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<h2>HL Paper 1</h2><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Prove by mathematical induction that <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msup><mo>d</mo><mi>n</mi></msup><mrow><mo>d</mo><msup><mi>x</mi><mi>n</mi></msup></mrow></mfrac><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><msup><mi mathvariant="normal">e</mi><mi>x</mi></msup></mrow></mfenced><mo>=</mo><mfenced open="[" close="]"><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>n</mi><mi>x</mi><mo>+</mo><mi>n</mi><mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></mfenced></mrow></mfenced><msup><mi mathvariant="normal">e</mi><mi>x</mi></msup></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>∈</mo><msup><mi mathvariant="normal">ℤ</mi><mo>+</mo></msup></math>.</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>Hence or otherwise, determine the Maclaurin series of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><msup><mi mathvariant="normal">e</mi><mi>x</mi></msup></math> in ascending powers of <math xmlns="http://www.w3.org/1998/Math/MathML"><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>.</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>Hence or otherwise, determine the value of <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 open="[" close="]"><mfrac><msup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><msup><mi mathvariant="normal">e</mi><mi>x</mi></msup><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced><mn>3</mn></msup><msup><mi>x</mi><mn>9</mn></msup></mfrac></mfenced></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</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_n}(x) = (\cos 2x)(\cos 4x) \ldots (\cos {2^n}x),{\text{ }}n \in {\mathbb{Z}^ + }">
<mrow>
<msub>
<mi>f</mi>
<mi>n</mi>
</msub>
</mrow>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo stretchy="false">(</mo>
<mi>cos</mi>
<mo><!-- --></mo>
<mn>2</mn>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mi>cos</mi>
<mo><!-- --></mo>
<mn>4</mn>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>…<!-- … --></mo>
<mo stretchy="false">(</mo>
<mi>cos</mi>
<mo><!-- --></mo>
<mrow>
<msup>
<mn>2</mn>
<mi>n</mi>
</msup>
</mrow>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>n</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<msup>
<mrow>
<mi mathvariant="double-struck">Z</mi>
</mrow>
<mo>+</mo>
</msup>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine whether <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f_n}"> <mrow> <msub> <mi>f</mi> <mi>n</mi> </msub> </mrow> </math></span> is an odd or even function, justifying your answer.</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>By using mathematical induction, prove that</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f_n}(x) = \frac{{\sin {2^{n + 1}}x}}{{{2^n}\sin 2x}},{\text{ }}x \ne \frac{{m\pi }}{2}"> <mrow> <msub> <mi>f</mi> <mi>n</mi> </msub> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mfrac> <mrow> <mi>sin</mi> <mo></mo> <mrow> <msup> <mn>2</mn> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </mrow> <mi>x</mi> </mrow> <mrow> <mrow> <msup> <mn>2</mn> <mi>n</mi> </msup> </mrow> <mi>sin</mi> <mo></mo> <mn>2</mn> <mi>x</mi> </mrow> </mfrac> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mi>x</mi> <mo>≠</mo> <mfrac> <mrow> <mi>m</mi> <mi>π</mi> </mrow> <mn>2</mn> </mfrac> </math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="m \in \mathbb{Z}"> <mi>m</mi> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> </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>Hence or otherwise, find an expression for the derivative of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f_n}(x)"> <mrow> <msub> <mi>f</mi> <mi>n</mi> </msub> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></span> with respect to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>.</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>Show that, for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n > 1"> <mi>n</mi> <mo>></mo> <mn>1</mn> </math></span>, the equation of the tangent to the curve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = {f_n}(x)"> <mi>y</mi> <mo>=</mo> <mrow> <msub> <mi>f</mi> <mi>n</mi> </msub> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></span> at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \frac{\pi }{4}"> <mi>x</mi> <mo>=</mo> <mfrac> <mi>π</mi> <mn>4</mn> </mfrac> </math></span> is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="4x - 2y - \pi = 0"> <mn>4</mn> <mi>x</mi> <mo>−</mo> <mn>2</mn> <mi>y</mi> <mo>−</mo> <mi>π</mi> <mo>=</mo> <mn>0</mn> </math></span>.</p>
<div class="marks">[8]</div>
<div class="question_part_label">d.</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{e}}^x}\,{\text{cos}}{\,^2}x">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mi>x</mi>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cos</mtext>
</mrow>
<mrow>
<msup>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
</msup>
</mrow>
<mi>x</mi>
</math></span>, where 0 ≤ <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> ≤ 5. The curve <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> is shown on the following graph which has local maximum points at A and C and touches the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>-axis at B and D.</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAnkAAAEhCAYAAAADPLamAAAgAElEQVR4Ae3dC3RV5Zn/8V+oSulgqqiDwSAsDAmCYuQmlwhEmABrEDuBsQ5/Li0X21pQqUqrrTKg1kvRioItSkCwglDKVGkHCSalBazFcNGKkBQpLpBoFcZiqo4jyX89W3dIQi7nJOeyL9+9luZc9nn3+37eTfKc95pSVVVVJQ4EEEAAAQQQQACBQAm0ClRpKAwCCCCAAAIIIICAI0CQx42AAAIIIIAAAggEUIAgL4CVSpEQQAABBBBAAIGogryPP/5Yu3fvjlLtkNZNzlBKSopSUkZr/o4Pvvj8+yr+fh91mLpOb1dGmSSnI4AAAggggAACCDQqEHGQZwHejTfeqMsvv1zbtm1rNNHab3ZU/vL9OlFaoDzt1KZX39HnMV2qLh5xtbIOf6B/1P4AzxBAAAEEEEAAAQRaKBBxkFdQUKAlS5Y4l8vJyYky0JNadblM/3JRud489o8vgrwz1D69s7r8yyXqEHEuWlhaPo4AAggggAACCIREIKLwauHChZo5c2YtEgv0ysrKar3W6JNW/6R2F6XpzT+/pffsxMq39OuH39C//ke22jb6Qd5EAAEEEEAAAQQQiFagySDPDfD69u1bnfbWrVudxxMmTNDhw4erX2/0Qat/0lnnf+WLUypVsWuDSvJu0NcuOKPRj/EmAggggAACCCCAQPQCjQZ569atc1rwLMCzx+4xaNAgWaD3yiuvKD8/P8JAr43OOv8sadsBHTpUpAd/labvfq2TGs2Ae0F+IoAAAggggAACCEQl0GCMZZMrxo4dKzfAS09Pr5WwBXrLly+PItD7sr563pnSR9u06JFXNXTmKF3Q4NVrXYonCCCAAAIIIIAAAlEKNBhmPfXUUw0GeO41Jk2apMcee8wJ9IqLi92XG/h5ms5sd56kdhoy/XpdlUY3bQNQvIwAAggggAACCLRYIKWhvWttyZTS0lJlZ2dXX8TWurOj7na31upnLXuNH5+obOkP9VTn7+meqy6gm7ZxLN5FAAEEEEAAAQRaJNBgkFdfqg0FefWdW/u1SlXseEJ3v5qjO6dcwmza2jg8QwABBBBAAAEEYi5wWsxTrE7QArsFuvrqtzRxWVeVvtFXd84iwKvm4QECCCCAAAIIIBBHgQbH5MXimpV/f0+l5XtV+k5f3XxTP1rwYoFKGggggAACCCCAQAQCCequjSAnnIIAAggggAACCCAQM4G4tuTFLJckhAACCCCAAAIIIBCVAEFeVFycjAACCCCAAAII+EOAIM8f9UQuEUAAAQQQQACBqAQI8qLi4mQEEEAAAQQQQMAfAgR5/qgncokAAggggAACCEQlQJAXFRcnI4AAAggggAAC/hAgyPNHPZFLBBBAAAEEEEAgKgGCvKi4OBkBBBBAAAEEEPCHAEGeP+qJXCKAAAIIIIAAAlEJEORFxcXJCCCAAAIIIICAPwQI8vxRT+QSAQQQQAABBBCISoAgLyouTkYAAQQQQAABBPwhQJDnj3oilwgggAACCCCAQFQCBHlRcXEyAggggAACCCDgDwGCPH/UE7lEAAEEEEAAAQSiEiDIi4qLkxFAAAEEEEAAAX8IEOT5o57IJQIIIIAAAgggEJUAQV5UXJyMAAIIIIAAAgj4Q4Agzx/1RC4RQAABBBBAAIGoBAjyouLiZAQQQAABBBBAwB8CBHn+qCdyiQACCCCAAAIIRCVAkBcVFycjgAACCCCAAAL+ECDI80c9kUsEEEAAAQQQQCAqAYK8qLg4GQEEEEAAAQQQ8IcAQZ4/6olcIoAAAggggAACUQkQ5EXFxckIIIAAAggggIA/BAjy/FFP5BIBBBAItkBluXY8t1LzJ1+qlJQUpaR0UO73n9RzO8r1WdlGPVf2SbDLT+kQiIMAQV4cUEkSAQQQQCBygcrybfrpN/N09boj6nJjoU5UVamq6oiKbu6lE7/7gS7MKtDRyJPjTAQQ+EIgpaqqqipSDft2ZUcUH4k0ac5DAAEEEAijQOVbWjd9tMYemKKS9Tepd9u6bQ8faMf8m7Sm10N64KpzwyhEmRFotsBpzf4kH0QAAQQQQKBFApU6vnmxZixtrdlFE+sJ8Czxs9T7+u/orZ0tuhAfRiCUAnW/MoUSgUIjgAACCCRBoLJMax94SuXqoqz0tg1nILW/8ofSitcwEO8gUL8AQV79LryKAAIIIBBvgYojKv1zuZSWoc7nnxHvq5E+AqETIMgLXZVTYAQQQAABBBAIgwBBXhhqmTIigAACXhRo20FZl6Z5MWfkCYFACBDkBaIaKQQCCCDgQ4FWnZXz9RypvFAbS475sABkGQFvCxDkebt+yB0CCCAQYIEvK3PctzU7bYcevPtp7aiorLesleW7tLnseL3v8SICCDQsQJDXsA3vIIAAAgjEWyB1qO4sfkqTSr+nPlffrqXFZaqovuZxlRWv1FN/bK0+manVr/IAAQQiEyDIi8yJsxBAAAEE4iLQSm27TdayHSX69UTp6WFZOrN6W7PV2vvVXH0jv7saWWAlLrkiUQSCIMCOF0GoRcqAAAIIIIAAAgjUEaAlrw4ITxFAAAEEEEAAgSAIEOQFoRYpAwIIIIAAAgggUEeAIK8OCE8RQAABBBBAAIEgCJwWhEJQBgQQQACBYAhs27ZNv//975Wenq7Ro0erXbt2wSgYpUAgCQIEeUlA55IIIIAAAqcKLFy4UDNnzqx+o2/fvnrhhRcI9KpFeIBAdAJ010bnxdkIIIAAAnEQePPNN2sFeHaJV155RStWrIjD1UgSgXAIsIRKOOqZUiKAAAKeEigrK9OuXbucrtmf/exnTebt2muvVV5ennr37q3s7Owmz+cEBBCQCPK4CxBAAAEEEiJggd369eu1bNky7dmzxwnWLrzwQrVv315nnHGG/uu//ktHjhyplZdx48bpn//5n/W3v/1Nb7/9tv74xz+qR48e+uY3v6mrr75amZmZtc7nCQIInBQgyDtpwSMEEEAAgRgLfPzxx9q5c6fuuusuFRcXa8CAAU5glpaWptNOqz0s/NChQ9q0aVN1oDdy5Ej16dOnVo4+++wzlZeXywJGC/i+853v6OabbybYq6XEEwQ+FyDI405AAAEEEIi5gAV3GzZs0Jw5c3T8+HGn1a5r165q06ZNo9eyIM7Ot/OaOtfOs2DPJmcQ7DXKypshFSDIC2nFU2wEEEAgXgK2DIq13O3fv1/9+vVTRkbGKa12sby2BXs2ScNa9pYsWaLx48c3GSDG8vqkhYBXBQjyvFoz5AsBBBDwmcDhw4c1e/ZsrVq1SmPGjFH37t3jGtzV5Xn33Xed1kMbp2eTORivV1eI52ETYAmVsNU45UUAAQRiLGBdswUFBerYsaMOHjyoW265RT179kxogGdFsgkcEyZMUFVVlbKyspw8xbioJIeArwRoyfNVdZFZBBBAwFsCNiZuxowZ2rt3rwYPHqwuXbp4IoM2ieN3v/udk6cnnniC7ltP1AqZSLQAQV6ixbkeAgggEBCB1atX67rrrnNmzA4ZMiThLXdNMdpYvZdeekmffPKJ041rW6VxIBAmAbprw1TblBUBBBCIgYB1z9r2Yxbg2SSHYcOGeS7As2KmpqZq+PDhOvPMM52FlHfv3h2D0pMEAv4RoCXPP3VFThFAAIGkC1j37NixY1VZWekEUBZI+eEoKSlxllrZunWrBg0a5Icsk0cEWixAS16LCUkAAQQQCIeALY1iExrOO+88Z/asXwI8qx1bVNkWV87JyZGVgwOBMAgQ5IWhlikjAggg0EKBRx55xAmQbJsxawmru1tFC5NPyMct0LPuZQK9hHBzEQ8I0F3rgUogCwgggIBXBWz8na199/zzzys3N9dZJsWreY00Xzbzdvny5aLrNlIxzvOrAEGeX2uOfCOAAAJxFrAA7/rrr9fmzZuVn5/vTGSI8yUTlvy+ffu0du1aAr2EiXOhZAgQ5CVDnWsigAACHhew3StGjRrldMva+nd+Gn8XKS2TMSKV4jy/ChDk+bXmyDcCCCAQJ4GaAZ5NVvDj+LtIaSzQe+utt1RYWCjW0YtUjfP8IsDEC7/UFPlEAAEEEiAQpgDPOG0yxllnneW0Wh47diwBwlwCgcQJEOQlzporIYAAAp4WsADPWu6sRSvoLXg1K8LdrcO2Z7NxiBwIBEWAIC8oNUk5EEAAgRYIWIBnO1d07NjRad0KchdtXSYrq5V9y5Ytuv/+++u+zXMEfCtAkOfbqiPjCCCAQGwE3ACvS5cuToAXm1T9lUqbNm2cGcTz5s1TQUGBvzJPbhFoQICJFw3A8DICCCAQBgG3i9ZtwQtDmRsro7uG3q5du5Sdnd3YqbyHgOcFCPI8X0VkEAEEEIiPgAV4tkyKjcGzCQgcnwvYjFsL9v7whz+oXbt2sCDgWwG6a31bdWQcAQQQaL6AG+DZeDRarGo7msfpp5+um266qfYbPEPAZwIEeT6rMLKLAAIItFSgZoAXplm0kbpZ4GsLQFtLHuPzIlXjPC8KEOR5sVbIEwIIIBAnAXcvWgtkCPAaRrYdPmyv3mnTpmn37t0Nn8g7CHhYgDF5Hq4csoYAAgjEUsDdi3bnzp0aM2ZMoHeyiJXbtm3b9N5772n79u2yGbgcCPhJgJY8P9UWeUUAAQRaIHDvvfdq8+bNGj58OAFehI5XXHGFKisrdfvtt0f4CU5DwDsCBHneqQtyggACCMRN4JFHHtEzzzzjrAVnXZEckQlYt/bQoUO1YMECZ3/byD7FWQh4Q4DuWm/UA7lAAAEE4iZQWFioESNGaPLkyc6OFnG7UIATfu211/SXv/xFW7duZVmVANdz0IpGS17QapTyIIAAAjUEbEyZBXjjx48nwKvhEu3D7t27q3Xr1pozZ060H+V8BJImQEte0ui5MAIIIBBfAVsqJS8vT506dWKx4xhQHzt2TI8//rg2btzouMYgSZJAIK4CBHlx5SVxBBBAIDkCNpO2X79+OvPMMzVs2LDkZCKAV7Vu2zfffJPdMAJYt0EsEt21QaxVyoQAAqEWcJdKsVmhQ4YMCbVFrAtv3bZVVVXORIxYp016CMRagCAv1qKkhwACCCRZYPHixSyVEqc6cGfbzps3j0WS42RMsrEToLs2dpakhAACCCRdwJ1JO336dLVv3z7p+QlqBmxCy9GjR/Xyyy+zSHJQKzkA5aIlLwCVSBEQQAABEygrK3Nm0o4bN44AL863hC2SbBMxVq5cGecrkTwCzRegJa/5dnwSAQQQ8IyABRw2/u68887ToEGDPJOvIGfkwIEDTpB36NAhpaenB7molM2nArTk+bTiyDYCCCBQU+Cmm25ytiqzFiaOxAh06dJF2dnZeuCBBxJzQa6CQJQCBHlRgnE6Aggg4DUB27LsD3/4gwYPHsyetAmunIEDB2rhwoWyMXocCHhNgO5ar9UI+UEAAQSiELDgIicnhy3LojCL9aklJSU6ceKEnnvuOSZhxBqX9FokQEtei/j4MAIIIJA8AdvRwmbRjhw5ki3LklcNTpdtaWmpnn/++STmgksjcKoAQd6pJryCAAIIeF7AFjyePXu2s59qnz59PJ/fIGfQ1s6z3UXmzp3rzLgNclkpm78ECPL8VV/kFgEEEHAEbMHjLVu2sGWZR+6Hbt26KSUlRStWrPBIjsgGAhJj8rgLEEAAAZ8JuOPwWPDYWxVnS6ksX75c1nWbmZnprcyRm1AKEOSFstopNAII+FXA1sOzWbQXXXSRevbs6ddiBDbfRUVFTtetzXjmQCDZAnTXJrsGuD4CCCAQhYCth3f66aere/fuUXyKUxMlcPnll2vBggXO7iOJuibXQaAhAYK8hmR4HQEEEPCYQEFBAevheaxO6manXbt2GjBggG688ca6b/EcgYQLEOQlnJwLIoAAAtEL2L6006ZNc9bES01NjT4BPpEwAVsgeePGjSyQnDBxLtSQAGPyGpLhdQQQQMAjArZcSv/+/XXOOeewL61H6qSpbLgLJBcWFjZ1Ku8jEDcBWvLiRkvCCCCAQGwE7r//fmf9NfaljY1nIlLp0aOHNm3aRGteIrC5RoMCtOQ1SMMbCCCAQPIF3OVSbrjhBtl4Lw7/CFhr3qeffiqbccuBQDIEaMlLhjrXRAABBCIQsOVSvvWtb2nMmDEEeBF4ee0Ua80rLi4WXbZeq5nw5IcgLzx1TUkRQMBnAnPmzNH//d//sVyKz+rNzW6bNm00btw43XLLLbJxlRwIJFqAIC/R4lwPAQQQiEDAWn8WLlyokSNHyvZG5fCnQEZGho4fP64NGzb4swDk2tcCjMnzdfWReQQQCKKAddPm5OTIuvtsT1QOfwvs27fPmTjD2Dx/16Mfc09Lnh9rjTwjgECgBaybtnXr1gR4Aalla83bv38/M20DUp9+KgYteX6qLfKKAAKBF7Bu2hEjRjg7JrDocXCqm5m2walLP5WEljw/1RZ5RQCBQAtYN+3NN9/sDNYnwAtWVbszbW1JHA4EEiVAS16ipLkOAggg0ITAzJkztXXrVo0ePbqJM3nbjwK05vmx1vydZ1ry/F1/5B4BBAIi4M6mHTx4cEBKRDHqCtCaV1eE5/EWoCUv3sKkjwACCDQhwGzaJoAC9La15p199tlauXJlgEpFUbwqQEueV2uGfCGAQGgEmE0bmqpWZmamVq1apbKysvAUmpImTYAgL2n0XBgBBBCQdu/e7Sx6PHDgQDhCIGATagYMGKDHHnssBKWliMkWoLs22TXA9RFAILQCttVV3759ddFFF6lnz56hdQhbwa17/vHHH9ehQ4eUnp4etuJT3gQK0JKXQGwuhQACCNQUsD/0VVVV7E1bEyUEj9u1a6fs7GytXr06BKWliMkUoCUvmfpcGwEEQitgY7KysrI0ffp0tW/fPrQOYS24teItX75cR48elQV9HAjEQ4CWvHiokiYCCCDQhMCMGTOUm5tLgNeEU1Df7tixo9OCu2nTpqAWkXJ5QIAgzwOVQBYQQCBcAtZNV1paql69eoWr4JS2lkD37t11zz331HqNJwjEUoAgL5aapIUAAgg0IWCD7ufOnat+/fqpTZs2TZzN20EWyMjI0PHjx8VWZ0Gu5eSWjSAvuf5cHQEEQiYwb948tW7dWt26dQtZySluXYHTTjvN6bJ99NFH677FcwRiIkCQFxNGEkEAAQSaFrA18RYsWCDWxGvaKixn2FZna9asYXHksFR4gstJkJdgcC6HAALhFZg9e7ZGjhzJbMrw3gKnlNy67G1x5PXr15/yHi8g0FIBgryWCvJ5BBBAIAIBd7KFtdxwIFBTwLY6u/XWW2XjNTkQiKUAQV4sNUkLAQQQqEeAyRb1oPBStYAtp2I7X2zevLn6NR4gEAsBgrxYKJIGAggg0IiATbY4ceIEky0aMQr7W/3799eiRYvCzkD5YyxAkBdjUJJDAAEEagq4ky2GDx9e82UeI1BLoFOnTiouLpbdLxwIxEqAIC9WkqSDAAII1BH4+OOPxWSLOig8rVfAnYBRVFRU7/u8iEBzBAjymqPGZxBAAIEIBFauXKm9e/c6m9FHcDqnhFzAJmAsW7ZM9uWAA4FYCBDkxUKRNBBAAIE6AocPH9a0adM0ePBg2aK3HAg0JZCWlqY9e/Zo586dTZ3K+whEJECQFxETJyGAAALRCVg3ra1/1qVLl+g+yNmhFbAvA7m5udqwYUNoDSh4bAUI8mLrSWoIIICACgsLtWrVKvXt2xcNBKISuPDCC3XvvffSZRuVGic3JECQ15AMryOAAALNELA18W6++WaNGzdOqampzUiBj4RZwNbMs0Bvy5YtYWag7DESIMiLESTJIIAAAiZge9OmpKQoIyMDEASaJdC9e3c99dRTzfosH0KgpgBBXk0NHiOAAAItELA1zmzh46FDhzLZogWOYf+ojeO07n62OQv7ndDy8hPktdyQFBBAAAFnDNXEiRM1cuRItWvXDhEEmi1g94+15v3mN79pdhp8EAETIMjjPkAAAQRiIGBr4n3wwQesiRcDS5KQ092/fPlyKBBokQBBXov4+DACCCAglZWVOWvijRo1im5aboiYCHTt2tXZ5szuLQ4EmitAkNdcOT6HAAIISE437YwZM5z1zdq3b48JAjERsG3OsrOztX79+pikRyLhFCDIC2e9U2oEEIiRgHXTlpaWqlevXjFKkWQQ+FzAxuXZNmccCDRXgCCvuXJ8DgEEQi/gbl2Wk5Mja3nhQCCWArZenm1ztm3btlgmS1ohEiDIC1FlU1QEEIitAFuXxdaT1GoLsM1ZbQ+eRS9AkBe9GZ9AAAEEtHr1aqeFZeDAgWggEDcBW1Tbtjljzby4EQc6YYK8QFcvhUMAgXgIWDftddddJ7pp46FLmjUFbDJPenq6SkpKar7MYwQiEkipqqqqiuhMydmqx86N4iORJs15CCCAgG8Exo8fr4MHD2rYsGG+yTMZ9a/Aa6+95sziLiws9G8hyHlSBGjJSwo7F0UAAb8KWDetbTnVt29fvxaBfPtMoHPnztq0aZOsBZkDgWgECPKi0eJcBBAItYDbTTtu3DilpqaG2oLCJ07A7jVbM2/t2rWJuyhXCoQA3bWBqEYKgQACiRCwbtq9e/dq9OjRibgc10CgWuDAgQN69dVX9frrr1e/xgMEmhKgJa8pId5HAAEEJGc2rXXTDh48GA8EEi7AmnkJJw/EBQnyAlGNFAIBBOIpQDdtPHVJOxIBd828Z599NpLTOQcBR4DuWm4EBBBAoAkBZtM2AcTbCRF499139eSTT+ro0aNq165dQq7JRfwtQEuev+uP3COAQJwFWPQ4zsAkH7GArZln+9lu3rw54s9wYrgFaMkLd/1TegQQaETAumk7duwoa8nr0qVLI2fyFgKJEbA18yoqKlRUVJSYC3IVXwvQkufr6iPzCCAQL4GPP/5Y7E0bL13Sba5A165dVVxcrN27dzc3CT4XIgGCvBBVNkVFAIHIBVauXMnetJFzcWaCBNq0aaMBAwbQkpcgb79fhu5av9cg+UcAgZgLlJWVKSsri27amMuSYCwEDh06pOXLl+ujjz6SBX0cCDQkQEteQzK8jgACoRSwbtoZM2YoNzeXcXihvAO8X+i0tDTZunlbtmzxfmbJYVIFCPKSys/FEUDAawKLFy92drW44oorvJY18oOAI2Br5tks24KCAkQQaFSAIK9RHt5EAIEwCdhg9lmzZmnUqFGyP6QcCHhVIDMzU2vWrJHNAOdAoCEBgryGZHgdAQRCJWDdtBMmTNDIkSNl65FxIOBlgdTUVGVnZ2vt2rVeziZ5S7IAQV6SK4DLI4CANwRsuZSqqirnD6c3ckQuEGhc4LLLLtOSJUtkX1A4EKhPgCCvPhVeQwCBUAkUFhZq4cKFGjp0KN20oap5fxfWJmDs2bNHO3fu9HdByH3cBFhCJW60JIwAAn4QsDFNeXl56tGjh7p16+aHLJNHBKoFSkpKdPbZZ8vWdeRAoK4ALXl1RXiOAAKhErBu2tatWxPgharWg1NY225v1apVTMAITpXGtCQEeTHlJDEEEPCTgC1BsW3bNg0bNsxP2SavCFQLtGvXTjbTlgkY1STNfnDs2DHZ0I0gHQR5QapNyoIAAhEL2K4W06ZNcxY9ZteAiNk40YMCffr0YQJGDOpl8+bN+uUvfxmDlLyTBEGed+qCnCCAQIIEbDbi2LFjnQCvY8eOCboql0EgPgK2+wUTMFpuW1FR4YxvbHlK3kmBIM87dUFOEEAgQQK33367Kisrxa4WCQLnMnEVsIW7bX3HRYsWxfU6QU/89ddfV//+/QNVTIK8QFUnhUEAgaYEbMzNggULWC6lKSje95UAEzBaXl02hKNt27YtT8hDKRDkeagyyAoCCMRXwJZLGTFihMaNGycbsM6BQFAEmIDR8pp87rnn1Llz55Yn5KEUCPI8VBlkBQEE4idg4/BsuZQBAwawXEr8mEk5iQJ2b7MDRvMqwFrx7Dj33HObl4BHP0WQ59GKIVsIIBBbgcWLF2vLli0aOHBgbBMmNQQ8ImA7YHz44YfsgNGM+vjoo4+cTwWthZ8grxk3Ax9BAAF/CdhaeLNmzdKoUaPEcin+qjtyG7mATcDo3r275s6dG/mHONMROHDggG677bbAaRDkBa5KKRACCNQUsHF406dP15gxY9S+ffuab/EYgcAJ2MLImzZtktv9GLgCxqlA+/btky1FE7SDIC9oNUp5EECgWsAdh2fblvXs2bP6dR4gEFSB1NRUZ9zp+vXrg1rEuJTrr3/9qzp06BCXtJOZKEFeMvW5NgIIxFXAHYfHtmVxZSZxjwlYa96tt94q+5LDEZmATVi55JJLIjvZR2cR5PmossgqAghELsA4vMitODNYAraLS3p6ujZs2BCsgsWpNDakw46gzay1MqVUVVVVReqWkpLinBrFRyJNmvMQQACBmAnYL+28vDx17dqVbtqYqZKQnwRee+012TZdRUVFfsp2UvK6e/duXX755QpibENLXlJuKS6KAALxEmAcXrxkSddPAvYFp7i4WNaizdG4QFBn1lqpCfIar3veRQABnwkwDs9nFUZ24yJgSwXl5ubq2WefjUv6QUr05ZdfDuTMWqsjgrwg3amUBYGQCzAOL+Q3AMWvJXDxxRdr4cKFOnbsWK3XeVJbwJabsckqQTwI8oJYq5QJgRAK2Di8nJwc1sMLYd1T5PoF3P1sV6xYUf8JvOrMQA7inrVu1TLxwpXgJwII+FbAxuFdc801zkBzlkvxbTWS8TgIHDp0SNu3b9eOHTvY7aUeX2vFy8rKkm1rFsTdcGjJq6fSeQkBBPwlMHv2bO3du1dDhgzxV8bJLQJxFrD9bP/xj384+zbH+VK+TP7gwYPOF8QgBnhWIQR5vrwtyTQCCLgCq1evdsYd5efny/bu5EAAgZMC9m8iOztbDzzwwMkXeVQtYC15/fr1q34etAcEeUGrUcqDQIgEbH2r6667TpMnT5Zt58SBAAKnCrjLqWFT/0UAABmRSURBVNi/F47aAq+++qq6detW+8UAPWNMXoAqk6IgECYBm2gxcuRI2er+ffr0CVPRKSsCUQvYzPMePXrosccei/qzQf6AbfKwa9cup7UziOWkJS+ItUqZEAi4gLvg8emnnx7YX84Br0KKl2ABllM5Fdy6au248MILT30zIK8Q5AWkIikGAmESuP/++52B5DaTlnF4Yap5ytpcAZZTOVXOJl307dtXZhPUgyAvqDVLuRAIqIBNtJg3b56zHl5QZ8QFtOooVpIFBgwYoFmzZjlrwyU5K564vLXkDR061BN5iVcmCPLiJUu6CCAQcwEbV+ROtAjyt++Yw5EgApIzfjU9PV0bNmzAQ5JNuujfv3+gLZh4EejqpXAIBEfAJlrk5eWpU6dOTLQITrVSkgQL7Nu3z9nmrKioKMFX9t7lbNJFaWlpYLc0M3Fa8rx335EjBBCoI2ATLWyZlNatWxPg1bHhKQLRCGRkZKi4uFjWKh7mw510YbPzg3wQ5AW5dikbAgERuP766/XOO+84S6YEpEgUA4GkCNhEJVt6aNGiRUm5vlcu+vrrrwd6pwvXmSDPleAnAgh4UuCRRx7R5s2bNXz4cGbSerKGyJTfBGy9vFWrVsltzfJb/mORX+u2tt8pQT8I8oJew5QPAR8L/Pa3v3VmA44aNYodLXxcj2TdWwI2K91m2oZ5YeTt27cHeiyee8cx8cKV4CcCCHhKwMYM5eTkaPz48erSpYun8kZmEPC7wLFjx/T444/r6NGjgV4nrr56srKfc845gZ90YWWnJa++O4DXEEAgqQI2k3bq1KnO2CECvKRWBRcPqIAtQZSdna1ly5YFtIQNF2vv3r3Om5mZmQ2fFJB3CPICUpEUA4GgCNi3bOuetT9C7EkblFqlHF4UuOyyy3Trrbc6S6p4MX/xytObb76p2267LV7JeypdgjxPVQeZQSDcArZUyowZM5wJFkOGDAk3BqVHIM4CtnxI9+7dtWnTpjhfyVvJb9myJfCLILvijMlzJfiJAAJJF5g4caIzk3bChAnMpE16bZCBMAgcOnRINglhx44dCsM2gfZF8itf+Yp27drldFcHvY5pyQt6DVM+BHwi4C6Vkp+fT4Dnkzojm/4XsNa8Dz/8MDRbndkOF3ZkZWX5v/IiKAFBXgRInIIAAvEVsADPNk4fM2YMS6XEl5rUEThFwPZvnTNnzimvB/GF1157TdOmTQtFq6XVH0FeEO9iyoSAjwTctfBs2zKbbMGBAAKJFbCtzo4fPx6Krc5sPN6VV16ZWOAkXo0xeUnE59IIhF3AXQvPAryg7yEZ9rqm/N4WsB0gbGZ7UVGRtzPagtyFbTyeUdGS14Ibho8igEDzBdwAz/bRJMBrviOfRCAWAtaaV1xcHOjWvLCNx7P7gpa8WPzrIA0EEIhKwBY7tsDOAjzWwouKjpMRiJtASUmJPv3008C25q1YsULWXfvkk0/GzdBrCdOS57UaIT8IBFzAAjwL7gjwAl7RFM93Aj169Ah0a17YxuPZDUhLnu/+GZJhBPwrYAGe7WZx3nnnadCgQf4tCDlHIKAC1ppXVVWlF154IVAldMfjWZdtGLYzcyuPljxXgp8IIBBXATfAO+2003TFFVfE9VokjgACzROw1ryNGzcGbmzezp07HZAwBXhWYFrymvfvgE8hgEAUAjUDPOumtUCPAwEEvCkQxNa8hQsXOsvE3HHHHd5Ej1OuaMmLEyzJIoDA5wIEeNwJCPhLIIiteS+++GIoJ3nRkuevf3vkFgFfCRDg+aq6yCwC1QJBas2z30M2m//o0aOhW3CdlrzqW5oHCCAQSwECvFhqkhYCiRUIUmveG2+8oWuuuSZ0AZ7dMQR5if13w9UQCIUAAV4oqplCBligTZs2zjJHd999t+9L+ctf/lL5+fm+L0dzCkCQV/mW1k3toxFL96myOYJ8BgEEagkQ4NXi4AkCvhUIQmueLZ2yZMkS9ezZ07f10JKMhz7Iq9xfpMVLd6hw9UvaT5TXknuJzyIgN8A788wznVYAZtFyUyDgXwG3Ne+uu+7ybSFs6ZS+ffsqOzvbt2VoScZDHuR9oF3P71HuT76rtMIXtHX/Jy2x5LMIhFrA9qK1wc3p6ekaNmwYy6SE+m6g8EERsNa8/fv3a926db4s0u9//3tNmjTJl3mPRabDPbv2eLG+P/6gpv6iswq6D9PTE4u074GrlBoLWdJAIEQCFuDl5OSwVVmI6pyihkdg3759sv+2b98ua93zy+HucrFr1y5a8vxSabHL5ycqW7tG+t4YZZ7VUyMm9lb50y+q5Dh9trEzJqUwCBQUFDgB3pgxY0K5DlUY6pgyhlsgIyPDWUj4+eef9xWEu8tFVlaWr/Idy8yGt7u28qC2buuua/u1k3Suhk69QXnlhdpYciyWvqSFQKAFHnnkEU2bNk2TJ08O7cDmQFcwhUNAcoZe9OvXT3PnztWxY/75G2ldtffee6+vWh9jfcOFNMir1PHNK7VuUJ4ub/s5QauMgfp63hE9/cwf9DaNebG+z0gvYALWDTJx4kQ99NBDmj59ujMWL2BFpDgIIFBDwFrzUlJStGLFihqveveh/Y764Q9/qCFDhng3kwnIWUiDvGMq2fjf+u3Ui/WllBTnxk350sWaWliu8qXPaiMTMBJw63EJvwrYDNrRo0fLukJs7an27dv7tSjkGwEEIhSwmfI2S3XWrFnOLPoIP5a009xZtb169UpaHrxw4VAGeZVlz+thPai/V1WpqsZ/Jw7/SlPStmr11oOsmeeFu5M8eE7AJljk5eU5XTY2Bi81lWlKnqskMoRAnARs9rwtRTJ//vw4XSF2ybqzav00USR2pT+ZUgiDPFs2pVT5UweeMou21QWD9f8mdmDNvJP3B48QqBZwJ1h06tTJacljDbxqGh4gEBqBgQMHasGCBdq9e7dny2zjBq2r1mb8h/0IWZD3qcqLF+jWh7+kzuefUU/dt1V6Vhep8Ef61o82qZyxefUY8VLYBOwXpo2/mzdvnjPBok+fPmEjoLwIIPCFQLt27ZSbm6sf/OAHnjXZvHlzqBdArlkxIQryPlHZ0gnqMOw/tbn8Pg376nVaWlZz8WN7f5Kypv5SUrk235enDul3qJglVWreLzwOmYB9Wx88eLBef/11fe1rX2OCRcjqn+IiUJ+AjXPbuHGjZxdItskhM2bMqC/roXst3Ishh666KTACkQnYzLT777/fab2zsXfdu3dnB4vI6DgLgVAI2OLIe/bs0datW2Wte145ysrKZOviHT161FP5SpZPiFrykkXMdRHwl4C13vXv319Lly51lkexjb0Zf+evOiS3CMRbwF1SxcbneelYu3atbrvtNgK8LyqFIM9Ldyd5iaFApY4X36EO7hI5p/y8VJPnr1Zx2fEYXtPfSVnr3Zw5c3T55ZfrnHPO0YQJE1gexd9VSu4RiJuAffEbNGiQ09pvrWdRHbalaIcvli875XdzilJyv6+l6zarrCK6gfHuhItrrrkmquwE+eRQB3nWYmGbLtu6XxxBE2il1Kt+rCNV76lodm8pr0ClJ9wlc/5XR0pm6/zfztKwobO0dB+Bni2NYmtg2bdgW9zYfnnTehe0fxOUB4HYCtgamQMGDHDGv9mXxIiP1Kv0wJET+nvR7UrTv6ug9OOTy5l9WKqiidLTY3OVlTk9qt/P7oQL+/3F8blAKIM8uxntD5m1WIwdO9YZTF5YWMg9ERqBM5TWe6LuW3yP8sqX6kfLShTWMM++4MycOdNZauCiiy6Sjb9jcePQ/EOgoAi0WMCWVCktLdXKlStbnJaTQNtMXTXlPq3f+5QmaammfqdAOyJo0XPHEXt51m9sgKJL5bToTvfu2dZM+/7779fK4EcffaQDBw7Ues2evPrqq1qyZEmt10eMGKHt27frq1/9qvP6ueeeS59+LSGeBEnAfiEuXrzYWb3evonfeOONLGwcpAqmLAgkSMAWG7b16GwPa/s7mp6eHoMrt1Lbbv+hHy98RZvG/kR3rBmlDVO6qbFWqQ0bNjjXHTVqVAyuH5wkfBHkuf39toyDHS+//LLz015/7rnnGqwN634aOnToKe/bStj1HbYBc0OHDeS046yzzlK3bt2cx5dcconz01YBD/uq2g6Ej/5XWb5dTxesVmHaFBV8s88pC2P7qChRZdWCO/tl+KMf/cjZzm/y5MksixKVICcjgEBdgS5dujg7YcyePTt2LXo6Q2mX9NKlWvT5BgXf6KbMRqI8WzZl0qRJ/C2uUzmeCvKs6+itt97Sm2++6azLVTeIs28KZ599tiy4atu2rYYPH64HH3zQKdJXvvKViL9BWNesfeOoe9Sccm15sZZAO2q2CB45csQJMv/nf/7H6eqtmYabP5uZaPnr3Lmz8weUALCmUpIeF05V1pem1rl4b80uekZTuoVjay4bd/ftb39bx48fl32hsdlxjLurc0vwFAEEmiVg62k++uijGjZsmKZOrfu7tllJqtX5nZWdJhX+eb8OV1QqM7X+KM9+t1mDj60IwFFbIGlBnnWv7t27tzqg+8lPfuLkzG19s0DODeJi3XV65ZVXOk3LNbtsly9fXqt7tm6Ts+3XV/d48sknnZcsGHUDwYqKCicIrBmg1izT+eefT/BXFzIRz23ixYYpX3wT/FTlO36jZx6do9uG5eqdglVaNOUStU1EPpJwDfsFeNddd6m4uNgZc8ead0moBC6JQMAFbB/r8ePHO39b7W9sZmZmQkpsvROzZs3SY489VutveEIu7oOLJCzIc4O6Xbt26cUXX6zuZrVuUGv5stcvvPDChFSStazZN45vfOMbevfdd51WjbpBXTR1597M9QWCFuy99957znWsm9lm87pdzDbN21pUrPvXglq6faNRb8m5NvEiX7cu665271ylqVPnaUjOCk3J/HJLEvXUZ91u2UWLFjnB3ciRI3XLLbfQleGpWiIzCARLwLpt3dm29neupb1Yle8c1O5ySXkZSm9bfyueOxbPAkyOUwXiGuRZl6e1HmzZsqV6osO9996rG264QQsXLoy4e/XUbLf8Fbv5EjHN2gJANwjMz893Mm5/gA8dOqSDBw/KgkC7SW2Wrx11Az/3sy0vMSmcItDqXHXO7iAVHlDp4QopAEGefZnatGmT7r77bn344YfOlwiCu1NqnhcQQCBOAjbbdvXq1c6OOXPnzm3BVT5V+es79WddoinfGqaMemI8+31nO/PYjFov7brRgkLH/KP1sLXsGha0WADn7nNpLVfWdGtTrKuqqnTHHXcoLy8vqQFey0rY8k9bgGnBmznY/nrW7Ws2ZmQBsB02iNS2ZklJSXGWe7Hntq6f3dQcMRKo/Ic+eP9/JXVRVrq/O2vt391DDz3kLGI8b948XXzxxU7XibUSt/TbdIy0SQYBBEIgYL9vbIar/R5q/tJklarYt0p3zFik8qFTdENex3pn1v785z9Xhw4dnOv5j/Yzla/7tvM3PiWlg3Lnb1eFPlX59kWabAtFZ8zXjs9aXqqYtOS5LXZuV6RNQLBgxQZBEl1HXkluq58Ff3ZYi58FfrYMjHX12kxIO6y1z8Yr2jp/9sccY4clqv9Vlu/Qr595VDOWHtXQ26drVIb/umrt/rBW8vnz5zutd9ZNwmzZqG4DTkYAgTgI2Fqb48aNcyY42t+wqHqkKspUvKZAd099UJuH/qeKVn5XvevpqrVGjx/+8IfOUC9/fpE9TWn5P1fVidu1bvpojX34Vyq87BX96dAQLTpSpeUxqpeUKmtCivCwViU77CP2B2bnzp166qmnnK5YCzxs+rItWULQESFoM06zFhtbSsY2h/71r3+tV155pVbQ16lTp1C3kp4ktW3NfqRuw+6TDemo9xg6WwU3X6tRV/dWWszbtOu9YkxetIkU9g3ZvinbOFYbC9q1a1da7GKiSyIIIBArAftdZWPSbdmyWnGBbWvWbZgebOCXc9qkn+jR/FwNauB3s8UfQ4YMcXoMrXfQ38fJv1Wa8iu98mS+Lojh36NmBXk2i8W6D+2wwM66ZlsyccHfFZTc3Fsr6htvvKGSkpLqoM9m81q90NKX3LqJ5dXtW2tRUZGWLVumPXv2KDc31wnwbLIOBwIIIOBFgc8++0wvvPCC0+NUUFAQsy+iP/7xj52/dxY8+rMVr05tOUHvBO2+p7jJRZ/rfLLJp80K8qzVzmbF9urVKxjATTL554T6gj66d/1Tf25O3ZZy66Z3AztrsbvsssuUlpbG+nYuFD8RQMDTAhbo/eIXv3B6+Z544okWxww2LMwmKkbdDexhpcq312l637FaemnNpb5OZtj+HsyZM8fZmSjaBrVmBXlR9PCezCWPkiLQUNBnS7f06dNHtmZatDdNUgoSgotaXdnWemvXrtWqVaucErstdgR2IbgBKCICARWwBdgtOLPhXC0J9Kz717ZQ27hxozNxMRBclW9p3Q336U//9Fc9uKq3ivbdo6vqLPrsbuBgvXTmGM3fbIK8QNwlkReivqDPPu2uV2jrHNni09HcRJFfnTNrCrjjK21sq42vtG5YC7ptJwobuHzOOefQYlcTjMcIIOBbgZYGem6AZ8PFbFWKYByf6u11d+pu3aCFlxTpX7MeV3bRSl376jq9de33lH/BGdXFtFVLZs6cKQv0rGU00sksBHnVhOF84C5S7W4l5+48YhruNm3uNnIWANr2cdFsIecVVWvutrUJk5V3C65tyz6bMLNjxw797Gc/c2gsqLPJExbUWXAdiPElXql08oEAAp4SsEDvpZdeku2OYbtM1WxMsL9F77//fr2/o92WrOAEeBXaMf9q9bntDM3+1QLdmd9NbSu2a/7VX9Nt+oZ+tfgHys88dbtNN9C1St26dWtEa/0S5Hnqn4A3MmMBiW3TZrN4bZs2+2lHzQDQzakbCNpzC1ZszSL3sOCw5hHpN4+an4nFY/uHYdve2ExkO6zV0hbpjEdAZa1z7hZ31kL3l7/8RWvWrHGuaz4XXHCB85/twWy/6OKRh1iYkQYCCCAQDwEbo/enP/1Jv/vd7/Tss886Wy3a70rrhnUPN5hzFzu2vz2RBjVuGkH9GW2g16wgL6h4lAsBBBBAAAEEEPCLQFPBL0GeX2qSfCKAAAIIIIAAAnUEdu3a5ayXWudl52lUO17YrFpbEJnZtfVR8poXBWwsno3Dq3vYfof/9m//Vvflep8naxxfvZnhRQQQQCBAAjY8qL71PhM5g9ZvcU3dLltbXquhI6qWPEvEbxgNFZzXwyNgC3e7W8JZqW12ki3QWWsF9vBwUFIEEEDAUwKzZ8+uNebbxno/+uijCRuz7Ke4pm6AN2jQoEbrkiCvUR7eDIqA/cN49913neKw9V5QapVyIIBAUARsBq1N9Gvbtq2uvPLKhAV45ueXIM+dZWx5bmosnntfEOS5EvxEAAEEEEAAgdAJ+CHIq7lOXjQLIkcd5IWu9ikwAggggAACCCCQJAG3BS/uO14kqXxcFgEEEEAAAQQQCKWATSD86U9/qkmTJtVaQDoSDFryIlHiHAQQQAABBBBAwGcCrXyWX7KLAAIIIIAAAgggEIEAQV4ESJyCAAIIIIAAAgj4TYAgz281Rn4RQAABBBBAAIEIBCII8j5V+Y6n9f3cDkpJuVSTf7pN5ZURpMwpCCCAAAIIIICAJwUqdbz4DnVISXHWybNlVFJGLFVZwOKbJoK8SlXsWKTxtx7QiJUHVXXiBU1+7z6Nf3i7KjxZaWQKAQQQQAABBBBoQqDykF58Zr3Kq09LU97XByqjiaio+nSfPGi8OJVlWnPHWvW787u6Ku0MqdUFuuoH31O/h+drTdknPiki2UQAAQQQQAABBFyBSlXsWqfF5y7Q36uqVOX8d0Qbp3RT40GR+3n//Gy0PJX7X9LqwguUld72ZIlSe2rExLe1eutBBaxV82QZeYQAAggggAACARU4pu1rnlHhgw/o3qXrVFx2PKDlVGNB6yfav/UFFaZlqPP5Z9QB+F8Vrn5J+4ny6rjwFAEEEEAAAQS8LFBZ9rweeHCHpEI9OHWshmX9u76/bl8gh6E10pJXocOlB6RLM5TetpHTvFyT5A0BBBBAAAEEEKgh0CpzijZWVenEkRL9umC2hlqwN/ZW/XzHBzXOCsZDordg1COlQAABBBBAAIEoBFql9dY1Ux5Q0ZFC3T50px5es1NB67htJMhrq/SsLtKf9+twBf2yUdw3nIoAAggggAACPhFolTZMP7jzG9LTL6rkeLDinUaCvC8rI2ek8upWUuX7Orj7g0BONa5bVJ4jgAACCCCAQNAFWqlteoYuDeDwtEaCPKlVxkB9/dLfa2PJsZM1XHFEpX/upa/ndG5s1sbJ83mEAAIIIIAAAgh4VqBSFYf/pt7fH6PMRqMizxagwYw1XpxWmbr2x+O0/e5FKi7/VKp8W8X3P6zt37tV12Z+ucFEeQMBBBBAAAEEEPCegO3i9d96bkf5F8vAfary7U/o3s29NXPoud7Lbgtz1HiQp1Zq2/u7WvnAuVreu7VSvjRFG7Pu0srv9VONlfNamAU+jgACCCCAAAIIJEjg76/okT4d9KWUFHWY/Ki2VVylO+f9i9KaiIgSlLuYXialypZ65kAAAQQQQAABBBAIlEAA49ZA1Q+FQQABBBBAAAEEmiVAkNcsNj6EAAIIIIAAAgh4W4Agz9v1Q+4QQAABBBBAAIFmCRDkNYuNDyGAAAIIIIAAAt4WIMjzdv2QOwQQQAABBBBAoFkC/x8g4RaISFl+aQAAAABJRU5ErkJggg=="></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use integration by parts to show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {{{\text{e}}^x}\,{\text{cos}}\,2x{\text{d}}x = } \frac{{2{{\text{e}}^x}}}{5}{\text{sin}}\,2x + \frac{{{{\text{e}}^x}}}{5}{\text{cos}}\,2x + c"> <mo>∫</mo> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> <mo>=</mo> </mrow> <mfrac> <mrow> <mn>2</mn> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> </mrow> <mn>5</mn> </mfrac> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mfrac> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> </mrow> <mn>5</mn> </mfrac> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mi>c</mi> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c \in \mathbb{R}"> <mi>c</mi> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">R</mi> </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>Hence, show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {{{\text{e}}^x}\,{\text{cos}}{\,^2}x{\text{d}}x = } \frac{{{{\text{e}}^x}}}{5}{\text{sin}}\,2x + \frac{{{{\text{e}}^x}}}{{10}}{\text{cos}}\,2x + \frac{{{{\text{e}}^x}}}{2} + c"> <mo>∫</mo> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>cos</mtext> </mrow> <mrow> <msup> <mspace width="thinmathspace"></mspace> <mn>2</mn> </msup> </mrow> <mi>x</mi> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> <mo>=</mo> </mrow> <mfrac> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> </mrow> <mn>5</mn> </mfrac> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mfrac> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> </mrow> <mrow> <mn>10</mn> </mrow> </mfrac> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mfrac> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> </mrow> <mn>2</mn> </mfrac> <mo>+</mo> <mi>c</mi> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c \in \mathbb{R}"> <mi>c</mi> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> </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>Find the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>-coordinates of A and of C , giving your answers in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a + {\text{arctan}}\,b"> <mi>a</mi> <mo>+</mo> <mrow> <mtext>arctan</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>b</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 \in \mathbb{R}"> <mi>b</mi> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> </math></span>.</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>Find the area enclosed by the curve and the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>-axis between B and D, as shaded on the diagram.</p>
<div class="marks">[5]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><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="{\left( {{\text{sin}}\,x + {\text{cos}}\,x} \right)^2} = 1 + {\text{sin}}\,2x"> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>+</mo> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</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>Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{sec}}\,2x + {\text{tan}}\,2x = \frac{{{\text{cos}}\,x + {\text{sin}}\,x}}{{{\text{cos}}\,x - {\text{sin}}\,x}}"> <mrow> <mtext>sec</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mrow> <mtext>tan</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> <mo>=</mo> <mfrac> <mrow> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>+</mo> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> <mrow> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>−</mo> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> </mfrac> </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>Hence or otherwise find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_0^{\frac{\pi }{6}} {\left( {{\text{sec}}\,2x + {\text{tan}}\,2x} \right)} {\text{d}}x"> <msubsup> <mo>∫</mo> <mn>0</mn> <mrow> <mfrac> <mi>π</mi> <mn>6</mn> </mfrac> </mrow> </msubsup> <mrow> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>sec</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mrow> <mtext>tan</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>2</mn> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </math></span> in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ln}}\left( {a + \sqrt b } \right)"> <mrow> <mtext>ln</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mi>a</mi> <mo>+</mo> <msqrt> <mi>b</mi> </msqrt> </mrow> <mo>)</mo> </mrow> </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 \in \mathbb{Z}"> <mi>b</mi> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> </math></span>.</p>
<div class="marks">[9]</div>
<div class="question_part_label">c.</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">
<mi>f</mi>
</math></span> defined by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = {x^2} - {a^2},{\text{ }}x \in \mathbb{R}">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−<!-- − --></mo>
<mrow>
<msup>
<mi>a</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>x</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span> is a positive constant.</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) = x\sqrt {f(x)} ">
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>x</mi>
<msqrt>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</msqrt>
</math></span> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left| x \right| > a">
<mrow>
<mo>|</mo>
<mi>x</mi>
<mo>|</mo>
</mrow>
<mo>></mo>
<mi>a</mi>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Showing any <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> intercepts, any maximum or minimum points and any asymptotes, sketch the following curves on separate axes.</p>
