mrow>
<msqrt>
<mn>1</mn>
<mo>+</mo>
<mrow>
<msup>
<mrow>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo>−<!-- − --></mo>
<mn>10</mn>
<mo stretchy="false">)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
</mrow>
</mfrac>
<mo>,</mo>
</mrow>
</mtd>
<mtd>
<mrow>
<mi>t</mi>
<mo>></mo>
<mn>10</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo fence="true" stretchy="true" symmetric="true"></mo>
</mrow>
</math></span></p>
</div>
<div class="specification">
<p>His velocity when he reaches the ground is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2.8{\text{ m}}{{\text{s}}^{ - 1}}">
<mn>2.8</mn>
<mrow>
<mtext> m</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>s</mtext>
</mrow>
<mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find his velocity when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t = 15">
<mi>t</mi>
<mo>=</mo>
<mn>15</mn>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate the vertical distance Xavier travelled in the first 10 seconds.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="h">
<mi>h</mi>
</math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v(15) = \frac{{98}}{{\sqrt {1 + {{(15 - 10)}^2}} }}">
<mi>v</mi>
<mo stretchy="false">(</mo>
<mn>15</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mrow>
<mn>98</mn>
</mrow>
<mrow>
<msqrt>
<mn>1</mn>
<mo>+</mo>
<mrow>
<msup>
<mrow>
<mo stretchy="false">(</mo>
<mn>15</mn>
<mo>−</mo>
<mn>10</mn>
<mo stretchy="false">)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
</mrow>
</mfrac>
</math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v(15) = 19.2{\text{ }}({\text{m}}{{\text{s}}^{ - 1}})">
<mi>v</mi>
<mo stretchy="false">(</mo>
<mn>15</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>19.2</mn>
<mrow>
<mtext> </mtext>
</mrow>
<mo stretchy="false">(</mo>
<mrow>
<mtext>m</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>s</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mo stretchy="false">)</mo>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int\limits_0^{10} {9.8t\,{\text{d}}t} ">
<munderover>
<mo>∫</mo>
<mn>0</mn>
<mrow>
<mn>10</mn>
</mrow>
</munderover>
<mrow>
<mn>9.8</mn>
<mi>t</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 490{\text{ }}({\text{m}})">
<mo>=</mo>
<mn>490</mn>
<mrow>
<mtext> </mtext>
</mrow>
<mo stretchy="false">(</mo>
<mrow>
<mtext>m</mtext>
</mrow>
<mo stretchy="false">)</mo>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{98}}{{\sqrt {1 + {{(t - 10)}^2}} }} = 2.8">
<mfrac>
<mrow>
<mn>98</mn>
</mrow>
<mrow>
<msqrt>
<mn>1</mn>
<mo>+</mo>
<mrow>
<msup>
<mrow>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo>−</mo>
<mn>10</mn>
<mo stretchy="false">)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
</mrow>
</mfrac>
<mo>=</mo>
<mn>2.8</mn>
</math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t = 44.985 \ldots {\text{ }}({\text{s}})">
<mi>t</mi>
<mo>=</mo>
<mn>44.985</mn>
<mo>…</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mo stretchy="false">(</mo>
<mrow>
<mtext>s</mtext>
</mrow>
<mo stretchy="false">)</mo>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="h = 490 + \int\limits_{10}^{44.9...} {\frac{{98}}{{\sqrt {1 + {{(t - 10)}^2}} }}{\text{d}}t} ">
<mi>h</mi>
<mo>=</mo>
<mn>490</mn>
<mo>+</mo>
<munderover>
<mo>∫</mo>
<mrow>
<mn>10</mn>
</mrow>
<mrow>
<mn>44.9...</mn>
</mrow>
</munderover>
<mrow>
<mfrac>
<mrow>
<mn>98</mn>
</mrow>
<mrow>
<msqrt>
<mn>1</mn>
<mo>+</mo>
<mrow>
<msup>
<mrow>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo>−</mo>
<mn>10</mn>
<mo stretchy="false">)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
</mrow>
</mfrac>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</math></span> <strong><em>(M1)(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="h = 906{\text{ (m}})">
<mi>h</mi>
<mo>=</mo>
<mn>906</mn>
<mrow>
<mtext> (m</mtext>
</mrow>
<mo stretchy="false">)</mo>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[5 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mfrac><mrow><mtext>arctan</mtext><mfenced><mrow><mi>cos</mi><mo> </mo><mi>x</mi></mrow></mfenced><mo>-</mo><mi>k</mi></mrow><msup><mi>x</mi><mn>2</mn></msup></mfrac></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that a finite limit only exists for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Using l’Hôpital’s rule, show algebraically that the value of the limit is <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></math>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>(as <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><msup><mi>x</mi><mn>2</mn></msup><mo>=</mo><mn>0</mn></math>, the indeterminate form <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>0</mn><mn>0</mn></mfrac></math> is required for the limit to exist)</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mfenced><mrow><mtext>arctan</mtext><mfenced><mrow><mi>cos</mi><mo> </mo><mi>x</mi></mrow></mfenced><mo>-</mo><mi>k</mi></mrow></mfenced><mo>=</mo><mn>0</mn></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>arctan</mtext><mo> </mo><mn>1</mn><mo>-</mo><mi>k</mi><mo>=</mo><mn>0</mn><mo> </mo><mo> </mo><mfenced><mrow><mi>k</mi><mo>=</mo><mtext>arctan</mtext><mo> </mo><mn>1</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p>so <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac></math> <em><strong>AG</strong></em></p>
<p><strong><br>Note:</strong> Award <em><strong>M1A0</strong></em> for using <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac></math> to show the limit is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>0</mn><mn>0</mn></mfrac></math>.</p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mfrac><mrow><mtext>arctan</mtext><mfenced><mrow><mi>cos</mi><mo> </mo><mi>x</mi></mrow></mfenced><mo>-</mo><mstyle displaystyle="true"><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac></mstyle></mrow><msup><mi>x</mi><mn>2</mn></msup></mfrac><mfenced><mrow><mo>=</mo><mfrac><mn>0</mn><mn>0</mn></mfrac></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mfrac><mstyle displaystyle="true"><mfrac><mrow><mo>-</mo><mi>sin</mi><mo> </mo><mi>x</mi></mrow><mrow><mn>1</mn><mo>+</mo><msup><mi>cos</mi><mn>2</mn></msup><mo> </mo><mi>x</mi></mrow></mfrac></mstyle><mrow><mn>2</mn><mi>x</mi></mrow></mfrac></math> <em><strong>A1A1</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>A1</strong> </em>for a correct numerator and <em><strong>A1</strong> </em>for a correct denominator.</p>
<p><br>recognises to apply l’Hôpital’s rule again <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mfrac><mstyle displaystyle="true"><mfrac><mrow><mo>-</mo><mi>sin</mi><mo> </mo><mi>x</mi></mrow><mrow><mn>1</mn><mo>+</mo><msup><mi>cos</mi><mn>2</mn></msup><mo> </mo><mi>x</mi></mrow></mfrac></mstyle><mrow><mn>2</mn><mi>x</mi></mrow></mfrac><mo> </mo><mfenced><mrow><mo>=</mo><mfrac><mn>0</mn><mn>0</mn></mfrac></mrow></mfenced></math></p>
<p><strong><br>Note:</strong> Award <em><strong>M0</strong></em> if their limit is not the indeterminate form <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>0</mn><mn>0</mn></mfrac></math>.</p>
<p><strong><br>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mfrac><mstyle displaystyle="true"><mfrac><mrow><mo>-</mo><mi>cos</mi><mo> </mo><mi>x</mi><mfenced><mrow><mn>1</mn><mo>+</mo><msup><mi>cos</mi><mn>2</mn></msup><mo> </mo><mi>x</mi></mrow></mfenced><mo>-</mo><mn>2</mn><mo> </mo><msup><mi>sin</mi><mn>2</mn></msup><mo> </mo><mi>x</mi><mo> </mo><mi>cos</mi><mo> </mo><mi>x</mi></mrow><msup><mfenced><mrow><mn>1</mn><mo>+</mo><msup><mi>cos</mi><mn>2</mn></msup><mo> </mo><mi>x</mi></mrow></mfenced><mn>2</mn></msup></mfrac></mstyle><mn>2</mn></mfrac><mo> </mo></math> <em><strong>A1A1</strong></em></p>
<p><strong><br>Note:</strong> Award <em><strong>A1</strong> </em>for a correct first term in the numerator and <em><strong>A1</strong> </em>for a correct second term in the numerator.</p>
<p><br><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mfrac><mrow><mo>-</mo><mi>cos</mi><mo> </mo><mi>x</mi></mrow><mrow><mn>2</mn><mfenced><mrow><mn>1</mn><mo>+</mo><msup><mi>cos</mi><mn>2</mn></msup><mo> </mo><mi>x</mi></mrow></mfenced><mo>-</mo><mn>4</mn><mi>x</mi><mo> </mo><mi>sin</mi><mo> </mo><mi>x</mi><mo> </mo><mi>cos</mi><mo> </mo><mi>x</mi></mrow></mfrac><mo> </mo></math> <em><strong>A1A1</strong></em></p>
<p><strong><br>Note:</strong> Award <em><strong>A1</strong> </em>for a correct numerator and <em><strong>A1</strong> </em>for a correct denominator.</p>
<p><strong><br>THEN</strong></p>
<p>substitutes <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn></math> into the correct expression to evaluate the limit <em><strong>A1</strong></em></p>
<p><strong><br>Note:</strong> The final <em><strong>A1</strong> </em>is dependent on all previous marks.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>Part (a) Many candidates recognised the indeterminate form and provided a nice algebraic proof. Some verified by substituting the given value. Therefore, there is a need to teach the candidates the difference between proof and verification. Only a few candidates were able to give a complete 'show that' proof.</p>
<p>Part (b) Many candidates realised that they needed to apply the L'Hôpital's rule twice. There were many mistakes in differentiation using the chain rule. Not all candidates clearly showed the final substitution.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>A particle <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> moves in a straight line such that after time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> seconds, its velocity, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi></math> in <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>m s</mtext><mrow><mo>-</mo><mn>1</mn></mrow></msup></math>, is given by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mo>=</mo><msup><mtext>e</mtext><mrow><mo>−</mo><mn>3</mn><mi>t</mi></mrow></msup><mo> </mo><mi>sin</mi><mo> </mo><mn>6</mn><mo> </mo><mi>t</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo><</mo><mi>t</mi><mo><</mo><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></math>.</p>
</div>
<div class="specification">
<p>At time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> has displacement <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mo>(</mo><mi>t</mi><mo>)</mo></math>; at time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mo>(</mo><mn>0</mn><mo>)</mo><mo>=</mo><mn>0</mn></math>.</p>
</div>
<div class="specification">
<p>At successive times when the acceleration of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> is<math xmlns="http://www.w3.org/1998/Math/MathML"><mo> </mo><mn>0</mn><mo> </mo><msup><mtext>m s</mtext><mrow><mo>−</mo><mn>2</mn></mrow></msup><mo> </mo></math>, the velocities of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> form a geometric sequence. The acceleration of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> is zero at times <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>t</mi><mn>1</mn></msub><mo>,</mo><mo> </mo><msub><mi>t</mi><mn>2</mn></msub><mo>,</mo><mo> </mo><msub><mi>t</mi><mn>3</mn></msub></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>t</mi><mn>1</mn></msub><mo><</mo><msub><mi>t</mi><mn>2</mn></msub><mo><</mo><msub><mi>t</mi><mn>3</mn></msub></math> and the respective velocities are <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>v</mi><mn>1</mn></msub><mo>,</mo><mo> </mo><msub><mi>v</mi><mn>2</mn></msub><mo>,</mo><mo> </mo><msub><mi>v</mi><mn>3</mn></msub></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the times when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> comes to instantaneous rest.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>.</p>
<div class="marks">[7]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the maximum displacement of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math>, in metres, from its initial position.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the total distance travelled by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> in the first <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.</mo><mn>5</mn></math> seconds of its motion.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that, at these times, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tan</mi><mo> </mo><mn>6</mn><mi>t</mi><mo>=</mo><mn>2</mn></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">e.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msub><mi>v</mi><mn>2</mn></msub><msub><mi>v</mi><mn>1</mn></msub></mfrac><mo>=</mo><mfrac><msub><mi>v</mi><mn>3</mn></msub><msub><mi>v</mi><mn>2</mn></msub></mfrac><mo>=</mo><mo>-</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></mrow></msup></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">e.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac><mfenced><mrow><mo>=</mo><mn>0</mn><mo>.</mo><mn>524</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi mathvariant="normal">π</mi><mn>3</mn></mfrac><mfenced><mrow><mo>=</mo><mn>1</mn><mo>.</mo><mn>05</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><em><strong><br>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to use integration by parts <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mo>=</mo><mo>∫</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mi>t</mi></mrow></msup><mtext> </mtext><mi>sin</mi><mo> </mo><mn>6</mn><mi>t</mi><mo> </mo><mtext>d</mtext><mi>t</mi></math></p>
<p><strong><br>EITHER</strong></p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><mfrac><mrow><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mi>t</mi></mrow></msup><mtext> </mtext><mi>sin</mi><mo> </mo><mn>6</mn><mi>t</mi></mrow><mn>3</mn></mfrac><mo>-</mo><mo>∫</mo><mo>-</mo><mn>2</mn><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mi>t</mi></mrow></msup><mtext> cos</mtext><mo> </mo><mn>6</mn><mi>t</mi><mo> </mo><mtext>d</mtext><mi>t</mi></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><mfrac><mrow><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mi>t</mi></mrow></msup><mtext> </mtext><mi>sin</mi><mo> </mo><mn>6</mn><mi>t</mi></mrow><mn>3</mn></mfrac><mo>-</mo><mfenced><mrow><mfrac><mrow><mn>2</mn><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mi>t</mi></mrow></msup><mtext> cos</mtext><mo> </mo><mn>6</mn><mi>t</mi></mrow><mn>3</mn></mfrac><mo>-</mo><mo>∫</mo><mo>-</mo><mn>4</mn><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mi>t</mi></mrow></msup><mtext> sin</mtext><mo> </mo><mn>6</mn><mi>t</mi><mo> </mo><mtext>d</mtext><mi>t</mi></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><mfrac><mrow><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mi>t</mi></mrow></msup><mtext> </mtext><mi>sin</mi><mo> </mo><mn>6</mn><mi>t</mi></mrow><mn>3</mn></mfrac><mo>-</mo><mfenced><mrow><mfrac><mrow><mn>2</mn><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mi>t</mi></mrow></msup><mtext> cos</mtext><mo> </mo><mn>6</mn><mi>t</mi></mrow><mn>3</mn></mfrac><mo>+</mo><mn>4</mn><mi>s</mi></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mi>s</mi><mo>=</mo><mfrac><mrow><mo>-3</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mi>t</mi></mrow></msup><mtext> </mtext><mi>sin</mi><mo> </mo><mn>6</mn><mi>t</mi><mo>-</mo><mn>6</mn><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mi>t</mi></mrow></msup><mtext> cos</mtext><mo> </mo><mn>6</mn><mi>t</mi></mrow><mn>9</mn></mfrac></math> <em><strong>M1</strong></em></p>
<p><br><strong>OR</strong></p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><mfrac><mrow><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mi>t</mi></mrow></msup><mtext> cos</mtext><mo> </mo><mn>6</mn><mi>t</mi></mrow><mn>6</mn></mfrac><mo>-</mo><mo>∫</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mi>t</mi></mrow></msup><mtext> cos</mtext><mo> </mo><mn>6</mn><mi>t</mi><mo> </mo><mtext>d</mtext><mi>t</mi></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><mfrac><mrow><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mi>t</mi></mrow></msup><mtext> cos</mtext><mo> </mo><mn>6</mn><mi>t</mi></mrow><mn>6</mn></mfrac><mo>-</mo><mfenced><mrow><mfrac><mrow><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mi>t</mi></mrow></msup><mtext> sin</mtext><mo> </mo><mn>6</mn><mi>t</mi></mrow><mn>12</mn></mfrac><mo>+</mo><mo>∫</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mi>t</mi></mrow></msup><mtext> sin</mtext><mo> </mo><mn>6</mn><mi>t</mi><mo> </mo><mtext>d</mtext><mi>t</mi></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><mfrac><mrow><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mi>t</mi></mrow></msup><mtext> cos</mtext><mo> </mo><mn>6</mn><mi>t</mi></mrow><mn>6</mn></mfrac><mo>-</mo><mfenced><mrow><mfrac><mrow><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mi>t</mi></mrow></msup><mtext> sin</mtext><mo> </mo><mn>6</mn><mi>t</mi></mrow><mn>12</mn></mfrac><mo>+</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mi>s</mi></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>5</mn><mn>4</mn></mfrac><mi>s</mi><mo>=</mo><mfrac><mrow><mo>-</mo><mn>2</mn><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mi>t</mi></mrow></msup><mtext> cos</mtext><mo> </mo><mn>6</mn><mi>t</mi><mo>-</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mi>t</mi></mrow></msup><mtext> sin</mtext><mo> </mo><mn>6</mn><mi>t</mi></mrow><mn>12</mn></mfrac></math> <em><strong>M1</strong></em></p>
<p><br><strong>THEN</strong></p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mo>=</mo><mo>-</mo><mfrac><mrow><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mi>t</mi></mrow></msup><mtext> </mtext><mfenced><mrow><mtext> sin</mtext><mo> </mo><mn>6</mn><mi>t</mi><mo>+</mo><mn>2</mn><mo> </mo><mtext>cos</mtext><mo> </mo><mn>6</mn><mi>t</mi></mrow></mfenced></mrow><mn>15</mn></mfrac><mfenced><mrow><mo>+</mo><mi>c</mi></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p>at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn><mo>,</mo><mo> </mo><mi>s</mi><mo>=</mo><mn>0</mn><mo>⇒</mo><mn>0</mn><mo>=</mo><mo>-</mo><mfrac><mn>2</mn><mn>15</mn></mfrac><mo>+</mo><mi>c</mi></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>=</mo><mfrac><mn>2</mn><mn>15</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mo>=</mo><mfrac><mn>2</mn><mn>15</mn></mfrac><mo>-</mo><mfrac><mrow><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mi>t</mi></mrow></msup><mtext> </mtext><mfenced><mrow><mtext> sin</mtext><mo> </mo><mn>6</mn><mi>t</mi><mo>+</mo><mn>2</mn><mo> </mo><mtext>cos</mtext><mo> </mo><mn>6</mn><mi>t</mi></mrow></mfenced></mrow><mn>15</mn></mfrac></math></p>
<p><em><strong><br>[7 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p>substituting <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></math> into their equation for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>s</mi><mo>=</mo><mfrac><mn>2</mn><mn>15</mn></mfrac><mo>-</mo><mfrac><mrow><msup><mtext>e</mtext><mrow><mo>-</mo><mstyle displaystyle="true"><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></mstyle></mrow></msup><mtext> </mtext><mfenced><mrow><mtext> sin</mtext><mo> </mo><mi mathvariant="normal">π</mi><mo>+</mo><mn>2</mn><mo> </mo><mtext>cos</mtext><mo> </mo><mi mathvariant="normal">π</mi></mrow></mfenced></mrow><mn>15</mn></mfrac></mrow></mfenced></math></p>
<p><br><strong>OR</strong><br><br></p>
<p>using GDC to find maximum value <em><strong>(M1)</strong></em><br><br></p>
<p><strong>OR</strong></p>
<p>evaluating <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mn>0</mn><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></msubsup><mi>v</mi><mtext>d</mtext><mi>t</mi></math> <em><strong>(M1)</strong></em></p>
<p><br><strong>THEN</strong></p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>0</mn><mo>.</mo><mn>161</mn><mfenced><mrow><mo>=</mo><mfrac><mn>2</mn><mn>15</mn></mfrac><mfenced><mrow><mn>1</mn><mo>+</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mstyle displaystyle="false"><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></mstyle></mrow></msup></mrow></mfenced></mrow></mfenced></math> <em><strong>A1</strong></em> </p>
<p><em><strong><br>[2 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1 </strong></p>
<p><strong><br>EITHER</strong></p>
<p>distance required <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><munderover><mo>∫</mo><mn>0</mn><mrow><mn>1</mn><mo>.</mo><mn>5</mn></mrow></munderover><mfenced open="|" close="|"><mrow><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mi>t</mi></mrow></msup><mo> </mo><mi>sin</mi><mo> </mo><mn>6</mn><mi>t</mi></mrow></mfenced><mo> </mo><mtext>d</mtext><mi>t</mi></math> <em><strong>(M1)</strong></em></p>
<p><br><strong>OR</strong></p>
<p>distance required <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><munderover><mo>∫</mo><mn>0</mn><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></munderover><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mi>t</mi></mrow></msup><mo> </mo><mi>sin</mi><mo> </mo><mn>6</mn><mi>t</mi><mo> </mo><mtext>d</mtext><mi>t</mi><mo>+</mo><mfenced open="|" close="|"><mrow><munderover><mo>∫</mo><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac><mfrac><mi mathvariant="normal">π</mi><mn>3</mn></mfrac></munderover><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mi>t</mi></mrow></msup><mo> </mo><mi>sin</mi><mo> </mo><mn>6</mn><mi>t</mi><mo> </mo><mtext>d</mtext><mi>t</mi></mrow></mfenced><mo>+</mo><munderover><mo>∫</mo><mfrac><mi mathvariant="normal">π</mi><mn>3</mn></mfrac><mrow><mn>1</mn><mo>.</mo><mn>5</mn></mrow></munderover><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mi>t</mi></mrow></msup><mo> </mo><mi>sin</mi><mo> </mo><mn>6</mn><mi>t</mi><mo> </mo><mtext>d</mtext><mi>t</mi></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>=</mo><mn>0</mn><mo>.</mo><mn>16105</mn><mo>…</mo><mo>+</mo><mn>0</mn><mo>.</mo><mn>033479</mn><mo>…</mo><mo>+</mo><mn>0</mn><mo>.</mo><mn>006806</mn><mo>…</mo></mrow></mfenced></math></p>
<p><br><strong>THEN</strong></p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>0</mn><mo>.</mo><mn>201</mn><mo> </mo><mfenced><mtext>m</mtext></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><br>using successive minimum and maximum values on the displacement graph <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>16105</mn><mo>…</mo><mo>+</mo><mfenced><mrow><mn>0</mn><mo>.</mo><mn>16105</mn><mo>…</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>12757</mn><mo>…</mo></mrow></mfenced><mo>+</mo><mfenced><mrow><mn>0</mn><mo>.</mo><mn>13453</mn><mo>…</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>12757</mn><mo>…</mo></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>0</mn><mo>.</mo><mn>201</mn><mo> </mo><mfenced><mtext>m</mtext></mfenced></math> <em><strong>A1</strong></em></p>
<p><em><strong><br>[2 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>valid attempt to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mtext>d</mtext><mi>v</mi></mrow><mrow><mtext>d</mtext><mi>t</mi></mrow></mfrac></math> using product rule and set <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mtext>d</mtext><mi>v</mi></mrow><mrow><mtext>d</mtext><mi>t</mi></mrow></mfrac><mo>=</mo><mn>0</mn></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mtext>d</mtext><mi>v</mi></mrow><mrow><mtext>d</mtext><mi>t</mi></mrow></mfrac><mo>=</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mi>t</mi></mrow></msup><mn>6</mn><mo> </mo><mi>cos</mi><mo> </mo><mn>6</mn><mi>t</mi><mo>-</mo><mn>3</mn><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mi>t</mi></mrow></msup><mo> </mo><mi>sin</mi><mo> </mo><mn>6</mn><mi>t</mi></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mtext>d</mtext><mi>v</mi></mrow><mrow><mtext>d</mtext><mi>t</mi></mrow></mfrac><mo>=</mo><mn>0</mn><mo>⇒</mo><mi>tan</mi><mo> </mo><mn>6</mn><mi>t</mi><mo>=</mo><mn>2</mn></math> <em><strong>AG</strong></em></p>
<p><em><strong><br>[2 marks]</strong></em></p>
<div class="question_part_label">e.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to evaluate <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>t</mi><mn>1</mn></msub><mo>,</mo><mo> </mo><msub><mi>t</mi><mn>2</mn></msub><mo>,</mo><mo> </mo><msub><mi>t</mi><mn>3</mn></msub></math> in exact form <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><msub><mi>t</mi><mn>1</mn></msub><mo>=</mo><mtext>arctan</mtext><mo> </mo><mn>2</mn><mfenced><mrow><mo>⇒</mo><msub><mi>t</mi><mn>1</mn></msub><mo>=</mo><mfrac><mn>1</mn><mn>6</mn></mfrac><mo> </mo><mtext>arctan</mtext><mo> </mo><mn>2</mn></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><msub><mi>t</mi><mn>2</mn></msub><mo>=</mo><mi mathvariant="normal">π</mi><mo>+</mo><mtext>arctan</mtext><mo> </mo><mn>2</mn><mfenced><mrow><mo>⇒</mo><msub><mi>t</mi><mn>2</mn></msub><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac><mo>+</mo><mfrac><mn>1</mn><mn>6</mn></mfrac><mo> </mo><mtext>arctan</mtext><mo> </mo><mn>2</mn></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><msub><mi>t</mi><mn>3</mn></msub><mo>=</mo><mn>2</mn><mi mathvariant="normal">π</mi><mo>+</mo><mtext>arctan</mtext><mo> </mo><mn>2</mn><mfenced><mrow><mo>⇒</mo><msub><mi>t</mi><mn>3</mn></msub><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>3</mn></mfrac><mo>+</mo><mfrac><mn>1</mn><mn>6</mn></mfrac><mo> </mo><mtext>arctan</mtext><mo> </mo><mn>2</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> The <em><strong>A1</strong></em> is for any two consecutive correct, or showing that <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><msub><mi>t</mi><mn>2</mn></msub><mo>=</mo><mi mathvariant="normal">π</mi><mo>+</mo><mn>6</mn><msub><mi>t</mi><mn>1</mn></msub></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><msub><mi>t</mi><mn>3</mn></msub><mo>=</mo><mi mathvariant="normal">π</mi><mo>+</mo><mn>6</mn><msub><mi>t</mi><mn>2</mn></msub></math>.</p>
<p><br>showing that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mo> </mo><mn>6</mn><msub><mi>t</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><mo>-</mo><mi>sin</mi><mo> </mo><mn>6</mn><msub><mi>t</mi><mi>n</mi></msub></math></p>
<p>eg <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tan</mi><mo> </mo><mn>6</mn><mi>t</mi><mo>=</mo><mn>2</mn><mo>⇒</mo><mi>sin</mi><mo> </mo><mn>6</mn><mi>t</mi><mo>=</mo><mo>±</mo><mfrac><mn>2</mn><msqrt><mn>5</mn></msqrt></mfrac></math> <em><strong>M1A1</strong></em></p>
<p>showing that <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><msub><mi>t</mi><mrow><mi>n</mi><mo>+1</mo></mrow></msub></mrow></msup><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><msub><mi>t</mi><mi>n</mi></msub></mrow></msup></mfrac><mo>=</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></mrow></msup></math> <em><strong>M1</strong></em></p>
<p>eg <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mfenced><mrow><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac><mo>+</mo><mi>k</mi></mrow></mfenced></mrow></msup><mo>÷</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mi>k</mi></mrow></msup><mo>=</mo><msup><mtext>e</mtext><mrow><mi>-</mi><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></mrow></msup></math></p>
<p><br><strong>Note:</strong> Award the <em><strong>A1</strong></em> for any two consecutive terms.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msub><mi>v</mi><mn>3</mn></msub><msub><mi>v</mi><mn>2</mn></msub></mfrac><mo>=</mo><mfrac><msub><mi>v</mi><mn>2</mn></msub><msub><mi>v</mi><mn>1</mn></msub></mfrac><mo>=</mo><mo>-</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></mrow></msup></math> <em><strong>AG</strong></em></p>
<p><em><strong><br>[5 marks]</strong></em></p>
<div class="question_part_label">e.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>The following diagram shows part of the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2{x^2} = {\text{si}}{{\text{n}}^3}\,y">
<mn>2</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mrow>
<mtext>si</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mn>3</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>y</mi>
</math></span> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0 \leqslant y \leqslant \pi ">
<mn>0</mn>
<mo>⩽<!-- ⩽ --></mo>
<mi>y</mi>
<mo>⩽<!-- ⩽ --></mo>
<mi>π<!-- π --></mi>
</math></span>.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<div class="specification">
<p>The shaded region <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="R">
<mi>R</mi>
</math></span> is the area bounded by the curve, the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span>-axis and the lines <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 0">
<mi>y</mi>
<mo>=</mo>
<mn>0</mn>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = \pi ">
<mi>y</mi>
<mo>=</mo>
<mi>π<!-- π --></mi>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Using implicit differentiation, find an expression for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}}"> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>y</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mfrac> </math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the equation of the tangent to the curve at the point <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\frac{1}{4}{\text{, }}\frac{{5\pi }}{6}} \right)"> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> <mrow> <mtext>, </mtext> </mrow> <mfrac> <mrow> <mn>5</mn> <mi>π</mi> </mrow> <mn>6</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the area of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="R"> <mi>R</mi> </math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The region <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="R"> <mi>R</mi> </math></span> is now rotated about the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y"> <mi>y</mi> </math></span>-axis, through <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2\pi "> <mn>2</mn> <mi>π</mi> </math></span> radians, to form a solid.</p>
<p>By writing <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{{\text{si}}{{\text{n}}^3}\,y}"> <mrow> <mrow> <mtext>si</mtext> </mrow> <mrow> <msup> <mrow> <mtext>n</mtext> </mrow> <mn>3</mn> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mi>y</mi> </mrow> </math></span> as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {1 - {\text{co}}{{\text{s}}^2}\,y} \right){\text{sin}}\,y"> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow> <mtext>co</mtext> </mrow> <mrow> <msup> <mrow> <mtext>s</mtext> </mrow> <mn>2</mn> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mi>y</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>y</mi> </math></span>, show that the volume of the solid formed is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{2\pi }}{3}"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mn>3</mn> </mfrac> </math></span>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>valid attempt to differentiate implicitly <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="4x = 3\,{\text{si}}{{\text{n}}^2}\,y\,{\text{cos}}\,y\frac{{{\text{d}}y}}{{{\text{d}}x}}"> <mn>4</mn> <mi>x</mi> <mo>=</mo> <mn>3</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>si</mtext> </mrow> <mrow> <msup> <mrow> <mtext>n</mtext> </mrow> <mn>2</mn> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mi>y</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>y</mi> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>y</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mfrac> </math></span> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}} = \frac{{4x}}{{3\,{\text{si}}{{\text{n}}^2}\,y\,{\text{cos}}\,y}}"> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>y</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mn>4</mn> <mi>x</mi> </mrow> <mrow> <mn>3</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>si</mtext> </mrow> <mrow> <msup> <mrow> <mtext>n</mtext> </mrow> <mn>2</mn> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mi>y</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>y</mi> </mrow> </mfrac> </math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\frac{1}{4}{\text{, }}\frac{{5\pi }}{6}} \right){\text{, }}\frac{{{\text{d}}y}}{{{\text{d}}x}} = \frac{{4x}}{{3\,{\text{si}}{{\text{n}}^2}\,y\,{\text{cos}}\,y}} = \frac{1}{{3{{\left( {\frac{1}{2}} \right)}^2}\left( { - \frac{{\sqrt 3 }}{2}} \right)}}"> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> <mrow> <mtext>, </mtext> </mrow> <mfrac> <mrow> <mn>5</mn> <mi>π</mi> </mrow> <mn>6</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow> <mtext>, </mtext> </mrow> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>y</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mn>4</mn> <mi>x</mi> </mrow> <mrow> <mn>3</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>si</mtext> </mrow> <mrow> <msup> <mrow> <mtext>n</mtext> </mrow> <mn>2</mn> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mi>y</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>y</mi> </mrow> </mfrac> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mn>3</mn> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mfrac> <mrow> <msqrt> <mn>3</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow \frac{{{\text{d}}y}}{{{\text{d}}x}} = - \frac{8}{{3\sqrt 3 }}\left( { = - 1.54} \right)"> <mo stretchy="false">⇒</mo> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>y</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mfrac> <mo>=</mo> <mo>−</mo> <mfrac> <mn>8</mn> <mrow> <mn>3</mn> <msqrt> <mn>3</mn> </msqrt> </mrow> </mfrac> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mo>−</mo> <mn>1.54</mn> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p>hence equation of tangent is</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y - \frac{{5\pi }}{6} = - 1.54\left( {x - \frac{1}{4}} \right)"> <mi>y</mi> <mo>−</mo> <mfrac> <mrow> <mn>5</mn> <mi>π</mi> </mrow> <mn>6</mn> </mfrac> <mo>=</mo> <mo>−</mo> <mn>1.54</mn> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> <strong>OR</strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = - 1.54x + 3.00"> <mi>y</mi> <mo>=</mo> <mo>−</mo> <mn>1.54</mn> <mi>x</mi> <mo>+</mo> <mn>3.00</mn> </math></span> <em><strong>(M1)A1</strong></em></p>