<p><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>;</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>Showing any <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> intercepts, any maximum or minimum points and any asymptotes, sketch the following curves on separate axes.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = \frac{1}{{f(x)}}"> <mi>y</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </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>Showing any <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> intercepts, any maximum or minimum points and any asymptotes, sketch the following curves on separate axes.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = \left| {\frac{1}{{f(x)}}} \right|"> <mi>y</mi> <mo>=</mo> <mrow> <mo>|</mo> <mrow> <mfrac> <mn>1</mn> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>|</mo> </mrow> </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>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {f(x)\cos x{\text{d}}x} "> <mo>∫</mo> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mi>cos</mi> <mo></mo> <mi>x</mi> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </math></span>.</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>By finding <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> explain why <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g"> <mi>g</mi> </math></span> is an increasing function.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the functions <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g">
<mi>g</mi>
</math></span> defined on the domain <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0 < x < 2\pi ">
<mn>0</mn>
<mo><</mo>
<mi>x</mi>
<mo><</mo>
<mn>2</mn>
<mi>π<!-- π --></mi>
</math></span> by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = 3\,{\text{cos}}\,2x">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>3</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mi>x</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g\left( x \right) = 4 - 11\,{\text{cos}}\,x">
<mi>g</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>4</mn>
<mo>−<!-- − --></mo>
<mn>11</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</math></span>.</p>
<p>The following diagram shows the graphs 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> and <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></p>
<p style="text-align: center;"><img 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AqoK/IDVapUSeWr79q1awFH82vTCehF4kOHDklaWprp1WX9LCVAS8BSwUWrts4PhMVFFvsJ6EViLVf7W8QWmEiAloCJUkmgTtolFJGnzA+UAEBDT9m8ebNMnDhRTfFhs3oWEnCaAC0Bp4n6dD3tEsr8QD4JwKXbQp7I+4TNgFhIwA0CVAJuUPX4mnAlpEuox9A9uh3yPWGRH7uQMd20R9BDdhtOB1kucLqEWi7AGKuPdNMHDhyQTZs2Ma9QjMx4WGwEaAnExsnYo3SWULqEGisiRyp22223SVZWFvMKOUKTF4kkQEsgkoZlr5kl1DKBJVndt99+W+bPn8/NZ5LkyNMvJUBL4FIeVr1bsmSJqi+3jLRKbAlX9pZbblHnzpkzJ+Fr8EQSyEmAlkBOIpa81y6hffv2lXbt2llSa1YzWQIbNmyQGTNmMMtosiB5/kUCtAQuorDrxdy5c6VGjRrSsmVLuyrO2iZFAHmhIHfIn4UEnCBAJeAERY+vgQXC6dOnq2yT3DLSY/g+3w7yxvQf5I9+wEICyRKgEkiWoA/njx49Wu0VwMAwH+AbcEtEhCOADP2AhQSSJUAlkCxBj8/H/rMMDPMYuoG3g0sw+gE3pjdQOJZViQvDFglMB4ZddtllMnz4cItqzqq6QQB7DpQtW1ZFE7txfV4zHARoCVgkZx0YhlzzLCTQrVs3QewA00mwLyRDgJZAMvQ8PBdWwE033SRVq1YVKgEPwRt+K6STOHjwoJoWKlq0qOG1ZfVMJEBLwESpRKkTdgzbsWOHIH0ACwloAugP27dvF1iJLCSQCAFaAolQ8/gcnR6CgWEeg7fkdrAG9u/fLx9++CGTy1kiM5OqSUvAJGnkURedHkKnDcjjMH4cUgKwBmAl0hoIaQdIstm0BJIE6PbpTA/hNuFgXJ/WQDDk6EcraAn4QT2Oe86bN4/pIeLgFdZDaQ2EVfLJt5uWQPIMXbuCtgK4b7BriAN14aVLl8onn3zCtYFASdX9xtAScJ9xwndAMFD16tWF6SESRhiqE1u3bs21gVBJ3JnG0hJwhqPjV0E6gCZNmsiYMWOkXr16jl+fFwwmAQSPbdmyhdZAMMXrSqtoCbiCNfmLwgpAkjAqgORZhukK8CCjp1CYJJ58W6kEkmfo+BVgBaxatUq4b7DjaAN/weLFiwviSZ544glBlDkLCRREgEqgIEIef48fbv/+/dVmMbVq1fL47rxdEAjQGgiCFL1rA5WAd6xjupNOEkcrICZcPCgKAVgDnTp1ojUQhQ0/yk2ASiA3E98+gRUAMx4/4NTUVN/qwRvbT4BxA/bL0KsWUAl4RTqG+8AKYJK4GEDxkAIJYBBBa6BATDxARKgEDOkGSBLXpUsXufvuu2kFGCIT26tBa8B2CXpTfyoBbzgXeBedJA4BPywk4AQBbQ08+eST9BRyAmhAr0ElYIBgYQU88sgjyrUPi3osJOAUAVgD2G/g/fffd+qSvE7ACFAJGCBQbQUwVbQBwghYFWANtGzZUp5++umAtYzNcYoAlYBTJBO8DpLEwQpAkjhaAQlC5Gn5EsjIyFB7ESMIkYUEchKgEshJxOP3OlU0k8R5DD5Et0PQIVKQ/PnPfw5Rq9nUWAlQCcRKyoXjYAVMnTpVOnfuLEWKFHHhDrwkCVwggODDlStXqg3pyYQEIglQCUTS8Pj13LlzmSraY+ZhvZ22BpYtWxZWBGx3HgSoBPIA4/bHWVlZMn36dBUXQCvAbdq8Pgg0b95c9TlYoCwkoAlQCWgSHv8fPXo0U0V7zDzst8O6U40aNeTVV18NOwq2P4IAlUAEDK9ewksD87NMEucVcd4HBGBxtm3bVjAAQWwKCwmAAJWAD/1g/Pjxygpgqmgf4If8ltdff70i8OKLL4acBJuvCXB7SU3Co/9r1qwRpIaYNm2aVK1a1aO7JnKb07Lr+Sfksdc/i35yqWbSq0+GNGxwjaRezrFEdEhmfootKOfPny9nz56VokWLmllJ1sozAo4rgSpVqki1atVUA+CbjNHutddeK3Xq1JHLL7/cs4aZeCOkim7cuLGAUa9evUysYu46nd0lzz88Xfb3GCujW1a8YDpmfyOfvfeWLJrxuhxuNkDG/q6pVHRBERw+fFgOHDggx44dky+++OKSupUpU0ZKlSollSpVkvLly0u5cuUu+Z5v8iZw8uRJ6dOnj8BTCO7JYSznzp2TXbt2yT//+U/ZvXu32pcZHJ577jm56qqrQoWksNOtxYPuxIkTMnHiRIEHDEYdzz//vLoNOlyzZs2kdu3aTt/WiuvpDWPwA7S6FCol1Zp3lwf//zHpN3uVrGtxnfSsU8KRJv3000/qR7lz507BwwoFEa/dunW75Pqff/656l8bNmxQnxcrVkyuvvpqqVmzJhXCJaRyv4ECRZppJJbDGkGYrIF//OMf8tFHH8ny5csVmLJly6p9vB999FH1HgOLsBXHlQAUAP7atWun/gD0yJEjsnHjRnnttdcUfPxQMzMz5YYbbggNb24Yk7+o9cMfP1BMU8CdsU2bNgVakHpE9+GHH8qKFStk69atgocc5r4xorvsssvyv3FIv7311ltl4MCBsm3bNmWdBhkD+sj69etl0aJF8tVXXwke/A8++KA0adJEKlSoEOSmx9Q2x6eDUlJS1I3Pnz8ftQLQxAsWLBD4KodJGWDkgf0CZs+ebdd+AdGmg+Q/8s1nG+WtRc/Lu2XvTXo6aP/+/bJu3Tr18Meov3v37gmZ5PixQ4m88MILsm/fPoF1cOONN6opSSqD3D/HCRMmyK9//Wt56623cn8ZgE9yPvyRSA9ZVcM0+IxFjJ4rAV2pSGUA4dx3332B1cpwx2vRooXyCIKFZFVRSuAxef2bKLUu1UWemNVD6hRLbGH41KlTsmnTJvXArl69ugwbNiyhh3+Umsknn3wir7/+umC6CMoAGVrT09OjHRrazzAXPnz4cNmzZ48akAUJBJ4vkydPViN/PF8wnRi2uf5Y5embEkAFc2pqzMtZ95CMgfSsWbOkf//+am3Eukyh0SwBLAxveU9Wzlss70pL6Tf2PmlZsVgMJH45BItyeEBjDhZTg27JHWsHSJwGyyAtLU0pAy4iX5DDjz/+KCNHjpQ77rhDpkyZ8otwLH6Fqee//OUvsnbtWqXYHn744dCuQcYqRl+VgK7kN998I88884wSHDyKkFo5KHN1sAIwR923b1/XHnSaoyv/oymBn2+U/eVaGdtvtmTV6yezR7eU1BgMAsz3v/POO8rbB1M/Dz30kFIErtQ94qLvvfeezJw5UwVJXXfddcoq4xSRKEU8Y8YMtQhfunTpCGL2vYQTCpQZ5vx79+5t5+/NB+wx/GzdrxVGgyNGjJAxY8YIRm7YZxc/2iCUIG8YU6jklVIVzhQHj8rJc9kFigtz/whSgrsnrD6MQr3yxmjatKlAFvCKweLx4sWLBS6oYS9BCB7D6B9TiVAAmPp59tlnqQDi6NhGKAFdX/xQIUBYA3/6058EkbWYMrK14GEHq2bAgAGB3DAm+9x38i2EU6W8lMknTgCeP0iVgQVIWHhYuHVr+ie/voI4FcgCI1+8hp/8xx9/LKhfWAumJzHogn88PNhsKxgsov4YPGIQicGkVwML21jlVV+jlAAqCQFiDhcuXJjX+8Mf/qBcTPNqgMmfP/XUUyphFwLEglXgHbRZ3vzry7L2m3rSpXMDKZ9HT8L0Dx7+GH3fc889akrG76k+xKloqwAL06gfFqnDWm6++WbZsWOHII7FloLBIdbaMFjEoBGDRwwiWeInYMSaQF7VhofHE088oVb4EXxmk2sXXGDhjYJtIxs1apRXEw3+vIC0EVJKat/ZXTo0vl6ur1YqahIqxIu88cYbyvUTozQTf6TwIsGg44cfflBBaWan8nCvu8B1GQnmVq1a5d5NHLoypn/w8MdvDIPF3/72tw5dOZyXMVoJQCRYNIYC2LJli1UCRyAORpjjxo0L5a5hmP9H+7HYiAVZv0f/+f288VDBDm8InMKoWM+T53dO0L7T7qII6jTZctWun+A/ePBgqwaGpvYZ45UAwMH0w5wlAq6QegJpF0zOQ2S/FRB/d/32228Ff5hjRsoHTP/AC2fs2LFGy0q3FH0M+z0j6hgxCwgqCpv3EILHrrnmGuViqbmY9B/z/7AAEGQKy9LkgYVJ3AqqixVKQDcCaSfgSoo5wMcff9zYh0uHDh3k+PHjKhBH1z3I/zHqnzNnjvLFRzuxeUm/fv2kR48exsooL3loN0PEFCBtBQLNwlI2b96srG4Tg8eQ8gE5yEz/7dvYV6xSAgAMcxDz7KaOBuAFg5wkGFWFZb8AtBXTdZElr7QhkceY+lr3MSiArl27SsmSJU2tqqP1QvDYqFGjpGPHjsYEj0XOAvTs2VP5/zvaaF5MrFMCkJleMMbrp59+2iizsH379mohG+H4YSnRdkizWQlAblgnQLQpFowRURuWKGNtDRw6dEhFWPvZh6EAYPHbth7oJ7NE7p2HY18il/LuHLj44eGPAj98/GBNKNgwBtMJ0R6KJtTPjTrAtRL7RUQW61Nli6iBBVwQsfb00ksvCaa8wlC0LHWqZb/aHKkA4BhCDyD3JGGlJaBx4OEPJYD0sH67kFq5YYwGmeB/RNzCtxwLqLfffrtKRFa5cmWVCygoOerxMMIUCTyHsNYTBhdSDGSw8xj2c/AjlUROBWCTa3iCPyVfT7NaCYAcFAGsApiMfioCa1NFJ9j9bHIBTbCJF08LmyKAlxfm3+HaiykxLwsVgJe0L9zLeiWAZvjdcWAF3HTTTWqUaM22kUn0Na0A4AWE3alMdtdNopmXnIo+BrdELBqHwSLwwxrw+3d8icBD9MbKNYGc8sFDCAtIcB+D5xB+qF4WuK4i7B6+5UEvWgHARA+LAoBM0cfgo452Iwgu6GsE2H8BBQn/vCqIBfLboveqrSbdJxCWgAbqx0hCp4pGEiu4Ewa5RCoAPBDDYAHklCf6WFjWCLy0BnQcANNA5Oxx7r8PhCWgMflhEehU0a1bt9bVCOR/KoALYkUfgwWEqTBYBEFOR+2VNYBIYASCYR2CXkDePz4CZQlofJEWAUxMt7aVQ6roSpUq2bthjAZWwH9EkK5evVpNhYTVAsiJCH0Mu8XBMSHIAWVuWwOI+UF6b0zlIpEfi/cEAmUJaHzaIsAOQ9hswq04AqSKRtEjJn3/IP2HBUAFkFui6GPIcIv/S5cuDWwqat233VgbQHJIMET0P9b0WAogcGS59EpJkRT8NZ8uW89kS/aRD2R6rzqSktJcpmw9U8AFon8dSEtAN1XHEeC905HFYUgSxykg3ZPy/o8+BjfKQoUKyV133RXIpHNuWQMYoGEzGKd/m3lLKwjf/CiHlw+UG7pslnuXjZbyH52WtqPvllpXJD6ed00JmII7NTVVbeyC+vzrX/8S5LhnKZgARmcVK1ZUaRNgsp8+fbrgk0J6BKYbq1SpQlZxyL9u3boqEO3o0aPy6aefxnFmeA7NM/XK6fUytFaGTJJ+suwf06RzxSJJQUlcfSR1W+9O/vLLL+Xf//63GqnVqFFDSpQo4d3NLb0TFUB8gsNoFgvEv/rVr1Qa6vjODt/RiCpHJPKZM2eoABIRf4nfSOt7rxOp00BqV0hOAeD2hROpQyzn5KnFYjnZhWN0Zki4oCWThhqBYTD7kSp60KBBgdswhlNAiXc+7eY4cuTIwG1Mo6OIsT8EXGQTLVhQhxcQCqaZsKbCEieB7O/k1Fc/iKx5Rzbu6yE1aybHMPCWgMaLIB+9Q9nLL7+sP477P3LlrFy5Urp16xZYBQD3R3oBxd01VJrjTp06CfYthjINUsFmQQjEHD16tNrWMdG2TZs2TeX6gicQFUAiFH+UwyvmyKqyTaWZfC57vkhsMTjyzqFRAmg0FAEsAfgkY9QWb0FgGHzE8UMPWiIxbQGEKRVEvPKP5fgHHnhA7agWxBgC9A1Mqc6dOzcWFLmOwW9u7dq1Kv2GW27buW4asA+yD6+UsWvGmtoAAA2rSURBVGtulCfGPSz3tvpS/rp6uxzeOldGLD8o2Qm2NVRKAIwQjAJzNBFFgM6/fft2lTEzQd5GnkYF4JxYMLrFlEm1atVUhlWk2g5KwUb02N51+vTpkpWVFVezMB2rA8KaNm0a17k8WOQ/W6dI9ZRUyZghMnBKJ6lYuLzUva2BHJk0RWYcbCYjOleRRB/mrnkHmbYmkLMj6flbKITevXvn/DrXe+0SisCW5s2b5/re1g+oANyRHFxHEUyGOfAgBZNh97GpU6eqTXYwLRpL0etxDAiLhZb3xySqPLyvqcN3xIM/Hotg8ODBkp6eLo0bN3a4Jv5djgrAPfbYBB2pmOExhGCyn376yb2beXhlWAPYNGnVqlWCTZQKKpEKgAFhBdHy5/vQKgHgjlURYK+AN998Ux2PH0EQClJBYN6aawDuSROKYPLkyXL27FnFOiiKAHtnt2zZUkXjw1sur5JTAXAhOC9S/n4eaiUA9AUpAiwGd+nSRf0FZeP4Dz744GIqiDClg/bjp4YFUHilIc8UlG5QFAGsAayPIY16tEIFEI2KmZ+FXglALDkVAeZxddFJrbDZeBAKFMDWrVslIyODbqAeCVS7JwdJESASH+nTsYkS2hVZqAAiaZj/OrQLw9FEoxeLsYCF+Us8LJs0aaL8oxs1ahTtFGs+wwgUI1H8YO+55x65//77ral7UCqqH45paWlqdzLszWxzQQAZPKE6duwoU6ZMUU2BZfDMM8+orKDJBGXazMW2ulMJ5JAYcpsjUApz5e+++65a2MOisM0Fborr169XCoCbdvgryaApgs2bN6vpLvxHDiC4gcKNtE+fPgwG87erxXx3KoEoqPBDRUQwcsJMmDBBbF4LQMK8N954Qy1OYm4aUxMs/hLQiqBYsWKBcB/FuhIGGrBwMLWKPxZ7CHBNIIqsYKZDAVx//fWyceNGa1MAwAPopZdeUiMybK5DBRBF2D58BDlAHtp91OaAMgwykFLis88+k2bNmlEB+NCfkr0lLYEcBOHy1r17d/UpIiPHjx+vOjgCxOrUqZPjaDPfYv4fG3ZjTeO6666TESNGSKlSpcysbIhrFRlQhoV621KR7Nq1SzZs2KAygiKdBJwoDh06pCyCEIvVuqZTCeQQGeY04fEA97d69eqpiE9ESK5bt05q166tdhEzeUEvcv4fC8A9evTg3GwOGZv0FooA/Wvbtm0qEt2GgQbiHt555x21xgTlhWy6+KxNmzbKGpg0aZJJiFmXAghQCUQA0qkhxo0bp0bPEV8pf2h4PZQpU0bat28vJUuWjPzaiNeY/nn//ffVD3LMmDHCHC1GiKXASsAl2ZaBRmQfe/TRR6Vdu3YX24cI4tatW6sYlFatWl38nC/MJkAl8LN89DQQNqGBh1DRokVzSQ47bMHtDQFkJk0PYfoHD3/UD15NGJkhWpXFLgLavRILrC1atDBqoIGRPrzl9u3bp6YYBw4cGLWP9e3bV3bs2KEsBWwcw2I+ASqBn2U0a9YslfBLTwPlJTpsjj1nzhw1PVS9enVl/sLLw6+CHa2wxwF+pHD/7NChA6d//BKGA/eF5xDm1tHPMKpGviq/i577Rz1yjv5z1g1xKJUqVRK4VXNaKCcdM99TCYiotLj169dXCb+waXgsBbsi6QAZP6yCyJEZ0hYPHz5cmKM9FsmZfwzWCbD5Chb2/RxoIMEgNsg5efKkijBHgGEsFiZybSHVSkEDKvMlEY4ahl4JYBoIc+cIg4c7ZbRpoLy6An6sCxcuVFYB1gowD1quXLm8Dnfkc0z97N69W3ll4IIc/TuC1ciL6IEGLM0bb7zRM+80uH3CIsHUDwYWv/vd7+J2L9bTQnlNrRoJPKSVCr0SgAso9oTFghc2WE+k4AeDQCysFWDkBj9wN5RB5KIcvDJiHZkl0iaeYwYBDDQQUwBXTAw0br75ZtdcSSMf/pjPR//C2kQi2T/1tFA0JwszyLIWmkColYD2Zli8ePHFza81mHj/w8MDqRkWLFig5nOdUgZ65P/RRx+peX9cd9iwYZz6iVdAlh+Pgcazzz6rghidVAboX3hg62mfZB/+kZj1tNDq1auVlRz5HV+bQyC0SgAdHzlO6tatK/Pnz3dMIljQwz6q2Mwer/GDReQxFsviWUDGgu+BAwfUvDAqh3UH+GEz6tcxUVl5oUhlgP6EWBYMDOJxWcaD//jx45f0L6wrIa4E006JjPzzgqmnhRBXQG+hvCj5+3lolYDunBitwCXP6QLLAKP3JUuWqIhjXB/3qVy5slIMV1xxRa5bHj16VGCSI2UFFn4R5Qtvn1tvvZUj/1y0wv0BlAEerJgmQsFgA4u2mIYsX758LjhnzpwR/CGiF3P9KF70Lz0tRG+hXCIx5oNQKgEdFeyVmYp5XXhKwNtD/2jz6gFYiEP6algPiFBmIYH8CMDa3Llzp/LN//vf/67WpfI7HnEkGO1fffXVnvUvPe26bNkyZX3nVz9+5z2B0CmBrKwsgTuonwtWUArRtuWDh5KTprj33Yl39JsALFAEPOYs8HqLxb0z53lOvR8yZIjaajMZBwyn6sLrXEogVEoA3juYV0/EHfRSbHxHAiQQD4FkXLHjuQ+PjZ9AqFJJDx06VPk/Izo4nniA+LHyDBIggUgC+L3NmzdP7W2B7Lws5hAIjRLAOgD8rbEO4MZCsDkiZU1IwEwC8GSCOzbicrBOwGIGgVBMB2FzdSy2+rkOYIa4WQsS8J8APPMwIOPeA/7LAjUIvBLQ8QBcBzCjw7EWJKDX5kCCaSX87w+Bng7CYhQ2jUfhOoD/nY01IAEQQNAY4mcQ6zBgwABC8ZlAYZ/v7+rtsQAFsxP7BHMdwFXUvDgJxEUAebqwPod02YjajzV7b1w34cExEQjsdJDOW+JEXqCYSPIgEiCBuAnofTwwUGvcuHHc5/OE5AkE0hLAQjDymSNUvWfPnslT4hVIgARcIZCZmamineG4wUAyVxAXeNHAWQJ6IRgm5owZMxgPUGAX4AEk4C8BHUiGWriVy8vfFpp990ApgUivA3Ymszsea0cCkQT04I1efJFUvHkdGCWA0QQ8Deh/7E3H4V1IwGkCOp6nT58+tOKdhpvP9QKzJqAVAD2B8pE2vyIBgwlgYRi/X6wPoDi5z4fBzfa9aoGIE4CHgU4JQQ8D3/sUK0ACCRPA7xceffg943fN4j4B6y0B7WI2c+ZMbmHnfn/hHUjAdQLw6Dt9+rT0799f3YsxBO4it1oJRCoAdhR3OwqvTgJeEtC/ZyoC96lbqwSQhRAdBEnhdIdxHxfvQAIk4BUB/bumInCXuJVKAF4ECDeHF8Ef//hHdwnx6iRAAr4RoCJwH711SoBuZO53Ct6BBEwiQEXgrjSsUgI6HxD9iN3tFLw6CZhGgIrAPYlYowT0IjAVgHudgVcmAZMJRCqCHTt2MKDMIWFZoQS0AoAbKBJOcX9gh6TPy5CAZQSgCJBaAgkiUSZOnKj2J7CsGUZV1+i0EUgFMWbMGJk8ebJAAeiRgFEEWRkSIAHPCei1wRtuuIFJ55Kkb2zEMJLBIRUEFMCyZcuoAJIUNE8ngSARQGQxUk+jVKpUSaAUWBIjYKQS2Lt3r7Rp0+birmCdO3dOrHU8iwRIILAEsDsZnEWwToh8Q5g2ZomfgHFKAEFg6enpqiWHDh3ibkPxy5RnkEBoCGDbWOwbgqBRBJX17dtXkJaaJXYCxigBzP8PGTJEBYFhR7D33nuP+wLHLkceSQKhJQBHkREjRqg9i5F4DjMHGEyyxEbACCWQlZUlTZs2vTj/P2nSJHoAxSY/HkUCJPAzgVatWglmD7CrIDIKYFCJtUWW/An4qgQgoPHjx0v9+vWV2xcWejj/n7/A+C0JkEDeBDA9hH0I4EwCpxKsLWLdgCVvAr4oAUz9QDAQ0MiRI1X+8Jdeekmw0MNCAiRAAskSwGASVkGzZs1UTEGnTp3oQZQHVE+VgH74Y+oHwR4QEASF/OEMAMtDQvyYBEggIQKwCjC1jN3KUOBBRGWQG6UnSgCr9XDfinz4Y+oHAoKgWEiABEjALQKIKVixYkUuZYDZCK4ZiLgWMQztu337dlm7dq288cYbSr569y9O+7jV3XldEiCBggggsAzPJKwZoCDO4JZbbpHf/OY3UrZs2dANTF1TAloQ8N+FBdCgQQNO+Wgo/E8CJOA7AVgBH3/8sbz22msqMDWyQnBTx0xFGIprSgBz/ZzqCUMXYhtJIBgEkKngxIkTcuzYMbnyyitDE6jqmhI4f/58MHoGW0ECJEACASbgycJwgPmxaSRAAiRgNQEqAavFx8qTAAmQQHIEqASS48ezSYAESMBqAlQCVouPlScBEiCB5AhQCSTHj2eTAAmQgNUEqASsFh8rTwIkQALJEaASSI4fzyYBEiABqwlQCVgtPlaeBEiABJIjQCWQHD+eTQIkQAJWE6ASsFp8rDwJkAAJJEeASiA5fjybBEiABKwmQCVgtfhYeRIgARJIjgCVQHL8eDYJkAAJWE2ASsBq8bHyJEACJJAcASqB5PjxbBIgARKwmgCVgNXiY+VJgARIIDkCVALJ8ePZJEACJGA1ASoBq8XHypMACZBAcgSoBJLjx7NJgARIwGoChZ2uPfcWdpoor0cCJEAC7hGgJeAeW16ZBEiABIwnQCVgvIhYQRIgARJwj8D/ASh3tvom3EfWAAAAAElFTkSuQmCC"></p>
</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>-coordinates of the points of intersection of the two graphs.</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 exact area of the shaded region, giving your answer in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p\pi + q\sqrt 3 "> <mi>p</mi> <mi>π</mi> <mo>+</mo> <mi>q</mi> <msqrt> <mn>3</mn> </msqrt> </math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p"> <mi>p</mi> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q \in \mathbb{Q}"> <mi>q</mi> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">Q</mi> </mrow> </math></span>.</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>At the points A and B on the diagram, the gradients of the two graphs are equal.</p>
<p>Determine the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y"> <mi>y</mi> </math></span>-coordinate of A on the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g"> <mi>g</mi> </math></span>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p>A continuous random variable <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> has the probability density function</p>
<p style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfenced open="{" close><mtable><mtr><mtd><mfrac><mn>2</mn><mrow><mfenced><mrow><mi>b</mi><mo>-</mo><mi>a</mi></mrow></mfenced><mfenced><mrow><mi>c</mi><mo>-</mo><mi>a</mi></mrow></mfenced></mrow></mfrac><mfenced><mrow><mi>x</mi><mo>-</mo><mi>a</mi></mrow></mfenced><mo>,</mo></mtd><mtd><mi>a</mi><mo>≤</mo><mi>x</mi><mo>≤</mo><mi>c</mi></mtd></mtr><mtr><mtd><mfrac><mn>2</mn><mrow><mfenced><mrow><mi>b</mi><mo>-</mo><mi>a</mi></mrow></mfenced><mfenced><mrow><mi>b</mi><mo>-</mo><mi>c</mi></mrow></mfenced></mrow></mfrac><mfenced><mrow><mi>b</mi><mo>-</mo><mi>x</mi></mrow></mfenced><mo>,</mo></mtd><mtd><mi>c</mi><mo><</mo><mi>x</mi><mo>≤</mo><mi>b</mi></mtd></mtr><mtr><mtd><mn>0</mn><mo>,</mo></mtd><mtd><mtext>otherwise</mtext></mtd></mtr></mtable></mfenced></math>.</p>
<p>The following diagram shows the 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> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>≤</mo><mi>x</mi><mo>≤</mo><mi>b</mi></math>.</p>
<p><img style="display:block;margin-left:auto;margin-right:auto;" src="data:image/png;base64,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"></p>
<p>Given that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>≥</mo><mfrac><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow><mn>2</mn></mfrac></math>, find an expression for the median of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>,</mo><mo> </mo><mi>b</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math>.</p>
</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">
<mi>f</mi>
</math></span> defined by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = {{\text{e}}^x}\sin x,{\text{ }}0 \leqslant x \leqslant \pi ">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mi>x</mi>
</msup>
</mrow>
<mi>sin</mi>
<mo><!-- --></mo>
<mi>x</mi>
<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>The curvature at any point <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(x,{\text{ }}y)">
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>y</mi>
<mo stretchy="false">)</mo>
</math></span> on a graph is defined as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\kappa = \frac{{\left| {\frac{{{{\text{d}}^2}y}}{{{\text{d}}{x^2}}}} \right|}}{{{{\left( {1 + {{\left( {\frac{{{\text{d}}y}}{{{\text{d}}x}}} \right)}^2}} \right)}^{\frac{3}{2}}}}}">
<mi>κ<!-- κ --></mi>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<mo>|</mo>
<mrow>
<mfrac>
<mrow>
<mrow>
<msup>
<mrow>
<mtext>d</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</mrow>
<mo>|</mo>
</mrow>
</mrow>
<mrow>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mrow>
<msup>
<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>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mfrac>
<mn>3</mn>
<mn>2</mn>
</mfrac>
</mrow>
</msup>
</mrow>
</mrow>
</mfrac>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f"> <mi>f</mi> </math></span> has a local maximum value when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \frac{{3\pi }}{4}"> <mi>x</mi> <mo>=</mo> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>4</mn> </mfrac> </math></span>.</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 <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>-coordinate of the point of inflexion of the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f"> <mi>f</mi> </math></span>.</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>Sketch the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f"> <mi>f</mi> </math></span>, clearly indicating the position of the local maximum point, the point of inflexion and the axes intercepts.</p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the area of the region enclosed by the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f"> <mi>f</mi> </math></span> and the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>-axis.</p>
<p>The curvature at any point <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(x,{\text{ }}y)"> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mi>y</mi> <mo stretchy="false">)</mo> </math></span> on a graph is defined as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\kappa = \frac{{\left| {\frac{{{{\text{d}}^2}y}}{{{\text{d}}{x^2}}}} \right|}}{{{{\left( {1 + {{\left( {\frac{{{\text{d}}y}}{{{\text{d}}x}}} \right)}^2}} \right)}^{\frac{3}{2}}}}}"> <mi>κ</mi> <mo>=</mo> <mfrac> <mrow> <mrow> <mo>|</mo> <mrow> <mfrac> <mrow> <mrow> <msup> <mrow> <mtext>d</mtext> </mrow> <mn>2</mn> </msup> </mrow> <mi>y</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mrow> </mfrac> </mrow> <mo>|</mo> </mrow> </mrow> <mrow> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow> <msup> <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> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> </msup> </mrow> </mrow> </mfrac> </math></span>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of the curvature of the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f"> <mi>f</mi> </math></span> at the local maximum point.</p>
<div class="marks">[3]</div>
<div class="question_part_label">g.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\kappa "> <mi>κ</mi> </math></span> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \frac{\pi }{2}"> <mi>x</mi> <mo>=</mo> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </math></span> and comment on its meaning with respect to the shape of the graph.</p>
<div class="marks">[2]</div>
<div class="question_part_label">h.</div>
</div>
<br><hr><br><div class="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = \frac{{2 - 3{x^5}}}{{2{x^3}}},\,\,x \in \mathbb{R},\,\,x \ne 0">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mn>2</mn>
<mo>−<!-- − --></mo>
<mn>3</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>5</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mn>2</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
</mrow>
</mfrac>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>≠<!-- ≠ --></mo>