<p><strong>Note:</strong> Accept <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = - 1.54x + 3"> <mi>y</mi> <mo>=</mo> <mo>−</mo> <mn>1.54</mn> <mi>x</mi> <mo>+</mo> <mn>3</mn> </math></span>. </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \sqrt {\frac{1}{2}{\text{si}}{{\text{n}}^3}\,y} "> <mi>x</mi> <mo>=</mo> <msqrt> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mrow> <mtext>si</mtext> </mrow> <mrow> <msup> <mrow> <mtext>n</mtext> </mrow> <mn>3</mn> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mi>y</mi> </msqrt> </math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_0^\pi {\sqrt {\frac{1}{2}{\text{si}}{{\text{n}}^3}\,y\,{\text{d}}y} } "> <msubsup> <mo>∫</mo> <mn>0</mn> <mi>π</mi> </msubsup> <mrow> <msqrt> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mrow> <mtext>si</mtext> </mrow> <mrow> <msup> <mrow> <mtext>n</mtext> </mrow> <mn>3</mn> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mi>y</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>d</mtext> </mrow> <mi>y</mi> </msqrt> </mrow> </math></span> <em><strong>(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 1.24"> <mo>=</mo> <mn>1.24</mn> </math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>use of volume <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \int {\pi {x^2}} \,{\text{d}}y"> <mo>=</mo> <mo>∫</mo> <mrow> <mi>π</mi> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>d</mtext> </mrow> <mi>y</mi> </math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \int_0^\pi {\frac{1}{2}} \pi \,{\text{si}}{{\text{n}}^3}\,y\,{\text{d}}y"> <mo>=</mo> <msubsup> <mo>∫</mo> <mn>0</mn> <mi>π</mi> </msubsup> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mi>π</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>si</mtext> </mrow> <mrow> <msup> <mrow> <mtext>n</mtext> </mrow> <mn>3</mn> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mi>y</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>d</mtext> </mrow> <mi>y</mi> </math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{2}\pi \int_0^\pi {\left( {{\text{sin}}\,y - {\text{sin}}\,y\,{\text{co}}{{\text{s}}^2}\,y} \right){\text{d}}y} "> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mi>π</mi> <msubsup> <mo>∫</mo> <mn>0</mn> <mi>π</mi> </msubsup> <mrow> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>y</mi> <mo>−</mo> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>y</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>co</mtext> </mrow> <mrow> <msup> <mrow> <mtext>s</mtext> </mrow> <mn>2</mn> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mi>y</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mtext>d</mtext> </mrow> <mi>y</mi> </mrow> </math></span></p>
<p><strong>Note:</strong> Condone absence of limits up to this point.</p>
<p>reasonable attempt to integrate <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{2}\pi \left[ { - {\text{cos}}\,y + \frac{1}{3}{\text{co}}{{\text{s}}^3}\,y} \right]_0^\pi "> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mi>π</mi> <msubsup> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>y</mi> <mo>+</mo> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> <mrow> <mtext>co</mtext> </mrow> <mrow> <msup> <mrow> <mtext>s</mtext> </mrow> <mn>3</mn> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mi>y</mi> </mrow> <mo>]</mo> </mrow> <mn>0</mn> <mi>π</mi> </msubsup> </math></span> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for correct limits (not to be awarded if previous <em><strong>M1</strong></em> has not been awarded) and <em><strong>A1</strong></em> for correct integrand.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{2}\pi \left( {1 - \frac{1}{3}} \right) - \frac{1}{2}\pi \left( { - 1 + \frac{1}{3}} \right)"> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mi>π</mi> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mi>π</mi> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>1</mn> <mo>+</mo> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </math> <em><strong>A1</strong></em></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{2\pi }}{3}"> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mn>3</mn> </mfrac> </math></span> <em><strong>AG</strong></em></p>
<p><strong>Note:</strong> Do not accept decimal answer equivalent to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{2\pi }}{3}"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mn>3</mn> </mfrac> </math></span>.</p>
<p><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>The following diagram shows part of the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>p</mi><mo>+</mo><mi>q</mi><mo> </mo><mi>sin</mi><mo> </mo><mo>(</mo><mi>r</mi><mi>x</mi><mo>)</mo></math> . The graph has a local maximum point at <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>-</mo><mfrac><mrow><mn>9</mn><mi mathvariant="normal">π</mi></mrow><mn>4</mn></mfrac><mo>,</mo><mo> </mo><mn>5</mn></mrow></mfenced></math> and a local minimum point at <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>-</mo><mfrac><mrow><mn>3</mn><mi mathvariant="normal">π</mi></mrow><mn>4</mn></mfrac><mo>,</mo><mo> </mo><mo>-</mo><mn>1</mn></mrow></mfenced></math>.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find the area of the shaded region.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p>the principal axis is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>5</mn><mo>+</mo><mfenced><mrow><mo>-</mo><mn>1</mn></mrow></mfenced></mrow><mn>2</mn></mfrac><mfenced><mrow><mo>=</mo><mn>2</mn></mrow></mfenced></math></p>
<p>so <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mn>2</mn></math> <em><strong>A1</strong></em></p>
<p>the amplitude is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>5</mn><mo>-</mo><mfenced><mrow><mo>-</mo><mn>1</mn></mrow></mfenced></mrow><mn>2</mn></mfrac><mfenced><mrow><mo>=</mo><mn>3</mn></mrow></mfenced></math></p>
<p>so <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi><mo>=</mo><mn>3</mn></math> <em><strong>A1</strong></em></p>
<p><br><strong>EITHER</strong></p>
<p>one period is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mfenced><mrow><mo>-</mo><mfrac><mrow><mn>3</mn><mi mathvariant="normal">π</mi></mrow><mn>4</mn></mfrac><mo>-</mo><mfenced><mrow><mo>-</mo><mfrac><mrow><mn>9</mn><mi mathvariant="normal">π</mi></mrow><mn>4</mn></mfrac></mrow></mfenced></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>3</mn><mi mathvariant="normal">π</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mfrac><mrow><mn>2</mn><mi mathvariant="normal">π</mi></mrow><mi>r</mi></mfrac><mo>=</mo><mn>3</mn><mi mathvariant="normal">π</mi></math></p>
<p><strong><br>OR</strong></p>
<p>Substituting a point eg <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>1</mn><mo>=</mo><mn>2</mn><mo>+</mo><mi>sin</mi><mo> </mo><mfenced><mrow><mo>-</mo><mfrac><mrow><mn>3</mn><mi mathvariant="normal">π</mi></mrow><mn>4</mn></mfrac><mi>r</mi></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mo> </mo><mfenced><mrow><mo>-</mo><mfrac><mrow><mn>3</mn><mi mathvariant="normal">π</mi></mrow><mn>4</mn></mfrac><mi>r</mi></mrow></mfenced><mo>=</mo><mo>-</mo><mn>1</mn><mo>⇒</mo><mo>-</mo><mfrac><mrow><mn>3</mn><mi mathvariant="normal">π</mi></mrow><mn>4</mn></mfrac><mi>r</mi><mo>=</mo><mo>…</mo><mo>-</mo><mfrac><mrow><mn>5</mn><mi mathvariant="normal">π</mi></mrow><mn>2</mn></mfrac><mo>,</mo><mo> </mo><mo>-</mo><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac><mo>,</mo><mo> </mo><mfrac><mrow><mn>3</mn><mi mathvariant="normal">π</mi></mrow><mn>2</mn></mfrac><mo>,</mo><mo>…</mo></math></p>
<p>Choice of correct solution <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mrow><mn>3</mn><mi mathvariant="normal">π</mi></mrow><mn>4</mn></mfrac><mi>r</mi><mo>=</mo><mo>-</mo><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></math> <em><strong>(M1)</strong></em></p>
<p><br><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>r</mi><mo>=</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>⇒</mo><mi>y</mi><mo>=</mo><mn>2</mn><mo>+</mo><mn>3</mn><mo> </mo><mi>sin</mi><mo> </mo><mfenced><mfrac><mrow><mn>2</mn><mi>x</mi></mrow><mn>3</mn></mfrac></mfenced></mrow></mfenced></math></p>
<p><strong><br>Note:</strong> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math> can be both given as negatives for full marks</p>
<p><em><strong><br>[4 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>roots are <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mo>-</mo><mn>1</mn><mo>.</mo><mn>09459</mn><mo>…</mo><mo>,</mo><mo> </mo><mi>x</mi><mo>=</mo><mo>-</mo><mn>3</mn><mo>.</mo><mn>617797</mn><mo>…</mo></math><em><strong> (A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mrow><mo>-</mo><mn>3</mn><mo>.</mo><mn>617797</mn><mo>…</mo></mrow><mrow><mo>-</mo><mn>1</mn><mo>.</mo><mn>09459</mn><mo>…</mo></mrow></msubsup><mfenced><mrow><mn>2</mn><mo>+</mo><mn>3</mn><mo> </mo><mi>sin</mi><mo> </mo><mfenced><mfrac><mrow><mn>2</mn><mi>x</mi></mrow><mn>3</mn></mfrac></mfenced></mrow></mfenced><mtext>d</mtext><mi>x</mi></math><em><strong> (M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><mn>1</mn><mo>.</mo><mn>66</mn><mfenced><mrow><mo>=</mo><mo>-</mo><mn>1</mn><mo>.</mo><mn>66179</mn><mo>…</mo></mrow></mfenced></math><em><strong> (A1)</strong></em></p>
<p>so area <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>1</mn><mo>.</mo><mn>66</mn><mo> </mo><mo> </mo><mfenced><msup><mtext>units</mtext><mn>2</mn></msup></mfenced></math><em><strong> A1</strong></em></p>
<p><em><strong><br>[4 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>A function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> satisfies the conditions <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( 0 \right) = - 4">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mn>0</mn>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mo>−</mo>
<mn>4</mn>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( 1 \right) = 0">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mn>1</mn>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span> and its second derivative is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f''\left( x \right) = 15\sqrt x + \frac{1}{{{{\left( {x + 1} \right)}^2}}}">
<msup>
<mi>f</mi>
<mo>″</mo>
</msup>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>15</mn>
<msqrt>
<mi>x</mi>
</msqrt>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> ≥ 0.</p>
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right)">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f'\left( x \right) = \int {\left( {15\sqrt x + \frac{1}{{{{\left( {x + 1} \right)}^2}}}} \right)} \,{\text{d}}x = 10{x^{\frac{3}{2}}} - \frac{1}{{x + 1}}\left( { + c} \right)">
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mo>∫</mo>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>15</mn>
<msqrt>
<mi>x</mi>
</msqrt>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
<mo>=</mo>
<mn>10</mn>
<mrow>
<msup>
<mi>x</mi>
<mrow>
<mfrac>
<mn>3</mn>
<mn>2</mn>
</mfrac>
</mrow>
</msup>
</mrow>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mo>+</mo>
<mi>c</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>(M1)A1A1</strong></em></p>
<p><strong>Note:</strong> <em><strong>A1</strong></em> for first term, <em><strong>A1</strong></em> for second term. Withhold one <em><strong>A1</strong></em> if extra terms are seen.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = \int {\left( {10{x^{\frac{3}{2}}} - \frac{1}{{x + 1}} + c} \right)} \,{\text{d}}x = 4{x^{\frac{5}{2}}} - {\text{ln}}\left( {x + 1} \right) + cx + d">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mo>∫</mo>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>10</mn>
<mrow>
<msup>
<mi>x</mi>
<mrow>
<mfrac>
<mn>3</mn>
<mn>2</mn>
</mfrac>
</mrow>
</msup>
</mrow>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</mfrac>
<mo>+</mo>
<mi>c</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
<mo>=</mo>
<mn>4</mn>
<mrow>
<msup>
<mi>x</mi>
<mrow>
<mfrac>
<mn>5</mn>
<mn>2</mn>
</mfrac>
</mrow>
</msup>
</mrow>
<mo>−</mo>
<mrow>
<mtext>ln</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mi>c</mi>
<mi>x</mi>
<mo>+</mo>
<mi>d</mi>
</math></span> <em><strong> A1</strong></em></p>
<p><strong>Note:</strong> Allow FT from incorrect <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f'\left( x \right)">
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span> if it is of the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f'\left( x \right) = A{x^{\frac{3}{2}}} + \frac{B}{{x + 1}} + c">
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mi>A</mi>
<mrow>
<msup>
<mi>x</mi>
<mrow>
<mfrac>
<mn>3</mn>
<mn>2</mn>
</mfrac>
</mrow>
</msup>
</mrow>
<mo>+</mo>
<mfrac>
<mi>B</mi>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</mfrac>
<mo>+</mo>
<mi>c</mi>
</math></span>.</p>
<p>Accept <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ln}}\left| {x + 1} \right|">
<mrow>
<mtext>ln</mtext>
</mrow>
<mrow>
<mo>|</mo>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>|</mo>
</mrow>
</math></span>.</p>
<p> </p>
<p>attempt to use at least one boundary condition in their <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right)">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span> <em><strong> (M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 0">
<mi>x</mi>
<mo>=</mo>
<mn>0</mn>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = - 4">
<mi>y</mi>
<mo>=</mo>
<mo>−</mo>
<mn>4</mn>
</math></span></p>
<p>⇒ <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d = - 4">
<mi>d</mi>
<mo>=</mo>
<mo>−</mo>
<mn>4</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 1">
<mi>x</mi>
<mo>=</mo>
<mn>1</mn>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 0">
<mi>y</mi>
<mo>=</mo>
<mn>0</mn>
</math></span></p>
<p>⇒ <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0 = 4 - {\text{ln}}\,2 + c - 4">
<mn>0</mn>
<mo>=</mo>
<mn>4</mn>
<mo>−</mo>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mo>+</mo>
<mi>c</mi>
<mo>−</mo>
<mn>4</mn>
</math></span></p>
<p>⇒ <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c = {\text{ln}}\,2\left( { = 0.693} \right)">
<mi>c</mi>
<mo>=</mo>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mn>0.693</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = 4{x^{\frac{5}{2}}} - {\text{ln}}\left( {x + 1} \right) + x\,{\text{ln}}\,2 - 4">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>4</mn>
<mrow>
<msup>
<mi>x</mi>
<mrow>
<mfrac>
<mn>5</mn>
<mn>2</mn>
</mfrac>
</mrow>
</msup>
</mrow>
<mo>−</mo>
<mrow>
<mtext>ln</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mi>x</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mo>−</mo>
<mn>4</mn>
</math></span></p>
<p> </p>
<p><em><strong>[7 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>A point P moves in a straight line with velocity <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v">
<mi>v</mi>
</math></span> ms<sup>−1</sup> given by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v\left( t \right) = {{\text{e}}^{ - t}} - 8{t^2}{{\text{e}}^{ - 2t}}">
<mi>v</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−<!-- − --></mo>
<mi>t</mi>
</mrow>
</msup>
</mrow>
<mo>−<!-- − --></mo>
<mn>8</mn>
<mrow>
<msup>
<mi>t</mi>
<mn>2</mn>
</msup>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−<!-- − --></mo>
<mn>2</mn>
<mi>t</mi>
</mrow>
</msup>
</mrow>
</math></span> at time <em>t</em> seconds, where <em>t</em> ≥ 0.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the first time <em>t</em><sub>1</sub> at which P has zero velocity.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an expression for the acceleration of P at time <em>t</em>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of the acceleration of P at time <em>t</em><sub>1</sub>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p>attempt to solve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v\left( t \right) = 0">
<mi>v</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span> for <em>t</em> or equivalent <em><strong>(M1)</strong></em></p>
<p><em>t</em><sub>1</sub> = 0.441(s) <em><strong> A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a\left( t \right) = \frac{{{\text{d}}v}}{{{\text{d}}t}} = - {{\text{e}}^{ - t}} - 16t{{\text{e}}^{ - 2t}} + 16{t^2}{{\text{e}}^{ - 2t}}">
<mi>a</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>v</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mo>−</mo>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mi>t</mi>
</mrow>
</msup>
</mrow>
<mo>−</mo>
<mn>16</mn>
<mi>t</mi>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mn>2</mn>
<mi>t</mi>
</mrow>
</msup>
</mrow>
<mo>+</mo>
<mn>16</mn>
<mrow>
<msup>
<mi>t</mi>
<mn>2</mn>
</msup>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mn>2</mn>
<mi>t</mi>
</mrow>
</msup>
</mrow>
</math></span> <em><strong>M1A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1</strong> </em>for attempting to differentiate using the product rule.</p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a\left( {{t_1}} \right) = - 2.28">
<mi>a</mi>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msub>
<mi>t</mi>
<mn>1</mn>
</msub>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mo>−</mo>
<mn>2.28</mn>
</math></span> (ms<sup>−2</sup>) <em><strong>A1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<br><hr><br><div class="question">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="l">
<mi>l</mi>
</math></span> be the tangent to the curve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = x{{\text{e}}^{2x}}">
<mi>y</mi>
<mo>=</mo>
<mi>x</mi>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mn>2</mn>
<mi>x</mi>
</mrow>
</msup>
</mrow>
</math></span> at the point (1, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{{\text{e}}^2}">
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</math></span>).</p>
<p>Find the coordinates of the point where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="l">
<mi>l</mi>
</math></span> meets the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>-axis.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><strong>METHOD 1</strong></p>
<p>equation of tangent is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 22.167 \ldots x - 14.778 \ldots ">
<mi>y</mi>
<mo>=</mo>
<mn>22.167</mn>
<mo>…</mo>
<mi>x</mi>
<mo>−</mo>
<mn>14.778</mn>
<mo>…</mo>
</math></span> <strong>OR</strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = - 7.389 \ldots = 22.167 \ldots \left( {x - 1} \right)">
<mi>y</mi>
<mo>=</mo>
<mo>−</mo>
<mn>7.389</mn>
<mo>…</mo>
<mo>=</mo>
<mn>22.167</mn>
<mo>…</mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>(M1)(A1)</strong></em></p>
<p>meets the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>-axis when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 0">
<mi>y</mi>
<mo>=</mo>
<mn>0</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 0.667">
<mi>x</mi>
<mo>=</mo>
<mn>0.667</mn>
</math></span></p>
<p>meets <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>-axis at (0.667, 0)<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( { = \left( {\frac{2}{3},\,\,0} \right)} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>2</mn>
<mn>3</mn>
</mfrac>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mn>0</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \frac{2}{3}">
<mi>x</mi>
<mo>=</mo>
<mfrac>
<mn>2</mn>
<mn>3</mn>
</mfrac>
</math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 0.667">
<mi>x</mi>
<mo>=</mo>
<mn>0.667</mn>
</math></span> seen and <em><strong>A1</strong></em> for coordinates (<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>, 0) given.</p>
<p> </p>
<p><strong>METHOD 1</strong></p>
<p>Attempt to differentiate <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}} = {{\text{e}}^{2x}} + 2x{{\text{e}}^{2x}}">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mn>2</mn>
<mi>x</mi>
</mrow>
</msup>
</mrow>
<mo>+</mo>
<mn>2</mn>
<mi>x</mi>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mn>2</mn>
<mi>x</mi>
</mrow>
</msup>
</mrow>
</math></span></p>
<p>when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 1">
<mi>x</mi>
<mo>=</mo>
<mn>1</mn>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}} = 3{{\text{e}}^2}">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mn>3</mn>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</math></span> <em><strong>(M1)</strong></em></p>
<p>equation of the tangent is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y - {{\text{e}}^2} = 3{{\text{e}}^2}\left( {x - 1} \right)">
<mi>y</mi>
<mo>−</mo>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>3</mn>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 3{{\text{e}}^2}x - 2{{\text{e}}^2}">
<mi>y</mi>
<mo>=</mo>
<mn>3</mn>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mi>x</mi>
<mo>−</mo>
<mn>2</mn>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</math></span></p>
<p>meets <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>-axis at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \frac{2}{3}">
<mi>x</mi>
<mo>=</mo>
<mfrac>
<mn>2</mn>
<mn>3</mn>
</mfrac>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {\frac{2}{3},\,\,0} \right)}">
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>2</mn>
<mn>3</mn>
</mfrac>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mn>0</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
</math></span> <em><strong>A1A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \frac{2}{3}">
<mi>x</mi>
<mo>=</mo>
<mfrac>
<mn>2</mn>
<mn>3</mn>
</mfrac>
</math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 0.667">
<mi>x</mi>
<mo>=</mo>
<mn>0.667</mn>
</math></span> seen and <em><strong>A1</strong></em> for coordinates (<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>, 0) given.</p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>A body moves in a straight line such that its velocity, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v\,{\text{m}}{{\text{s}}^{ - 1}}">
<mi>v</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>m</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>s</mtext>
</mrow>
<mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</math></span>, after <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> seconds is given by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v = 2\,{\text{sin}}\left( {\frac{t}{{10}} + \frac{\pi }{5}} \right)\csc \left( {\frac{t}{{30}} + \frac{\pi }{4}} \right)">
<mi>v</mi>
<mo>=</mo>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mi>t</mi>
<mrow>
<mn>10</mn>
</mrow>
</mfrac>
<mo>+</mo>
<mfrac>
<mi>π<!-- π --></mi>
<mn>5</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mi>csc</mi>
<mo><!-- --></mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mi>t</mi>
<mrow>
<mn>30</mn>
</mrow>
</mfrac>
<mo>+</mo>
<mfrac>
<mi>π<!-- π --></mi>
<mn>4</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0 \leqslant t \leqslant 60">
<mn>0</mn>
<mo>⩽<!-- ⩽ --></mo>
<mi>t</mi>
<mo>⩽<!-- ⩽ --></mo>
<mn>60</mn>
</math></span>.</p>
<p>The following diagram shows the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v">
<mi>v</mi>
</math></span> against <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span>. Point <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{A}}">
<mrow>
<mtext>A</mtext>
</mrow>
</math></span> is a local maximum and point <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{B}}">
<mrow>
<mtext>B</mtext>
</mrow>
</math></span> is a local minimum.</p>
<p style="text-align: center;"><img 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"></p>
</div>
<div class="specification">
<p>The body first comes to rest at time <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t = {t_1}">
<mi>t</mi>
<mo>=</mo>
<mrow>
<msub>
<mi>t</mi>
<mn>1</mn>
</msub>
</mrow>
</math></span>. Find</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the coordinates of point <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{A}}"> <mrow> <mtext>A</mtext> </mrow> </math></span> and the coordinates of point <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{B}}"> <mrow> <mtext>B</mtext> </mrow> </math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, write down the maximum speed of the body.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{t_1}"> <mrow> <msub> <mi>t</mi> <mn>1</mn> </msub> </mrow> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>the distance travelled between <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t = 0"> <mi>t</mi> <mo>=</mo> <mn>0</mn> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t = {t_1}"> <mi>t</mi> <mo>=</mo> <mrow> <msub> <mi>t</mi> <mn>1</mn> </msub> </mrow> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>the acceleration when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t = {t_1}"> <mi>t</mi> <mo>=</mo> <mrow> <msub> <mi>t</mi> <mn>1</mn> </msub> </mrow> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the distance travelled in the first 30 seconds.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{A}}\left( {7.47{\text{, }}2.28} \right)"> <mrow> <mtext>A</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mn>7.47</mn> <mrow> <mtext>, </mtext> </mrow> <mn>2.28</mn> </mrow> <mo>)</mo> </mrow> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{B}}\left( {43.4{\text{,}} - 2.45} \right)"> <mrow> <mtext>B</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mn>43.4</mn> <mrow> <mtext>,</mtext> </mrow> <mo>−</mo> <mn>2.45</mn> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A1</strong></em><em><strong>A1</strong></em><em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>maximum speed is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2.45\,\left( {{\text{m}}{{\text{s}}^{ - 1}}} \right)"> <mn>2.45</mn> <mspace width="thinmathspace"></mspace> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>m</mtext> </mrow> <mrow> <msup> <mrow> <mtext>s</mtext> </mrow> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v = 0 \Rightarrow {t_1} = 25.1\,\left( {\text{s}} \right)"> <mi>v</mi> <mo>=</mo> <mn>0</mn> <mo stretchy="false">⇒</mo> <mrow> <msub> <mi>t</mi> <mn>1</mn> </msub> </mrow> <mo>=</mo> <mn>25.1</mn> <mspace width="thinmathspace"></mspace> <mrow> <mo>(</mo> <mrow> <mtext>s</mtext> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>(M1)</strong><strong>A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_0^{{t_1}} {v\,{\text{d}}t} "> <msubsup> <mo>∫</mo> <mn>0</mn> <mrow> <mrow> <msub> <mi>t</mi> <mn>1</mn> </msub> </mrow> </mrow> </msubsup> <mrow> <mi>v</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 41.0\,\left( {\text{m}} \right)"> <mo>=</mo> <mn>41.0</mn> <mspace width="thinmathspace"></mspace> <mrow> <mo>(</mo> <mrow> <mtext>m</mtext> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a = \frac{{{\text{d}}v}}{{{\text{d}}t}}"> <mi>a</mi> <mo>=</mo> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>v</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </mfrac> </math></span> at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t = {t_1} = 25.1"> <mi>t</mi> <mo>=</mo> <mrow> <msub> <mi>t</mi> <mn>1</mn> </msub> </mrow> <mo>=</mo> <mn>25.1</mn> </math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a = - 0.200\,\,\left( {{\text{m}}{{\text{s}}^{ - 2}}} \right)"> <mi>a</mi> <mo>=</mo> <mo>−</mo> <mn>0.200</mn> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>m</mtext> </mrow> <mrow> <msup> <mrow> <mtext>s</mtext> </mrow> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </msup> </mrow> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Accept <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a = - 0.2"> <mi>a</mi> <mo>=</mo> <mo>−</mo> <mn>0.2</mn> </math></span>.</p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to integrate between 0 and 30 <em><strong>(M1)</strong></em></p>
<p><strong>Note:</strong> An unsupported answer of 38.6 can imply integrating from 0 to 30.</p>
<p> </p>
<p><strong>EITHER</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_0^{30} {\left| v \right|} \,{\text{d}}t"> <msubsup> <mo>∫</mo> <mn>0</mn> <mrow> <mn>30</mn> </mrow> </msubsup> <mrow> <mrow> <mo>|</mo> <mi>v</mi> <mo>|</mo> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>d</mtext> </mrow> <mi>t</mi> </math></span> <em><strong>(A1)</strong></em></p>
<p> </p>
<p><strong>OR</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="41.0 - \int_{{t_1}}^{30} {v\,{\text{d}}t} "> <mn>41.0</mn> <mo>−</mo> <msubsup> <mo>∫</mo> <mrow> <mrow> <msub> <mi>t</mi> <mn>1</mn> </msub> </mrow> </mrow> <mrow> <mn>30</mn> </mrow> </msubsup> <mrow> <mi>v</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </math></span> <em><strong>(A1)</strong></em></p>
<p> </p>
<p><strong>THEN</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 43.3\,\left( {\text{m}} \right)"> <mo>=</mo> <mn>43.3</mn> <mspace width="thinmathspace"></mspace> <mrow> <mo>(</mo> <mrow> <mtext>m</mtext> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>A water trough which is 10 metres long has a uniform cross-section in the shape of a semicircle with radius 0.5 metres. It is partly filled with water as shown in the following diagram of the cross-section. The centre of the circle is O and the angle KOL is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\theta ">
<mi>θ<!-- θ --></mi>
</math></span> radians.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-09_om_11.09.30.png" alt="M17/5/MATHL/HP2/ENG/TZ1/08"></p>
</div>
<div class="specification">
<p>The volume of water is increasing at a constant rate of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0.0008{\text{ }}{{\text{m}}^3}{{\text{s}}^{ - 1}}">
<mn>0.0008</mn>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>m</mtext>
</mrow>
<mn>3</mn>
</msup>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>s</mtext>
</mrow>
<mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an expression for the volume of water <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="V{\text{ }}({{\text{m}}^3})">
<mi>V</mi>
<mrow>
<mtext> </mtext>
</mrow>
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mrow>
<mtext>m</mtext>
</mrow>
<mn>3</mn>
</msup>
</mrow>
<mo stretchy="false">)</mo>
</math></span> in the trough in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\theta ">
<mi>θ</mi>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}\theta }}{{{\text{d}}t}}">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>θ</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
</math></span> when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\theta = \frac{\pi }{3}">
<mi>θ</mi>
<mo>=</mo>
<mfrac>
<mi>π</mi>
<mn>3</mn>
</mfrac>
</math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p>area of segment <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{2} \times {0.5^2} \times (\theta - \sin \theta )">
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mo>×</mo>
<mrow>
<msup>
<mn>0.5</mn>
<mn>2</mn>
</msup>
</mrow>
<mo>×</mo>
<mo stretchy="false">(</mo>
<mi>θ</mi>
<mo>−</mo>
<mi>sin</mi>
<mo></mo>
<mi>θ</mi>
<mo stretchy="false">)</mo>
</math></span> <strong><em>M1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="V = {\text{area of segment}} \times 10">
<mi>V</mi>
<mo>=</mo>
<mrow>
<mtext>area of segment</mtext>
</mrow>
<mo>×</mo>
<mn>10</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="V = \frac{5}{4}(\theta - \sin \theta )">
<mi>V</mi>
<mo>=</mo>
<mfrac>
<mn>5</mn>
<mn>4</mn>
</mfrac>
<mo stretchy="false">(</mo>
<mi>θ</mi>
<mo>−</mo>
<mi>sin</mi>
<mo></mo>
<mi>θ</mi>
<mo stretchy="false">)</mo>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}V}}{{{\text{d}}t}} = \frac{5}{4}(1 - \cos \theta )\frac{{{\text{d}}\theta }}{{{\text{d}}t}}">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>V</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mn>5</mn>
<mn>4</mn>
</mfrac>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>−</mo>
<mi>cos</mi>
<mo></mo>
<mi>θ</mi>
<mo stretchy="false">)</mo>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>θ</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
</math></span> <strong><em>M1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0.0008 = \frac{5}{4}\left( {1 - \cos \frac{\pi }{3}} \right)\frac{{{\text{d}}\theta }}{{{\text{d}}t}}">
<mn>0.0008</mn>
<mo>=</mo>
<mfrac>
<mn>5</mn>
<mn>4</mn>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>cos</mi>
<mo></mo>
<mfrac>
<mi>π</mi>
<mn>3</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>θ</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
</math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}\theta }}{{{\text{d}}t}} = 0.00128{\text{ }}({\text{rad}}\,{s^{ - 1}})">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>θ</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mn>0.00128</mn>
<mrow>
<mtext> </mtext>
</mrow>
<mo stretchy="false">(</mo>
<mrow>
<mtext>rad</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<msup>
<mi>s</mi>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mo stretchy="false">)</mo>
</math></span> <strong><em>A1</em></strong></p>
<p><strong>METHOD 2</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}\theta }}{{{\text{d}}t}} = \frac{{{\text{d}}\theta }}{{{\text{d}}V}} \times \frac{{{\text{d}}V}}{{{\text{d}}t}}">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>θ</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>θ</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>V</mi>
</mrow>
</mfrac>
<mo>×</mo>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>V</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
</math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}V}}{{{\text{d}}\theta }} = \frac{5}{4}(1 - \cos \theta )">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>V</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>θ</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mn>5</mn>
<mn>4</mn>
</mfrac>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>−</mo>
<mi>cos</mi>
<mo></mo>
<mi>θ</mi>
<mo stretchy="false">)</mo>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}\theta }}{{{\text{d}}t}} = \frac{{4 \times 0.0008}}{{5\left( {1 - \cos \frac{\pi }{3}} \right)}}">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>θ</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mn>4</mn>
<mo>×</mo>
<mn>0.0008</mn>
</mrow>
<mrow>
<mn>5</mn>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>cos</mi>
<mo></mo>
<mfrac>
<mi>π</mi>
<mn>3</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
</mfrac>
</math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}\theta }}{{{\text{d}}t}} = 0.00128\left( {\frac{4}{{3125}}} \right)({\text{rad }}{s^{ - 1}})">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>θ</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mn>0.00128</mn>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>4</mn>
<mrow>
<mn>3125</mn>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo stretchy="false">(</mo>
<mrow>
<mtext>rad </mtext>
</mrow>
<mrow>
<msup>
<mi>s</mi>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mo stretchy="false">)</mo>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> is defined by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>2</mn></mrow><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow></mfrac></math>, for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>≠</mo><mi>p</mi></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>≠</mo><mi>q</mi></math>.</p>
</div>
<div class="specification">
<p>The graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> has exactly one point of inflexion.</p>
</div>
<div class="specification">
<p>The function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math> is defined by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mfrac></math>, for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi><mo>,</mo><mo> </mo><mi>x</mi><mo>≠</mo><mo>-</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math> and the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-coordinate of the point of inflexion.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>3</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mn>3</mn></math>, showing the values of any axes intercepts, the coordinates of any local maxima and local minima, and giving the equations of any asymptotes.</p>
<div class="marks">[5]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the equations of all the asymptotes on the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By considering the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>-</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math>, or otherwise, solve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo><</mo><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">f.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>attempt to solve <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn><mo>=</mo><mn>0</mn></math> e.g. by factorising <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>,</mo><mo> </mo><mi>q</mi><mo>=</mo><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math> or vice versa <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to use quotient rule or product rule <em><strong>(M1)</strong></em></p>
<p> </p>
<p><strong>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mrow><mn>3</mn><mfenced><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow></mfenced><mo>-</mo><mn>8</mn><mi>x</mi><mfenced><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mfenced></mrow><msup><mfenced><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup></mfrac><mfenced><mrow><mo>=</mo><mfrac><mrow><mo>-</mo><mn>12</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>16</mn><mi>x</mi><mo>-</mo><mn>3</mn></mrow><msup><mfenced><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup></mfrac></mrow></mfenced></math> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for each term in the numerator with correct signs, provided correct denominator is seen.</p>
<p> </p>
<p><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mo>-</mo><mn>8</mn><mi>x</mi><mfenced><mrow><mn>3</mn><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mfenced><msup><mfenced><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow></mfenced><mrow><mo>-</mo><mn>2</mn></mrow></msup><mo>+</mo><mn>3</mn><msup><mfenced><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow></mfenced><mrow><mo>-</mo><mn>1</mn></mrow></msup></math> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for each term.</p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to find the local min point on <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced></math> OR solve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>''</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>0</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mo>-</mo><mn>1</mn><mo>.</mo><mn>60</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img src="data:image/png;base64,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"> <em><strong>A1A1A1A1A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for both vertical asymptotes with their equations, award <em><strong>A1</strong></em> for horizontal asymptote with equation, award <em><strong>A1</strong></em> for each correct branch including asymptotic behaviour, coordinates of minimum and maximum points (may be seen next to the graph) and values of axes intercepts.<br>If vertical asymptotes are absent (or not vertical) and the branches overlap as a consequence, award maximum <em><strong>A0A1A0A1A1</strong></em>.</p>
<p> </p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mo>-</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><mfenced><mrow><mo>=</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>667</mn></mrow></mfenced></math> <em><strong> A1</strong></em></p>
<p>(oblique asymptote has) gradient <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>4</mn><mn>3</mn></mfrac><mfenced><mrow><mo>=</mo><mn>1</mn><mo>.</mo><mn>33</mn></mrow></mfenced></math> <em><strong> (A1)</strong></em></p>
<p>appropriate method to find complete equation of oblique asymptote <em><strong> M1</strong></em></p>
<p> <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mi>x</mi><mo>+</mo><mn>2</mn><mover><menclose><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>0</mn><mi>x</mi><mo>-</mo><mn>1</mn></menclose><mrow><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mfrac><mn>4</mn><mn>3</mn></mfrac><mi>x</mi><mo>-</mo><mfrac><mn>8</mn><mn>9</mn></mfrac></mrow></mover></math></p>
<p style="padding-left:60px;"> <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mstyle displaystyle="true"><mfrac><mn>8</mn><mn>3</mn></mfrac></mstyle><mi>x</mi></mrow><mrow><mo>-</mo><mstyle displaystyle="true"><mfrac><mn>8</mn><mn>3</mn></mfrac></mstyle><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfrac></math></p>
<p style="padding-left:60px;"> <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mn>8</mn><mn>3</mn></mfrac><mi>x</mi><mo>-</mo><mfrac><mn>16</mn><mstyle displaystyle="true"><mfrac><mn>9</mn><mstyle displaystyle="true"><mfrac><mn>7</mn><mn>9</mn></mfrac></mstyle></mfrac></mstyle></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfrac><mn>4</mn><mn>3</mn></mfrac><mi>x</mi><mo>-</mo><mfrac><mn>8</mn><mn>9</mn></mfrac><mfenced><mrow><mo>=</mo><mn>1</mn><mo>.</mo><mn>33</mn><mi>x</mi><mo>-</mo><mn>0</mn><mo>.</mo><mn>889</mn></mrow></mfenced></math> <em><strong> A1</strong></em></p>
<p><strong>Note:</strong> Do not award the final<em><strong> A1</strong></em> if the answer is not given as an equation.</p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempting to find at least one critical value <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>x</mi><mo>=</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>568729</mn><mo>…</mo><mo>,</mo><mo> </mo><mi>x</mi><mo>=</mo><mn>1</mn><mo>.</mo><mn>31872</mn><mo>…</mo></mrow></mfenced></math> <em><strong> (M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><mo><</mo><mi>x</mi><mo><</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>569</mn></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>0</mn><mo>.</mo><mn>5</mn><mo><</mo><mi>x</mi><mo><</mo><mn>0</mn><mo>.</mo><mn>5</mn></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>></mo><mn>1</mn><mo>.</mo><mn>32</mn></math> <em><strong>A1</strong></em><em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Only penalize once for use of <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>≤</mo></math> rather than <math xmlns="http://www.w3.org/1998/Math/MathML"><mo><</mo></math>.</p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">f.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">f.</div>
</div>
<br><hr><br><div class="specification">
<p>The region <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A">
<mi>A</mi>
</math></span> is enclosed by the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 2\arcsin (x - 1) - \frac{\pi }{4}">
<mi>y</mi>
<mo>=</mo>
<mn>2</mn>
<mi>arcsin</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>−<!-- − --></mo>
<mfrac>
<mi>π<!-- π --></mi>
<mn>4</mn>
</mfrac>
</math></span>, the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span>-axis and the line <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = \frac{\pi }{4}">
<mi>y</mi>
<mo>=</mo>
<mfrac>
<mi>π<!-- π --></mi>
<mn>4</mn>
</mfrac>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down a definite integral to represent the area of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A">
<mi>A</mi>
</math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate the area of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A">
<mi>A</mi>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><strong>METHOD 1</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2\arcsin (x - 1) - \frac{\pi }{4} = \frac{\pi }{4}">
<mn>2</mn>
<mi>arcsin</mi>
<mo></mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>−</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>−</mo>
<mfrac>
<mi>π</mi>
<mn>4</mn>
</mfrac>
<mo>=</mo>
<mfrac>
<mi>π</mi>
<mn>4</mn>
</mfrac>
</math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 1 + \frac{1}{{\sqrt 2 }}\,\,\,( = 1.707 \ldots )">
<mi>x</mi>
<mo>=</mo>
<mn>1</mn>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mo stretchy="false">(</mo>
<mo>=</mo>
<mn>1.707</mn>
<mo>…</mo>
<mo stretchy="false">)</mo>
</math></span> <strong><em>(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int\limits_0^{1 + \frac{1}{{\sqrt 2 }}} {\frac{\pi }{4} - \left( {2\arcsin \left( {x - 1} \right) - \frac{\pi }{4}} \right)dx} ">
<munderover>
<mo>∫</mo>
<mn>0</mn>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
</mrow>
</munderover>
<mrow>
<mfrac>
<mi>π</mi>
<mn>4</mn>
</mfrac>
<mo>−</mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mi>arcsin</mi>
<mo></mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mfrac>
<mi>π</mi>
<mn>4</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mi>d</mi>
<mi>x</mi>
</mrow>
</math></span> <strong><em>M1A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>M1 </em></strong>for an attempt to find the difference between two functions, <strong><em>A1 </em></strong>for all correct.</p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 0,{\text{ }}y = \frac{{ - 5\pi }}{4}\,\,\,( = - 3.93)">
<mi>x</mi>
<mo>=</mo>
<mn>0</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>y</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mo>−</mo>
<mn>5</mn>
<mi>π</mi>
</mrow>
<mn>4</mn>
</mfrac>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mo stretchy="false">(</mo>
<mo>=</mo>
<mo>−</mo>
<mn>3.93</mn>
<mo stretchy="false">)</mo>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 1 + \sin \left( {\frac{{4y + \pi }}{8}} \right)">
<mi>x</mi>
<mo>=</mo>
<mn>1</mn>
<mo>+</mo>
<mi>sin</mi>
<mo></mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mn>4</mn>
<mi>y</mi>
<mo>+</mo>
<mi>π</mi>
</mrow>
<mn>8</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>M1A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>M1 </em></strong>for an attempt to find the inverse function.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_{\frac{{ - 5\pi }}{4}}^{\frac{\pi }{4}} {\left( {1 + \sin \left( {\frac{{4y + \pi }}{8}} \right)} \right){\text{d}}y} ">
<msubsup>
<mo>∫</mo>
<mrow>
<mfrac>
<mrow>
<mo>−</mo>
<mn>5</mn>
<mi>π</mi>
</mrow>
<mn>4</mn>
</mfrac>
</mrow>
<mrow>
<mfrac>
<mi>π</mi>
<mn>4</mn>
</mfrac>
</mrow>
</msubsup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>sin</mi>
<mo></mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mn>4</mn>
<mi>y</mi>
<mo>+</mo>
<mi>π</mi>
</mrow>
<mn>8</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
</math></span> <strong><em>A1</em></strong></p>
<p><strong>METHOD 3</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_0^{1.38...} {\left( {2\arcsin \left( {x - 1} \right) - \frac{\pi }{4}} \right){\text{d}}x} \left| + \right.\int\limits_0^{1.71...} {\frac{\pi }{4}{\text{d}}x - \int\limits_{1.38...}^{1.71...} {\left( {2\arcsin \left( {x - 1} \right) - \frac{\pi }{4}} \right)dx} } ">
<msubsup>
<mo>∫</mo>
<mn>0</mn>
<mrow>
<mn>1.38...</mn>
</mrow>
</msubsup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mi>arcsin</mi>
<mo></mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mfrac>
<mi>π</mi>
<mn>4</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
<mrow>
<mo>|</mo>
<mo>+</mo>
<mo fence="true" stretchy="true" symmetric="true"></mo>
</mrow>
<munderover>
<mo>∫</mo>
<mn>0</mn>
<mrow>
<mn>1.71...</mn>
</mrow>
</munderover>
<mrow>
<mfrac>
<mi>π</mi>
<mn>4</mn>
</mfrac>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
<mo>−</mo>
<munderover>
<mo>∫</mo>
<mrow>
<mn>1.38...</mn>
</mrow>
<mrow>
<mn>1.71...</mn>
</mrow>
</munderover>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mi>arcsin</mi>
<mo></mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mfrac>
<mi>π</mi>
<mn>4</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mi>d</mi>
<mi>x</mi>
</mrow>
</mrow>
</math></span> <strong><em>M1A1A1A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>M1 </em></strong>for considering the area below the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>-axis and above the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>-axis and <strong><em>A1 </em></strong>for each correct integral.</p>
<p> </p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{area}} = 3.30{\text{ (square units)}}">
<mrow>
<mtext>area</mtext>
</mrow>
<mo>=</mo>
<mn>3.30</mn>
<mrow>
<mtext> (square units)</mtext>
</mrow>
</math></span> <strong><em>A2</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The following graph shows the two parts of the curve defined by the equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^2}y = 5 - {y^4}">
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mi>y</mi>
<mo>=</mo>
<mn>5</mn>
<mo>−<!-- − --></mo>
<mrow>
<msup>
<mi>y</mi>
<mn>4</mn>
</msup>
</mrow>
</math></span>, and the normal to the curve at the point P(2 , 1).</p>
<p style="text-align: center;"><img 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"></p>
<p> </p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that there are exactly two points on the curve where the gradient is zero.</p>
<div class="marks">[7]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the equation of the normal to the curve at the point P.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The normal at P cuts the curve again at the point Q. Find the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>-coordinate of Q.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The shaded region is rotated by 2<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\pi "> <mi>π</mi> </math></span> about the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y"> <mi>y</mi> </math></span>-axis. Find the volume of the solid formed.</p>
<div class="marks">[7]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p>differentiating implicitly: <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2xy + {x^2}\frac{{{\text{d}}y}}{{{\text{d}}x}} = - 4{y^3}\frac{{{\text{d}}y}}{{{\text{d}}x}}"> <mn>2</mn> <mi>x</mi> <mi>y</mi> <mo>+</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>y</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mfrac> <mo>=</mo> <mo>−</mo> <mn>4</mn> <mrow> <msup> <mi>y</mi> <mn>3</mn> </msup> </mrow> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>y</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mfrac> </math></span> <em><strong>A1A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for each side.</p>
<p>if <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}} = 0"> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>y</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mfrac> <mo>=</mo> <mn>0</mn> </math></span> then either <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 0"> <mi>x</mi> <mo>=</mo> <mn>0</mn> </math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 0"> <mi>y</mi> <mo>=</mo> <mn>0</mn> </math></span> <em><strong> M1A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 0 \Rightarrow "> <mi>x</mi> <mo>=</mo> <mn>0</mn> <mo stretchy="false">⇒</mo> </math></span> two solutions for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y\left( {y = \pm \sqrt[4]{5}} \right)"> <mi>y</mi> <mrow> <mo>(</mo> <mrow> <mi>y</mi> <mo>=</mo> <mo>±</mo> <mroot> <mn>5</mn> <mn>4</mn> </mroot> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong> R1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 0"> <mi>y</mi> <mo>=</mo> <mn>0</mn> </math></span> not possible (as 0 ≠ 5) <em><strong>R1</strong></em></p>
<p>hence exactly two points <strong><em>AG</em></strong></p>
<p><strong>Note:</strong> For a solution that only refers to the graph giving two solutions at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 0"> <mi>x</mi> <mo>=</mo> <mn>0</mn> </math></span> and no solutions for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 0"> <mi>y</mi> <mo>=</mo> <mn>0</mn> </math></span> award <strong><em>R1</em></strong> only.</p>
<p><em><strong>[7 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>at (2, 1) <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="4 + 4\frac{{{\text{d}}y}}{{{\text{d}}x}} = - 4\frac{{{\text{d}}y}}{{{\text{d}}x}}"> <mn>4</mn> <mo>+</mo> <mn>4</mn> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>y</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mfrac> <mo>=</mo> <mo>−</mo> <mn>4</mn> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>y</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mfrac> </math></span> <em><strong> M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}} = - \frac{1}{2}"> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>y</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mfrac> <mo>=</mo> <mo>−</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </math></span> <em><strong>(A1)</strong></em></p>
<p>gradient of normal is 2 <em><strong>M1</strong></em></p>
<p>1 = 4 + <em>c</em> <em><strong> (M1)</strong></em></p>
<p>equation of normal is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 2x - 3"> <mi>y</mi> <mo>=</mo> <mn>2</mn> <mi>x</mi> <mo>−</mo> <mn>3</mn> </math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>substituting <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^2}\left( {2x - 3} \right) = 5 - {\left( {2x - 3} \right)^4}"> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>x</mi> <mo>−</mo> <mn>3</mn> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>5</mn> <mo>−</mo> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>x</mi> <mo>−</mo> <mn>3</mn> </mrow> <mo>)</mo> </mrow> <mn>4</mn> </msup> </mrow> </math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {\frac{{y + 3}}{2}} \right)^2}\,y = 5 - {y^4}"> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mi>y</mi> <mo>+</mo> <mn>3</mn> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mi>y</mi> <mo>=</mo> <mn>5</mn> <mo>−</mo> <mrow> <msup> <mi>y</mi> <mn>4</mn> </msup> </mrow> </math></span> <em><strong>(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 0.724"> <mi>x</mi> <mo>=</mo> <mn>0.724</mn> </math></span> <em><strong> A1</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>recognition of two volumes <em><strong>(M1)</strong></em></p>
<p>volume <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1 = \pi \int_1^{\sqrt[4]{5}} {\frac{{5 - {y^4}}}{y}} {\text{d}}y\left( { = 101\pi = 3.178 \ldots } \right)"> <mn>1</mn> <mo>=</mo> <mi>π</mi> <msubsup> <mo>∫</mo> <mn>1</mn> <mrow> <mroot> <mn>5</mn> <mn>4</mn> </mroot> </mrow> </msubsup> <mrow> <mfrac> <mrow> <mn>5</mn> <mo>−</mo> <mrow> <msup> <mi>y</mi> <mn>4</mn> </msup> </mrow> </mrow> <mi>y</mi> </mfrac> </mrow> <mrow> <mtext>d</mtext> </mrow> <mi>y</mi> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mn>101</mn> <mi>π</mi> <mo>=</mo> <mn>3.178</mn> <mo>…</mo> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong> M1A1A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1</strong></em> for attempt to use <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\pi \int {{x^2}} {\text{d}}y"> <mi>π</mi> <mo>∫</mo> <mrow> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mrow> <mrow> <mtext>d</mtext> </mrow> <mi>y</mi> </math></span>, <em><strong>A1</strong></em> for limits, <em><strong>A1</strong></em> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\frac{{5 - {y^4}}}{y}}"> <mrow> <mfrac> <mrow> <mn>5</mn> <mo>−</mo> <mrow> <msup> <mi>y</mi> <mn>4</mn> </msup> </mrow> </mrow> <mi>y</mi> </mfrac> </mrow> </math></span> Condone omission of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\pi "> <mi>π</mi> </math></span> at this stage.</p>
<p>volume 2</p>
<p><strong>EITHER</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{3}\pi \times {2^2} \times 4\left( { = 16.75 \ldots } \right)"> <mo>=</mo> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> <mi>π</mi> <mo>×</mo> <mrow> <msup> <mn>2</mn> <mn>2</mn> </msup> </mrow> <mo>×</mo> <mn>4</mn> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mn>16.75</mn> <mo>…</mo> </mrow> <mo>)</mo> </mrow> </math></span> <strong> <em>(M1)(A1)</em></strong></p>
<p><strong>OR</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \pi \int_{ - 3}^1 {{{\left( {\frac{{y + 3}}{2}} \right)}^2}} {\text{d}}y\left( { = \frac{{16\pi }}{3} = 16.75 \ldots } \right)"> <mo>=</mo> <mi>π</mi> <msubsup> <mo>∫</mo> <mrow> <mo>−</mo> <mn>3</mn> </mrow> <mn>1</mn> </msubsup> <mrow> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mi>y</mi> <mo>+</mo> <mn>3</mn> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> </mrow> <mrow> <mtext>d</mtext> </mrow> <mi>y</mi> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mfrac> <mrow> <mn>16</mn> <mi>π</mi> </mrow> <mn>3</mn> </mfrac> <mo>=</mo> <mn>16.75</mn> <mo>…</mo> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>(M1)(A1)</strong></em></p>
<p><strong>THEN</strong></p>
<p>total volume = 19.9 <em><strong>A1</strong></em></p>
<p><em><strong>[7 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>The function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> has a derivative given by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mn>1</mn><mrow><mi>x</mi><mfenced><mrow><mi>k</mi><mo>-</mo><mi>x</mi></mrow></mfenced></mrow></mfrac><mo>,</mo><mo> </mo><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi><mo>,</mo><mo> </mo><mi>x</mi><mo>≠</mo><mi>o</mi><mo>,</mo><mo> </mo><mi>x</mi><mo>≠</mo><mi>k</mi></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> is a positive constant.</p>
</div>
<div class="specification">
<p>Consider <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math>, the population of a colony of ants, which has an initial value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1200</mn></math>.</p>
<p>The rate of change of the population can be modelled by the differential equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>P</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>P</mi><mfenced><mrow><mi>k</mi><mo>-</mo><mi>P</mi></mrow></mfenced></mrow><mrow><mn>5</mn><mi>k</mi></mrow></mfrac></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> is the time measured in days, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>≥</mo><mn>0</mn></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> is the upper bound for the population.</p>
</div>
<div class="specification">
<p>At <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>10</mn></math> the population of the colony has doubled in size from its initial value.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>′</mo><mo>(</mo><mi>x</mi><mo>)</mo></math> can be written in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>a</mi><mi>x</mi></mfrac><mo>+</mo><mfrac><mi>b</mi><mrow><mi>k</mi><mo>-</mo><mi>x</mi></mrow></mfrac></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>,</mo><mo> </mo><mi>b</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>. Find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, find an expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By solving the differential equation, show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>=</mo><mfrac><mrow><mn>1200</mn><mi>k</mi></mrow><mrow><mfenced><mrow><mi>k</mi><mo>-</mo><mn>1200</mn></mrow></mfenced><msup><mtext>e</mtext><mrow><mo>-</mo><mstyle displaystyle="true"><mfrac><mi>t</mi><mn>5</mn></mfrac></mstyle></mrow></msup><mo>+</mo><mn>1200</mn></mrow></mfrac></math>.</p>
<div class="marks">[8]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math>, giving your answer correct to four significant figures.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> when the rate of change of the population is at its maximum.</p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mrow><mi>x</mi><mfenced><mrow><mi>k</mi><mo>-</mo><mi>x</mi></mrow></mfenced></mrow></mfrac><mo>≡</mo><mfrac><mi>a</mi><mi>x</mi></mfrac><mo>+</mo><mfrac><mi>b</mi><mrow><mi>k</mi><mo>-</mo><mi>x</mi></mrow></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mfenced><mrow><mi>k</mi><mo>-</mo><mi>x</mi></mrow></mfenced><mo>+</mo><mi>b</mi><mi>x</mi><mo>=</mo><mn>1</mn></math> <em><strong> (A1)</strong></em></p>
<p>attempt to compare coefficients OR substitute <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mi>k</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn></math> and solve <em><strong> (M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mfrac><mn>1</mn><mi>k</mi></mfrac></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>=</mo><mfrac><mn>1</mn><mi>k</mi></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mfrac><mn>1</mn><mrow><mi>k</mi><mi>x</mi></mrow></mfrac><mo>+</mo><mfrac><mn>1</mn><mrow><mi>k</mi><mfenced><mrow><mi>k</mi><mo>-</mo><mi>x</mi></mrow></mfenced></mrow></mfrac></math></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to integrate their <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>a</mi><mi>x</mi></mfrac><mo>+</mo><mfrac><mi>b</mi><mrow><mi>k</mi><mo>-</mo><mi>x</mi></mrow></mfrac></math> <em><strong> (M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mfrac><mn>1</mn><mi>k</mi></mfrac><mo>∫</mo><mfenced><mrow><mfrac><mn>1</mn><mi>x</mi></mfrac><mo>+</mo><mfrac><mn>1</mn><mrow><mi>k</mi><mo>-</mo><mi>x</mi></mrow></mfrac></mrow></mfenced><mo>d</mo><mi>x</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>1</mn><mi>k</mi></mfrac><mfenced><mrow><mi>ln</mi><mfenced open="|" close="|"><mi>x</mi></mfenced><mo>-</mo><mi>ln</mi><mfenced open="|" close="|"><mrow><mi>k</mi><mo>-</mo><mi>x</mi></mrow></mfenced></mrow></mfenced><mfenced><mrow><mo>+</mo><mi>c</mi></mrow></mfenced><mfenced><mrow><mo>=</mo><mfrac><mn>1</mn><mi>k</mi></mfrac><mi>ln</mi><mfenced open="|" close="|"><mfrac><mi>x</mi><mrow><mi>k</mi><mo>-</mo><mi>x</mi></mrow></mfrac></mfenced><mfenced><mrow><mo>+</mo><mi>c</mi></mrow></mfenced></mrow></mfenced></math> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for each correct term. Award <em><strong>A1A0</strong></em> for a correct answer without modulus signs. Condone the absence of <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>+</mo><mi>c</mi></math>.</p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to separate variables and integrate both sides <em><strong> M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mi>k</mi><mo>∫</mo><mfrac><mn>1</mn><mrow><mi>P</mi><mfenced><mrow><mi>k</mi><mo>-</mo><mi>P</mi></mrow></mfenced></mrow></mfrac><mo>d</mo><mi>P</mi><mo>=</mo><mo>∫</mo><mn>1</mn><mo>d</mo><mi>t</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mfenced><mrow><mi>ln</mi><mo> </mo><mi>P</mi><mo>-</mo><mi>ln</mi><mfenced><mrow><mi>k</mi><mo>-</mo><mi>P</mi></mrow></mfenced></mrow></mfenced><mo>=</mo><mi>t</mi><mo>+</mo><mi>c</mi></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> There are variations on this which should be accepted, such as <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mi>k</mi></mfrac><mfenced><mrow><mi>ln</mi><mo> </mo><mi>P</mi><mo>-</mo><mi>ln</mi><mfenced><mrow><mi>k</mi><mo>-</mo><mi>P</mi></mrow></mfenced></mrow></mfenced><mo>=</mo><mfrac><mn>1</mn><mrow><mn>5</mn><mi>k</mi></mrow></mfrac><mi>t</mi><mo>+</mo><mi>c</mi></math>. Subsequent marks for these variations should be awarded as appropriate.</p>
<p> </p>
<p><strong>EITHER</strong></p>
<p>attempt to substitute <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn><mo>,</mo><mo> </mo><mi>P</mi><mo>=</mo><mn>1200</mn></math> into an equation involving <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math> <em><strong> M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>=</mo><mn>5</mn><mfenced><mrow><mi>ln</mi><mo> </mo><mn>1200</mn><mo>-</mo><mi>ln</mi><mfenced><mrow><mi>k</mi><mo>-</mo><mn>1200</mn></mrow></mfenced></mrow></mfenced><mfenced><mrow><mo>=</mo><mn>5</mn><mo> </mo><mi>ln</mi><mfenced><mfrac><mn>1200</mn><mrow><mi>k</mi><mo>-</mo><mn>1200</mn></mrow></mfrac></mfenced></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mfenced><mrow><mi>ln</mi><mo> </mo><mi>P</mi><mo>-</mo><mi>ln</mi><mfenced><mrow><mi>k</mi><mo>-</mo><mi>P</mi></mrow></mfenced></mrow></mfenced><mo>=</mo><mi>t</mi><mo>+</mo><mn>5</mn><mfenced><mrow><mi>ln</mi><mo> </mo><mn>1200</mn><mo>-</mo><mi>ln</mi><mfenced><mrow><mi>k</mi><mo>-</mo><mn>1200</mn></mrow></mfenced></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mfenced><mfrac><mrow><mi>P</mi><mfenced><mrow><mi>k</mi><mo>-</mo><mn>1200</mn></mrow></mfenced></mrow><mrow><mn>1200</mn><mfenced><mrow><mi>k</mi><mo>-</mo><mi>P</mi></mrow></mfenced></mrow></mfrac></mfenced><mo>=</mo><mfrac><mi>t</mi><mn>5</mn></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>P</mi><mfenced><mrow><mi>k</mi><mo>-</mo><mn>1200</mn></mrow></mfenced></mrow><mrow><mn>1200</mn><mfenced><mrow><mi>k</mi><mo>-</mo><mi>P</mi></mrow></mfenced></mrow></mfrac><mo>=</mo><msup><mtext>e</mtext><mfrac><mi>t</mi><mn>5</mn></mfrac></msup></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mfenced><mfrac><mi>P</mi><mrow><mi>k</mi><mo>-</mo><mi>P</mi></mrow></mfrac></mfenced><mo>=</mo><mfrac><mrow><mi>t</mi><mo>+</mo><mi>c</mi></mrow><mn>5</mn></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>P</mi><mrow><mi>k</mi><mo>-</mo><mi>P</mi></mrow></mfrac><mo>=</mo><mi>A</mi><msup><mtext>e</mtext><mfrac><mi>t</mi><mn>5</mn></mfrac></msup></math> <em><strong>A1</strong></em></p>
<p>attempt to substitute <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn><mo>,</mo><mo> </mo><mi>P</mi><mo>=</mo><mn>1200</mn></math> <em><strong> M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1200</mn><mrow><mi>k</mi><mo>-</mo><mn>1200</mn></mrow></mfrac><mo>=</mo><mi>A</mi></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>P</mi><mrow><mi>k</mi><mo>-</mo><mi>P</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>1200</mn><msup><mtext>e</mtext><mstyle displaystyle="true"><mfrac><mi>t</mi><mn>5</mn></mfrac></mstyle></msup></mrow><mrow><mi>k</mi><mo>-</mo><mn>1200</mn></mrow></mfrac></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>THEN</strong></p>