<mn>0</mn>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>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> has a local maximum at A. Find the coordinates of A.</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>Show that there is exactly one point of inflexion, B, 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">[5]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The coordinates of B can be expressed in the form B<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {{2^a},\,b \times {2^{ - 3a}}} \right)"> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mn>2</mn> <mi>a</mi> </msup> </mrow> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mi>b</mi> <mo>×</mo> <mrow> <msup> <mn>2</mn> <mrow> <mo>−</mo> <mn>3</mn> <mi>a</mi> </mrow> </msup> </mrow> </mrow> <mo>)</mo> </mrow> </math></span> where <em>a</em>, <em>b</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \in \mathbb{Q}"> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">Q</mi> </mrow> </math></span>. Find the value of <em>a</em> and the value of <em>b</em>.</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>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 the position of the points A and B.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p>By using the substitution <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi><mo>=</mo><mtext>sec</mtext><mo> </mo><mi>x</mi></math> or otherwise, find an expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∫</mo><mn>0</mn><mfrac><mi>π</mi><mn>3</mn></mfrac></munderover><msup><mtext>sec</mtext><mi>n</mi></msup><mo> </mo><mi>x</mi><mo> </mo><mi>tan</mi><mo> </mo><mi>x</mi><mo> </mo><mo>d</mo><mi>x</mi></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> is a non-zero real number.</p>
</div>
<br><hr><br><div class="specification">
<p>The continuous random variable <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> has probability density function</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfenced open="{" close><mtable columnspacing="1.4ex" columnalign="left"><mtr><mtd><mfrac><mi>k</mi><msqrt><mn>4</mn><mo>-</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup></msqrt></mfrac><mo>,</mo></mtd><mtd><mn>0</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mn>1</mn></mtd></mtr><mtr><mtd><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mn>0</mn><mo>,</mo></mtd><mtd><mtext>otherwise</mtext><mo>.</mo></mtd></mtr></mtable></mfenced></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>k</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>Find <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>E</mtext><mo>(</mo><mi>X</mi><mo>)</mo></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</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><mi>x</mi><msqrt><mn>1</mn><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup></msqrt></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>1</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mn>1</mn></math>.</p>
<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> is shown below.</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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"></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 odd 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>The range of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>≤</mo><mi>y</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">ℝ</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">[6]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the function 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><mi>x</mi><mo>-</mo><mn>5</mn></mrow><mrow><mi>x</mi><mo>-</mo><mi>k</mi></mrow></mfrac></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo> </mo><mo>∈</mo><mo> </mo><mi mathvariant="normal">ℝ</mi><mo> </mo><mo>\</mo><mo> </mo><mfenced open="{" close="}"><mi>k</mi></mfenced></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>k</mi><mn>2</mn></msup><mo>≠</mo><mn>5</mn></math>. </p>
</div>
<div class="specification">
<p>Consider the case where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mn>3</mn></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State the equation of the vertical asymptote on 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>.</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>State the equation of the horizontal asymptote on 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>.</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>Use an algebraic method to determine whether <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> is a self-inverse function.</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>y</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math>, stating clearly the equations of any asymptotes and the coordinates of any points of intersections with the coordinate axes.</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>The region bounded by the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis, 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>, and the lines <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>5</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>7</mn></math> is rotated through <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi mathvariant="normal">π</mi></math> about the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis. Find the volume of the solid generated, giving your answer in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">π</mi><mo>(</mo><mi>a</mi><mo>+</mo><mi>b</mi><mo> </mo><mi>ln</mi><mo> </mo><mn>2</mn><mo>)</mo><mo> </mo></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>,</mo><mo> </mo><mi>b</mi><mo> </mo><mo>∈</mo><mo> </mo><mi mathvariant="normal">ℤ</mi></math>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">e.</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 given by the equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = x\,{\text{tan}}\left( {\frac{{\pi xy}}{4}} \right)">
<mi>y</mi>
<mo>=</mo>
<mi>x</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>tan</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mi>π<!-- π --></mi>
<mi>x</mi>
<mi>y</mi>
</mrow>
<mn>4</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>At the point (1, 1) , show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}} = \frac{{2 + \pi }}{{2 - \pi }}"> <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>2</mn> <mo>+</mo> <mi>π</mi> </mrow> <mrow> <mn>2</mn> <mo>−</mo> <mi>π</mi> </mrow> </mfrac> </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>Hence find the equation of the normal to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="C"> <mi>C</mi> </math></span> at the point (1, 1).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</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) = \frac{1}{{{x^2} + 3x + 2}},{\text{ }}x \in \mathbb{R},{\text{ }}x \ne - 2,{\text{ }}x \ne - 1">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>3</mn>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
</mrow>
</mfrac>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>x</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>x</mi>
<mo>≠<!-- ≠ --></mo>
<mo>−<!-- − --></mo>
<mn>2</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>x</mi>
<mo>≠<!-- ≠ --></mo>
<mo>−<!-- − --></mo>
<mn>1</mn>
</math></span>.</p>
</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="{x^2} + 3x + 2"> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>3</mn> <mi>x</mi> <mo>+</mo> <mn>2</mn> </math></span> in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{(x + h)^2} + k"> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>+</mo> <mi>h</mi> <msup> <mo stretchy="false">)</mo> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mi>k</mi> </math></span>.</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>Factorize <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^2} + 3x + 2"> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>3</mn> <mi>x</mi> <mo>+</mo> <mn>2</mn> </math></span>.</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>Sketch the graph of <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>, indicating on it the equations of the asymptotes, the coordinates of the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y"> <mi>y</mi> </math></span>-intercept and the local maximum.</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>Hence find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p"> <mi>p</mi> </math></span> if <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_0^1 {f(x){\text{d}}x = \ln (p)} "> <msubsup> <mo>∫</mo> <mn>0</mn> <mn>1</mn> </msubsup> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> <mo>=</mo> <mi>ln</mi> <mo></mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</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( {\left| x \right|} \right)"> <mi>y</mi> <mo>=</mo> <mi>f</mi> <mrow> <mo>(</mo> <mrow> <mrow> <mo>|</mo> <mi>x</mi> <mo>|</mo> </mrow> </mrow> <mo>)</mo> </mrow> </math></span>.</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>Determine the area of the region enclosed between the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f\left( {\left| x \right|} \right)"> <mi>y</mi> <mo>=</mo> <mi>f</mi> <mrow> <mo>(</mo> <mrow> <mrow> <mo>|</mo> <mi>x</mi> <mo>|</mo> </mrow> </mrow> <mo>)</mo> </mrow> </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 with equations <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> and <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>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">f.</div>
</div>
<br><hr><br><div class="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = {\text{arccos}}\left( {\frac{x}{2}} \right)">
<mi>y</mi>
<mo>=</mo>
<mrow>
<mtext>arccos</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mi>x</mi>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</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>.</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 <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_0^1 {{\text{arccos}}\left( {\frac{x}{2}} \right){\text{d}}x} ">
<msubsup>
<mo>∫</mo>
<mn>0</mn>
<mn>1</mn>
</msubsup>
<mrow>
<mrow>