<p>attempt to rearrange and isolate <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> <em><strong> M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mi>k</mi><mo>-</mo><mn>1200</mn><mi>P</mi><mo>=</mo><mn>1200</mn><mi>k</mi><msup><mtext>e</mtext><mfrac><mi>t</mi><mn>5</mn></mfrac></msup><mo>-</mo><mn>1200</mn><mi>P</mi><msup><mtext>e</mtext><mfrac><mi>t</mi><mn>5</mn></mfrac></msup></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mi>k</mi><msup><mtext>e</mtext><mrow><mi>-</mi><mfrac><mi>t</mi><mn>5</mn></mfrac></mrow></msup><mo>-</mo><mn>1200</mn><mi>P</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mfrac><mi>t</mi><mn>5</mn></mfrac></mrow></msup><mo> </mo><mi mathvariant="normal">=</mi><mn>1200</mn><mi>k</mi><mo>-</mo><mn>1200</mn><mi>P</mi></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>k</mi><mi>P</mi></mfrac><mo>-</mo><mn>1</mn><mo>=</mo><mfrac><mrow><mi>k</mi><mo>-</mo><mn>1200</mn></mrow><mrow><mn>1200</mn><msup><mtext>e</mtext><mstyle displaystyle="true"><mfrac><mi>t</mi><mn>5</mn></mfrac></mstyle></msup></mrow></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mfenced><mrow><mi>k</mi><mo>-</mo><mn>1200</mn><mo>+</mo><mn>1200</mn><msup><mtext>e</mtext><mfrac><mi>t</mi><mn>5</mn></mfrac></msup></mrow></mfenced><mo>=</mo><mn>1200</mn><mi>k</mi><msup><mtext>e</mtext><mfrac><mi>t</mi><mn>5</mn></mfrac></msup></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mfenced><mrow><mi>k</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mfrac><mi>t</mi><mn>5</mn></mfrac></mrow></msup><mo>-</mo><mn>1200</mn><msup><mtext>e</mtext><mrow><mo>-</mo><mfrac><mi>t</mi><mn>5</mn></mfrac></mrow></msup><mo>+</mo><mn>1200</mn></mrow></mfenced><mo>=</mo><mn>1200</mn><mi>k</mi></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>=</mo><mfrac><mrow><mn>1200</mn><mi>k</mi></mrow><mrow><mfenced><mrow><mi>k</mi><mo>-</mo><mn>1200</mn></mrow></mfenced><msup><mtext>e</mtext><mrow><mo>-</mo><mstyle displaystyle="true"><mfrac><mi>t</mi><mn>5</mn></mfrac></mstyle></mrow></msup><mo>+</mo><mn>1200</mn></mrow></mfrac></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[8 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to substitute <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>10</mn><mo>,</mo><mo> </mo><mi>P</mi><mo>=</mo><mn>2400</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2400</mn><mo>=</mo><mfrac><mrow><mn>1200</mn><mi>k</mi></mrow><mrow><mfenced><mrow><mi>k</mi><mo>-</mo><mn>1200</mn></mrow></mfenced><msup><mtext>e</mtext><mrow><mo>-</mo><mn>2</mn></mrow></msup><mo>+</mo><mn>1200</mn></mrow></mfrac></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mn>2845</mn><mo>.</mo><mn>34</mn><mo>…</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mn>2845</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>(M1)(A1)A0</strong></em> for any other value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> which rounds to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2850</mn></math></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to find the maximum of the first derivative graph OR zero of the second derivative graph OR that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>=</mo><mfrac><mi>k</mi><mn>2</mn></mfrac><mfenced><mrow><mo>=</mo><mn>1422</mn><mo>.</mo><mn>67</mn><mo>…</mo></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>1</mn><mo>.</mo><mn>57814</mn><mo>…</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>1</mn><mo>.</mo><mn>58</mn></math> (days) <em><strong>A2</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Accept any value which rounds to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.</mo><mn>6</mn></math>.</p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the first three terms of the binomial expansion of <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>1</mn><mo>+</mo><mi>t</mi><msup><mo>)</mo><mrow><mo>-</mo><mn>1</mn></mrow></msup></math> in ascending powers of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By using the Maclaurin series for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mo> </mo><mi>x</mi></math> and the result from part (a), show that the Maclaurin series for <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>sec</mtext><mo> </mo><mi>x</mi></math> up to and including the term in <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>4</mn></msup></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>+</mo><mfrac><msup><mi>x</mi><mn>2</mn></msup><mn>2</mn></mfrac><mo>+</mo><mfrac><mrow><mn>5</mn><msup><mi>x</mi><mn>4</mn></msup></mrow><mn>24</mn></mfrac></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By using the Maclaurin series for <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>arctan</mtext><mo> </mo><mi>x</mi></math> and the result from part (b), find <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mfenced><mfrac><mrow><mi>x</mi><mtext> arctan</mtext><mo> </mo><mn>2</mn><mi>x</mi></mrow><mrow><mtext>sec</mtext><mo> </mo><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfrac></mfenced></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>-</mo><mi>t</mi><mo>+</mo><msup><mi>t</mi><mn>2</mn></msup></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>,</mo><mo> </mo><mo>-</mo><mi>t</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>t</mi><mn>2</mn></msup></math>.</p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>sec</mtext><mo> </mo><mi>x</mi><mo>=</mo><mfrac><mn>1</mn><mrow><mn>1</mn><mo>-</mo><mstyle displaystyle="true"><mfrac><msup><mi>x</mi><mn>2</mn></msup><mrow><mn>2</mn><mo>!</mo></mrow></mfrac></mstyle><mo>+</mo><mstyle displaystyle="true"><mfrac><msup><mi>x</mi><mn>4</mn></msup><mrow><mn>4</mn><mo>!</mo></mrow></mfrac></mstyle><mfenced><mrow><mo>-</mo><mo>…</mo></mrow></mfenced></mrow></mfrac><mo> </mo><mfenced><mrow><mo>=</mo><msup><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><msup><mi>x</mi><mn>2</mn></msup><mrow><mn>2</mn><mo>!</mo></mrow></mfrac><mo>+</mo><mfenced><mrow><mfrac><msup><mi>x</mi><mn>4</mn></msup><mrow><mn>4</mn><mo>!</mo></mrow></mfrac><mfenced><mrow><mo>-</mo><mo>…</mo></mrow></mfenced></mrow></mfenced></mrow></mfenced><mrow><mo>-</mo><mn>1</mn></mrow></msup></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mi>cos</mi><mo> </mo><mi>x</mi><mo>-</mo><mn>1</mn></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>sec</mtext><mo> </mo><mi>x</mi><mo>=</mo><mn>1</mn><mo>-</mo><mfenced><mrow><mi>cos</mi><mo> </mo><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mo>+</mo><msup><mfenced><mrow><mi>cos</mi><mo> </mo><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>1</mn><mo>-</mo><mfenced><mrow><mo>-</mo><mfrac><msup><mi>x</mi><mn>2</mn></msup><mrow><mn>2</mn><mo>!</mo></mrow></mfrac><mo>+</mo><mfrac><msup><mi>x</mi><mn>4</mn></msup><mrow><mn>4</mn><mo>!</mo></mrow></mfrac><mfenced><mrow><mo>-</mo><mo>…</mo></mrow></mfenced></mrow></mfenced><mo>+</mo><msup><mfenced><mrow><mo>-</mo><mfrac><msup><mi>x</mi><mn>2</mn></msup><mrow><mn>2</mn><mo>!</mo></mrow></mfrac><mo>+</mo><mfrac><msup><mi>x</mi><mn>4</mn></msup><mrow><mn>4</mn><mo>!</mo></mrow></mfrac><mfenced><mrow><mo>-</mo><mo>…</mo></mrow></mfenced></mrow></mfenced><mn>2</mn></msup></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>1</mn><mo>+</mo><mfrac><msup><mi>x</mi><mn>2</mn></msup><mn>2</mn></mfrac><mo>-</mo><mfrac><msup><mi>x</mi><mn>4</mn></msup><mn>24</mn></mfrac><mo>+</mo><mfrac><msup><mi>x</mi><mn>4</mn></msup><mn>4</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p>so the Maclaurin series for <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>sec</mtext><mo> </mo><mi>x</mi></math> up to and including the term in <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>4</mn></msup></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>+</mo><mfrac><msup><mi>x</mi><mn>2</mn></msup><mn>2</mn></mfrac><mo>+</mo><mfrac><mrow><mn>5</mn><msup><mi>x</mi><mn>4</mn></msup></mrow><mn>24</mn></mfrac></math> <em><strong>AG</strong></em></p>
<p><strong><br>Note:</strong> Condone the absence of ‘…’ </p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>arctan</mtext><mo> </mo><mn>2</mn><mi>x</mi><mo>=</mo><mn>2</mn><mi>x</mi><mo>-</mo><mfrac><msup><mfenced><mrow><mn>2</mn><mi>x</mi></mrow></mfenced><mn>3</mn></msup><mn>3</mn></mfrac><mo>+</mo><mo>…</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mfenced><mfrac><mrow><mi>x</mi><mtext> arctan</mtext><mo> </mo><mn>2</mn><mi>x</mi></mrow><mrow><mtext>sec</mtext><mo> </mo><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfrac></mfenced><mo>=</mo><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mfenced><mfrac><mrow><mi>x</mi><mfenced><mrow><mn>2</mn><mi>x</mi><mo>-</mo><mstyle displaystyle="true"><mfrac><msup><mfenced><mrow><mn>2</mn><mi>x</mi></mrow></mfenced><mn>3</mn></msup><mn>3</mn></mfrac></mstyle><mo>+</mo><mo>…</mo></mrow></mfenced></mrow><mrow><mfenced><mrow><mn>1</mn><mo>+</mo><mstyle displaystyle="true"><mfrac><msup><mi>x</mi><mn>2</mn></msup><mn>2</mn></mfrac></mstyle><mo>+</mo><mstyle displaystyle="true"><mfrac><mrow><mn>5</mn><msup><mi>x</mi><mn>4</mn></msup></mrow><mn>24</mn></mfrac></mstyle></mrow></mfenced><mo>-</mo><mn>1</mn></mrow></mfrac></mfenced></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mfenced><mfrac><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mstyle displaystyle="true"><mfrac><mrow><mn>8</mn><msup><mi>x</mi><mn>4</mn></msup></mrow><mn>3</mn></mfrac></mstyle><mo>+</mo><mo>…</mo></mrow><mrow><mstyle displaystyle="true"><mfrac><msup><mi>x</mi><mn>2</mn></msup><mn>2</mn></mfrac></mstyle><mo>+</mo><mstyle displaystyle="true"><mfrac><mrow><mn>5</mn><msup><mi>x</mi><mn>4</mn></msup></mrow><mn>24</mn></mfrac></mstyle></mrow></mfrac></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mfenced><mfrac><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mfenced><mrow><mn>1</mn><mo>-</mo><mstyle displaystyle="true"><mfrac><mrow><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup></mrow><mn>3</mn></mfrac></mstyle></mrow></mfenced></mrow><mstyle displaystyle="true"><mfrac><msup><mi>x</mi><mn>2</mn></msup><mn>2</mn></mfrac><mfenced><mrow><mn>1</mn><mo>+</mo><mfrac><mrow><mn>5</mn><msup><mi>x</mi><mn>2</mn></msup></mrow><mn>12</mn></mfrac></mrow></mfenced></mstyle></mfrac></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>4</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Condone missing ‘lim’ and errors in higher derivatives.<br>Do not award <em><strong>M1</strong></em> unless <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> is replaced by <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>x</mi></math> in <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>arctan</mtext></math>.</p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>A function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> is defined by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mrow><mi>k</mi><msup><mtext>e</mtext><mstyle displaystyle="true"><mfrac><mi>x</mi><mn>2</mn></mfrac></mstyle></msup></mrow><mrow><mn>1</mn><mo>+</mo><msup><mtext>e</mtext><mi>x</mi></msup></mrow></mfrac></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi><mo>,</mo><mo> </mo><mi>x</mi><mo>≥</mo><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>∈</mo><msup><mi mathvariant="normal">ℝ</mi><mo>+</mo></msup></math>.</p>
<p>The region enclosed by the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math>, the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis, the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-axis and the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mi>ln</mi><mo> </mo><mn>16</mn></math> is rotated <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>360</mn><mo>°</mo></math> about the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis to form a solid of revolution.</p>
</div>
<div class="specification">
<p>Pedro wants to make a small bowl with a volume of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>300</mn><mo> </mo><msup><mtext>cm</mtext><mn>3</mn></msup></math> based on the result from part (a). Pedro’s design is shown in the following diagrams.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<p style="text-align: left;">The vertical height of the bowl, <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>BO</mtext></math>, is measured along the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis. The radius of the bowl’s top is <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>OA</mtext></math> and the radius of the bowl’s base is <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>BC</mtext></math>. All lengths are measured in <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>cm</mtext></math>.</p>
</div>
<div class="specification">
<p>For design purposes, Pedro investigates how the cross-sectional radius of the bowl changes.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the volume of the solid formed is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>15</mn><msup><mi>k</mi><mn>2</mn></msup><mi mathvariant="normal">π</mi></mrow><mn>34</mn></mfrac></math> cubic units.</p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> that satisfies the requirements of Pedro’s design.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>OA</mtext></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>BC</mtext></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By sketching the graph of a suitable derivative of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math>, find where the cross-sectional radius of the bowl is decreasing most rapidly.</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State the cross-sectional radius of the bowl at this point.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>attempt to use <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mo>=</mo><mi mathvariant="normal">π</mi><munderover><mo>∫</mo><mi>a</mi><mi>b</mi></munderover><msup><mfenced><mrow><mi>f</mi><mfenced><mi>x</mi></mfenced></mrow></mfenced><mn>2</mn></msup><mo>d</mo><mi>x</mi></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mo>=</mo><mi mathvariant="normal">π</mi><munderover><mo>∫</mo><mn>0</mn><mrow><mi>ln</mi><mo> </mo><mn>16</mn></mrow></munderover><msup><mfenced><mfrac><mrow><mi>k</mi><msup><mtext>e</mtext><mstyle displaystyle="true"><mfrac><mi>x</mi><mn>2</mn></mfrac></mstyle></msup></mrow><mrow><mn>1</mn><mo>+</mo><msup><mtext>e</mtext><mi>x</mi></msup></mrow></mfrac></mfenced><mn>2</mn></msup><mo>d</mo><mi>x</mi><mo> </mo><mo> </mo><mfenced><mrow><mi>V</mi><mo>=</mo><msup><mi>k</mi><mn>2</mn></msup><mi mathvariant="normal">π</mi><munderover><mo>∫</mo><mn>0</mn><mrow><mi>ln</mi><mo> </mo><mn>16</mn></mrow></munderover><mfrac><msup><mtext>e</mtext><mi>x</mi></msup><msup><mfenced><mrow><mn>1</mn><mo>+</mo><msup><mtext>e</mtext><mi>x</mi></msup></mrow></mfenced><mn>2</mn></msup></mfrac><mo>d</mo><mi>x</mi></mrow></mfenced></math></p>
<p><br><strong>EITHER</strong></p>
<p>applying integration by recognition <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mi>k</mi><mn>2</mn></msup><mi mathvariant="normal">π</mi><msubsup><mfenced open="[" close="]"><mrow><mo>-</mo><mfrac><mn>1</mn><mrow><mn>1</mn><mo>+</mo><msup><mtext>e</mtext><mi>x</mi></msup></mrow></mfrac></mrow></mfenced><mn>0</mn><mrow><mi>ln</mi><mo> </mo><mn>16</mn></mrow></msubsup></math> <em><strong>A3</strong></em></p>
<p><br><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi><mo>=</mo><mn>1</mn><mo>+</mo><msup><mtext>e</mtext><mi>x</mi></msup><mo>⇒</mo><mo>d</mo><mi>u</mi><mo>=</mo><msup><mtext>e</mtext><mi>x</mi></msup><mo>d</mo><mi>x</mi></math> <em><strong>(A1)</strong></em></p>
<p>attempt to express the integral in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi></math> <em><strong>(M1)</strong></em></p>
<p>when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn><mo>,</mo><mo> </mo><mi>u</mi><mo>=</mo><mn>2</mn></math> and when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mi>ln</mi><mo> </mo><mn>16</mn><mo>,</mo><mo> </mo><mi>u</mi><mo>=</mo><mn>17</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mo>=</mo><msup><mi>k</mi><mn>2</mn></msup><mi mathvariant="normal">π</mi><munderover><mo>∫</mo><mn>2</mn><mn>17</mn></munderover><mfrac><mn>1</mn><msup><mi>u</mi><mn>2</mn></msup></mfrac><mo>d</mo><mi>u</mi></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mi>k</mi><mn>2</mn></msup><mi mathvariant="normal">π</mi><msubsup><mfenced open="[" close="]"><mrow><mo>-</mo><mfrac><mn>1</mn><mi>u</mi></mfrac></mrow></mfenced><mn>2</mn><mn>17</mn></msubsup></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi><mo>=</mo><msup><mtext>e</mtext><mi>x</mi></msup><mo>⇒</mo><mo>d</mo><mi>u</mi><mo>=</mo><msup><mtext>e</mtext><mi>x</mi></msup><mo>d</mo><mi>x</mi></math> <em><strong>(A1)</strong></em></p>
<p>attempt to express the integral in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi></math> <em><strong>(M1)</strong></em></p>
<p>when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn><mo>,</mo><mo> </mo><mi>u</mi><mo>=</mo><mn>1</mn></math> and when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mi>ln</mi><mo> </mo><mn>16</mn><mo>,</mo><mo> </mo><mi>u</mi><mo>=</mo><mn>16</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mo>=</mo><msup><mi>k</mi><mn>2</mn></msup><mi mathvariant="normal">π</mi><munderover><mo>∫</mo><mn>1</mn><mn>16</mn></munderover><mfrac><mn>1</mn><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>u</mi></mrow></mfenced><mn>2</mn></msup></mfrac><mo>d</mo><mi>u</mi></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mi>k</mi><mn>2</mn></msup><mi mathvariant="normal">π</mi><msubsup><mfenced open="[" close="]"><mrow><mo>-</mo><mfrac><mn>1</mn><mrow><mn>1</mn><mo>+</mo><mi>u</mi></mrow></mfrac></mrow></mfenced><mn>1</mn><mn>16</mn></msubsup></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Accept equivalent working with indefinite integrals and original limits for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>.</p>
<p> </p>
<p><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mi>k</mi><mn>2</mn></msup><mi mathvariant="normal">π</mi><mfenced><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>-</mo><mfrac><mn>1</mn><mn>17</mn></mfrac></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p>so the volume of the solid formed is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>15</mn><msup><mi>k</mi><mn>2</mn></msup><mi mathvariant="normal">π</mi></mrow><mn>34</mn></mfrac></math> cubic units <em><strong>AG</strong></em></p>
<p><strong><br>Note:</strong> Award <em><strong>(M1)(A0)(M0)(A0)(A0)(A1)</strong></em> when <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>15</mn><mn>34</mn></mfrac></math> is obtained from GDC</p>
<p> </p>
<p><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>a valid algebraic or graphical attempt to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>k</mi><mn>2</mn></msup><mo>=</mo><mfrac><mrow><mn>300</mn><mo>×</mo><mn>34</mn></mrow><mrow><mn>15</mn><mi mathvariant="normal">π</mi></mrow></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mn>14</mn><mo>.</mo><mn>7</mn><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mn>2</mn><msqrt><mfrac><mn>170</mn><mi mathvariant="normal">π</mi></mfrac></msqrt><mo>=</mo><msqrt><mfrac><mn>680</mn><mi mathvariant="normal">π</mi></mfrac></msqrt></mrow></mfenced></math> (as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>∈</mo><msup><mi mathvariant="normal">ℝ</mi><mo>+</mo></msup></math>) <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Candidates may use their GDC numerical solve feature.</p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempting to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>OA</mtext><mo>=</mo><mi>f</mi><mfenced><mn>0</mn></mfenced><mo>=</mo><mfrac><mi>k</mi><mn>2</mn></mfrac></math></p>
<p>with <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mn>14</mn><mo>.</mo><mn>712</mn><mo>…</mo><mo> </mo><mfenced><mrow><mo>=</mo><mn>2</mn><msqrt><mfrac><mn>170</mn><mi mathvariant="normal">π</mi></mfrac></msqrt><mo>=</mo><msqrt><mfrac><mn>680</mn><mi mathvariant="normal">π</mi></mfrac></msqrt></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>OA</mtext><mo>=</mo><mn>7</mn><mo>.</mo><mn>36</mn><mo> </mo><mfenced><mrow><mo>=</mo><msqrt><mfrac><mn>170</mn><mi mathvariant="normal">π</mi></mfrac></msqrt></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempting to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>BC</mtext><mo>=</mo><mi>f</mi><mfenced><mrow><mi>ln</mi><mo> </mo><mn>16</mn></mrow></mfenced><mo>=</mo><mfrac><mrow><mn>4</mn><mi>k</mi></mrow><mn>17</mn></mfrac></math></p>
<p>with <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mn>14</mn><mo>.</mo><mn>712</mn><mo>…</mo><mo> </mo><mfenced><mrow><mo>=</mo><mn>2</mn><msqrt><mfrac><mn>170</mn><mi mathvariant="normal">π</mi></mfrac></msqrt><mo>=</mo><msqrt><mfrac><mn>680</mn><mi mathvariant="normal">π</mi></mfrac></msqrt></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>BC</mtext><mo>=</mo><mn>3</mn><mo>.</mo><mn>46</mn><mo> </mo><mfenced><mrow><mo>=</mo><mfrac><mn>8</mn><mn>17</mn></mfrac><msqrt><mfrac><mn>170</mn><mi mathvariant="normal">π</mi></mfrac></msqrt><mo>=</mo><mfrac><mrow><mn>8</mn><msqrt><mn>10</mn></msqrt></mrow><msqrt><mn>17</mn><mi mathvariant="normal">π</mi></msqrt></mfrac></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p>recognising to graph <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><strong>Note:</strong> Award <strong>M1</strong> for attempting to use quotient rule or product rule differentiation. <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mrow><mi>k</mi><msup><mtext>e</mtext><mstyle displaystyle="true"><mfrac><mi>x</mi><mn>2</mn></mfrac></mstyle></msup><mfenced><mrow><mn>1</mn><mo>-</mo><msup><mtext>e</mtext><mi>x</mi></msup></mrow></mfenced></mrow><mrow><mn>2</mn><msup><mfenced><mrow><mn>1</mn><mo>+</mo><msup><mtext>e</mtext><mi>x</mi></msup></mrow></mfenced><mn>2</mn></msup></mrow></mfrac></math></p>
<p><img style="display:block;margin-left:auto;margin-right:auto;" src="data:image/png;base64,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"><br>for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>></mo><mn>0</mn></math> graph decreasing to the local minimum <em><strong>A1</strong></em></p>
<p>before increasing towards the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>OR</strong></p>
<p>recognising to graph <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo mathvariant="italic">=</mo><mi>f</mi><mo mathvariant="italic">''</mo><mfenced><mi mathvariant="italic">x</mi></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1</strong></em> for attempting to use quotient rule or product rule differentiation. <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>''</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mrow><mi>k</mi><msup><mtext>e</mtext><mstyle displaystyle="true"><mfrac><mi>x</mi><mn>2</mn></mfrac></mstyle></msup><mfenced><mrow><msup><mtext>e</mtext><mrow><mn>2</mn><mi>x</mi></mrow></msup><mo>-</mo><mn>6</mn><msup><mtext>e</mtext><mi>x</mi></msup><mi>+1</mi></mrow></mfenced></mrow><mrow><mn>4</mn><msup><mfenced><mrow><mn>1</mn><mo>+</mo><msup><mtext>e</mtext><mi>x</mi></msup></mrow></mfenced><mn>3</mn></msup></mrow></mfrac></math></p>
<p><img style="display:block;margin-left:auto;margin-right:auto;" 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">for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>></mo><mn>0</mn></math>, graph increasing towards and beyond the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-intercept <em><strong>A1</strong></em></p>
<p>recognising <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>''</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>0</mn></math> for maximum rate <em><strong>(A1)</strong></em></p>
<p> </p>
<p><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>1</mn><mo>.</mo><mn>76</mn><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mi>ln</mi><mfenced><mrow><mn>2</mn><msqrt><mn>2</mn></msqrt><mo>+</mo><mn>3</mn></mrow></mfenced></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note</strong>: Only award <em><strong>A</strong> </em>marks if either graph is seen.</p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempting to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mrow><mn>1</mn><mo>.</mo><mn>76</mn><mo>…</mo></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p>the cross-sectional radius at this point is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mo>.</mo><mn>20</mn><mo> </mo><mfenced><msqrt><mfrac><mn>85</mn><mi mathvariant="normal">π</mi></mfrac></msqrt></mfenced><mo> </mo><mfenced><mtext>cm</mtext></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">d.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.ii.</div>
</div>
<br><hr><br><div class="question">
<p>The function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> is defined by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = {\left( {x - 1} \right)^2}">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> ≥ 1 and the function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g">
<mi>g</mi>
</math></span> is defined by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g\left( x \right) = {x^2} + 1">
<mi>g</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>1</mn>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> ≥ 0.</p>
<p>The region <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="R">
<mi>R</mi>
</math></span> is bounded by the curves <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f\left( x \right)">
<mi>y</mi>
<mo>=</mo>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = g\left( x \right)">
<mi>y</mi>
<mo>=</mo>
<mi>g</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span> and the lines <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 0">
<mi>y</mi>
<mo>=</mo>
<mn>0</mn>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 0">
<mi>x</mi>
<mo>=</mo>
<mn>0</mn>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 9">
<mi>y</mi>
<mo>=</mo>
<mn>9</mn>
</math></span> as shown on the following diagram.</p>
<p style="text-align: center;"><img 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"></p>
<p style="text-align: left;">The shape of a clay vase can be modelled by rotating the region <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="R">
<mi>R</mi>
</math></span> through 360˚ about the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span>-axis.</p>
<p style="text-align: left;">Find the volume of clay used to make the vase.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p>volume <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \pi {\int_0^9 {\left( {{y^{\frac{1}{2}}} + 1} \right)} ^2}{\text{d}}y - \pi \int_1^9 {\left( {y - 1} \right)} {\text{d}}y">
<mo>=</mo>
<mi>π</mi>
<mrow>
<msubsup>
<mo>∫</mo>
<mn>0</mn>
<mn>9</mn>
</msubsup>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msup>
<mi>y</mi>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
</msup>
</mrow>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
<mo>−</mo>
<mi>π</mi>
<msubsup>
<mo>∫</mo>
<mn>1</mn>
<mn>9</mn>
</msubsup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>y</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</math></span> <em><strong>(M1)</strong></em><em><strong>(M1)</strong></em><em><strong>(M1)(A1)(A1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for use of formula for rotating about <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span>-axis, <em><strong>(M1)</strong></em> for finding at least one inverse, <em><strong>(M1)</strong></em> for subtracting volumes, <em><strong>(A1)</strong></em><em><strong>(A1)</strong></em>for each correct expression, including limits.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 268.6 \ldots - 100.5 \ldots \left( {85.5\pi - 32\pi } \right)">
<mo>=</mo>
<mn>268.6</mn>
<mo>…</mo>
<mo>−</mo>
<mn>100.5</mn>
<mo>…</mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>85.5</mn>
<mi>π</mi>
<mo>−</mo>
<mn>32</mn>
<mi>π</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 168\left( { = 53.5\pi } \right)">
<mo>=</mo>
<mn>168</mn>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mn>53.5</mn>
<mi>π</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A2</strong></em></p>
<p><em><strong>[7 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>Consider the function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = \frac{{\sqrt x }}{{\sin x}},{\text{ }}0 < x < \pi ">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mrow>
<msqrt>
<mi>x</mi>
</msqrt>
</mrow>
<mrow>
<mi>sin</mi>
<mo><!-- --></mo>
<mi>x</mi>
</mrow>
</mfrac>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>0</mn>
<mo><</mo>
<mi>x</mi>
<mo><</mo>
<mi>π<!-- π --></mi>
</math></span>.</p>
</div>
<div class="specification">
<p>Consider the region bounded by the curve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f(x)">
<mi>y</mi>
<mo>=</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span>, the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>-axis and the lines <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \frac{\pi }{6},{\text{ }}x = \frac{\pi }{3}">
<mi>x</mi>
<mo>=</mo>
<mfrac>
<mi>π<!-- π --></mi>
<mn>6</mn>
</mfrac>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>x</mi>
<mo>=</mo>
<mfrac>
<mi>π<!-- π --></mi>
<mn>3</mn>
</mfrac>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>-coordinate of the minimum point on the curve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f(x)"> <mi>y</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></span> satisfies the equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\tan x = 2x"> <mi>tan</mi> <mo></mo> <mi>x</mi> <mo>=</mo> <mn>2</mn> <mi>x</mi> </math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> for which <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x)"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></span> is a decreasing function.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f(x)"> <mi>y</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></span> showing clearly the minimum point and any asymptotic behaviour.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the coordinates of the point on the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f"> <mi>f</mi> </math></span> where the normal to the graph is parallel to the line <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = - x"> <mi>y</mi> <mo>=</mo> <mo>−</mo> <mi>x</mi> </math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>This region is now rotated through <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2\pi "> <mn>2</mn> <mi>π</mi> </math></span> radians about the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>-axis. Find the volume of revolution.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>attempt to use quotient rule or product rule <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f’(x) = \frac{{\sin x\left( {\frac{1}{2}{x^{ - \frac{1}{2}}}} \right) - \sqrt x \cos x}}{{{{\sin }^2}x}}{\text{ }}\left( { = \frac{1}{{2\sqrt x \sin x}} - \frac{{\sqrt x \cos x}}{{{{\sin }^2}x}}} \right)"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mfrac> <mrow> <mi>sin</mi> <mo></mo> <mi>x</mi> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mrow> <msup> <mi>x</mi> <mrow> <mo>−</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </msup> </mrow> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <msqrt> <mi>x</mi> </msqrt> <mi>cos</mi> <mo></mo> <mi>x</mi> </mrow> <mrow> <mrow> <msup> <mrow> <mi>sin</mi> </mrow> <mn>2</mn> </msup> </mrow> <mi>x</mi> </mrow> </mfrac> <mrow> <mtext> </mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <msqrt> <mi>x</mi> </msqrt> <mi>sin</mi> <mo></mo> <mi>x</mi> </mrow> </mfrac> <mo>−</mo> <mfrac> <mrow> <msqrt> <mi>x</mi> </msqrt> <mi>cos</mi> <mo></mo> <mi>x</mi> </mrow> <mrow> <mrow> <msup> <mrow> <mi>sin</mi> </mrow> <mn>2</mn> </msup> </mrow> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> <strong><em>A1A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>A1 </em></strong>for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{{2\sqrt x \sin x}}"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <msqrt> <mi>x</mi> </msqrt> <mi>sin</mi> <mo></mo> <mi>x</mi> </mrow> </mfrac> </math></span> or equivalent and <strong><em>A1 </em></strong>for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - \frac{{\sqrt x \cos x}}{{{{\sin }^2}x}}"> <mo>−</mo> <mfrac> <mrow> <msqrt> <mi>x</mi> </msqrt> <mi>cos</mi> <mo></mo> <mi>x</mi> </mrow> <mrow> <mrow> <msup> <mrow> <mi>sin</mi> </mrow> <mn>2</mn> </msup> </mrow> <mi>x</mi> </mrow> </mfrac> </math></span> or equivalent.</p>
<p> </p>
<p>setting <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f’(x) = 0"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{\sin x}}{{2\sqrt x }} - \sqrt x \cos x = 0"> <mfrac> <mrow> <mi>sin</mi> <mo></mo> <mi>x</mi> </mrow> <mrow> <mn>2</mn> <msqrt> <mi>x</mi> </msqrt> </mrow> </mfrac> <mo>−</mo> <msqrt> <mi>x</mi> </msqrt> <mi>cos</mi> <mo></mo> <mi>x</mi> <mo>=</mo> <mn>0</mn> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{\sin x}}{{2\sqrt x }} = \sqrt x \cos x"> <mfrac> <mrow> <mi>sin</mi> <mo></mo> <mi>x</mi> </mrow> <mrow> <mn>2</mn> <msqrt> <mi>x</mi> </msqrt> </mrow> </mfrac> <mo>=</mo> <msqrt> <mi>x</mi> </msqrt> <mi>cos</mi> <mo></mo> <mi>x</mi> </math></span> or equivalent <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\tan x = 2x"> <mi>tan</mi> <mo></mo> <mi>x</mi> <mo>=</mo> <mn>2</mn> <mi>x</mi> </math></span> <strong><em>AG</em></strong></p>
<p><strong><em>[5 marks]</em></strong></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 1.17"> <mi>x</mi> <mo>=</mo> <mn>1.17</mn> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0 < x \leqslant 1.17"> <mn>0</mn> <mo><</mo> <mi>x</mi> <mo>⩽</mo> <mn>1.17</mn> </math></span> <strong><em>A1A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>A1 </em></strong>for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0 < x"> <mn>0</mn> <mo><</mo> <mi>x</mi> </math></span> and <strong><em>A1 </em></strong>for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x \leqslant 1.17"> <mi>x</mi> <mo>⩽</mo> <mn>1.17</mn> </math></span>. Accept <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x < 1.17"> <mi>x</mi> <mo><</mo> <mn>1.17</mn> </math></span>.</p>
<p> </p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img src="images/Schermafbeelding_2018-02-08_om_16.19.25.png" alt="N17/5/MATHL/HP2/ENG/TZ0/10.b/M"></p>
<p>concave up curve over correct domain with one minimum point above the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>-axis. <strong><em>A1</em></strong></p>
<p>approaches <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 0"> <mi>x</mi> <mo>=</mo> <mn>0</mn> </math></span> asymptotically <strong><em>A1</em></strong></p>
<p>approaches <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \pi "> <mi>x</mi> <mo>=</mo> <mi>π</mi> </math></span> asymptotically <strong><em>A1</em></strong></p>
<p> </p>
<p>Note: For the final <strong><em>A1 </em></strong>an asymptote must be seen, and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\pi "> <mi>π</mi> </math></span> must be seen on the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>-axis or in an equation.</p>
<p> </p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f’(x){\text{ }}\left( { = \frac{{\sin x\left( {\frac{1}{2}{x^{ - \frac{1}{2}}}} \right) - \sqrt x \cos x}}{{{{\sin }^2}x}}} \right) = 1"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow> <mtext> </mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mfrac> <mrow> <mi>sin</mi> <mo></mo> <mi>x</mi> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mrow> <msup> <mi>x</mi> <mrow> <mo>−</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </msup> </mrow> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <msqrt> <mi>x</mi> </msqrt> <mi>cos</mi> <mo></mo> <mi>x</mi> </mrow> <mrow> <mrow> <msup> <mrow> <mi>sin</mi> </mrow> <mn>2</mn> </msup> </mrow> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>1</mn> </math></span> <strong><em>(A1)</em></strong></p>
<p>attempt to solve for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 1.96"> <mi>x</mi> <mo>=</mo> <mn>1.96</mn> </math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f(1.96 \ldots )"> <mi>y</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mn>1.96</mn> <mo>…</mo> <mo stretchy="false">)</mo> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 1.51"> <mo>=</mo> <mn>1.51</mn> </math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="V = \pi \int_{\frac{\pi }{6}}^{\frac{\pi }{3}} {\frac{{x{\text{d}}x}}{{{{\sin }^2}x}}} "> <mi>V</mi> <mo>=</mo> <mi>π</mi> <msubsup> <mo>∫</mo> <mrow> <mfrac> <mi>π</mi> <mn>6</mn> </mfrac> </mrow> <mrow> <mfrac> <mi>π</mi> <mn>3</mn> </mfrac> </mrow> </msubsup> <mrow> <mfrac> <mrow> <mi>x</mi> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> <mrow> <mrow> <msup> <mrow> <mi>sin</mi> </mrow> <mn>2</mn> </msup> </mrow> <mi>x</mi> </mrow> </mfrac> </mrow> </math></span> <strong><em>(M1)(A1)</em></strong></p>
<p> </p>
<p><strong>Note:</strong> <strong><em>M1 </em></strong>is for an integral of the correct squared function (with or without limits and/or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\pi "> <mi>π</mi> </math></span>).</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 2.68{\text{ }}( = 0.852\pi )"> <mo>=</mo> <mn>2.68</mn> <mrow> <mtext> </mtext> </mrow> <mo stretchy="false">(</mo> <mo>=</mo> <mn>0.852</mn> <mi>π</mi> <mo stretchy="false">)</mo> </math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the differential equation</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mi>f</mi><mfenced><mfrac><mi>y</mi><mi>x</mi></mfrac></mfenced><mo>,</mo><mo> </mo><mi>x</mi><mo>></mo><mn>0</mn></math></p>
</div>
<div class="specification">
<p>The curve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mfenced><mi>x</mi></mfenced></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>></mo><mn>0</mn></math> has a gradient function given by</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn><mi>x</mi><mi>y</mi><mo>+</mo><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup></mrow><msup><mi>x</mi><mn>2</mn></msup></mfrac></math>.</p>
<p>The curve passes through the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>1</mn><mo>,</mo><mo> </mo><mo>-</mo><mn>1</mn></mrow></mfenced></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use the substitution <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>v</mi><mi>x</mi></math> to show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><mfrac><mrow><mo>d</mo><mi>v</mi></mrow><mrow><mi>f</mi><mfenced><mi>v</mi></mfenced><mo>-</mo><mi>v</mi></mrow></mfrac><mo>=</mo><mi>ln</mi><mo> </mo><mi>x</mi><mo>+</mo><mi>C</mi></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math> is an arbitrary constant.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By using the result from part (a) or otherwise, solve the differential equation and hence show that the curve has equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>x</mi><mfenced><mrow><mi>tan</mi><mo> </mo><mfenced><mrow><mi>ln</mi><mo> </mo><mi>x</mi></mrow></mfenced><mo>-</mo><mn>1</mn></mrow></mfenced></math>.</p>
<div class="marks">[9]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The curve has a point of inflexion at <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><msub><mi>x</mi><mn>1</mn></msub><mo>,</mo><mo> </mo><msub><mi>y</mi><mn>1</mn></msub></mrow></mfenced></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>e</mtext><mrow><mo>-</mo><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></mrow></msup><mo><</mo><msub><mi>x</mi><mn>1</mn></msub><mo><</mo><msup><mtext>e</mtext><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></msup></math>. Determine the coordinates of this point of inflexion.</p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use the differential equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn><mi>x</mi><mi>y</mi><mo>+</mo><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup></mrow><msup><mi>x</mi><mn>2</mn></msup></mfrac></math> to show that the points of zero gradient on the curve lie on two straight lines of the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>m</mi><mi>x</mi></math> where the values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math> are to be determined.</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color:#999;font-size:90%;font-style:italic;">* This sample question was produced by experienced DP mathematics senior examiners to aid teachers in preparing for external assessment in the new MAA course. There may be minor differences in formatting compared to formal exam papers.</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>v</mi><mi>x</mi><mo>⇒</mo><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mi>v</mi><mo>+</mo><mi>x</mi><mfrac><mrow><mo>d</mo><mi>v</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac></math> <strong>M1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mo>+</mo><mi>x</mi><mfrac><mrow><mo>d</mo><mi>v</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mi>f</mi><mfenced><mi>v</mi></mfenced></math> <strong>A1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><mfrac><mrow><mo>d</mo><mi>v</mi></mrow><mrow><mi>f</mi><mfenced><mi>v</mi></mfenced><mo>-</mo><mi>v</mi></mrow></mfrac><mo>=</mo><mo>∫</mo><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mi>x</mi></mfrac></math> <strong>A1</strong></p>
<p>integrating the RHS, <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><mfrac><mrow><mo>d</mo><mi>v</mi></mrow><mrow><mi>f</mi><mfenced><mi>v</mi></mfenced><mo>-</mo><mi>v</mi></mrow></mfrac><mo>=</mo><mi>ln</mi><mo> </mo><mi>x</mi><mo>+</mo><mi>C</mi></math> <strong>AG</strong></p>
<p> </p>
<p><strong>[3 marks]</strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p>attempts to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>v</mi></mfenced></math> <strong>M1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>v</mi></mfenced><mo>=</mo><msup><mi>v</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn><mi>v</mi><mo>+</mo><mn>2</mn></math> <strong>(A1)</strong></p>
<p>substitutes their <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>v</mi></mfenced></math> into <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><mfrac><mrow><mo>d</mo><mi>v</mi></mrow><mrow><mi>f</mi><mfenced><mi>v</mi></mfenced><mo>-</mo><mi>v</mi></mrow></mfrac></math> <strong>M1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><mfrac><mrow><mo>d</mo><mi>v</mi></mrow><mrow><mi>f</mi><mfenced><mi>v</mi></mfenced><mo>-</mo><mi>v</mi></mrow></mfrac><mo>=</mo><mo>∫</mo><mfrac><mrow><mo>d</mo><mi>v</mi></mrow><mrow><msup><mi>v</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>v</mi><mo>+</mo><mn>2</mn></mrow></mfrac></math></p>
<p>attempts to complete the square <strong>(M1)</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><mfrac><mrow><mo>d</mo><mi>v</mi></mrow><mrow><msup><mfenced><mrow><mi>v</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfrac></math> <strong>A1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>arctan</mtext><mo> </mo><mfenced><mrow><mi>v</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo> </mo><mfenced><mrow><mo>=</mo><mi>ln</mi><mo> </mo><mi>x</mi><mo>+</mo><mi>C</mi></mrow></mfenced></math> <strong>A1</strong></p>
<p> </p>
<p><strong>OR</strong></p>
<p>attempts to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>v</mi></mfenced></math> <strong>M1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mo>+</mo><mi>x</mi><mfrac><mrow><mo>d</mo><mi>v</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><msup><mi>v</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn><mi>v</mi><mo>+</mo><mn>2</mn></math> <strong>A1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><mfrac><mrow><mo>d</mo><mi>v</mi></mrow><mrow><msup><mi>v</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>v</mi><mo>+</mo><mn>2</mn></mrow></mfrac><mo>=</mo><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mi>x</mi></mfrac></math> <strong>M1</strong></p>
<p>attempts to complete the square <strong>(M1)</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><mfrac><mrow><mo>d</mo><mi>v</mi></mrow><mrow><msup><mfenced><mrow><mi>v</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfrac><mfenced><mrow><mo>=</mo><mo>∫</mo><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mi>x</mi></mfrac></mrow></mfenced></math> <strong>A1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>arctan</mtext><mo> </mo><mfenced><mrow><mi>v</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo> </mo><mfenced><mrow><mo>=</mo><mi>ln</mi><mo> </mo><mi>x</mi><mo>+</mo><mi>C</mi></mrow></mfenced></math> <strong>A1</strong></p>
<p> </p>
<p><strong>THEN</strong></p>
<p>when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>1</mn></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mo>=</mo><mo>-</mo><mn>1</mn></math> (or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>-</mo><mn>1</mn></math>) and so <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mo>=</mo><mn>0</mn></math> <strong>M1</strong></p>
<p>substitutes for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi></math> into their expression <strong>M1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>arctan</mtext><mo> </mo><mfenced><mrow><mfrac><mi>y</mi><mi>x</mi></mfrac><mo>+</mo><mn>1</mn></mrow></mfenced><mo>=</mo><mi>ln</mi><mo> </mo><mi>x</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>y</mi><mi>x</mi></mfrac><mo>+</mo><mn>1</mn><mo>=</mo><mi>tan</mi><mo> </mo><mfenced><mrow><mi>ln</mi><mo> </mo><mi>x</mi></mrow></mfenced></math> <strong>A1</strong></p>
<p>so <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>x</mi><mfenced><mrow><mi>tan</mi><mo> </mo><mfenced><mrow><mi>ln</mi><mo> </mo><mi>x</mi></mrow></mfenced><mo>-</mo><mn>1</mn></mrow></mfenced></math> <strong>AG</strong></p>
<p> </p>
<p><strong>[9 marks]</strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p><strong>EITHER</strong></p>
<p>a correct graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced></math> (for approximately <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>e</mtext><mrow><mo>-</mo><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></mrow></msup><mo><</mo><mi>x</mi><mo><</mo><msup><mtext>e</mtext><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></msup></math>) with a local minimum point below the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis <strong>A2</strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong>M1A1</strong> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mi>tan</mi><mo> </mo><mfenced><mrow><mi>ln</mi><mo> </mo><mi>x</mi></mrow></mfenced><mo>+</mo><msup><mtext>sec</mtext><mn>2</mn></msup><mo> </mo><mfenced><mrow><mi>ln</mi><mo> </mo><mi>x</mi></mrow></mfenced><mo>-</mo><mn>1</mn></math>.</p>
<p> </p>
<p>attempts to find the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-coordinate of the local minimum point on the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced></math> <strong>(M1)</strong></p>
<p style="text-align:center;"><img 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"></p>
<p style="text-align:left;"><strong>OR</strong></p>
<p style="text-align:left;">a correct graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mo>''</mo><mfenced><mi>x</mi></mfenced></math> (for approximately <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>e</mtext><mrow><mo>-</mo><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></mrow></msup><mo><</mo><mi>x</mi><mo><</mo><msup><mtext>e</mtext><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></msup></math>) showing the location of the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-intercept <strong>A2</strong></p>
<p style="text-align:left;"> </p>
<p style="text-align:left;"><strong>Note:</strong> Award <strong>M1A1</strong> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>y</mi></mrow><mrow><mo>d</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mfrac><mrow><msup><mtext>sec</mtext><mn>2</mn></msup><mo> </mo><mfenced><mrow><mi>ln</mi><mo> </mo><mi>x</mi></mrow></mfenced></mrow><mi>x</mi></mfrac><mo>+</mo><mfrac><mrow><mn>2</mn><mo> </mo><msup><mtext>sec</mtext><mn>2</mn></msup><mo> </mo><mfenced><mrow><mi>ln</mi><mo> </mo><mi>x</mi></mrow></mfenced><mo> </mo><mi>tan</mi><mo> </mo><mfenced><mrow><mi>ln</mi><mo> </mo><mi>x</mi></mrow></mfenced></mrow><mi>x</mi></mfrac></math>.</p>
<p> </p>
<p>attempts to find the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-intercept <strong>(M1)</strong></p>
<p style="text-align:center;"><img src="data:image/png;base64,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"></p>
<p style="text-align:left;"><strong>THEN</strong></p>
<p style="text-align:left;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>629</mn><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mtext>arctan</mtext><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup></mrow></mfenced></math> <strong>A1</strong></p>
<p>attempts to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mrow><mn>0</mn><mo>.</mo><mn>629</mn></mrow></mfenced><mo> </mo><mfenced><mrow><mi>f</mi><mfenced><msup><mtext>e</mtext><mrow><mo>-</mo><mtext>arctan</mtext><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup></mfenced></mrow></mfenced></math> <strong>(M1)</strong></p>
<p>the coordinates are <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>0</mn><mo>.</mo><mn>629</mn><mo>,</mo><mo> </mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>943</mn></mrow></mfenced><mo> </mo><mfenced><mrow><msup><mtext>e</mtext><mrow><mo>-</mo><mtext>arctan</mtext><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup><mi>,</mi><mo> </mo><mo>-</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mo> </mo><msup><mtext>e</mtext><mrow><mo>-</mo><mtext>arctan</mtext><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup></mrow></mfenced></math> <strong>A1</strong></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>attempts implicit differentiation on <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac></math> to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>y</mi></mrow><mrow><mo>d</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></math> <strong>M1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>y</mi></mrow><mrow><mo>d</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mfrac><mrow><mfenced><mrow><mn>2</mn><mi>y</mi><mo>+</mo><mn>3</mn><mi>x</mi></mrow></mfenced><mfenced><mrow><mi>x</mi><mstyle displaystyle="true"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac></mstyle><mo>-</mo><mi>y</mi></mrow></mfenced></mrow><msup><mi>x</mi><mn>3</mn></msup></mfrac></math> (or equivalent)</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>y</mi></mrow><mrow><mo>d</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mn>0</mn><mo>⇒</mo><mi>y</mi><mo>=</mo><mo>-</mo><mfrac><mrow><mn>3</mn><mi>x</mi></mrow><mn>2</mn></mfrac></math> (<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>≠</mo><mfrac><mi>y</mi><mi>x</mi></mfrac></math>) <strong>A1</strong></p>
<p>attempts to solve <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mrow><mn>3</mn><mi>x</mi></mrow><mn>2</mn></mfrac><mo>=</mo><mi>x</mi><mfenced><mrow><mi>tan</mi><mo> </mo><mfenced><mrow><mi>ln</mi><mo> </mo><mi>x</mi></mrow></mfenced><mo>-</mo><mn>1</mn></mrow></mfenced></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>e</mtext><mrow><mo>-</mo><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></mrow></msup><mo><</mo><mi>x</mi><mo><</mo><msup><mtext>e</mtext><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></msup></math> <strong>M1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>629</mn><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mtext>arctan</mtext><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup></mrow></mfenced></math> <strong>A1</strong></p>
<p>attempts to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mrow><mn>0</mn><mo>.</mo><mn>629</mn></mrow></mfenced><mo> </mo><mfenced><mrow><mi>f</mi><mfenced><mrow><mo>=</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mtext>arctan</mtext><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup></mrow></mfenced></mrow></mfenced></math> <strong>(M1)</strong></p>
<p>the coordinates are <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>0</mn><mo>.</mo><mn>629</mn><mo>,</mo><mo> </mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>943</mn></mrow></mfenced><mo> </mo><mfenced><mrow><msup><mtext>e</mtext><mrow><mo>-</mo><mtext>arctan</mtext><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup><mi>,</mi><mo> </mo><mo>-</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mo> </mo><msup><mtext>e</mtext><mrow><mo>-</mo><mtext>arctan</mtext><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup></mrow></mfenced></math> <strong>A1</strong></p>
<p> </p>
<p><strong>[6 marks]</strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mn>0</mn><mo>⇒</mo><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn><mi>x</mi><mi>y</mi><mo>+</mo><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>=</mo><mn>0</mn></math> <strong>M1</strong> </p>
<p>attempts to solve <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn><mi>x</mi><mi>y</mi><mo>+</mo><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>=</mo><mn>0</mn></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> <strong>M1</strong> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>y</mi><mo>+</mo><mn>2</mn><mi>x</mi></mrow></mfenced><mfenced><mrow><mi>y</mi><mo>+</mo><mi>x</mi></mrow></mfenced><mo>=</mo><mn>0</mn></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfrac><mrow><mo>-</mo><mn>3</mn><mi>x</mi><mo>±</mo><msqrt><msup><mfenced><mrow><mn>3</mn><mi>x</mi></mrow></mfenced><mn>2</mn></msup><mo>-</mo><mn>4</mn><mfenced><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced></msqrt></mrow><mn>2</mn></mfrac><mo> </mo><mfenced><mrow><mo>=</mo><mfrac><mrow><mo>-</mo><mn>3</mn><mi>x</mi><mo>±</mo><mi>x</mi></mrow><mn>2</mn></mfrac><mo>,</mo><mo> </mo><mfenced><mrow><mi>x</mi><mo>></mo><mn>0</mn></mrow></mfenced></mrow></mfenced></math> <strong>A1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>-</mo><mn>2</mn><mi>x</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>-</mo><mi>x</mi><mo> </mo><mfenced><mrow><mi>m</mi><mo>=</mo><mo>-</mo><mn>2</mn><mo>,</mo><mo> </mo><mo>-</mo><mn>1</mn></mrow></mfenced></math> <strong>A1</strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong>M1</strong> for stating <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mn>0</mn></math>, <strong>M1</strong> for substituting <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>m</mi><mi>x</mi></math> into <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mfenced><mrow><mo>=</mo><mn>0</mn></mrow></mfenced></math>, <strong>A1</strong> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>m</mi><mo>+</mo><mn>2</mn></mrow></mfenced><mfenced><mrow><mi>m</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>=</mo><mn>0</mn></math> and <strong>A1</strong> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mo>-</mo><mn>2</mn><mo>,</mo><mo> </mo><mo>-</mo><mn>1</mn><mo>⇒</mo><mi>y</mi><mo>=</mo><mo>-</mo><mn>2</mn><mi>x</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>-</mo><mi>x</mi></math>.</p>
<p> </p>
<p><strong>[4 marks]</strong></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the curve defined by the equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="4{x^2} + {y^2} = 7">
<mn>4</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>7</mn>
</math></span>.</p>
</div>
<div class="question">
<p>Find the volume of the solid formed when the region bounded by the curve, the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>-axis for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x \geqslant 0"> <mi>x</mi> <mo>⩾</mo> <mn>0</mn> </math></span> and the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y"> <mi>y</mi> </math></span>-axis for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y \geqslant 0"> <mi>y</mi> <mo>⩾</mo> <mn>0</mn> </math></span> is rotated through <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2\pi "> <mn>2</mn> <mi>π</mi> </math></span> about the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>-axis.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p>Use of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="V = \pi \int\limits_0^{\frac{{\sqrt 7 }}{2}} {{y^2}{\text{d}}x} "> <mi>V</mi> <mo>=</mo> <mi>π</mi> <munderover> <mo>∫</mo> <mn>0</mn> <mrow> <mfrac> <mrow> <msqrt> <mn>7</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> </mrow> </munderover> <mrow> <mrow> <msup> <mi>y</mi> <mn>2</mn> </msup> </mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="V = \pi \int\limits_0^{\frac{{\sqrt 7 }}{2}} {\left( {7 - 4{x^2}} \right){\text{d}}x} "> <mi>V</mi> <mo>=</mo> <mi>π</mi> <munderover> <mo>∫</mo> <mn>0</mn> <mrow> <mfrac> <mrow> <msqrt> <mn>7</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> </mrow> </munderover> <mrow> <mrow> <mo>(</mo> <mrow> <mn>7</mn> <mo>−</mo> <mn>4</mn> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mrow> <mo>)</mo> </mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </math></span> <strong><em>(M1)(A1)</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Condone absence of limits or incorrect limits for <strong><em>M </em></strong>mark.</p>
<p>Do not condone absence of or multiples of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\pi "> <mi>π</mi> </math></span>.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 19.4\,\,\,\left( { = \frac{{7\sqrt 7 \pi }}{3}} \right)"> <mo>=</mo> <mn>19.4</mn> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mfrac> <mrow> <mn>7</mn> <msqrt> <mn>7</mn> </msqrt> <mi>π</mi> </mrow> <mn>3</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>A function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> is defined by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mtext>arcsin</mtext><mfenced><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfrac></mfenced><mo>,</mo><mo> </mo><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
</div>
<div class="specification">
<p>A function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math> is defined by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mtext>arcsin</mtext><mfenced><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfrac></mfenced><mo>,</mo><mo> </mo><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi><mo>,</mo><mo> </mo><mi>x</mi><mo>≥</mo><mn>0</mn></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> is an even function.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By considering limits, show that the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> has a horizontal asymptote and state its equation.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mrow><mn>2</mn><mi>x</mi></mrow><mrow><msqrt><msup><mi>x</mi><mn>2</mn></msup></msqrt><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfenced></mrow></mfrac></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi><mo>,</mo><mo> </mo><mi>x</mi><mo>≠</mo><mn>0</mn></math>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By using the expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced></math> and the result <math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><msup><mi>x</mi><mn>2</mn></msup></msqrt><mo>=</mo><mfenced open="|" close="|"><mi>x</mi></mfenced></math>, show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> is decreasing for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo><</mo><mn>0</mn></math>.</p>
<p> </p>
<div class="marks">[3]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>g</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo>(</mo><mi>x</mi><mo>)</mo></math>, justifying your answer.</p>
<div class="marks">[5]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State the domain of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>g</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><msup><mi>g</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo>(</mo><mi>x</mi><mo>)</mo></math>, clearly indicating any asymptotes with their equations and stating the values of any axes intercepts.</p>
<div class="marks">[3]</div>
<div class="question_part_label">f.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mrow><mo>-</mo><mi>x</mi></mrow></mfenced><mo>=</mo><mtext>arcsin</mtext><mfenced><mfrac><mrow><msup><mfenced><mrow><mo>-</mo><mi>x</mi></mrow></mfenced><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow><mrow><msup><mfenced><mrow><mo>-</mo><mi>x</mi></mrow></mfenced><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfrac></mfenced><mo>=</mo><mtext>arcsin</mtext><mfenced><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfrac></mfenced><mo>=</mo><mi>f</mi><mfenced><mi>x</mi></mfenced></math> <em><strong>R1</strong></em></p>
<p><br><strong>OR</strong></p>
<p>a sketch graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mfenced><mi>x</mi></mfenced></math> with line symmetry in the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-axis indicated <em><strong>R1</strong></em></p>
<p><br><strong>THEN</strong></p>
<p>so <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced></math> is an even function. <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>→</mo><mo>±</mo><mo>∞</mo><mo>,</mo><mo> </mo><mo> </mo><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>→</mo><mtext>arcsin</mtext><mo> </mo><mn>1</mn><mfenced><mrow><mo>→</mo><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p>so the horizontal asymptote is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></math> <em><strong>A1</strong></em> </p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempting to use the quotient rule to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mtext>d</mtext><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mfenced><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfrac></mfenced></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mtext>d</mtext><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mfenced><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfrac></mfenced><mo>=</mo><mfrac><mrow><mn>2</mn><mi>x</mi><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfenced><mo>-</mo><mn>2</mn><mi>x</mi><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow></mfenced></mrow><msup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup></mfrac><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mfrac><mrow><mn>4</mn><mi>x</mi></mrow><msup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup></mfrac></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p>attempting to use the chain rule to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mtext>d</mtext><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mfenced><mrow><mtext>arcsin</mtext><mfenced><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfrac></mfenced></mrow></mfenced></math> <em><strong>M1</strong></em></p>
<p>let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi><mo>=</mo><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfrac></math> and so <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mtext>arcsin</mtext><mo> </mo><mi>u</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>u</mi></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><msqrt><mn>1</mn><mo>-</mo><msup><mi>u</mi><mn>2</mn></msup></msqrt></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mn>1</mn><msqrt><mn>1</mn><mo>-</mo><msup><mfenced><mstyle displaystyle="true"><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfrac></mstyle></mfenced><mn>2</mn></msup></msqrt></mfrac><mo>×</mo><mfrac><mrow><mn>4</mn><mi>x</mi></mrow><msup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup></mfrac></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mn>4</mn><mi>x</mi></mrow><msqrt><msup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup><mo>-</mo><msup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup></msqrt></mfrac><mo>×</mo><mfrac><mn>1</mn><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfenced></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mn>4</mn><mi>x</mi></mrow><msqrt><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup></msqrt></mfrac><mo>×</mo><mfrac><mn>1</mn><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfenced></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mn>2</mn><mi>x</mi></mrow><mrow><msqrt><msup><mi>x</mi><mn>2</mn></msup></msqrt><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfenced></mrow></mfrac></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mrow><mn>2</mn><mi>x</mi></mrow><mrow><mfenced open="|" close="|"><mi>x</mi></mfenced><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfenced></mrow></mfrac></math></p>
<p><br><strong>EITHER</strong></p>
<p>for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo><</mo><mn>0</mn><mo>,</mo><mo> </mo><mfenced open="|" close="|"><mi>x</mi></mfenced><mo>=</mo><mo>-</mo><mi>x</mi></math> <em><strong>(A1)</strong></em></p>
<p>so <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mo>-</mo><mfrac><mrow><mn>2</mn><mi>x</mi></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfrac></math> <em><strong>A1</strong></em></p>
<p><br><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mi>x</mi></mfenced><mo>></mo><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn><mo>></mo><mn>0</mn></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>x</mi><mo><</mo><mn>0</mn><mo>,</mo><mo> </mo><mi>x</mi><mo><</mo><mn>0</mn></math> <em><strong>A1</strong></em></p>
<p><br><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo><</mo><mn>0</mn></math> <em><strong>R1</strong></em></p>
<p><strong><br>Note:</strong> Award <em><strong>R1</strong></em> for stating that in <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced></math>, the numerator is negative, and the denominator is positive.</p>
<p><br>so <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> is decreasing for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo><</mo><mn>0</mn></math> <em><strong>AG</strong></em></p>
<p><strong><br>Note:</strong> Do not accept a graphical solution</p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mtext>arcsin</mtext><mfenced><mfrac><mrow><msup><mi>y</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow><mrow><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfrac></mfenced></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>sin</mtext><mo> </mo><mi>x</mi><mo>=</mo><mfrac><mrow><msup><mi>y</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow><mrow><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfrac><mo>⇒</mo><msup><mi>y</mi><mn>2</mn></msup><mo> </mo><mtext>sin</mtext><mo> </mo><mi>x</mi><mo>+</mo><mtext>sin</mtext><mo> </mo><mi>x</mi><mo>=</mo><msup><mi>y</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mfrac><mrow><mn>1</mn><mo>+</mo><mtext>sin</mtext><mo> </mo><mi>x</mi></mrow><mrow><mn>1</mn><mo>-</mo><mtext>sin</mtext><mo> </mo><mi>x</mi></mrow></mfrac></math> <em><strong>A1</strong></em></p>
<p>domain of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi><mo>,</mo><mo> </mo><mi>x</mi><mo>≥</mo><mn>0</mn></math> and so the range of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>g</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math> must be <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi><mo>,</mo><mo> </mo><mi>y</mi><mo>≥</mo><mn>0</mn></math></p>
<p>hence the positive root is taken (or the negative root is rejected) <em><strong>R1</strong></em></p>
<p><br><strong>Note:</strong> The <em><strong>R1</strong></em> is dependent on the above<em><strong> A1</strong></em>.</p>
<p><br>so <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><msup><mi>g</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mi>x</mi></mfenced><mo>=</mo></mrow></mfenced><msqrt><mfrac><mrow><mn>1</mn><mo>+</mo><mtext>sin</mtext><mo> </mo><mi>x</mi></mrow><mrow><mn>1</mn><mo>-</mo><mtext>sin</mtext><mo> </mo><mi>x</mi></mrow></mfrac></msqrt></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> The final <em><strong>A1</strong></em> is not dependent on <em><strong>R1</strong></em> mark.</p>
<p> </p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>domain is <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac><mo>≤</mo><mi>x</mi><mo><</mo><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Accept correct alternative notations, for example, <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⌊</mo><mo>-</mo><mstyle displaystyle="false"><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></mstyle><mo>,</mo><mo> </mo><mstyle displaystyle="false"><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></mstyle><mo>⌊</mo></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⌊</mo><mo>-</mo><mstyle displaystyle="false"><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></mstyle><mo>,</mo><mo> </mo><mstyle displaystyle="false"><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac><mo>)</mo></mstyle></math>.<br>Accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>[</mo><mo>-</mo><mn>1</mn><mo>.</mo><mn>57</mn><mo>,</mo><mo> </mo><mn>1</mn><mo>.</mo><mn>57</mn><mo>[</mo></math> if correct to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math> s.f.</p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="padding-left:60px;"><img 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"> <em><strong>A1</strong></em><em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p><strong>Note:<em> A1</em></strong> for correct domain and correct range and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-intercept at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>1</mn></math><br><em><strong> A1</strong></em> for asymptotic behaviour <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>→</mo><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></math><br><em><strong> A1</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></math><br> Coordinates are not required. <br> Do not accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>1</mn><mo>.</mo><mn>57</mn></math> or other inexact values.</p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">f.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">f.</div>
</div>
<br><hr><br><div class="specification">
<p>The curve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="C">
<mi>C</mi>
</math></span> is defined by equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="xy - \ln y = 1,{\text{ }}y > 0">
<mi>x</mi>
<mi>y</mi>
<mo>−<!-- − --></mo>
<mi>ln</mi>
<mo><!-- --></mo>
<mi>y</mi>
<mo>=</mo>
<mn>1</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>y</mi>
<mo>></mo>
<mn>0</mn>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}}">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
</math></span> in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the equation of the tangent to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="C">
<mi>C</mi>
</math></span> at the point <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\frac{2}{{\text{e}}},{\text{ e}}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>2</mn>
<mrow>
<mtext>e</mtext>
</mrow>
</mfrac>
<mo>,</mo>
<mrow>
<mtext> e</mtext>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y + x\frac{{{\text{d}}y}}{{{\text{d}}x}} - \frac{1}{y}\frac{{{\text{d}}y}}{{{\text{d}}x}} = 0">
<mi>y</mi>
<mo>+</mo>
<mi>x</mi>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mi>y</mi>
</mfrac>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mn>0</mn>
</math></span> <strong><em>M1A1A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>A1 </em></strong>for the first two terms, <strong><em>A1 </em></strong>for the third term and the 0.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}} = \frac{{{y^2}}}{{1 - xy}}">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>x</mi>
<mi>y</mi>
</mrow>
</mfrac>
</math></span> <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Accept <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{ - {y^2}}}{{\ln y}}">
<mfrac>
<mrow>
<mo>−</mo>
<mrow>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mi>ln</mi>
<mo></mo>
<mi>y</mi>
</mrow>
</mfrac>
</math></span>.</p>
<p> </p>
<p><strong>Note:</strong> Accept <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{ - y}}{{x - \frac{1}{y}}}">
<mfrac>
<mrow>
<mo>−</mo>
<mi>y</mi>
</mrow>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mi>y</mi>
</mfrac>
</mrow>
</mfrac>
</math></span>.</p>