<mtext>arccos</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mi>x</mi>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</math></span>.</p>
<div class="marks">[7]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><msqrt><mn>1</mn><mo>+</mo><mi>x</mi></msqrt></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>></mo><mo>-</mo><mn>1</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><mo>''</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mo>-</mo><mfrac><mn>1</mn><mrow><mn>4</mn><msqrt><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mn>3</mn></msup></msqrt></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>Use mathematical induction to prove that <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>f</mi><mfenced><mi>n</mi></mfenced></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><msup><mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><mfrac><mrow><mfenced><mrow><mn>2</mn><mi>n</mi><mo>-</mo><mn>3</mn></mrow></mfenced><mo>!</mo></mrow><mrow><mfenced><mrow><mi>n</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mo>!</mo></mrow></mfrac><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfenced><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>-</mo><mi>n</mi></mrow></msup></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>∈</mo><mi mathvariant="normal">ℤ</mi><mo>,</mo><mo> </mo><mi>n</mi><mo>≥</mo><mn>2</mn></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>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><msup><mtext>e</mtext><mrow><mi>m</mi><mi>x</mi></mrow></msup><mo>,</mo><mo> </mo><mi>m</mi><mo>∈</mo><mi mathvariant="normal">ℚ</mi></math>.</p>
<p>Consider the function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math> defined by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>×</mo><mi>g</mi><mfenced><mi>x</mi></mfenced></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>></mo><mo>-</mo><mn>1</mn></math>.</p>
<p>It is given that the <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup></math> term in the Maclaurin series for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> has a coefficient of <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>7</mn><mn>4</mn></mfrac></math>.</p>
<p>Find the possible values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math>.</p>
<div class="marks">[8]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = \frac{{2x - 4}}{{{x^2} - 1}}{\text{, }} - 1 < x < 1">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mn>2</mn>
<mi>x</mi>
<mo>−<!-- − --></mo>
<mn>4</mn>
</mrow>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</mfrac>
<mrow>
<mtext>, </mtext>
</mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo><</mo>
<mi>x</mi>
<mo><</mo>
<mn>1</mn>
</math></span>.</p>
</div>
<div class="specification">
<p>For 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>
<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> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </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>Show that, if <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f'\left( x \right) = 0"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </math></span>, then <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 2 - \sqrt 3 "> <mi>x</mi> <mo>=</mo> <mn>2</mn> <mo>−</mo> <msqrt> <mn>3</mn> </msqrt> </math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>find the coordinates of the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y"> <mi>y</mi> </math></span>-intercept.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>show that there are no <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>-intercepts.</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>sketch the graph, showing clearly any asymptotic behaviour.</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>Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{3}{{x + 1}} - \frac{1}{{x - 1}} = \frac{{2x - 4}}{{{x^2} - 1}}"> <mfrac> <mn>3</mn> <mrow> <mi>x</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> <mo>−</mo> <mfrac> <mn>1</mn> <mrow> <mi>x</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> <mi>x</mi> <mo>−</mo> <mn>4</mn> </mrow> <mrow> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </math></span>.</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>The area enclosed by 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> and the line <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 4"> <mi>y</mi> <mo>=</mo> <mn>4</mn> </math></span> can be expressed as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ln}}\,v"> <mrow> <mtext>ln</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>v</mi> </math></span>. Find the 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">[7]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_{ - 2}^2 {f\left( x \right){\text{d}}x = 10} ">
<msubsup>
<mo>∫<!-- ∫ --></mo>
<mrow>
<mo>−<!-- − --></mo>
<mn>2</mn>
</mrow>
<mn>2</mn>
</msubsup>
<mrow>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
<mo>=</mo>
<mn>10</mn>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_0^2 {f\left( x \right){\text{d}}x = 12} ">
<msubsup>
<mo>∫<!-- ∫ --></mo>
<mn>0</mn>
<mn>2</mn>
</msubsup>
<mrow>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
<mo>=</mo>
<mn>12</mn>
</mrow>
</math></span>, find</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_{ - 2}^0 {\left( {f\left( x \right){\text{ + 2}}} \right){\text{d}}x} ">
<msubsup>
<mo>∫</mo>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
<mn>0</mn>
</msubsup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mrow>
<mtext> + 2</mtext>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_{ - 2}^0 {f\left( {x{\text{ + 2}}} \right){\text{d}}x} ">
<msubsup>
<mo>∫</mo>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
<mn>0</mn>
</msubsup>
<mrow>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mrow>
<mtext> + 2</mtext>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>Using the substitution <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u = {\text{sin}}\,x">
<mi>u</mi>
<mo>=</mo>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</math></span>, find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {\frac{{{\text{co}}{{\text{s}}^3}x\,{\text{d}}x}}{{\sqrt {{\text{sin}}\,x} }}} ">
<mo>∫</mo>
<mrow>
<mfrac>
<mrow>
<mrow>
<mtext>co</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>s</mtext>
</mrow>
<mn>3</mn>
</msup>
</mrow>
<mi>x</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
<mrow>
<msqrt>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</msqrt>
</mrow>
</mfrac>
</mrow>
</math></span>.</p>
</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><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><msup><mtext>e</mtext><mi>x</mi></msup><mo> </mo><mi>sin</mi><mo> </mo><mi>x</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</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><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><msup><mtext>e</mtext><mi>x</mi></msup><mo> </mo><mi>cos</mi><mo> </mo><mi>x</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the Maclaurin series for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></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.</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 an approximate value for <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mn>0</mn><mn>1</mn></msubsup><msup><mtext>e</mtext><msup><mi>x</mi><mn>2</mn></msup></msup><mo> </mo><mi>sin</mi><mfenced><msup><mi>x</mi><mn>2</mn></msup></mfenced><mo>d</mo><mi>x</mi></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>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> satisfies the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>''</mo><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mn>2</mn><mo>(</mo><mi>g</mi><mo>'</mo><mo>(</mo><mi>x</mi><mo>)</mo><mo>-</mo><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo></math>.</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>Hence, deduce that <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>g</mi><mfenced><mn>4</mn></mfenced></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>2</mn><mfenced><mrow><mi>g</mi><mo>'''</mo><mfenced><mi>x</mi></mfenced><mo>-</mo><mi>g</mi><mo>''</mo><mfenced><mi>x</mi></mfenced></mrow></mfenced></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 result from part (c), find the Maclaurin series for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo></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.</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>Hence, or otherwise, determine the value of <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><msup><mtext>e</mtext><mi>x</mi></msup><mo> </mo><mi>cos</mi><mo> </mo><mi>x</mi><mo>-</mo><mn>1</mn><mo>-</mo><mi>x</mi></mrow><msup><mi>x</mi><mn>3</mn></msup></mfrac></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p>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>
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span>, 0 ≤ <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> ≤ 5 is shown in the following diagram. The curve intercepts the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>-axis at (1, 0) and (4, 0) and has a local minimum at (3, −1).</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<div class="specification">
<p>The shaded area enclosed by the curve <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>, the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>-axis and the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span>-axis is 0.5. Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( 0 \right) = 3">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mn>0</mn>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>3</mn>
</math></span>,</p>
</div>
<div class="specification">
<p>The area enclosed by the curve <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> and the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>-axis between <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 = 4">
<mi>x</mi>
<mo>=</mo>
<mn>4</mn>
</math></span> is 2.5 .</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>-coordinate 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">[1]</div>