<p> </p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{m_T} = \frac{{{{\text{e}}^2}}}{{1 - {\text{e}} \times \frac{2}{{\text{e}}}}}">
<mrow>
<msub>
<mi>m</mi>
<mi>T</mi>
</msub>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mrow>
<mtext>e</mtext>
</mrow>
<mo>×</mo>
<mfrac>
<mn>2</mn>
<mrow>
<mtext>e</mtext>
</mrow>
</mfrac>
</mrow>
</mfrac>
</math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{m_T} = - {{\text{e}}^2}">
<mrow>
<msub>
<mi>m</mi>
<mi>T</mi>
</msub>
</mrow>
<mo>=</mo>
<mo>−</mo>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</math></span> <strong><em>(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y - {\text{e}} = - {{\text{e}}^2}x + 2{\text{e}}">
<mi>y</mi>
<mo>−</mo>
<mrow>
<mtext>e</mtext>
</mrow>
<mo>=</mo>
<mo>−</mo>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
<mrow>
<mtext>e</mtext>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - {{\text{e}}^2}x - y + 3{\text{e}} = 0">
<mo>−</mo>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mi>x</mi>
<mo>−</mo>
<mi>y</mi>
<mo>+</mo>
<mn>3</mn>
<mrow>
<mtext>e</mtext>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span> or equivalent <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Accept <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = - 7.39x + 8.15">
<mi>y</mi>
<mo>=</mo>
<mo>−</mo>
<mn>7.39</mn>
<mi>x</mi>
<mo>+</mo>
<mn>8.15</mn>
</math></span>.</p>
<p> </p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>Differentiate from first principles the function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = 3{x^3} - x">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>3</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>−</mo>
<mi>x</mi>
</math></span>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><strong>METHOD 1</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{f\left( {x + h} \right) - f\left( x \right)}}{h}">
<mfrac>
<mrow>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mi>h</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</mrow>
<mi>h</mi>
</mfrac>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{\left( {3{{\left( {x + h} \right)}^3} - \left( {x + h} \right)} \right) - \left( {3{x^3} - x} \right)}}{h}">
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>3</mn>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mi>h</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>3</mn>
</msup>
</mrow>
<mo>−</mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mi>h</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>3</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>−</mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mi>h</mi>
</mfrac>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{3\left( {{x^3} + 3{x^2}h + 3x{h^2} + {h^3}} \right) - x - h - {3x^3} + x}}{h}">
<mo>=</mo>
<mfrac>
<mrow>
<mn>3</mn>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>3</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mi>h</mi>
<mo>+</mo>
<mn>3</mn>
<mi>x</mi>
<mrow>
<msup>
<mi>h</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mi>h</mi>
<mn>3</mn>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mi>x</mi>
<mo>−</mo>
<mi>h</mi>
<mo>−</mo>
<mrow>
<mn>3</mn>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>+</mo>
<mi>x</mi>
</mrow>
<mi>h</mi>
</mfrac>
</math></span> <em><strong>(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{9{x^2}h + 9x{h^2} + 3{h^3} - h}}{h}">
<mo>=</mo>
<mfrac>
<mrow>
<mn>9</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mi>h</mi>
<mo>+</mo>
<mn>9</mn>
<mi>x</mi>
<mrow>
<msup>
<mi>h</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>3</mn>
<mrow>
<msup>
<mi>h</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>−</mo>
<mi>h</mi>
</mrow>
<mi>h</mi>
</mfrac>
</math></span> <em><strong>A1</strong></em></p>
<p>cancelling <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="h">
<mi>h</mi>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 9{x^2} + 9xh + 3{h^2} - 1">
<mo>=</mo>
<mn>9</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>9</mn>
<mi>x</mi>
<mi>h</mi>
<mo>+</mo>
<mn>3</mn>
<mrow>
<msup>
<mi>h</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>1</mn>
</math></span></p>
<p>then <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mathop {{\text{lim}}}\limits_{h \to 0} \left( {9{x^2} + 9xh + 3{h^2} - 1} \right)">
<munder>
<mrow>
<mrow>
<mtext>lim</mtext>
</mrow>
</mrow>
<mrow>
<mi>h</mi>
<mo stretchy="false">→</mo>
<mn>0</mn>
</mrow>
</munder>
<mo></mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>9</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>9</mn>
<mi>x</mi>
<mi>h</mi>
<mo>+</mo>
<mn>3</mn>
<mrow>
<msup>
<mi>h</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 9{x^2} - 1">
<mo>=</mo>
<mn>9</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>1</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Final <em><strong>A1</strong> </em>dependent on all previous marks.</p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{f\left( {x + h} \right) - f\left( x \right)}}{h}">
<mfrac>
<mrow>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mi>h</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</mrow>
<mi>h</mi>
</mfrac>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{\left( {3{{\left( {x + h} \right)}^3} - \left( {x + h} \right)} \right) - \left( {3{x^3} - x} \right)}}{h}">
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>3</mn>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mi>h</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>3</mn>
</msup>
</mrow>
<mo>−</mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mi>h</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>3</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>−</mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mi>h</mi>
</mfrac>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{3\left( {{{\left( {x + h} \right)}^3} - {x^3}} \right) + \left( {x - \left( {x + h} \right)} \right)}}{h}">
<mo>=</mo>
<mfrac>
<mrow>
<mn>3</mn>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mi>h</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>3</mn>
</msup>
</mrow>
<mo>−</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mi>h</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mi>h</mi>
</mfrac>
</math></span> <em><strong>(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{3h\left( {{{\left( {x + h} \right)}^2} + x\left( {x + h} \right) + {x^2}} \right) - h}}{h}">
<mo>=</mo>
<mfrac>
<mrow>
<mn>3</mn>
<mi>h</mi>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mi>h</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mi>x</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mi>h</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mi>h</mi>
</mrow>
<mi>h</mi>
</mfrac>
</math></span> <em><strong>A1</strong></em></p>
<p>cancelling <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="h">
<mi>h</mi>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 3\left( {{{\left( {x + h} \right)}^2} + x\left( {x + h} \right) + {x^2}} \right) - 1">
<mo>=</mo>
<mn>3</mn>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mi>h</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mi>x</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mi>h</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mn>1</mn>
</math></span></p>
<p>then <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mathop {{\text{lim}}}\limits_{h \to 0} \left( {3\left( {{{\left( {x + h} \right)}^2} + x\left( {x + h} \right) + {x^2}} \right) - 1} \right)">
<munder>
<mrow>
<mrow>
<mtext>lim</mtext>
</mrow>
</mrow>
<mrow>
<mi>h</mi>
<mo stretchy="false">→</mo>
<mn>0</mn>
</mrow>
</munder>
<mo></mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>3</mn>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mi>h</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mi>x</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mi>h</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 9{x^2} - 1">
<mo>=</mo>
<mn>9</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>1</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Final <em><strong>A1</strong> </em>dependent on all previous marks.</p>
<p> </p>
<p><em><strong>[5 marks]</strong></em></p>
<p> </p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="question">
<p>By using the substitution <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^2} = 2\sec \theta ">
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>2</mn>
<mi>sec</mi>
<mo></mo>
<mi>θ</mi>
</math></span>, show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {\frac{{{\text{d}}x}}{{x\sqrt {{x^4} - 4} }} = \frac{1}{4}\arccos \left( {\frac{2}{{{x^2}}}} \right) + c} ">
<mo>∫</mo>
<mrow>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
<mrow>
<mi>x</mi>
<msqrt>
<mrow>
<msup>
<mi>x</mi>
<mn>4</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>4</mn>
</msqrt>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>4</mn>
</mfrac>
<mi>arccos</mi>
<mo></mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>2</mn>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mi>c</mi>
</mrow>
</math></span>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><strong>EITHER</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^2} = 2\sec \theta ">
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>2</mn>
<mi>sec</mi>
<mo></mo>
<mi>θ</mi>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2x\frac{{{\text{d}}x}}{{{\text{d}}\theta }} = 2\sec \theta \tan \theta ">
<mn>2</mn>
<mi>x</mi>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>θ</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mn>2</mn>
<mi>sec</mi>
<mo></mo>
<mi>θ</mi>
<mi>tan</mi>
<mo></mo>
<mi>θ</mi>
</math></span> <strong><em>M1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {\frac{{{\text{d}}x}}{{x\sqrt {{x^4} - 4} }}} ">
<mo>∫</mo>
<mrow>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
<mrow>
<mi>x</mi>
<msqrt>
<mrow>
<msup>
<mi>x</mi>
<mn>4</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>4</mn>
</msqrt>
</mrow>
</mfrac>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \int {\frac{{\sec \theta \tan \theta {\text{d}}\theta }}{{2\sec \theta \sqrt {4{{\sec }^2}\theta - 4} }}} ">
<mo>=</mo>
<mo>∫</mo>
<mrow>
<mfrac>
<mrow>
<mi>sec</mi>
<mo></mo>
<mi>θ</mi>
<mi>tan</mi>
<mo></mo>
<mi>θ</mi>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>θ</mi>
</mrow>
<mrow>
<mn>2</mn>
<mi>sec</mi>
<mo></mo>
<mi>θ</mi>
<msqrt>
<mn>4</mn>
<mrow>
<msup>
<mrow>
<mi>sec</mi>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mi>θ</mi>
<mo>−</mo>
<mn>4</mn>
</msqrt>
</mrow>
</mfrac>
</mrow>
</math></span> <strong><em>M1A1</em></strong></p>
<p><strong>OR</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \sqrt 2 {(\sec \theta )^{\frac{1}{2}}}{\text{ }}\left( { = \sqrt 2 {{(\cos \theta )}^{ - \frac{1}{2}}}} \right)">
<mi>x</mi>
<mo>=</mo>
<msqrt>
<mn>2</mn>
</msqrt>
<mrow>
<mo stretchy="false">(</mo>
<mi>sec</mi>
<mo></mo>
<mi>θ</mi>
<msup>
<mo stretchy="false">)</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
</msup>
</mrow>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<msqrt>
<mn>2</mn>
</msqrt>
<mrow>
<msup>
<mrow>
<mo stretchy="false">(</mo>
<mi>cos</mi>
<mo></mo>
<mi>θ</mi>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}x}}{{{\text{d}}\theta }} = \frac{{\sqrt 2 }}{2}{(\sec \theta )^{\frac{1}{2}}}\tan \theta {\text{ }}\left( { = \frac{{\sqrt 2 }}{2}{{(\cos \theta )}^{ - \frac{3}{2}}}\sin \theta } \right)">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>θ</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
<mn>2</mn>
</mfrac>
<mrow>
<mo stretchy="false">(</mo>
<mi>sec</mi>
<mo></mo>
<mi>θ</mi>
<msup>
<mo stretchy="false">)</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
</msup>
</mrow>
<mi>tan</mi>
<mo></mo>
<mi>θ</mi>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mfrac>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
<mn>2</mn>
</mfrac>
<mrow>
<msup>
<mrow>
<mo stretchy="false">(</mo>
<mi>cos</mi>
<mo></mo>
<mi>θ</mi>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<mo>−</mo>
<mfrac>
<mn>3</mn>
<mn>2</mn>
</mfrac>
</mrow>
</msup>
</mrow>
<mi>sin</mi>
<mo></mo>
<mi>θ</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>M1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {\frac{{{\text{d}}x}}{{x\sqrt {{x^4} - 4} }}} ">
<mo>∫</mo>
<mrow>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
<mrow>
<mi>x</mi>
<msqrt>
<mrow>
<msup>
<mi>x</mi>
<mn>4</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>4</mn>
</msqrt>
</mrow>
</mfrac>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \int {\frac{{\sqrt 2 {{(\sec \theta )}^{\frac{1}{2}}}\tan \theta {\text{d}}\theta }}{{2\sqrt 2 {{(\sec \theta )}^{\frac{1}{2}}}\sqrt {4{{\sec }^2}\theta - 4} }}{\text{ }}\left( { = \int {\frac{{\sqrt 2 {{(\cos \theta )}^{ - \frac{3}{2}}}\sin \theta {\text{d}}\theta }}{{2\sqrt 2 {{(\cos \theta )}^{ - \frac{1}{2}}}\sqrt {4{{\sec }^2}\theta - 4} }}} } \right)} ">
<mo>=</mo>
<mo>∫</mo>
<mrow>
<mfrac>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
<mrow>
<msup>
<mrow>
<mo stretchy="false">(</mo>
<mi>sec</mi>
<mo></mo>
<mi>θ</mi>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
</msup>
</mrow>
<mi>tan</mi>
<mo></mo>
<mi>θ</mi>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>θ</mi>
</mrow>
<mrow>
<mn>2</mn>
<msqrt>
<mn>2</mn>
</msqrt>
<mrow>
<msup>
<mrow>
<mo stretchy="false">(</mo>
<mi>sec</mi>
<mo></mo>
<mi>θ</mi>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
</msup>
</mrow>
<msqrt>
<mn>4</mn>
<mrow>
<msup>
<mrow>
<mi>sec</mi>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mi>θ</mi>
<mo>−</mo>
<mn>4</mn>
</msqrt>
</mrow>
</mfrac>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mo>∫</mo>
<mrow>
<mfrac>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
<mrow>
<msup>
<mrow>
<mo stretchy="false">(</mo>
<mi>cos</mi>
<mo></mo>
<mi>θ</mi>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<mo>−</mo>
<mfrac>
<mn>3</mn>
<mn>2</mn>
</mfrac>
</mrow>
</msup>
</mrow>
<mi>sin</mi>
<mo></mo>
<mi>θ</mi>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>θ</mi>
</mrow>
<mrow>
<mn>2</mn>
<msqrt>
<mn>2</mn>
</msqrt>
<mrow>
<msup>
<mrow>
<mo stretchy="false">(</mo>
<mi>cos</mi>
<mo></mo>
<mi>θ</mi>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
</msup>
</mrow>
<msqrt>
<mn>4</mn>
<mrow>
<msup>
<mrow>
<mi>sec</mi>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mi>θ</mi>
<mo>−</mo>
<mn>4</mn>
</msqrt>
</mrow>
</mfrac>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
</math></span> <strong><em>M1A1</em></strong></p>
<p><strong>THEN</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{2}\int {\frac{{\tan \theta {\text{d}}\theta }}{{2\tan \theta }}} ">
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mo>∫</mo>
<mrow>
<mfrac>
<mrow>
<mi>tan</mi>
<mo></mo>
<mi>θ</mi>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>θ</mi>
</mrow>
<mrow>
<mn>2</mn>
<mi>tan</mi>
<mo></mo>
<mi>θ</mi>
</mrow>
</mfrac>
</mrow>
</math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{4}\int {{\text{d}}\theta } ">
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>4</mn>
</mfrac>
<mo>∫</mo>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>θ</mi>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{\theta }{4} + c">
<mo>=</mo>
<mfrac>
<mi>θ</mi>
<mn>4</mn>
</mfrac>
<mo>+</mo>
<mi>c</mi>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^2} = 2\sec \theta \Rightarrow \cos \theta = \frac{2}{{{x^2}}}">
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>2</mn>
<mi>sec</mi>
<mo></mo>
<mi>θ</mi>
<mo stretchy="false">⇒</mo>
<mi>cos</mi>
<mo></mo>
<mi>θ</mi>
<mo>=</mo>
<mfrac>
<mn>2</mn>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</math></span> <strong><em>M1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>This <strong><em>M1 </em></strong>may be seen anywhere, including a sketch of an appropriate triangle.</p>
<p> </p>
<p>so <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{\theta }{4} + c = \frac{1}{4}\arccos \left( {\frac{2}{{{x^2}}}} \right) + c">
<mfrac>
<mi>θ</mi>
<mn>4</mn>
</mfrac>
<mo>+</mo>
<mi>c</mi>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>4</mn>
</mfrac>
<mi>arccos</mi>
<mo></mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>2</mn>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mi>c</mi>
</math></span> <strong><em>AG</em></strong></p>
<p><strong><em>[7 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>A continuous random variable <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> has the probability density function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> given by</p>
<p style="padding-left: 210px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfenced open="{" close><mtable columnalign="left"><mtr><mtd><mfrac><mi>x</mi><msqrt><msup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>k</mi></mrow></mfenced><mn>3</mn></msup></msqrt></mfrac><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mn>0</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mn>4</mn></mtd></mtr><mtr><mtd><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mn>0</mn><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mtext>otherwise</mtext></mtd></mtr></mtable></mfenced></math></p>
<p>where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>∈</mo><msup><mi mathvariant="normal">ℝ</mi><mo>+</mo></msup></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mn>16</mn><mo>+</mo><mi>k</mi></msqrt><mo>-</mo><msqrt><mi>k</mi></msqrt><mo>=</mo><msqrt><mi>k</mi></msqrt><msqrt><mn>16</mn><mo>+</mo><mi>k</mi></msqrt></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>recognition of the need to integrate <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>x</mi><msqrt><msup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>k</mi></mrow></mfenced><mn>3</mn></msup></msqrt></mfrac></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><mfrac><mi>x</mi><msqrt><msup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>k</mi></mrow></mfenced><mn>3</mn></msup></msqrt></mfrac><mo>d</mo><mi>x</mi><mfenced><mrow><mo>=</mo><mn>1</mn></mrow></mfenced></math></p>
<p> </p>
<p><strong>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>k</mi><mo>⇒</mo><mfrac><mrow><mo>d</mo><mi>u</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mn>2</mn><mi>x</mi></math> (or equivalent) <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><mfrac><mi>x</mi><msqrt><msup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>k</mi></mrow></mfenced><mn>3</mn></msup></msqrt></mfrac><mo>d</mo><mi>x</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>∫</mo><msup><mi>u</mi><mrow><mo>-</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow></msup><mo>d</mo><mi>u</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><msup><mi>u</mi><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup><mfenced><mrow><mo>+</mo><mi>c</mi></mrow></mfenced><mo> </mo><mfenced><mrow><mo>=</mo><mo>-</mo><msup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>k</mi></mrow></mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup><mfenced><mrow><mo>+</mo><mi>c</mi></mrow></mfenced></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><mfrac><mi>x</mi><msqrt><msup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>k</mi></mrow></mfenced><mn>3</mn></msup></msqrt></mfrac><mo>d</mo><mi>x</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>∫</mo><mfrac><mrow><mn>2</mn><mi>x</mi></mrow><msqrt><msup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>k</mi></mrow></mfenced><mn>3</mn></msup></msqrt></mfrac><mo>d</mo><mi>x</mi></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><msup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>k</mi></mrow></mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup><mfenced><mrow><mo>+</mo><mi>c</mi></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>THEN</strong></p>
<p>attempt to use correct limits for their integrand and set equal to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mfenced open="[" close="]"><mrow><mo>-</mo><msup><mi>u</mi><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup></mrow></mfenced><mi>k</mi><mrow><mn>16</mn><mo>+</mo><mi>k</mi></mrow></msubsup><mo>=</mo><mn>1</mn></math> <strong>OR </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mfenced open="[" close="]"><mrow><mo>-</mo><msup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>k</mi></mrow></mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup></mrow></mfenced><mn>0</mn><mn>4</mn></msubsup><mo>=</mo><mn>1</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><msup><mfenced><mrow><mn>16</mn><mo>+</mo><mi>k</mi></mrow></mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup><mo>+</mo><msup><mi>k</mi><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup><mo>=</mo><mn>1</mn><mfenced><mrow><mo>⇒</mo><mfrac><mn>1</mn><msqrt><mi>k</mi></msqrt></mfrac><mo>-</mo><mfrac><mn>1</mn><msqrt><mn>16</mn><mo>+</mo><mi>k</mi></msqrt></mfrac><mo>=</mo><mn>1</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mn>16</mn><mo>+</mo><mi>k</mi></msqrt><mo>-</mo><msqrt><mi>k</mi></msqrt><mo>=</mo><msqrt><mi>k</mi></msqrt><msqrt><mn>16</mn><mo>+</mo><mi>k</mi></msqrt></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to solve <math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mn>16</mn><mo>+</mo><mi>k</mi></msqrt><mo>-</mo><msqrt><mi>k</mi></msqrt><mo>=</mo><msqrt><mi>k</mi></msqrt><msqrt><mn>16</mn><mo>+</mo><mi>k</mi></msqrt></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>645038</mn><mo>…</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>0</mn><mo>.</mo><mn>645</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The curve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math> has equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>e</mtext><mrow><mn>2</mn><mi>y</mi></mrow></msup><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mi>y</mi></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup></mrow><mrow><mn>2</mn><msup><mtext>e</mtext><mrow><mn>2</mn><mi>y</mi></mrow></msup><mo>-</mo><mn>1</mn></mrow></mfrac></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The tangent to <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math> at the point Ρ is parallel to the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-axis.</p>
<p>Find the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-coordinate of Ρ.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color:#999;font-size:90%;font-style:italic;">* This sample question was produced by experienced DP mathematics senior examiners to aid teachers in preparing for external assessment in the new MAA course. There may be minor differences in formatting compared to formal exam papers.</p>
<p>attempts implicit differentiation on both sides of the equation <strong>M1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><msup><mtext>e</mtext><mrow><mn>2</mn><mi>y</mi></mrow></msup><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac></math> <strong>A1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>2</mn><msup><mtext>e</mtext><mrow><mn>2</mn><mi>y</mi></mrow></msup><mo>-</mo><mn>1</mn></mrow></mfenced><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup></math> <strong>A1</strong></p>
<p>so <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup></mrow><mrow><mn>2</mn><msup><mtext>e</mtext><mrow><mn>2</mn><mi>y</mi></mrow></msup><mo>-</mo><mn>1</mn></mrow></mfrac></math> <strong>AG</strong></p>
<p> </p>
<p><strong>[3 marks]</strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempts to solve <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><msup><mtext>e</mtext><mrow><mn>2</mn><mi>y</mi></mrow></msup><mo>-</mo><mn>1</mn><mo>=</mo><mn>0</mn></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> <strong>(M1)</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>346</mn><mo>…</mo><mo> </mo><mfenced><mrow><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>ln</mi><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mfenced></math> <strong>A1</strong></p>
<p>attempts to solve <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>e</mtext><mrow><mn>2</mn><mi>y</mi></mrow></msup><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mi>y</mi></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> given their value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> <strong>(M1)</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>946</mn><mo> </mo><mfenced><mrow><mo>=</mo><msup><mfenced><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mfenced><mrow><mn>1</mn><mo>-</mo><mi>ln</mi><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mfenced></mrow></mfenced><mfrac><mn>1</mn><mn>3</mn></mfrac></msup></mrow></mfenced></math> <strong>A1</strong></p>
<p> </p>
<p><strong>[4 marks]</strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2{x^3} - 3x + 1">
<mn>2</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>3</mn>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
</math></span> can be expressed in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="Ax\left( {{x^2} + 1} \right) + Bx + C">
<mi>A</mi>
<mi>x</mi>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mi>B</mi>
<mi>x</mi>
<mo>+</mo>
<mi>C</mi>
</math></span>, find the values of the constants <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A">
<mi>A</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="B">
<mi>B</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="C">
<mi>C</mi>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {\frac{{2{x^3} - 3x + 1}}{{{x^2} + 1}}} {\text{d}}x">
<mo>∫</mo>
<mrow>
<mfrac>
<mrow>
<mn>2</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>3</mn>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>1</mn>
</mrow>
</mfrac>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2{x^3} - 3x + 1 = Ax\left( {{x^2} + 1} \right) + Bx + C">
<mn>2</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>3</mn>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
<mo>=</mo>
<mi>A</mi>
<mi>x</mi>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mi>B</mi>
<mi>x</mi>
<mo>+</mo>
<mi>C</mi>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A = 2,\,\,C = 1,">
<mi>A</mi>
<mo>=</mo>
<mn>2</mn>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>C</mi>
<mo>=</mo>
<mn>1</mn>
<mo>,</mo>
</math></span><em><strong> A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A + B = - 3 \Rightarrow B = - 5">
<mi>A</mi>
<mo>+</mo>
<mi>B</mi>
<mo>=</mo>
<mo>−</mo>
<mn>3</mn>
<mo stretchy="false">⇒</mo>
<mi>B</mi>
<mo>=</mo>
<mo>−</mo>
<mn>5</mn>
</math></span><em><strong> A1</strong></em></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {\frac{{2{x^3} - 3x + 1}}{{{x^2} + 1}}} {\text{d}}x = \int {\left( {2x - \frac{{5x}}{{{x^2} + 1}} + \frac{1}{{{x^2} + 1}}} \right)} {\text{d}}x">
<mo>∫</mo>
<mrow>
<mfrac>
<mrow>
<mn>2</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>3</mn>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>1</mn>
</mrow>
</mfrac>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
<mo>=</mo>
<mo>∫</mo>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mi>x</mi>
<mo>−</mo>
<mfrac>
<mrow>
<mn>5</mn>
<mi>x</mi>
</mrow>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>1</mn>
</mrow>
</mfrac>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>1</mn>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</math></span> <em><strong>M1M1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1</strong></em> for dividing by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {{x^2} + 1} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span> to get <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2x">
<mn>2</mn>
<mi>x</mi>
</math></span>,<em><strong> M1</strong></em> for separating the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="5x">
<mn>5</mn>
<mi>x</mi>
</math></span> and 1.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {x^2} - \frac{5}{2}{\text{ln}}\left( {{x^2} + 1} \right) + {\text{arctan}}\,x\left( { + c} \right)">
<mo>=</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mfrac>
<mn>5</mn>
<mn>2</mn>
</mfrac>
<mrow>
<mtext>ln</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mrow>
<mtext>arctan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mrow>
<mo>(</mo>
<mrow>
<mo>+</mo>
<mi>c</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>(M1)A1A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(M1)A1</strong></em> for integrating <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\frac{{5x}}{{{x^2} + 1}}}">
<mrow>
<mfrac>
<mrow>
<mn>5</mn>
<mi>x</mi>
</mrow>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>1</mn>
</mrow>
</mfrac>
</mrow>
</math></span>, <em><strong>A1</strong></em> for the other two terms.</p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>A particle moves along a horizontal line such that at time <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> seconds, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> ≥ 0, its acceleration <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span> is given by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span> = 2<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> − 1. When <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> = 6 , its displacement <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="s">
<mi>s</mi>
</math></span> from a fixed origin O is 18.25 m. When <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> = 15, its displacement from O is 922.75 m. Find an expression for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="s">
<mi>s</mi>
</math></span> in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p>attempt to integrate <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span> to find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v">
<mi>v</mi>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v = \int {a\,{\text{d}}t = \int {\left( {2t - 1} \right)} } \,{\text{d}}t">
<mi>v</mi>
<mo>=</mo>
<mo>∫</mo>
<mrow>
<mi>a</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
<mo>=</mo>
<mo>∫</mo>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mi>t</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {t^2} - t + c">
<mo>=</mo>
<mrow>
<msup>
<mi>t</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mi>t</mi>
<mo>+</mo>
<mi>c</mi>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="s = \int {v\,{\text{d}}t = \int {\left( {{t^2} - t + c} \right)} } \,{\text{d}}t">
<mi>s</mi>
<mo>=</mo>
<mo>∫</mo>
<mrow>
<mi>v</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
<mo>=</mo>
<mo>∫</mo>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msup>
<mi>t</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mi>t</mi>
<mo>+</mo>
<mi>c</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{{t^3}}}{3} - \frac{{{t^2}}}{2} + ct + d">
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>t</mi>
<mn>3</mn>
</msup>
</mrow>
</mrow>
<mn>3</mn>
</mfrac>
<mo>−</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>t</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mn>2</mn>
</mfrac>
<mo>+</mo>
<mi>c</mi>
<mi>t</mi>
<mo>+</mo>
<mi>d</mi>
</math></span> <em><strong>A1</strong></em></p>
<p>attempt at substitution of given values <em><strong>(M1)</strong></em></p>
<p>at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t = 6{\text{,}}\,\,\,18.25 = 72 - 18 + 6c + d">
<mi>t</mi>
<mo>=</mo>
<mn>6</mn>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mn>18.25</mn>
<mo>=</mo>
<mn>72</mn>
<mo>−</mo>
<mn>18</mn>
<mo>+</mo>
<mn>6</mn>
<mi>c</mi>
<mo>+</mo>
<mi>d</mi>
</math></span></p>
<p>at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t = 15{\text{,}}\,\,\,922.75 = 1125 - 112.5 + 15c + d">
<mi>t</mi>
<mo>=</mo>
<mn>15</mn>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mn>922.75</mn>
<mo>=</mo>
<mn>1125</mn>
<mo>−</mo>
<mn>112.5</mn>
<mo>+</mo>
<mn>15</mn>
<mi>c</mi>
<mo>+</mo>
<mi>d</mi>
</math></span></p>
<p>solve simultaneously: <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c = - 6{\text{,}}\,\,d = 0.25">
<mi>c</mi>
<mo>=</mo>
<mo>−</mo>
<mn>6</mn>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mo>=</mo>
<mn>0.25</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow s = \frac{{{t^3}}}{3} - \frac{{{t^2}}}{2} + - 6t + \frac{1}{4}">
<mo stretchy="false">⇒</mo>
<mi>s</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>t</mi>
<mn>3</mn>
</msup>
</mrow>
</mrow>
<mn>3</mn>
</mfrac>
<mo>−</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>t</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mn>2</mn>
</mfrac>
<mo>+</mo>
<mo>−</mo>
<mn>6</mn>
<mi>t</mi>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mn>4</mn>
</mfrac>
</math></span></p>
<p> </p>
<p><em><strong>[6 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>Consider the curve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math> given by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>x</mi><mo>-</mo><mi>x</mi><mi>y</mi><mo> </mo><mi>ln</mi><mo>(</mo><mi>x</mi><mi>y</mi><mo>)</mo></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>></mo><mn>0</mn><mo>,</mo><mo> </mo><mi>y</mi><mo>></mo><mn>0</mn></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>+</mo><mfenced><mrow><mi>x</mi><mfrac><mstyle displaystyle="true"><mo>d</mo><mi>y</mi></mstyle><mstyle displaystyle="true"><mo>d</mo><mi>x</mi></mstyle></mfrac><mo>+</mo><mi>y</mi></mrow></mfenced><mfenced><mrow><mn>1</mn><mo>+</mo><mi>ln</mi><mfenced><mrow><mi>x</mi><mi>y</mi></mrow></mfenced></mrow></mfenced><mo>=</mo><mn>1</mn></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find the equation of the tangent to <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math> at the point where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>1</mn></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>attempts to differentiate implicitly including at least one application of the product rule <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi><mo>=</mo><mi>x</mi><mi>y</mi><mo>,</mo><mo> </mo><mi>v</mi><mo>=</mo><mi>ln</mi><mfenced><mrow><mi>x</mi><mi>y</mi></mrow></mfenced><mo>,</mo><mo> </mo><mfrac><mrow><mo>d</mo><mi>u</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mi>x</mi><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>+</mo><mi>y</mi><mo>,</mo><mo> </mo><mfrac><mrow><mo>d</mo><mi>v</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mfenced><mrow><mi>x</mi><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>+</mo><mi>y</mi></mrow></mfenced><mfrac><mn>1</mn><mrow><mi>x</mi><mi>y</mi></mrow></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mn>1</mn><mo>-</mo><mfenced open="[" close="]"><mrow><mfrac><mrow><mi>x</mi><mi>y</mi></mrow><mrow><mi>x</mi><mi>y</mi></mrow></mfrac><mfenced><mrow><mi>x</mi><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>+</mo><mi>y</mi></mrow></mfenced><mo>+</mo><mfenced><mrow><mi>x</mi><mfrac><mstyle displaystyle="true"><mo>d</mo><mi>y</mi></mstyle><mstyle displaystyle="true"><mo>d</mo><mi>x</mi></mstyle></mfrac><mo>+</mo><mi>y</mi></mrow></mfenced><mi>ln</mi><mfenced><mrow><mi>x</mi><mi>y</mi></mrow></mfenced></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>(M1)A1</strong></em> for implicitly differentiating <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>x</mi><mfenced><mrow><mn>1</mn><mo>-</mo><mi>y</mi><mo> </mo><mi>ln</mi><mfenced><mrow><mi>x</mi><mi>y</mi></mrow></mfenced></mrow></mfenced></math> and obtaining <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mn>1</mn><mo>-</mo><mfenced open="[" close="]"><mrow><mfrac><mrow><mi>x</mi><mi>y</mi></mrow><mrow><mi>x</mi><mi>y</mi></mrow></mfrac><mfenced><mrow><mi>x</mi><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>+</mo><mi>y</mi></mrow></mfenced><mo>+</mo><mi>x</mi><mfrac><mstyle displaystyle="true"><mo>d</mo><mi>y</mi></mstyle><mstyle displaystyle="true"><mo>d</mo><mi>x</mi></mstyle></mfrac><mi>ln</mi><mfenced><mrow><mi>x</mi><mi>y</mi></mrow></mfenced><mo>+</mo><mi>y</mi><mo> </mo><mi>ln</mi><mfenced><mrow><mi>x</mi><mi>y</mi></mrow></mfenced></mrow></mfenced></math>.</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mn>1</mn><mo>-</mo><mfenced open="[" close="]"><mrow><mfenced><mrow><mi>x</mi><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>+</mo><mi>y</mi></mrow></mfenced><mo>+</mo><mfenced><mrow><mi>x</mi><mfrac><mstyle displaystyle="true"><mo>d</mo><mi>y</mi></mstyle><mstyle displaystyle="true"><mo>d</mo><mi>x</mi></mstyle></mfrac><mo>+</mo><mi>y</mi></mrow></mfenced><mi>ln</mi><mfenced><mrow><mi>x</mi><mi>y</mi></mrow></mfenced></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mn>1</mn><mo>-</mo><mfenced><mrow><mi>x</mi><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>+</mo><mi>y</mi></mrow></mfenced><mfenced><mrow><mn>1</mn><mo>+</mo><mi>ln</mi><mfenced><mrow><mi>x</mi><mi>y</mi></mrow></mfenced></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>+</mo><mfenced><mrow><mi>x</mi><mfrac><mstyle displaystyle="true"><mo>d</mo><mi>y</mi></mstyle><mstyle displaystyle="true"><mo>d</mo><mi>x</mi></mstyle></mfrac><mo>+</mo><mi>y</mi></mrow></mfenced><mfenced><mrow><mn>1</mn><mo>+</mo><mi>ln</mi><mfenced><mrow><mi>x</mi><mi>y</mi></mrow></mfenced></mrow></mfenced><mo>=</mo><mn>1</mn></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>x</mi><mo>-</mo><mi>x</mi><mi>y</mi><mo> </mo><mi>ln</mi><mo> </mo><mi>x</mi><mo>-</mo><mi>x</mi><mi>y</mi><mo> </mo><mi>ln</mi><mo> </mo><mi>y</mi></math></p>
<p>attempts to differentiate implicitly including at least one application of the product rule <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mn>1</mn><mo>-</mo><mfenced><mrow><mfrac><mrow><mi>x</mi><mi>y</mi></mrow><mi>x</mi></mfrac><mo>+</mo><mfenced><mrow><mi>x</mi><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>+</mo><mi>y</mi></mrow></mfenced><mi>ln</mi><mo> </mo><mi>x</mi></mrow></mfenced><mo>-</mo><mfenced><mrow><mfrac><mrow><mi>x</mi><mi>y</mi></mrow><mi>y</mi></mfrac><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>+</mo><mfenced><mrow><mi>x</mi><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>+</mo><mi>y</mi></mrow></mfenced><mi>ln</mi><mo> </mo><mi>y</mi></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p>or equivalent to the above, for example</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mn>1</mn><mo>-</mo><mfenced><mrow><mi>x</mi><mo> </mo><mi>ln</mi><mo> </mo><mi>x</mi><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>+</mo><mfenced><mrow><mn>1</mn><mo>+</mo><mi>ln</mi><mo> </mo><mi>x</mi></mrow></mfenced><mi>y</mi></mrow></mfenced><mo>-</mo><mfenced><mrow><mi>y</mi><mo> </mo><mi>ln</mi><mo> </mo><mi>y</mi><mo>+</mo><mi>x</mi><mfenced><mrow><mi>ln</mi><mo> </mo><mi>y</mi><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>+</mo><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac></mrow></mfenced></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mn>1</mn><mo>-</mo><mi>x</mi><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mfenced><mrow><mi>ln</mi><mo> </mo><mi>x</mi><mo>+</mo><mi>ln</mi><mo> </mo><mi>y</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>-</mo><mi>y</mi><mfenced><mrow><mi>ln</mi><mo> </mo><mi>x</mi><mo>+</mo><mi>ln</mi><mo> </mo><mi>y</mi><mo>+</mo><mn>1</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p>or equivalent to the above, for example</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mn>1</mn><mo>-</mo><mi>x</mi><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mfenced><mrow><mi>ln</mi><mo> </mo><mfenced><mrow><mi>x</mi><mi>y</mi></mrow></mfenced><mo>+</mo><mn>1</mn></mrow></mfenced><mo>-</mo><mi>y</mi><mfenced><mrow><mi>ln</mi><mo> </mo><mfenced><mrow><mi>x</mi><mi>y</mi></mrow></mfenced><mo>+</mo><mn>1</mn></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>+</mo><mfenced><mrow><mi>x</mi><mfrac><mstyle displaystyle="true"><mo>d</mo><mi>y</mi></mstyle><mstyle displaystyle="true"><mo>d</mo><mi>x</mi></mstyle></mfrac><mo>+</mo><mi>y</mi></mrow></mfenced><mfenced><mrow><mn>1</mn><mo>+</mo><mi>ln</mi><mfenced><mrow><mi>x</mi><mi>y</mi></mrow></mfenced></mrow></mfenced><mo>=</mo><mn>1</mn></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>METHOD 3</strong></p>
<p>attempt to differentiate implicitly including at least one application of the product rule <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi><mo>=</mo><mi>x</mi><mo> </mo><mi>ln</mi><mo> </mo><mfenced><mrow><mi>x</mi><mi>y</mi></mrow></mfenced><mo>,</mo><mo> </mo><mi>v</mi><mo>=</mo><mi>y</mi><mo>,</mo><mo> </mo><mfrac><mrow><mo>d</mo><mi>u</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mi>ln</mi><mfenced><mrow><mi>x</mi><mi>y</mi></mrow></mfenced><mo>+</mo><mfenced><mrow><mi>x</mi><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>+</mo><mi>y</mi></mrow></mfenced><mfrac><mi>x</mi><mrow><mi>x</mi><mi>y</mi></mrow></mfrac><mo>,</mo><mo> </mo><mfrac><mrow><mo>d</mo><mi>v</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mn>1</mn><mo>-</mo><mfenced><mrow><mi>x</mi><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mi>ln</mi><mfenced><mrow><mi>x</mi><mi>y</mi></mrow></mfenced><mo>+</mo><mi>y</mi><mo> </mo><mi>ln</mi><mfenced><mrow><mi>x</mi><mi>y</mi></mrow></mfenced><mo>+</mo><mfrac><mrow><mi>x</mi><mi>y</mi></mrow><mrow><mi>x</mi><mi>y</mi></mrow></mfrac><mfenced><mrow><mi>x</mi><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>+</mo><mi>y</mi></mrow></mfenced></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mn>1</mn><mo>-</mo><mi>x</mi><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mfenced><mrow><mi>ln</mi><mfenced><mrow><mi>x</mi><mi>y</mi></mrow></mfenced><mo>+</mo><mn>1</mn></mrow></mfenced><mo>-</mo><mi>y</mi><mfenced><mrow><mi>ln</mi><mfenced><mrow><mi>x</mi><mi>y</mi></mrow></mfenced><mo>+</mo><mn>1</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>+</mo><mfenced><mrow><mi>x</mi><mfrac><mstyle displaystyle="true"><mo>d</mo><mi>y</mi></mstyle><mstyle displaystyle="true"><mo>d</mo><mi>x</mi></mstyle></mfrac><mo>+</mo><mi>y</mi></mrow></mfenced><mfenced><mrow><mn>1</mn><mo>+</mo><mi>ln</mi><mfenced><mrow><mi>x</mi><mi>y</mi></mrow></mfenced></mrow></mfenced><mo>=</mo><mn>1</mn></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>METHOD 4</strong></p>
<p>lets <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi><mo>=</mo><mi>x</mi><mi>y</mi></math> and attempts to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>x</mi><mo>-</mo><mi>w</mi><mo> </mo><mi>ln</mi><mo> </mo><mi>w</mi></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mn>1</mn><mo>-</mo><mfenced><mrow><mfrac><mrow><mo>d</mo><mi>w</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>+</mo><mfrac><mstyle displaystyle="true"><mo>d</mo><mi>w</mi></mstyle><mstyle displaystyle="true"><mo>d</mo><mi>x</mi></mstyle></mfrac><mi>ln</mi><mo> </mo><mi>w</mi></mrow></mfenced><mo> </mo><mfenced><mrow><mo>=</mo><mn>1</mn><mo>-</mo><mfrac><mrow><mo>d</mo><mi>w</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mfenced><mrow><mn>1</mn><mo>+</mo><mi>ln</mi><mo> </mo><mi>w</mi></mrow></mfenced></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>w</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mi>x</mi><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>+</mo><mi>y</mi></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mn>1</mn><mo>-</mo><mfenced><mrow><mi>x</mi><mfrac><mstyle displaystyle="true"><mo>d</mo><mi>y</mi></mstyle><mstyle displaystyle="true"><mo>d</mo><mi>x</mi></mstyle></mfrac><mo>+</mo><mi>y</mi><mo>+</mo><mfenced><mrow><mi>x</mi><mfrac><mstyle displaystyle="true"><mo>d</mo><mi>y</mi></mstyle><mstyle displaystyle="true"><mo>d</mo><mi>x</mi></mstyle></mfrac><mo>+</mo><mi>y</mi></mrow></mfenced><mi>ln</mi><mfenced><mrow><mi>x</mi><mi>y</mi></mrow></mfenced></mrow></mfenced><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mn>1</mn><mo>-</mo><mfenced><mrow><mi>x</mi><mfrac><mstyle displaystyle="true"><mo>d</mo><mi>y</mi></mstyle><mstyle displaystyle="true"><mo>d</mo><mi>x</mi></mstyle></mfrac><mo>+</mo><mi>y</mi></mrow></mfenced><mfenced><mrow><mn>1</mn><mo>+</mo><mi>ln</mi><mfenced><mrow><mi>x</mi><mi>y</mi></mrow></mfenced></mrow></mfenced></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>+</mo><mfenced><mrow><mi>x</mi><mfrac><mstyle displaystyle="true"><mo>d</mo><mi>y</mi></mstyle><mstyle displaystyle="true"><mo>d</mo><mi>x</mi></mstyle></mfrac><mo>+</mo><mi>y</mi></mrow></mfenced><mfenced><mrow><mn>1</mn><mo>+</mo><mi>ln</mi><mfenced><mrow><mi>x</mi><mi>y</mi></mrow></mfenced></mrow></mfenced><mo>=</mo><mn>1</mn></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>substitutes <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>1</mn></math> into <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>x</mi><mo>-</mo><mi>x</mi><mi>y</mi><mo> </mo><mi>ln</mi><mfenced><mrow><mi>x</mi><mi>y</mi></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>1</mn><mo>-</mo><mi>y</mi><mo> </mo><mi>ln</mi><mo> </mo><mi>y</mi><mo>⇒</mo><mi>y</mi><mo>=</mo><mn>1</mn></math> <em><strong>A1</strong></em></p>
<p>substitutes <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>1</mn></math> and their non-zero value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> into <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>+</mo><mfenced><mrow><mi>x</mi><mfrac><mstyle displaystyle="true"><mo>d</mo><mi>y</mi></mstyle><mstyle displaystyle="true"><mo>d</mo><mi>x</mi></mstyle></mfrac><mo>+</mo><mi>y</mi></mrow></mfenced><mfenced><mrow><mn>1</mn><mo>+</mo><mi>ln</mi><mfenced><mrow><mi>x</mi><mi>y</mi></mrow></mfenced></mrow></mfenced><mo>=</mo><mn>1</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mn>0</mn><mo> </mo><mo> </mo><mfenced><mrow><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mn>0</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p>equation of the tangent is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>1</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>substitutes <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>1</mn></math> into <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>+</mo><mfenced><mrow><mi>x</mi><mfrac><mstyle displaystyle="true"><mo>d</mo><mi>y</mi></mstyle><mstyle displaystyle="true"><mo>d</mo><mi>x</mi></mstyle></mfrac><mo>+</mo><mi>y</mi></mrow></mfenced><mfenced><mrow><mn>1</mn><mo>+</mo><mi>ln</mi><mfenced><mrow><mi>x</mi><mi>y</mi></mrow></mfenced></mrow></mfenced><mo>=</mo><mn>1</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>+</mo><mfenced><mrow><mfrac><mstyle displaystyle="true"><mo>d</mo><mi>y</mi></mstyle><mstyle displaystyle="true"><mo>d</mo><mi>x</mi></mstyle></mfrac><mo>+</mo><mi>y</mi></mrow></mfenced><mfenced><mrow><mn>1</mn><mo>+</mo><mi>ln</mi><mfenced><mi>y</mi></mfenced></mrow></mfenced><mo>=</mo><mn>1</mn></math></p>
<p><br><strong>EITHER</strong></p>
<p>correctly substitutes <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mi>y</mi><mo>=</mo><mfrac><mrow><mn>1</mn><mo>-</mo><mi>y</mi></mrow><mi>y</mi></mfrac></math> into <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>+</mo><mfenced><mrow><mfrac><mstyle displaystyle="true"><mo>d</mo><mi>y</mi></mstyle><mstyle displaystyle="true"><mo>d</mo><mi>x</mi></mstyle></mfrac><mo>+</mo><mi>y</mi></mrow></mfenced><mfenced><mrow><mn>1</mn><mo>+</mo><mi>ln</mi><mfenced><mrow><mi>x</mi><mi>y</mi></mrow></mfenced></mrow></mfenced><mo>=</mo><mn>1</mn></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mfenced><mrow><mn>1</mn><mo>+</mo><mfrac><mn>1</mn><mi>y</mi></mfrac></mrow></mfenced><mo>=</mo><mn>0</mn><mo>⇒</mo><mfrac><mstyle displaystyle="true"><mo>d</mo><mi>y</mi></mstyle><mstyle displaystyle="true"><mo>d</mo><mi>x</mi></mstyle></mfrac><mo>=</mo><mn>0</mn><mo> </mo><mfenced><mrow><mi>y</mi><mo>=</mo><mn>1</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><strong><br>OR</strong></p>
<p>correctly substitutes <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>+</mo><mi>y</mi><mo> </mo><mi>ln</mi><mo> </mo><mi>y</mi><mo>=</mo><mn>1</mn></math> into <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>+</mo><mfenced><mrow><mfrac><mstyle displaystyle="true"><mo>d</mo><mi>y</mi></mstyle><mstyle displaystyle="true"><mo>d</mo><mi>x</mi></mstyle></mfrac><mo>+</mo><mi>y</mi></mrow></mfenced><mfenced><mrow><mn>1</mn><mo>+</mo><mi>ln</mi><mfenced><mrow><mi>x</mi><mi>y</mi></mrow></mfenced></mrow></mfenced><mo>=</mo><mn>1</mn></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mfenced><mrow><mn>2</mn><mo>+</mo><mi>ln</mi><mo> </mo><mi>y</mi></mrow></mfenced><mo>=</mo><mn>0</mn><mo>⇒</mo><mfrac><mstyle displaystyle="true"><mo>d</mo><mi>y</mi></mstyle><mstyle displaystyle="true"><mo>d</mo><mi>x</mi></mstyle></mfrac><mo>=</mo><mn>0</mn><mo> </mo><mfenced><mrow><mi>y</mi><mo>=</mo><mn>1</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><br><strong>THEN</strong></p>
<p>substitutes <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>1</mn></math> into <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>x</mi><mo>-</mo><mi>x</mi><mi>y</mi><mo> </mo><mi>ln</mi><mfenced><mrow><mi>x</mi><mi>y</mi></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>1</mn><mo>-</mo><mi>y</mi><mo> </mo><mi>ln</mi><mo> </mo><mi>y</mi><mo>⇒</mo><mi>y</mi><mo>=</mo><mn>1</mn></math></p>
<p>equation of the tangent is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>1</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the differential equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>y</mi><mo>-</mo><mi>x</mi></mrow><mrow><mi>y</mi><mo>+</mo><mi>x</mi></mrow></mfrac></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>,</mo><mo> </mo><mi>y</mi><mo>></mo><mn>0</mn></math>.</p>
<p>It is given that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>2</mn></math> when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>1</mn></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Solve the differential equation, giving your answer in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mrow><mi>x</mi><mo>,</mo><mo> </mo><mi>y</mi></mrow></mfenced><mo>=</mo><mn>0</mn></math>.</p>
<div class="marks">[9]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> against <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> has a local maximum between <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>2</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>3</mn></math>. Determine the coordinates of this local maximum.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that there are no points of inflexion on the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> against <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color:#999;font-size:90%;font-style:italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p>puts <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>v</mi><mi>x</mi></math> so that <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mi>v</mi><mo>+</mo><mi>x</mi><mfrac><mrow><mo>d</mo><mi>v</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mo>+</mo><mi>x</mi><mfrac><mrow><mo>d</mo><mi>v</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>v</mi><mi>x</mi><mo>-</mo><mi>x</mi></mrow><mrow><mi>v</mi><mi>x</mi><mo>+</mo><mi>x</mi></mrow></mfrac><mo> </mo><mfenced><mrow><mo>=</mo><mfrac><mrow><mi>v</mi><mo>-</mo><mn>1</mn></mrow><mrow><mi>v</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p>attempts to express <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mfrac><mrow><mo>d</mo><mi>v</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac></math> as a single rational fraction in <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mfrac><mrow><mo>d</mo><mi>v</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mo>-</mo><mfrac><mrow><msup><mi>v</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow><mrow><mi>v</mi><mo>+</mo><mn>1</mn></mrow></mfrac></math> <em><strong>M1</strong></em></p>
<p>attempts to separate variables <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><mfrac><mrow><mi>v</mi><mo>+</mo><mn>1</mn></mrow><mrow><msup><mi>v</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfrac><mo>d</mo><mi>v</mi><mo>=</mo><mo>-</mo><mo>∫</mo><mfrac><mn>1</mn><mi>x</mi></mfrac><mo>d</mo><mi>x</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>ln</mi><mfenced><mrow><msup><mi>v</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfenced><mo>+</mo><mtext>arctan</mtext><mo> </mo><mi>v</mi><mo>=</mo><mo>-</mo><mi>ln</mi><mo> </mo><mi>x</mi><mfenced><mrow><mo>+</mo><mi>C</mi></mrow></mfenced></math> <em><strong>A1A1</strong></em></p>
<p>substitutes <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>2</mn><mo>,</mo><mo> </mo><mi>x</mi><mo>=</mo><mn>1</mn></math> and attempts to find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>ln</mi><mo> </mo><mn>5</mn><mo>+</mo><mtext>arctan</mtext><mo> </mo><mn>2</mn></math> <em><strong>A1</strong></em></p>
<p>the solution is</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>ln</mi><mfenced><mrow><mfrac><msup><mi>y</mi><mn>2</mn></msup><msup><mi>x</mi><mn>2</mn></msup></mfrac><mo>+</mo><mn>1</mn></mrow></mfenced><mo>+</mo><mtext>arctan</mtext><mfenced><mfrac><mi>y</mi><mi>x</mi></mfrac></mfenced><mo>+</mo><mi>ln</mi><mo> </mo><mi>x</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>ln</mi><mo> </mo><mn>5</mn><mo>-</mo><mtext>arctan</mtext><mo> </mo><mn>2</mn><mo>=</mo><mn>0</mn></math> <em><strong>A1</strong></em></p>
<p><br><em><strong>[9 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>at a maximum, <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mn>0</mn></math> <em><strong>M1</strong></em></p>
<p>attempts to substitute <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>x</mi></math> into their solution <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>ln</mi><mo> </mo><mn>2</mn><mo>+</mo><mtext>arctan</mtext><mo> </mo><mn>1</mn><mo>+</mo><mi>ln</mi><mo> </mo><mi>x</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>ln</mi><mo> </mo><mn>5</mn><mo>+</mo><mtext>arctan</mtext><mo> </mo><mn>2</mn></math></p>
<p>attempts to solve for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>,</mo><mi>y</mi></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>2</mn><mo>.</mo><mn>18</mn><mo>,</mo><mo> </mo><mn>2</mn><mo>.</mo><mn>18</mn></mrow></mfenced><mo> </mo><mo> </mo><mfenced><mrow><mfrac><msqrt><mn>10</mn></msqrt><mn>2</mn></mfrac><msup><mtext>e</mtext><mrow><mtext>arctan</mtext><mo> </mo><mn>2</mn><mo>-</mo><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac></mrow></msup><mi>,</mi><mfrac><msqrt><mn>10</mn></msqrt><mn>2</mn></mfrac><msup><mtext>e</mtext><mrow><mtext>arctan</mtext><mo> </mo><mn>2</mn><mo>-</mo><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac></mrow></msup></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><strong><br>Note:</strong> Accept all answers that round to the correct <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo> </mo><mtext>sf</mtext></math> answer.<br>Accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>2</mn><mo>.</mo><mn>18</mn><mo>,</mo><mo> </mo><mi>y</mi><mo>=</mo><mn>2</mn><mo>.</mo><mn>18</mn></math>.</p>
<p><br><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>attempts (quotient rule) implicit differentiation <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>y</mi></mrow><mrow><mo>d</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mfrac><mrow><mfenced><mstyle displaystyle="true"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>-</mo><mn>1</mn></mstyle></mfenced><mstyle displaystyle="true"><mfenced><mrow><mi>y</mi><mo>+</mo><mi>x</mi></mrow></mfenced></mstyle><mstyle displaystyle="true"><mo>-</mo></mstyle><mstyle displaystyle="true"><mfenced><mrow><mi>y</mi><mo>-</mo><mi>x</mi></mrow></mfenced></mstyle><mstyle displaystyle="true"><mfenced><mrow><mfrac><mstyle displaystyle="true"><mo>d</mo><mi>y</mi></mstyle><mstyle displaystyle="true"><mo>d</mo><mi>x</mi></mstyle></mfrac><mo>+</mo><mn>1</mn></mrow></mfenced></mstyle></mrow><mstyle displaystyle="true"><msup><mfenced><mrow><mi>y</mi><mo>+</mo><mi>x</mi></mrow></mfenced><mn>2</mn></msup></mstyle></mfrac></math></p>
<p>correctly substitutes <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>y</mi><mo>-</mo><mi>x</mi></mrow><mrow><mi>y</mi><mo>+</mo><mi>x</mi></mrow></mfrac></math> into <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>y</mi></mrow><mrow><mo>d</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mfenced><mrow><mstyle displaystyle="true"><mfrac><mrow><mi>y</mi><mo>-</mo><mi>x</mi></mrow><mrow><mi>y</mi><mo>+</mo><mi>x</mi></mrow></mfrac></mstyle><mo>-</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mi>y</mi><mo>+</mo><mi>x</mi></mrow></mfenced><mo>-</mo><mfenced><mrow><mi>y</mi><mo>-</mo><mi>x</mi></mrow></mfenced><mfenced><mrow><mstyle displaystyle="true"><mfrac><mrow><mi>y</mi><mo>-</mo><mi>x</mi></mrow><mrow><mi>y</mi><mo>+</mo><mi>x</mi></mrow></mfrac></mstyle><mo>+</mo><mn>1</mn></mrow></mfenced></mrow><msup><mfenced><mrow><mi>y</mi><mo>+</mo><mi>x</mi></mrow></mfenced><mn>2</mn></msup></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><mfrac><mrow><mn>2</mn><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup></mrow></mfenced></mrow><msup><mfenced><mrow><mi>x</mi><mo>+</mo><mi>y</mi></mrow></mfenced><mn>3</mn></msup></mfrac></math> <em><strong>A1</strong></em></p>
<p>this expression can never be zero therefore no points of inflexion <em><strong>R1</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>attempts implicit differentiation on <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>y</mi><mo>+</mo><mi>x</mi></mrow></mfenced><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mi>y</mi><mo>-</mo><mi>x</mi></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>+</mo><mn>1</mn></mrow></mfenced><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>+</mo><mfenced><mrow><mi>y</mi><mo>+</mo><mi>x</mi></mrow></mfenced><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>y</mi></mrow><mrow><mo>d</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>-</mo><mn>1</mn></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>y</mi><mo>+</mo><mi>x</mi></mrow></mfenced><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>y</mi></mrow><mrow><mo>d</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>-</mo><mn>1</mn><mo>-</mo><msup><mfenced><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac></mfenced><mn>2</mn></msup><mo>-</mo><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><mn>1</mn><mo>-</mo><msup><mfenced><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac></mfenced><mn>2</mn></msup></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>1</mn><mo>-</mo><msup><mfenced><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac></mfenced><mn>2</mn></msup><mo><</mo><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>+</mo><mi>y</mi><mo>></mo><mn>0</mn><mo>,</mo><mo> </mo><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>y</mi></mrow><mrow><mo>d</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mo>≠</mo><mn>0</mn></math> therefore no points of inflexion <em><strong>R1</strong></em></p>
<p><strong><br>Note:</strong> Accept putting <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>y</mi></mrow><mrow><mo>d</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mn>0</mn></math> and obtaining contradiction.</p>
<p><br><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Assuming the Maclaurin series for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mo> </mo><mi>x</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo>(</mo><mn>1</mn><mo>+</mo><mi>x</mi><mo>)</mo></math>, show that the Maclaurin series for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mo>(</mo><mi>ln</mi><mo>(</mo><mn>1</mn><mo>+</mo><mi>x</mi><mo>)</mo><mo>)</mo></math> is</p>
<p style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mfrac><mn>5</mn><mn>12</mn></mfrac><msup><mi>x</mi><mn>4</mn></msup><mo>+</mo><mo>…</mo></math></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By differentiating the series in part (a), show that the Maclaurin series for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mo>(</mo><mi>ln</mi><mo>(</mo><mn>1</mn><mo>+</mo><mi>x</mi><mo>)</mo><mo>)</mo></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mfrac><mn>1</mn><mn>6</mn></mfrac><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mo>…</mo></math> .</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence determine the Maclaurin series for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tan</mi><mo>(</mo><mi>ln</mi><mo>(</mo><mn>1</mn><mo>+</mo><mi>x</mi><mo>)</mo><mo>)</mo></math> as far as the term in <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>3</mn></msup></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color:#999;font-size:90%;font-style:italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p><strong>METHOD 1</strong> </p>
<p>attempts to substitute <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mfenced><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>=</mo><mi>x</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><msup><mi>x</mi><mn>4</mn></msup><mo>+</mo><mo>…</mo></math> into</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mo> </mo><mi>x</mi><mo>=</mo><mn>1</mn><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mfrac><mn>1</mn><mn>24</mn></mfrac><msup><mi>x</mi><mn>4</mn></msup><mo>-</mo><mo>…</mo></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mo> </mo><mi>x</mi><mo>=</mo><mfenced><mrow><mi>ln</mi><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced></mrow></mfenced><mo>=</mo><mn>1</mn><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mfenced><mrow><mi>x</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mo>…</mo></mrow></mfenced><mn>2</mn></msup><mo>+</mo><mfrac><mn>1</mn><mn>24</mn></mfrac><msup><mfenced><mrow><mi>x</mi><mo>+</mo><mo>…</mo></mrow></mfenced><mn>4</mn></msup><mo>+</mo><mo>…</mo></math> <em><strong>A1</strong></em></p>
<p>attempts to expand the <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>RHS</mtext></math> up to and including the <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>4</mn></msup></math> term <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>1</mn><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><msup><mi>x</mi><mn>4</mn></msup><mo>+</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><msup><mi>x</mi><mn>4</mn></msup><mo>…</mo></mrow></mfenced><mo>+</mo><mfrac><mn>1</mn><mn>24</mn></mfrac><msup><mi>x</mi><mn>4</mn></msup><mo>+</mo><mo>…</mo></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>1</mn><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mfrac><mn>5</mn><mn>12</mn></mfrac><msup><mi>x</mi><mn>4</mn></msup><mo>+</mo><mo>…</mo></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>attempts to substitute <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mfenced><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfenced></math> into <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mo> </mo><mi>x</mi><mo>=</mo><mn>1</mn><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mfrac><mn>1</mn><mn>24</mn></mfrac><msup><mi>x</mi><mn>4</mn></msup><mo>-</mo><mo>…</mo></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mo> </mo><mfenced><mrow><mi>ln</mi><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced></mrow></mfenced><mo>=</mo><mn>1</mn><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mfenced><mrow><mi>ln</mi><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced></mrow></mfenced><mn>2</mn></msup><mo>+</mo><mfrac><mn>1</mn><mn>24</mn></mfrac><msup><mfenced><mrow><mi>ln</mi><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced></mrow></mfenced><mn>4</mn></msup><mo>-</mo><mo>…</mo></math></p>
<p>attempts to find the Maclaurin series for <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>ln</mi><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced></mrow></mfenced><mn>2</mn></msup></math> up to and including the <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>4</mn></msup></math> term <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>ln</mi><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced></mrow></mfenced><mn>2</mn></msup><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mfrac><mn>11</mn><mn>12</mn></mfrac><msup><mi>x</mi><mn>4</mn></msup><mo>-</mo><mo>…</mo></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>ln</mi><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced></mrow></mfenced><mn>2</mn></msup><mo>=</mo><msup><mi>x</mi><mn>4</mn></msup><mo>-</mo><mo>…</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>1</mn><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mfrac><mn>11</mn><mn>12</mn></mfrac><msup><mi>x</mi><mn>4</mn></msup><mo>+</mo><mo>…</mo></mrow></mfenced><mo>+</mo><mfrac><mn>1</mn><mn>24</mn></mfrac><msup><mi>x</mi><mn>4</mn></msup><mo>+</mo><mo>…</mo></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>1</mn><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mfrac><mn>5</mn><mn>12</mn></mfrac><msup><mi>x</mi><mn>4</mn></msup><mo>+</mo><mo>…</mo></math> <em><strong>AG</strong></em></p>
<p><br><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mi>sin</mi><mo>(</mo><mi>ln</mi><mo>(</mo><mn>1</mn><mo>+</mo><mi>x</mi><mo>)</mo><mo>)</mo><mo>×</mo><mfrac><mn>1</mn><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mo>-</mo><mi>x</mi><mo>+</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mfrac><mn>5</mn><mn>3</mn></mfrac><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mo>…</mo></math> <strong style="font-style:italic;">A1</strong><em><strong>A</strong><strong>1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mo>(</mo><mi>ln</mi><mo>(</mo><mn>1</mn><mo>+</mo><mi>x</mi><mo>)</mo><mo>)</mo><mo>=</mo><mo>-</mo><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mfenced><mrow><mo>-</mo><mi>x</mi><mo>+</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mfrac><mn>5</mn><mn>3</mn></mfrac><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mo>…</mo></mrow></mfenced></math></p>
<p>attempts to expand the <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>RHS</mtext></math> up to and including the <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>3</mn></msup></math> term <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi>x</mi><mo>-</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mfrac><mn>5</mn><mn>3</mn></mfrac><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mo>…</mo></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi>x</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mfrac><mn>1</mn><mn>6</mn></mfrac><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mo>…</mo></math> <em><strong>AG</strong></em></p>
<p><br><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tan</mi><mo>(</mo><mi>ln</mi><mo>(</mo><mn>1</mn><mo>+</mo><mi>x</mi><mo>)</mo><mo>)</mo><mo>=</mo><msub><mi>a</mi><mn>0</mn></msub><mo>+</mo><msub><mi>a</mi><mn>1</mn></msub><mi>x</mi><mo>+</mo><msub><mi>a</mi><mn>2</mn></msub><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msub><mi>a</mi><mn>3</mn></msub><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mo>…</mo></math></p>
<p>uses <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mo>(</mo><mi>ln</mi><mo>(</mo><mn>1</mn><mo>+</mo><mi>x</mi><mo>)</mo><mo>)</mo><mo>=</mo><mi>cos</mi><mo>(</mo><mi>ln</mi><mo>(</mo><mn>1</mn><mo>+</mo><mi>x</mi><mo>)</mo><mo>)</mo><mo>×</mo><mi>tan</mi><mo>(</mo><mi>ln</mi><mo>(</mo><mn>1</mn><mo>+</mo><mi>x</mi><mo>)</mo><mo>)</mo></math> to form <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mfrac><mn>1</mn><mn>6</mn></mfrac><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mo>…</mo><mo>=</mo><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mo>…</mo></mrow></mfenced><mfenced><mrow><msub><mi>a</mi><mn>0</mn></msub><mo>+</mo><msub><mi>a</mi><mn>1</mn></msub><mi>x</mi><mo>+</mo><msub><mi>a</mi><mn>2</mn></msub><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msub><mi>a</mi><mn>3</mn></msub><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mo>…</mo></mrow></mfenced></math> <strong style="font-style:italic;">A1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msub><mi>a</mi><mn>0</mn></msub><mo>+</mo><msub><mi>a</mi><mn>1</mn></msub><mi>x</mi><mo>+</mo><mfenced><mrow><msub><mi>a</mi><mn>2</mn></msub><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mi>a</mi><mn>0</mn></msub></mrow></mfenced><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mfenced><mrow><msub><mi>a</mi><mn>3</mn></msub><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mi>a</mi><mn>1</mn></msub><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mi>a</mi><mn>0</mn></msub></mrow></mfenced><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mo>…</mo></math> <strong style="font-style:italic;">(A1)</strong></p>
<p>attempts to equate coefficients,</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>a</mi><mn>0</mn></msub><mo>=</mo><mn>0</mn><mo>,</mo><mo> </mo><mo> </mo><msub><mi>a</mi><mn>1</mn></msub><mo>=</mo><mn>1</mn><mo>,</mo><mo> </mo><mo> </mo><msub><mi>a</mi><mn>2</mn></msub><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mi>a</mi><mn>0</mn></msub><mo>=</mo><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>,</mo><mo> </mo><mo> </mo><msub><mi>a</mi><mn>3</mn></msub><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mi>a</mi><mn>1</mn></msub><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mi>a</mi><mn>0</mn></msub><mo>=</mo><mfrac><mn>1</mn><mn>6</mn></mfrac></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>a</mi><mn>0</mn></msub><mo>=</mo><mn>0</mn><mo>,</mo><mo> </mo><mo> </mo><msub><mi>a</mi><mn>1</mn></msub><mo>=</mo><mn>1</mn><mo>,</mo><mo> </mo><mo> </mo><msub><mi>a</mi><mn>2</mn></msub><mo>=</mo><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>,</mo><mo> </mo><mo> </mo><msub><mi>a</mi><mn>3</mn></msub><mo>=</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></math> <strong style="font-style:italic;">A1</strong></p>
<p>so <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tan</mi><mo>(</mo><mi>ln</mi><mo>(</mo><mn>1</mn><mo>+</mo><mi>x</mi><mo>)</mo><mo>)</mo><mo> </mo><mo>=</mo><mi>x</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mo>…</mo></math></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>uses <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tan</mi><mo>(</mo><mi>ln</mi><mo>(</mo><mn>1</mn><mo>+</mo><mi>x</mi><mo>)</mo><mo>)</mo><mo>=</mo><mfrac><mrow><mi>sin</mi><mo>(</mo><mi>ln</mi><mo>(</mo><mn>1</mn><mo>+</mo><mi>x</mi><mo>)</mo><mo>)</mo></mrow><mrow><mi>cos</mi><mo>(</mo><mi>ln</mi><mo>(</mo><mn>1</mn><mo>+</mo><mi>x</mi><mo>)</mo><mo>)</mo></mrow></mfrac></math> to form <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced><mrow><mi>x</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mfrac><mn>1</mn><mn>6</mn></mfrac><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mo>…</mo></mrow></mfenced><msup><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mo>…</mo></mrow></mfenced><mrow><mo>-</mo><mn>1</mn></mrow></msup></math> <strong style="font-style:italic;">A1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mo>…</mo></mrow></mfenced><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo>=</mo><mn>1</mn><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mo>…</mo></math> <strong style="font-style:italic;">(A1)</strong></p>
<p>attempts to expand the <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>RHS</mtext></math> up to and including the <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>3</mn></msup></math> term <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced><mrow><mi>x</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mfrac><mn>1</mn><mn>6</mn></mfrac><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mo>…</mo></mrow></mfenced><mfenced><mrow><mn>1</mn><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mo>…</mo></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi>x</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mfrac><mn>1</mn><mn>6</mn></mfrac><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mo>…</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi>x</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mo>…</mo></math> <strong style="font-style:italic;">A1</strong></p>
<p><br><strong>Note:</strong> Accept use of long division.</p>
<p><br><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Two boats <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math> travel due north.</p>
<p>Initially, boat <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math> is positioned <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>50</mn></math> metres due east of boat <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math>.</p>
<p>The distances travelled by boat <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> and boat <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math>, after <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> seconds, are <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> metres and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> metres respectively. The angle <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi></math> is the radian measure of the bearing of boat <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math> from boat <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math>. This information is shown on the following diagram.</p>
<p style="text-align: center;"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAhwAAAFzCAYAAAB1tNBuAAAgAElEQVR4Ae2dzcsd133HA/KfYK9ShzZBjS2EF03kF0RJaxIJyyaLOrLskkVjB/yAA7WhttEm7sZKZGqDAxZE0qqxFrGyaawSL9SNBYIGtLDQVgbFWQvpL5jyneh7e5555l7duXfmzHn5HLiae+flvHx+c+98dM6Zeb7WkCAAAQhAAAIQgMDEBL42cf5kDwEIQAACEIAABBqEg5MAAhCAAAQgAIHJCSAckyOmAAhAAAIQgAAEEA7OAQhAAAIQgAAEJieAcEyOmAIgAAEIQAACEEA4OAcgAAEIQAACEJicAMIxOWIKgAAEIJAPgevXv2gefvjr7evOnTttxQ8ePNB+1jZSHQR2dl5tY37mzEejNRjhGA0lGUEAAhDInwDCsVkMT516t9FrVdLFWxfyHBLCkUOUqCMEIACBjAn0CUfGzYlSdV+cVwmHZEM9R8eOPROlTtsW4jbRw7EtSY6HAAQgAIGWgARDF0FdDDV0oovm/YZUbt261bz00ouL/bS/LlAeglHGFy583Oanbdq3e8G9dOnTxQXY5fuCrX09jKPjDx9+qs3PIfO2t99+a1F35aHyVa7rr3KXJZevvFw311XtC1O3PirXKeSl45VfN7ks10tLD09pGbJUO7T/qtStT5ePjlUe5up2hfG5Xx59whHGVHkqjyGJIZUhtNgXAhCAQEEEdGH1xTu8GPq9L1DexxfJ8ELmfbX0hbjvAus8dKxS3z66gPWtdxkWAefl9V7qwuv3Xi67KK4qJxQVtcl5hUu3YxvhWMVfF/e+tKrejs+VK5+vrPOqPMy4KxyhyIUcLIl9de2uQzi6RPgMAQhAoBICuhjr4qELtS804QV2mXB08fgYXaSU/D92X7iVjyXFF+rwoqcLpJLr0H6494/W+QLnC6qFw/m7HdrPguE6uE5hnnoflu9jnI/yV1J5LtsCoHUu38f54rzq4uu83X7l7+NUV7P2OsWkL/Xl092v23blbRkz6/CYPsauh9vo481Bx4iDWYX5LXuPcCwjw3oIQAAChRPwRSW8UIYXH18EfYH1BV9YdMHWcc5DF2a9V/L+2sfJ/0P2BdcX/GUXVuWtly90yt/ld/N3XtrHddaxYZ1cDy99THjBdM+A1/nibrHxsRYsLZXMQGUuS87L7dd+bkcoASF/ve+mUB7UPvFRuSFr5xuu6+ajzzpOrz7GbpPqrTJV1rJXWP++crwO4TAJlhCAAAQqI+CLSnihDC94vnj7AuYLvv8HreO0Tsvw4u79wwveMuEIL8DCbxFQGd2Lncvv5u9jtN6pWyev97LvGOWvdjgfS0JKwuH6q25qYygL7n3o8vExXrrtqxj73OjGoE86lN86CeFYhxL7QAACECiQgC+oumj5f9O+0OjC0iccYS+AtuslaQiFw0LiC3W4jwXDFz1/Nl4fq4upkuuo/GMLh9uqsn0xVx18Qfc6y43r7LaEy772uq1amrX5KybrJve46FilMF99Dvmrzt7u+vYxdj20Tcli48/tyoH/IBwDgbE7BCAAgVII6ELki6cuqt2XL4LeRxfbsAfE+3u7LmRKvrh6u5bex4LhffzZTH3x7DvWXffOS3mE5Wm9kyXAF2Gv99Llh8d0ezi0ry/OYX30Pqy3y3I7XUa4dHnOR20JhcbrvbTMhHnofSgH3tdLy8CyfC0x6zDuCseycp1nt559nxGOPiqsgwAEIFAJAV1kdfHURUsX3/Di2SccwqKLoS9yuiD3XajDC5QucP7sC7UvwP5s3CozvMjrONdJ75ViCofKU7kuU+1We8IkCXMPwKoLcNguMVPSMlwvHhapsIzwverj8lQfvTcb76c8HFftozLci7UO465wKN8uhzBPl7tqiXCsosM2CEAAAhAYTMAXK12kLS2+qC7rcRhcCAdkRwDhyC5kVBgCEIBA2gT0P2H3gHSX3f+Jp90SajcmAYRjTJrkBQEIQAACLYFu97t6O5CNuk8OhKPu+NN6CEAAAhCAQBQCCEcUzBQCAQhAAAIQqJsAwlF3/Gk9BCAAAQhAIAoBhCMKZgqBAAQgAAEI1E0A4ag7/rQeAhCAAAQgEIUAwhEFM4VAAAIQgAAE6iaAcNQdf1oPAQhAAAIQiEIA4YiCmUIgAAEIQAACdRNAOOqOP62HAAQgAAEIRCGAcETBTCEQgAAEIACBugkgHHXHn9ZDAAIQgAAEohBAOKJgphAIQAACEIBA3QQQjrrjT+shAAEIQAACUQggHFEwUwgEIAABCECgbgIIR93xp/UQgAAEIACBKAQQjiiYKQQCEIAABCBQNwGEo+7403oIQAACEIBAFAIIRxTMFAIBCEAAAhComwDCUXf8aT0EIAABCEAgCgGEIwpmCoEABCAAAQjUTQDhqDv+tB4CEIAABCAQhQDCEQUzhUAAAhAYl8Dhw081Dz/89fZ17NgzuzI/c+ajxTbtc+XK57u28wECcxBAOOagTpkQgAAERiJg8Th16t1dOV6//kVz8OCBXev4AIE5CSAcc9KnbAhAAAJbEpBw7Oy82vZoXLr06SK3W7duNdpGgkAqBBCOVCJBPSAAAQhsQEBScefOnVYu1KMh0VBCODaAySGTEkA4JsVL5hCAAASmJeBeDA+hWEAQjmm5k/twAgjHcGYcAQEIQCAZAhYOVUhDKpokqiEWhCOZEFGRewQQDk4FCEAAAhkTCIVDzdDkUUmHlt1tGTeTqhdAAOEoIIg0AQIQqJdAn1ToNllJR9+2eknR8rkJIBxzR4DyIQABCGxBIJwo6mw8iRThMBGWKRBAOFKIAnWAAAQgMJDAhQsft70Y6snQSw/7CpMmkSIcIRHez00A4Zg7ApQPAQhAAAIQqIAAwlFBkGkiBCAAAQhAYG4CCMfcEaB8CEAAAhCAQAUEEI4KgkwTIQCB8gl89dWfGr1IEEiVAMKRamSoFwQgAIE1CUg0Dhx4pH3dvXt3zaPYDQJxCSAccXlTGgQgAIHRCbzyysuLO1b0ngSBFAkgHClGhTpBAAIQWJPAZ5/9oZWN48efX0iH1pEgkBoBhCO1iFAfCEAAAmsS0PCJh1I0rKLncejzk08+0TC0siZEdotGAOGIhpqCIAABCIxL4J13ft5Kxvnz59qMJRx6r6W2kSCQEgGEI6VoUBcIQAACaxK4evVqKxZHj/5gcYREQ0nr9F77kCCQCgGEI5VIUA8IQAACAwho2ERScePGjcVRFg6t0/tQRhY78QYCMxFAOGYCT7EQgAAENiXwwQfvt0KhZZgsHFq3bJ9wf95DICYBhCMmbcqCAAQgsCUB9170TQwNhUOTRt0LwgPBtoTO4aMQQDhGwUgmEIAABOIQ8O2vffMzQuFQbTzPQ8eQIDA3AYRj7ghQPgQgAIEBBCQP3aEUH94VDq3XvurpIEFgbgIIx9wRoHwIQAACIxHoE46RsiYbCGxNAOHYGiEZQAACEEiDAMKRRhyoRT8BhKOfC2shAAEIZEcA4cguZFVVGOGoKtw0FgIQKJkAwlFydPNvG8KRfwxpAQQgAIGWAMLBiZAyAYQj5ehQNwhAAAIDCCAcA2Cxa3QCCEd05BQIAQhAYBoCCMc0XMl1HAIIxzgcyQUCEIDA7AQQjtlDQAVWEEA4VsBhEwQgAIGcCCAcOUWrvroiHPXFnBZDAAKFEkA4Cg1sIc1COAoJJM2AAAQggHBwDqRMAOFIOTrUDQIQgMAAAgjHAFjsGp0AwhEdOQVCAAIQmIYAwjENV3IdhwDCMQ5HcoEABCAwOwGEY/YQUIEVBBCOFXDYBAEIQCAnAghHTtGqr64IR30xp8UQgEChBBCOQgNbSLMQjkICSTMgAAEIIBycAykTQDhSjg51gwAEIDCAAMIxABa7RieAcERHToEQgAAEpiGAcEzDlVzHIYBwjMORXCAAAQjMTgDhmD0EVGAFAYRjBRw2QQACEMiJAMKRU7TqqyvCUV/MaTEEIFAoAYSj0MAW0iyEo5BA0gwIQAACCAfnQMoEEI6Uo0PdIAABCAwggHAMgMWu0QkgHNGRUyAEIACBaQggHNNwJddxCCAc43AkFwhAAAKzE0A4Zg8BFVhBAOFYAYdNEIAABHIigHDkFK366opw1BdzWgwBCBRKAOEoNLCFNAvhKCSQNAMCEIAAwsE5kDIBhCPl6FA3CEAAAgMIIBwDYLFrdAIIR3TkFAgBCEBgGgIIxzRcyXUcAlUIx4ULHzf6Ivp16dKnu+h5vZYvvfTirm18gAAEIJALAf2GkSCQKoEqhMPwLR4HDx5obt265dXtcmfn1ebMmY92reMDBCAAgZwIIBw5Rau+ulYnHBILfSmPHXtmV7TffvutRkJCggAEIJArAYQj18jVUe/qhEO9GHrpi3nq1LuLKCMcCxS8gQAEMiWAcGQauEqqXaVwKLYSDH05PYyCcFRyxtNMCBRMAOEoOLgFNK1a4VDsNKyiL+j161+0AsKQSgFnNE2AQMUEEI6Kg59B06sWDk0c1QTSw4efajS3A+HI4IylihCAwFICCMdSNGxIgEDVwiH+ukVWX1K9EI4EzkiqAAEIbExglXBozlp3svzGBa04UL+p3bsAtfudO3cWv7X+zfVS//Hz8PaKrNmUOYGqhEMntOZqdJMnkSIcXTJ8hgAEciIwt3BYKrrPOgoZSnxUT+3r5LsH+36fvQ/L/AlUIxwaNrFN930pGVLJ/2SmBRConUDfb5uZxOjhWEc4/B+8UDhURz10UT0dpHIJVCMc5YaQlkEAAhD4C4F1hMO9CdpX78OkoZBwu4Zgwt6K7nblIVGQPFg2tK4vb5ezSjh0XFdEfBzL/AkgHPnHkBZAAAIQaAnogr0seSjDwxaSB0+Y9zH6bIHQOu2rPHUnn5IEJPzzD54DJ4lQsnSEktJuCP7pEw6vc92C3XlbEAGEo6Bg0hQIQKBuAvcTDglFmDRvTcdIPsL34T46ZpUIaBjE24cIh8oNX926hXXgfRkEqhSOu3fvNp999ocyIkgrIACB6gncuHGj+eST37YXcL3vS+rhCHsntI96LnTRv3Ll8/bJy31zKDTEEt7dIjFRXnppfx3voZkhwhEOnaiHQ3lJOiQ/pDIJVCkcx48/335J9AUlQQACEMiVgP7z9MorL7e/Z2FvgdZpW5jGEA4Ji8TAwiFh0edthUP1ZFgljFaZ76sTjvPnzy2+nAcOPLLnS1lmmGkVBCBQIgH/5ymUDb/XtjBJErrDFuqtkDCot+F+Qyphb4jz1XFjCYfL9/CMy2BZDoGqhOOrr/7USDL08pdS/xMgQQACEMiNwNWrVxe/Y/496y61j5OEQ9u1VJJAdOdneFKohzt08dcx2ldDHXqvnggl7aMeD63r9nBIHpYl18NlaD9NMlVdXNayY1mfN4GqhMNdj5q/oRPb/zsIv5R5h5PaQwACtRD44IP3298x/ZYte2kfJ13oLRTe3/LhfSQBkgdvl1CEd5y4R8TbJR/aP+w5saSE8z6Uv/L2cX3LblmuE8tyCFQjHKFkKHw64dXjoeWTTz7B0Eo55zQtgUAVBIYKRxVQaGTSBKoQDk2e8lCKJENJoqHkL+077/y8/cw/EIAABFImoLtQ9Hv12GMHV/YY6DeOifEpR7K+ulUhHPpy6sunCaNOFg59Pnr0B+32ZbeT+RiWEIAABOYgoP806ffLv1XqldVwyKFD31kqHdqHBIGUCBQvHJ5YpS9qmELhWLZPuD/vIQABCMQmoB4Kzz1TL+0bb7zehP8x0vtwErx+1/TSunC/2PWmPAj0ESheOGT5+gJ2v3yhcAiMe0HCSVZ9wFgHAQhAYEoC+g+QxMIiIeFY9aBC935YTNQT0n0Gx5T1JW8IrEugaOHw/Iw+iegKh76gkhN9yT3PY12I7AcBCEBgGwL6zdHvlP+DpB7ZTcSh+7u2TZ04FgJjEyhaOPS/BFl/n+33fTH1vwh90RGOsU8z8oMABLoE9LukIZNwXoZ6Wrf5/en7XeuWy2cIzEWgaOFYBZUv5io6bIMABKYioP/YePhDPar6j9GqIZMh9eB3bQgt9o1NAOGITZzyIACB6ghoDpl6L8J5Gerd6Ot93QYOwrENPY6dmgDCMTVh8ocABKokoKERzcPwvAwt9XmbIZP7gUQ47keI7XMSQDjmpE/ZEIBAUQQ8LyMcMlHPRvcuuakajXBMRZZ8xyCAcIxBkTwgAIGqCWgOxpBbWaeChXBMRZZ8xyCAcIxBkTwgAIHqCGhoRL0XHjLR3SZTzMsYAhbhGEKLfWMTQDhiE6c8CEAgWwIaMtE8jPBWVj0/Y8p5GUNgIRxDaLFvbAIIR2zilAcBCGRHoO8R43oiaGoJ4UgtItQnJIBwhDR4DwEIQOAeAU307M7LkHiknBCOlKND3RAOzgEIQAAC9whoaGSMR4zPBRThmIs85a5DAOFYhxL7QAACxRLwrazHjz/f/qFHTQKNeSvrmGARjjFpktfYBBCOsYmSHwQgkAUB38qqi7ReYz5ifC4ACMdc5Cl3HQIIxzqU2AcCECiCQPcR4+rVmPtW1jHBIhxj0iSvsQkgHGMTJT8IQCApApqX0b2VdepHjM8FAOGYizzlrkMA4ViHEvtAAALZEei7lTXWI8bngoVwzEWectchgHCsQ4l9IACBLAjo2RjdW1nH+tPvOQBAOHKIUr11RDjqjT0th0ARBFJ8xPhcYBGOuchT7joEEI51KLEPBCCQFAHfyho+Yly3sqbyiPG5YCEcc5Gn3HUIIBzrUGIfCEAgCQIaHgn/9LuGT1J8xPhcsBCOuchT7joEEI51KLEPBCAwG4FljxhXLwdpNwGEYzcPPqVFAOFIKx7UBgIQaJp2aES3roZ/+r3UW1nHDDjCMSZN8hqbAMIxNlHygwAENiLgeRl+xPiBA49k+4jxjQCMcBDCMQJEspiMAMIxGVoyhgAE1iHgR4xLMHTBLOER4+u0e4p9EI4pqJLnWAQQjrFIkg8EILA2AT9iPBwyKekR42uDGHlHhGNkoGQ3KgGEY1ScZAYBCCwjoCGT7iPG9afga7+VdRmvTdYjHJtQ45hYBBCOWKQpBwKVEqjxEeNzhRrhmIs85a5DAOFYhxL7QAACgwj0PWJc4kGalgDCMS1fct+OAMKxHT+OhgAE7hHQ0IiGSMJ5GRpC4XkZ8U4RhCMea0oaTgDhGM6MIyAAgXsEfCsrjxhP45RAONKIA7XoJ4Bw9HNhLQQgsIKAb2XVBU63s3Ir6wpYETchHBFhU9RgAgjHYGQcAIE6CfhWVj8vQw/o4lbWtM4FhCOteFCb3QQQjt08+AQBCAQENC8jfMS45mfwiPEAUGJvEY7EAkJ1dhFAOHbh4AMEIOB5GeFfZdWfflcPByltAghH2vGpvXYIR+1nAO2HwD0Cfbeyaq4GKR8CCEc+saqxpghHjVGnzRC4R0BDJuq9CG9lZV5GvqcHwpFv7GqoOcJRQ5RpIwQCAjxiPIBR2FuEo7CAFtYchKOwgNIcCCwjoOGRcF6GbmXVMAqpHAIIRzmxLLElCEeJUaVNELhHQBM9JRa+lVXCwSPGyz09EI5yY1tCyxCOEqJIGyAQEFj2iHGtJ5VNAOEoO765tw7hyD2C1B8CTdP+vRL1XOhhXLroaBIot7LWd2ogHPXFPKcWIxw5RYu6QqBDIHzEuC42PGK8A6iyjwhHZQHPrLkIR2YBo7oQ8CPGfSsrjxjnnDABhMMkWKZIAOFIMSrUCQIdAn7EePhXWfWn4JmX0QFV+UeEo/ITIPHmIxyJB4jq1U1A8zK6t7LyiPG6z4lVrUc4VtFh29wEEI65I0D5EOgQ4BHjHSB8XJsAwrE2KnacgQDCMQN0ioRAl4CGRrqPGNdfZdVTQUkQWJcAwrEuKfabgwDCMQd1yoRAcCtrOC9D0sG8DE6PTQkgHJuS47gYBBCOGJQpAwIBAR4xHsDg7agEEI5RcZLZyAQQjpGBkh0E+gj4VtbuI8YZMumjxbpNCSAcm5LjuBgEEI4YlCmjSgK+ldXPy9BS8zIYMqnydIjSaIQjCmYK2ZAAwrEhOA6DQB8B9ViEjxhXjwaPGO8jxbopCCAcU1Alz7EIIBxjkSSfqgn4EeMeMuER41WfDrM1HuGYDT0Fr0EA4VgDErtAoI9A362s6t1gXkYfLdbFIIBwxKBMGZsSQDg2JcdxVRKQTGgeRngrK48Yr/JUSLLRCEeSYaFS9wggHJwKEFiDAI8YXwMSu8xOAOGYPQRUYAUBhGMFHDbVTUC3smouhudl6G+aSDxIEEiVAMKRamSolwggHJwHEAgIaF6Ghkh8K6uGTnjEeACIt0kTQDiSDk/1lUM4qj8FAOBbWcN5GdzKynmRIwGEI8eo1VNnhKOeWNPSDgHfyqofaQ2bcCtrBxAfsyOAcGQXsqoqjHBUFW4a233E+PHjz7fzMriVlXOjBAIIRwlRLLcNCEe5saVl9wj4EePhkAmPGOf0KJEAwlFiVMtpE8IxYSzPnj3X7N+/v9m374F2efPmlxOWRtYhAc/L0J0l4ZCJejhIECiVAMJRamTLaBfCMVEcT59+rzly5Ghz+/btRqLx4IMPNTs7OxOVRrYmcPXq1T23smquBgkCNRBAOGqIcr5tRDgmiJ1k49Chx3flLPnortu1Ax82JsAjxjdGx4GFEUA4CgtoYc1BOEYOqHozNIRy+fLlXTlLNhCOXUi2+tD3iHHdyir5IEGgVgIIR62Rz6PdCMfIcdKwieZthEnDKpKQEydeDFfzfgMCGh7pzsvQMAoJAhBo2vlKcIBAqgQQjhEjY7GQXPS9NNRCGk6AR4wPZ8YRdRKgh6POuOfSaoRjxEhJKCQa165d25Wr5m9ovYSEtB4B38rafcQ4Qybr8WOvOgkgHHXGPZdWIxwjRkpDJrobJUye08EdKiGV/ve+lVUP49IPp57+ySPG+1mxFgJ9BBCOPiqsS4UAwjFiJDR3oztPQ6Kh3g2ewbEcdPiIcf1g8ojx5azYAoFVBBCOVXTYNjcBhGPECEg4wp4MDa1INpi7sReyHzHuIRMeMb6XEWsgMJQAwjGUGPvHJIBwjEhbvRuSDs3V0G2xPOxrN9y+W1n1p+CZl7GbE58gsCkBhGNTchwXg0CVwqELnL6Yutjppb+roVsrt33stYZNJBzq1dBSjzYnNe0fR+veyrota7hCAAJ7CSAce5mwJh0CVQqH5EJfzOeee7Z9Pf74d9vPWqfXkSPfbycram6B/ldOGk6AR4wPZ8YRENiWAMKxLUGOn5JA1cLxP5cvN+HrdxcvNr/68MPmrTffbEXEAvLyy//S/i99ykCUkLd6jtRj5HkZ+uus6j1C2kqILm3IgQDCkUOU6q0jwtGRjlBA/vvSpeb0L3/ZnHjhhbbnQ7dpvv/+f3ABDb4vvpU1/NPvPGI8AMRbCEQkgHBEhE1RgwkgHCuEoysf6vl49NFvt6/axaPvEeNaR4IABOYjgHDMx56S708A4VhTOCwf6vWweKjH45NPfnt/yoXs4VtZ1W79sGkiqNrPkEkhAaYZ2RNAOLIPYdENQDgGCkcoHj977bX2wvujH/1Tsbd2al6G5mF4XoaW+sytrEX/LtC4TAkgHJkGrpJqIxwbCofF4+zZXze6y0X/6y9lSMHzMsJbWXnEeCW/CDQzawIIR9bhK77yCMeWwiHx0DDLyz/5Sdvbcfbs2WxPGgmTHiseDpmUIlHZBoWKQ2AAAYRjACx2jU4A4RhBONzb8e/vvNNKx+uv/2v0QG5aoIZG1HvhIRPdbcK8jE1pchwE5iWAcMzLn9JXE0A4RhQOiYduo9WXPmXp0JCJ5mGEt7LyiPHVXxS2QiAHAghHDlGqt44Ix8jCkbJ0qOcinJeh4RM9EZQEAQiUQQDhKCOOpbYC4ZhAOELp0PM65ky6lbU7L6OmW3nnZE/ZEIhNAOGITZzyhhBAOCYSjlA6Yk+85BHjQ74C7AuBcgggHOXEssSWIBwTCoekQ3ev6K6PqZ9b4VtZjx9/vp1Dokmg3Mpa4leWNkFgOQGEYzkbtsxPAOGYWDh0y+zTT/9jo4eDTZF8K6t+aPTS8EnsHpUp2kWeEIDAcAIIx3BmHBGPAMIxsXCol+M3v/nPVgZ0Z8gYqfuIcfVqcCvrGGTJAwJ5E0A48o5f6bVHOCIIh6TDf39l0787oiGZ7q2sPGK89K8n7YPAMAIIxzBe7B2XAMIRSTg0tKJHoA99Pkffrazq4SBBAAIQ6BJAOLpE+JwSAYQjknCEd63cbwKpno3RvZWVeRkpfW2oCwTSJIBwpBkXavUXAghHROGQdCzr5ZCE8IhxvpYQgMA2BBCObehx7NQEEI7IwqFHnz/66LcbzeXwrazhI8YlHffrAZn6pCB/CEAgTwIIR55xq6XWCEdk4dBcDv0ofP/7T7dLPaODR4zX8nWjnRCYlgDCMS1fct+OAMIRSTh0a6weAvbII3/bisbhw09xK+t25y5HQwACHQIIRwcIH5MigHBMKBy/u3ix0Z+sP3ToO61kPPbYweanP32l+eMf/zepk4DKQAACZRBAOMqIY6mtQDhGFg4NmWiexnPPPdtKxre+9TfNs88+0/z+9/9V6jlEuyAAgUQIIByJBIJq9BIYJBynTr3bHDv2TG9GY668dOnT5tatW2tleeXK5+2FXct1k2471RdTd42M9frVhx/uGjJ5+ul/aH7xi1PrVon9IAABCGxNAOHYGiEZTEggOeG4c+dOKwOSjnXS22+/1e6v5bppLOHQvIyfvfZa893v/l1bB00AffPNf2v+/Oev1q0K+0EAAhAYjQDCMRpKMpqAQNbCITk5ePBA89JLLzaahLlu2kY4NGSieRn6g2z6cusWV/0tE+ZlrEuf/SAAgakIIBxTkSXfMQhsJBw7O6+2F1ud3HofJg2FhNs1BNPtrdDQjI71S/tcv/5F494Nr+/mHZaj92fOfNSoZ+PChY/bvLrldPf3502EQ/MyTrzwQlvON7/51833vvf3zMswUJYQgEASBIU/WccAABAQSURBVPTbSYJAqgQGC4dOaA9fSC7UsxCKgT6rx0HyoOQhDwmFkmRD+3i789AxSpaOdeTBoqLj1NMR1qPNrOcf/W2SH//4n1tx0HCI7iRZNo/j7Nlf75qX8cQTj7dDJj3ZsgoCEIDA7AQQjtlDQAVWEBgsHN2hC/cuSBzC92GZOsaSEq73e4mC811XODRJNJzAqvwlHRYZ5+2lnurpJ3q6B8VL9V5YOiQg+suunpfhW1mZl2GSLCEAgVQJIBypRoZ6icBg4XBPhPGp50InuQRAvRe66HeThCKUAx2jffXSeh3v49YVDgmGhlScfLeKpKcv6WmeFoy+5Ruvv75rXoZuZWVeRh9J1kEAAqkS0G8bCQKpEoguHJIEfSkkDBIOCYKEZIhwSEq0f584dIVI4NW70bdvuO4b3/irhltZUz1NqRcEILAOAf2mkSCQKoHBwuGhDzdIwqCLvyRgnSEV7Rv2TCgf9XIMEQ5PFnUdvLTMaHgnTJ4kGgpG9/0Pf/hceAjvIQABCGRHAOHILmRVVXiwcOiEVs+EkoZGuvMzJA+rJo1q/7AXQnkpz65wLBsakUxo374HfWmb8grzVz3XEQ7d2kqCAAQgkDMBhCPn6JVf98HCYaHQia2X5cOo1NOhIRJv18U/vONEkuJ5G9pH+3Z7JjTcom3aL0yeL+K8w3w998PbtNQ6Jz2UK9zWfa8/C0+CAAQgkDMB/a6F6ezZc82+fQ+0r9On31tsOnTo8Sb87A1a9+CDD7XbtI+O1fLmzS+bEydeXOSlfEkQGEpgkHAMzTyl/c+fP7dUOCQjmudBggAEIJAzga5wuC2SCAmDk8SiKxyhnEgybt++3Vy8eHEhHZcvX24P379/f6MXCQJDCVQjHAIj6ej2dDz55BPNjRs3hnJjfwhAAALJEVgmHEeOHG17KlzhnZ2dttfCn720dIQ9GOrl0P5OykvrSBAYSqAq4RAc9WR4ToeWJAhAAAKlEFgmHBIG90pcu3ZtT++G249wmATLKQhUJxyGuOyL6e0sIQABCORGYNnv2smTJxe9EuHQSrd9CEeXCJ/HJIBwjEmTvCAAAQjMSGCZcGi+hoZBJB6aALoseT/P79A8Dh0XSoonk2obCQJDCCAcQ2ixLwQgAIGECSwTDvdcaBLosuR9JBh6WT78WXM3PH/D65blxXoI9BFAOPqosA4CEIBAhgRWCYeEggSBOQkgHHPSp2wIQAACIxIIhUMTRTWEopeHSEYsiqwgMJgAwjEYGQdAAAIQSJNAKBwa/tDzN/z8jDRrTK1qIoBw1BRt2goBCBRNIBSOohtK47IkgHBkGTYqDQEIQGAvAYRjLxPWpEMA4UgnFtQEAhCAwFYEEI6t8HHwxAQQjokBkz0EIACBWAQQjlikKWcTAgjHJtQ4BgIQgECCBBCOBINClRYEEI4FCt5AAAIQyJsAwpF3/EqvPcJReoRpHwQgUA0BhKOaUGfZUIQjy7BRaQhAAAJ7CSAce5mwJh0CCEc6saAmEIAABLYigHBshY+DJyaAcEwMmOwhAAEIxCKAcMQiTTmbEEA4NqHGMRCAAAQSJIBwJBgUqrQggHAsUPAGAhCAQN4EEI6841d67RGO0iNM+yAAgWoIIBzVhDrLhiIcWYaNSkMAAhDYSwDh2MuENekQQDjSiQU1gQAEILAVAYRjK3wcPDEBhGNiwGQPAQhAIBYBhCMWacrZhADCsQk1joEABCCQIAGEI8GgUKUFAYRjgYI3EIAABPImgHDkHb/Sa49wlB5h2gcBCFRDAOGoJtRZNhThyDJsVBoCEIDAXgIIx14mrEmHAMKRTiyoCQQgAIGtCCAcW+Hj4IkJIBwTAyZ7CEAAArEIIByxSFPOJgQQjk2ocQwEIACBBAkgHAkGhSotCCAcCxS8gQAEIJA3AYQj7/iVXnuEo/QI0z4IQKAaAghHNaHOsqEIR5Zho9IQgAAE9hJAOPYyYU06BBCOdGJBTSAAAQhsRQDh2AofB09MAOGYGDDZQwACEIhFAOGIRZpyNiGAcGxCjWMgAAEIJEgA4UgwKFRpQQDhWKDgDQQgAIG8CSAcecev9NojHKVHmPZBAALVEEA4qgl1lg1FOLIMG5WGAAQgsJcAwrGXCWvSIYBwpBMLagIBCEBgKwIIx1b4OHhiAgjHxIDJHgIQgEAsAghHLNKUswkBhGMTahwDAQhAIEECCEeCQaFKCwIIxwIFbyAAAQjkTQDhyDt+pdce4Sg9wrQPAhCohgDCUU2os2wowpFl2Kg0BCAAgb0EEI69TFiTDgGEI51YUBMIQAACWxFAOLbCx8ETE0A4JgZM9hCAAARiEUA4YpGmnE0IIBybUOMYCEAAAgkSQDgSDApVWhBAOBYoeAMBCEAgbwIIR97xK732CEfpEaZ9EIBANQQQjmpCnWVDEY4sw0alIQABCOwlgHDsZcKadAggHOnEgppAAAIQ2IoAwrEVPg6emADCMTFgsocABCAQiwDCEYs05WxCIAnh2LfvgUavkydPtm04ffq99vPOzs6eNp09e645dOjxRtv00nEPPvhQc/ny5cbHaV3fsWFmfDFDGryHAARKIMDvWglRLLcNSQiH8FoWtJRU9KWbN79sBcOSce3atUbrJBx66Vgli4i2LUt8MZeRYT0EIJArAX7Xco1cHfVORjhu3769q5djGX5LR9iDceTI0Wb//v2LQywv6vVYlvhiLiPDeghAIFcC/K7lGrk66p2McAi3xMHDKsvwIxzLyLAeAhConQDCUfsZkHb7kxGOixcvtr0Ump+xKiEcq+iwDQIQqJkAwlFz9NNvexLCIYlQ74bmZGh+hoZXPB+ji9D7nDjx4mKTJEVzOJzUS6J8JDHLEl/MZWRYDwEI5EqA37VcI1dHvWcXDomBhEHSoaS5GL7rpBsC927oGL00j0P7+7OWnr/hdcvmcfDF7NLlMwQgkDsBftdyj2DZ9Z9dOObCyxdzLvKUCwEITEWA37WpyJLvGAQQjjEokgcEIACBBAggHAkEgSosJYBwLEXDBghAAAJ5EUA48opXbbVFOGqLOO2FAASKJYBwFBvaIhqGcBQRRhoBAQhAoGkQDs6ClAkgHClHh7pBAAIQGEAA4RgAi12jE0A4oiOnQAhAAALTEEA4puFKruMQQDjG4UguEIAABGYngHDMHgIqsIIAwrECDpsgAAEI5EQA4cgpWvXVFeGoL+a0GAIQKJQAwlFoYAtpFsJRSCBpBgQgAAGEg3MgZQIIR8rRoW4QgAAEBhBAOAbAYtfoBBCO6MgpEAIQgMA0BBCOabiS6zgEEI5xOJILBCAAgdkJIByzh4AKrCCAcKyAwyYIQAACORFAOHKKVn11RTjqizkthgAECiWAcBQa2EKahXAUEkiaAQEIQADh4BxImQDCkXJ0qBsEIACBAQQQjgGw2DU6AYQjOnIKhAAEIDANAYRjGq7kOg4BhGMcjuQCAQhAYHYCCMfsIaACKwggHCvgsAkCEIBATgQQjpyiVV9dEY76Yk6LIQCBQgkgHIUGtpBmIRyFBJJmQAACEEA4OAdSJoBwpBwd6gYBCEBgAAGEYwAsdo1OAOGIjpwCIQABCExDAOGYhiu5jkMA4RiHI7lAAAIQmJ0AwjF7CKjACgIIxwo4bIIABCCQEwGEI6do1VdXhKO+mNNiCECgUAIIR6GBLaRZCEchgaQZEIAABBAOzoGUCSAcKUeHukEAAhAYQADhGACLXaMTQDiiI6dACEAAAtMQQDim4Uqu4xBAOMbhSC4QgAAEZieAcMweAiqwggDCsQIOmyAAAQjkRADhyCla9dUV4agv5rQYAhAolADCUWhgC2kWwlFIIGkGBCAAAYSDcyBlAghHytGhbhCAAAQGEEA4BsBi1+gEEI7oyCkQAhCAwDQEEI5puJLrOAQQjnE4kgsEIACB2QkgHLOHgAqsIIBwrIDDJghAAAI5EUA4copWfXVFOOqLOS2GAAQKJYBwFBrYQpqFcBQSSJoBAQhAAOHgHEiZAMKRcnSoGwQgAIEBBBCOAbDYNToBhCM6cgqEAAQgMA0BhGMaruQ6DgGEYxyO5AIBCEBgdgIIx+whoAIrCCAcK+CwCQIQgEBOBBCOnKJVX10RjvpiToshAIFCCSAchQa2kGYhHIUEkmZAAAIQQDg4B1ImgHCkHB3qBgEIQGAAAYRjACx2jU4A4YiOnAIhAAEITEMA4ZiGK7mOQ6A64bh69WrzwQfvN/pinj9/rrl79+44JMkFAhCAwEwEPvnkt4vfNb0nQSBFAlUJxxtvvN6KhmTDrwMHHmn4gqZ4alInCEDgfgRu3LjRHD36g8XvmX/XtE7bSBBIiUA1wvHOOz/f86X0l1PLr776U0pxoS4QgAAE7kugTzb8u6ZtJAikRKAK4dCwib+Ey5bq/SBBAAIQyIXAZ5/94b6/a9qHBIFUCFQhHJq3sUw0WP//w0uwgAXnQFnngOarkSCQCgGE4958juPHn08lJtQDAhCAwH0JePL7KklEOO6LkR0iEqhCOMRTk0NXfTEZUol41lEUBCCwNQGGVLZGSAaRCVQjHKv+NyAZYdJo5DOP4iAAga0JMGl0a4RkEJFANcIhpn13qkg2uH0s4hlHURCAwGgENCG+Tzo0RMwzhkbDTEYjEahKOMRMPRl64Jd6PPT8Db6UI51JZAMBCMxGQMMr+k3TS5PkSRBIkUB1wpFiEKgTBCAAAQhAoHQCCEfpEaZ9oxC4fPlys2/fA4vXkSNHF/nevn270Wdtf/DBh5qzZ88ttvEGAhCAAAT+QgDh4EyAwBoEJBSSjr504sSLzaFDjzcSj4sXL7bice3atb5dWQcBCECgWgIIR7Whp+HrEpBohD0a4XHu+ZBoOGnfZft7H5YQgAAEaiOAcNQWcdo7mIDkwcMp6snY2dlpezOU0cmTJ9tt6t1wOn36vT3rvI0lBCAAgVoJIBy1Rp52DyIgoZBoaPhE8iGpUNJnzdsIk+ZwaJ9wWEXrLCvKx/M91ENiQdE6bSNBAAIQKJEAwlFiVGnTpAT279/fyoMKUe+HPofJwuE5HzdvftkKhiVDIqJ1EhW9LC8WEW0jQQACECiNAMJRWkRpz+QENIxiyVAPh9+7YPdYhD0clo6wB6MrKz7OouL8WEIAAhAogQDCUUIUaUNUAurBsGT0zeHoW4dwRA0RhUEAAgkSQDgSDApVSpuAhEIvpb67VDRXo3uXCsKRdkypHQQgMD0BhGN6xpSQMQGJguVCzdCwh3o3wrtSJBeSDCXP3wiHU7RenzWHQ0MwTjomnHDqnpHwFlvvyxICEIBA7gQQjtwjSP0nJSBRkBRIFnwXSSgbKlyfJQ+eFCrpCJN7N8I8JC3+rKXnb3gd8zhCgryHAARKIIBwlBBF2gABCEAAAhBInADCkXiAqB4EIAABCECgBAIIRwlRpA0QgAAEIACBxAkgHIkHiOpBAAIQgAAESiCAcJQQRdoAAQhAAAIQSJwAwpF4gKgeBCAAAQhAoAQC/wcOQQiPZTRSwQAAAABJRU5ErkJggg=="></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>x</mi><mo>+</mo><mn>50</mn><mo> </mo><mtext>cot</mtext><mo> </mo><mi>θ</mi></math> .</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>At time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math>, the following conditions are true.</p>
<p style="padding-left:60px;">Boat <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math> has travelled <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn></math> metres further than boat <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math>.<br>Boat <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math> is travelling at double the speed of boat <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math>.<br>The rate of change of the angle <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>0</mn><mo>.</mo><mn>1</mn></math> radians per second.</p>
<p>Find the speed of boat <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> at time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tan</mi><mo> </mo><mi>θ</mi><mo>=</mo><mfrac><mn>50</mn><mrow><mi>y</mi><mo>-</mo><mi>x</mi></mrow></mfrac></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>cot</mtext><mo> </mo><mi>θ</mi><mo>=</mo><mfrac><mrow><mi>y</mi><mo>-</mo><mi>x</mi></mrow><mn>50</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>x</mi><mo>+</mo><mn>50</mn><mo> </mo><mtext>cot</mtext><mo> </mo><mi>θ</mi></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>Note:</strong> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>-</mo><mi>x</mi></math> may be identified as a length on a diagram, and not written explicitly.</p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to differentiate with respect to <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>-</mo><mn>50</mn><msup><mfenced><mrow><mtext>cosec</mtext><mo> </mo><mi>θ</mi></mrow></mfenced><mn>2</mn></msup><mfrac><mrow><mo>d</mo><mi>θ</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></math> <em><strong>A1</strong></em></p>
<p>attempt to set speed of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math> equal to double the speed of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>-</mo><mn>50</mn><msup><mfenced><mrow><mtext>cosec</mtext><mo> </mo><mi>θ</mi></mrow></mfenced><mn>2</mn></msup><mfrac><mrow><mo>d</mo><mi>θ</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mo>-</mo><mn>50</mn><msup><mfenced><mrow><mtext>cosec</mtext><mo> </mo><mi>θ</mi></mrow></mfenced><mn>2</mn></msup><mfrac><mrow><mo>d</mo><mi>θ</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi><mo>=</mo><mtext>arctan</mtext><mo> </mo><mn>5</mn><mfenced><mrow><mo>=</mo><mn>1</mn><mo>.</mo><mn>373</mn><mo>…</mo><mo>=</mo><mn>78</mn><mo>.</mo><mn>69</mn><mo>…</mo><mo>°</mo></mrow></mfenced></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>cosec</mtext><mn>2</mn></msup><mo> </mo><mi>θ</mi><mo>=</mo><mn>1</mn><mo>+</mo><msup><mtext>cot</mtext><mn>2</mn></msup><mo> </mo><mi>θ</mi><mo>=</mo><mn>1</mn><mo>+</mo><msup><mfenced><mfrac><mn>1</mn><mn>5</mn></mfrac></mfenced><mn>2</mn></msup><mo>=</mo><mfrac><mn>26</mn><mn>25</mn></mfrac></math> <em><strong>(A1)</strong></em></p>
<p> </p>
<p><strong>Note:</strong> This <em><strong>A1</strong></em> can be awarded independently of previous marks.</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mo>-</mo><mn>50</mn><mfenced><mfrac><mn>26</mn><mn>25</mn></mfrac></mfenced><mo>×</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>1</mn></math></p>
<p>So the speed of boat <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mo>.</mo><mn>2</mn><mo> </mo><mfenced><msup><mtext>ms</mtext><mrow><mo>-</mo><mn>1</mn></mrow></msup></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mo>.</mo><mn>20</mn></math> from the use of inexact values.</p>
<p> </p>
<p><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>A small bead is free to move along a smooth wire in the shape of the curve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfrac><mn>10</mn><mrow><mn>3</mn><mo>-</mo><mn>2</mn><msup><mtext>e</mtext><mrow><mo>-</mo><mn>0</mn><mo>.</mo><mn>5</mn><mi>x</mi></mrow></msup></mrow></mfrac><mfenced><mrow><mi>x</mi><mo>≥</mo><mn>0</mn></mrow></mfenced></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mtext>d</mtext><mi>y</mi></mrow><mrow><mtext>d</mtext><mi>x</mi></mrow></mfrac></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>At the point on the curve where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>4</mn></math>, it is given that <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mtext>d</mtext><mi>y</mi></mrow><mrow><mtext>d</mtext><mi>t</mi></mrow></mfrac><mo>=</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>1</mn><mo> </mo><msup><mtext>m s</mtext><mrow><mo>-</mo><mn>1</mn></mrow></msup></math></p>
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mtext>d</mtext><mi>x</mi></mrow><mrow><mtext>d</mtext><mi>t</mi></mrow></mfrac></math> at this exact same instant.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p>valid attempt to use chain rule or quotient rule <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mtext>d</mtext><mi>y</mi></mrow><mrow><mtext>d</mtext><mi>x</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mo>-</mo><mn>10</mn><msup><mtext>e</mtext><mrow><mo>-</mo><mn>0</mn><mo>.</mo><mn>5</mn><mi>x</mi></mrow></msup></mrow><msup><mfenced><mrow><mn>3</mn><mo>-</mo><mn>2</mn><msup><mtext>e</mtext><mrow><mo>-</mo><mn>0</mn><mo>.</mo><mn>5</mn><mi>x</mi></mrow></msup></mrow></mfenced><mn>2</mn></msup></mfrac></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mtext>d</mtext><mi>y</mi></mrow><mrow><mtext>d</mtext><mi>x</mi></mrow></mfrac><mo>=</mo><mo>-</mo><mn>10</mn><msup><mtext>e</mtext><mrow><mo>-</mo><mn>0</mn><mo>.</mo><mn>5</mn><mi>x</mi></mrow></msup><mtext> </mtext><msup><mfenced><mrow><mn>3</mn><mo>-</mo><mn>2</mn><msup><mtext>e</mtext><mrow><mo>-</mo><mn>0</mn><mo>.</mo><mn>5</mn><mi>x</mi></mrow></msup></mrow></mfenced><mrow><mo>-</mo><mn>2</mn></mrow></msup></math> <em><strong>A1A1</strong></em></p>
<p><em><strong><br>[3 marks]</strong></em></p>
<p><em><strong><br></strong></em><strong>Note: </strong>Award <em><strong>A1</strong></em> for numerator and <em><strong>A1</strong></em> for denominator, or <em><strong>A1</strong></em> for each part if the second alternative given.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>valid attempt to use chain rule <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mtext>eg </mtext><mo> </mo><mfrac><mrow><mtext>d</mtext><mi>y</mi></mrow><mrow><mtext>d</mtext><mi>t</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mtext>d</mtext><mi>y</mi></mrow><mrow><mtext>d</mtext><mi>x</mi></mrow></mfrac><mo>×</mo><mfrac><mrow><mtext>d</mtext><mi>x</mi></mrow><mrow><mtext>d</mtext><mi>t</mi></mrow></mfrac></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mtext>d</mtext><mi>x</mi></mrow><mrow><mtext>d</mtext><mi>t</mi></mrow></mfrac><mo>=</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>1</mn><mo>÷</mo><mfrac><mrow><mo>-</mo><mn>10</mn><msup><mtext>e</mtext><mrow><mo>-</mo><mn>2</mn></mrow></msup></mrow><msup><mfenced><mrow><mn>3</mn><mo>-</mo><mn>2</mn><msup><mtext>e</mtext><mrow><mo>-</mo><mn>2</mn></mrow></msup></mrow></mfenced><mn>2</mn></msup></mfrac><mo> </mo><mfenced><mrow><mo>=</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>1</mn><mo>÷</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>181676</mn><mo>…</mo></mrow></mfenced></math> or equivalent <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>0</mn><mo>.</mo><mn>550428</mn><mo>…</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mtext>d</mtext><mi>x</mi></mrow><mrow><mtext>d</mtext><mi>t</mi></mrow></mfrac><mo>=</mo><mn>0</mn><mo>.</mo><mn>550</mn><mo> </mo><mfenced><msup><mtext>ms</mtext><mrow><mo>-</mo><mn>1</mn></mrow></msup></mfenced></math> <em><strong>A1</strong></em></p>
<p><em><strong><br>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>An earth satellite moves in a path that can be described by the curve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="72.5{x^2} + 71.5{y^2} = 1">
<mn>72.5</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>71.5</mn>
<mrow>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>1</mn>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = x(t)">
<mi>x</mi>
<mo>=</mo>
<mi>x</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = y(t)">
<mi>y</mi>
<mo>=</mo>
<mi>y</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
</math></span> are in thousands of kilometres and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> is time in seconds.</p>
<p>Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}x}}{{{\text{d}}t}} = 7.75 \times {10^{ - 5}}">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mn>7.75</mn>
<mo>×</mo>
<mrow>
<msup>
<mn>10</mn>
<mrow>
<mo>−</mo>
<mn>5</mn>
</mrow>
</msup>
</mrow>
</math></span> when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 3.2 \times {10^{ - 3}}">
<mi>x</mi>
<mo>=</mo>
<mn>3.2</mn>
<mo>×</mo>
<mrow>
<msup>
<mn>10</mn>
<mrow>
<mo>−</mo>
<mn>3</mn>
</mrow>
</msup>
</mrow>
</math></span>, find the possible values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}t}}">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
</math></span>.</p>
<p>Give your answers in standard form.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><strong>METHOD 1</strong></p>
<p>substituting for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> and attempting to solve for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span> (or vice versa) <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = ( \pm )0.11821 \ldots ">
<mi>y</mi>
<mo>=</mo>
<mo stretchy="false">(</mo>
<mo>±</mo>
<mo stretchy="false">)</mo>
<mn>0.11821</mn>
<mo>…</mo>
</math></span> <strong><em>(A1)</em></strong></p>
<p><strong>EITHER</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="145x + 143y\frac{{{\text{d}}y}}{{{\text{d}}x}} = 0{\text{ }}\left( {\frac{{{\text{d}}y}}{{{\text{d}}x}} = - \frac{{145x}}{{143y}}} \right)">
<mn>145</mn>
<mi>x</mi>
<mo>+</mo>
<mn>143</mn>
<mi>y</mi>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mn>0</mn>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mo>−</mo>
<mfrac>
<mrow>
<mn>145</mn>
<mi>x</mi>
</mrow>
<mrow>
<mn>143</mn>
<mi>y</mi>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>M1A1</em></strong></p>
<p><strong>OR</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="145x\frac{{{\text{d}}x}}{{{\text{d}}t}} + 143y\frac{{{\text{d}}y}}{{{\text{d}}t}} = 0">
<mn>145</mn>
<mi>x</mi>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
<mo>+</mo>
<mn>143</mn>
<mi>y</mi>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mn>0</mn>
</math></span> <strong><em>M1A1</em></strong></p>
<p><strong>THEN</strong></p>
<p>attempting to find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}x}}{{{\text{d}}t}}{\text{ }}\left( {\frac{{{\text{d}}y}}{{{\text{d}}t}} = - \frac{{145(3.2 \times {{10}^{ - 3}})}}{{143\left( {( \pm )0.11821 \ldots } \right)}} \times (7.75 \times {{10}^{ - 5}})} \right)">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mo>−</mo>
<mfrac>
<mrow>
<mn>145</mn>
<mo stretchy="false">(</mo>
<mn>3.2</mn>
<mo>×</mo>
<mrow>
<msup>
<mrow>
<mn>10</mn>
</mrow>
<mrow>
<mo>−</mo>
<mn>3</mn>
</mrow>
</msup>
</mrow>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<mn>143</mn>
<mrow>
<mo>(</mo>
<mrow>
<mo stretchy="false">(</mo>
<mo>±</mo>
<mo stretchy="false">)</mo>
<mn>0.11821</mn>
<mo>…</mo>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
</mfrac>
<mo>×</mo>
<mo stretchy="false">(</mo>
<mn>7.75</mn>
<mo>×</mo>
<mrow>
<msup>
<mrow>
<mn>10</mn>
</mrow>
<mrow>
<mo>−</mo>
<mn>5</mn>
</mrow>
</msup>
</mrow>
<mo stretchy="false">)</mo>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}t}} = \pm 2.13 \times {10^{ - 6}}">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mo>±</mo>
<mn>2.13</mn>
<mo>×</mo>
<mrow>
<msup>
<mn>10</mn>
<mrow>
<mo>−</mo>
<mn>6</mn>
</mrow>
</msup>
</mrow>
</math></span> <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award all marks except the final <strong><em>A1 </em></strong>to candidates who do not consider ±.</p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = ( \pm )\sqrt {\frac{{1 - 72.5{x^2}}}{{71.5}}} ">
<mi>y</mi>
<mo>=</mo>
<mo stretchy="false">(</mo>
<mo>±</mo>
<mo stretchy="false">)</mo>
<msqrt>
<mfrac>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mn>72.5</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mn>71.5</mn>
</mrow>
</mfrac>
</msqrt>
</math></span> <strong><em>M1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}} = ( \pm )0.0274 \ldots ">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mo stretchy="false">(</mo>
<mo>±</mo>
<mo stretchy="false">)</mo>
<mn>0.0274</mn>
<mo>…</mo>
</math></span> <strong><em>(M1)(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}t}} = ( \pm )0.0274 \ldots \times 7.75 \times {10^{ - 5}}">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mo stretchy="false">(</mo>
<mo>±</mo>
<mo stretchy="false">)</mo>
<mn>0.0274</mn>
<mo>…</mo>
<mo>×</mo>
<mn>7.75</mn>
<mo>×</mo>
<mrow>
<msup>
<mn>10</mn>
<mrow>
<mo>−</mo>
<mn>5</mn>
</mrow>
</msup>
</mrow>
</math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}t}} = \pm 2.13 \times {10^{ - 6}}">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mo>±</mo>
<mn>2.13</mn>
<mo>×</mo>
<mrow>
<msup>
<mn>10</mn>
<mrow>
<mo>−</mo>
<mn>6</mn>
</mrow>
</msup>
</mrow>
</math></span> <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award all marks except the final <strong><em>A1 </em></strong>to candidates who do not consider ±.</p>
<p> </p>
<p><strong><em>[6 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br>