<div class="question_part_label">a.</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="f\left( 1 \right)"> <mi>f</mi> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </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>find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( 4 \right)"> <mi>f</mi> <mrow> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> </math></span>.</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 curve <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>, 0 ≤ <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> ≤ 5 indicating clearly the coordinates of the maximum and minimum points and any intercepts with the coordinate axes.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</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><mn>3</mn><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn></mrow></mfrac><mo>,</mo><mo> </mo><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
</div>
<div class="specification">
<p>The region <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math> is bounded by 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>, the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis and the lines <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>x</mi><mo>=</mo><msqrt><mn>6</mn></msqrt></math>. Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> be the area of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math>.</p>
</div>
<div class="specification">
<p>The line <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mi>k</mi></math> divides <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi></math> into two regions of equal area.</p>
</div>
<div class="specification">
<p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math> be the gradient of a tangent to 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>.</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 any asymptotes with their equations and stating the coordinates of any points of intersection with the axes.</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>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mfrac><mrow><msqrt><mn>2</mn></msqrt><mi mathvariant="normal">π</mi></mrow><mn>2</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>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</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>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mo>-</mo><mfrac><mrow><mn>6</mn><mi>x</mi></mrow><msup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn></mrow></mfenced><mn>2</mn></msup></mfrac></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>Show that the maximum value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>27</mn><mn>32</mn></mfrac><msqrt><mfrac><mn>2</mn><mn>3</mn></mfrac></msqrt></math>.</p>
<div class="marks">[7]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="question">
<p>Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{lo}}{{\text{g}}_{{r^2}}}x = \frac{1}{2}{\text{lo}}{{\text{g}}_r}\,x">
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mrow>
<mrow>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</msub>
</mrow>
<mi>x</mi>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mi>r</mi>
</msub>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r,\,x \in {\mathbb{R}^ + }">
<mi>r</mi>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>∈</mo>
<mrow>
<msup>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
<mo>+</mo>
</msup>
</mrow>
</math></span>.</p>
</div>
<br><hr><br><div class="question">
<p>The folium of Descartes is a curve defined by the equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^3} + {y^3} - 3xy = 0">
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mi>y</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>3</mn>
<mi>x</mi>
<mi>y</mi>
<mo>=</mo>
<mn>0</mn>
</math></span>, shown in the following diagram.</p>
<p><img src="images/Schermafbeelding_2018-02-07_om_18.23.15.png" alt="N17/5/MATHL/HP1/ENG/TZ0/07"></p>
<p>Determine the exact coordinates of the point P on the curve where the tangent line is parallel to the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span>-axis.</p>
</div>
<br><hr><br><div class="specification">
<p>A particle moves along a straight line. Its displacement, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="s">
<mi>s</mi>
</math></span> metres, at time <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="s = t + \cos 2t,{\text{ }}t \geqslant 0">
<mi>s</mi>
<mo>=</mo>
<mi>t</mi>
<mo>+</mo>
<mi>cos</mi>
<mo><!-- --></mo>
<mn>2</mn>
<mi>t</mi>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>t</mi>
<mo>⩾<!-- ⩾ --></mo>
<mn>0</mn>
</math></span>. The first two times when the particle is at rest are denoted by <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> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{t_2}">
<mrow>
<msub>
<mi>t</mi>
<mn>2</mn>
</msub>
</mrow>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{t_1} < {t_2}">
<mrow>
<msub>
<mi>t</mi>
<mn>1</mn>
</msub>
</mrow>
<mo><</mo>
<mrow>
<msub>
<mi>t</mi>
<mn>2</mn>
</msub>
</mrow>
</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="{t_1}">
<mrow>
<msub>
<mi>t</mi>
<mn>1</mn>
</msub>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{t_2}">
<mrow>
<msub>
<mi>t</mi>
<mn>2</mn>
</msub>
</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 displacement of the particle 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.</div>
</div>
<br><hr><br><div class="question">
<p>Use l’Hôpital’s rule to determine the value of</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mo> </mo><mfrac><mrow><mn>2</mn><mo> </mo><mi>sin</mi><mo> </mo><mi>x</mi><mo>-</mo><mi>sin</mi><mo> </mo><mn>2</mn><mi>x</mi></mrow><msup><mi>x</mi><mn>3</mn></msup></mfrac></math>.</p>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2x - {x^2}"> <mn>2</mn> <mi>x</mi> <mo>−</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </math></span> in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a{\left( {x - h} \right)^2} + k"> <mi>a</mi> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <mi>h</mi> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mi>k</mi> </math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a{\text{, }}h{\text{, }}k \in \mathbb{R}"> <mi>a</mi> <mrow> <mtext>, </mtext> </mrow> <mi>h</mi> <mrow> <mtext>, </mtext> </mrow> <mi>k</mi> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> </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 the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_{\frac{1}{2}}^{\frac{3}{2}} {\frac{1}{{\sqrt {2x - {x^2}} }}} {\text{d}}x"> <msubsup> <mo>∫</mo> <mrow> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> </msubsup> <mrow> <mfrac> <mn>1</mn> <mrow> <msqrt> <mn>2</mn> <mi>x</mi> <mo>−</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </msqrt> </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>
<br><hr><br><div class="specification">
<p>Consider the expression <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><msqrt><mn>1</mn><mo>+</mo><mi>a</mi><mi>x</mi></msqrt></mfrac><mo>-</mo><msqrt><mn>1</mn><mo>-</mo><mi>x</mi></msqrt></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>∈</mo><mi mathvariant="normal">ℚ</mi><mo>,</mo><mo> </mo><mi>a</mi><mo>≠</mo><mn>0</mn></math>.</p>
<p>The binomial expansion of this expression, in ascending powers of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>, as far as the term in <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mi>b</mi><mi>x</mi><mo>+</mo><mi>b</mi><msup><mi>x</mi><mn>2</mn></msup></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>∈</mo><mi mathvariant="normal">ℚ</mi></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>a</mi></math> and the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math>.</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>State the restriction which must be placed on <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> for this expansion to be valid.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Using the substitution <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \tan \theta ">
<mi>x</mi>
<mo>=</mo>
<mi>tan</mi>
<mo></mo>
<mi>θ</mi>
</math></span> show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int\limits_0^1 {\frac{1}{{{{\left( {{x^2} + 1} \right)}^2}}}{\text{d}}x = } \int\limits_0^{\frac{\pi }{4}} {{{\cos }^2}\theta {\text{d}}\theta } ">
<munderover>
<mo>∫</mo>
<mn>0</mn>
<mn>1</mn>
</munderover>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
<mo>=</mo>
</mrow>
<munderover>
<mo>∫</mo>
<mn>0</mn>
<mrow>
<mfrac>
<mi>π</mi>
<mn>4</mn>
</mfrac>
</mrow>
</munderover>
<mrow>
<mrow>
<msup>
<mrow>
<mi>cos</mi>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mi>θ</mi>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>θ</mi>
</mrow>
</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>Hence find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int\limits_0^1 {\frac{1}{{{{\left( {{x^2} + 1} \right)}^2}}}{\text{d}}x} ">
<munderover>
<mo>∫</mo>
<mn>0</mn>
<mn>1</mn>
</munderover>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>A curve has equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3x - 2{y^2}{{\text{e}}^{x - 1}} = 2">
<mn>3</mn>
<mi>x</mi>
<mo>−<!-- − --></mo>
<mn>2</mn>
<mrow>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mi>x</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mo>=</mo>
<mn>2</mn>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>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> 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">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the equations of the tangents to this curve at the points where the curve intersects the line <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>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mn>1</mn><mn>9</mn></msubsup><mfenced><mfrac><mrow><mn>3</mn><msqrt><mi>x</mi></msqrt><mo>-</mo><mn>5</mn></mrow><msqrt><mi>x</mi></msqrt></mfrac></mfenced><mo>d</mo><mi>x</mi></math>.</p>
</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><mn>1</mn><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>x</mi><mo>-</mo><mn>3</mn></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><mo>-</mo><mn>1</mn><mo>,</mo><mo> </mo><mi>x</mi><mo>≠</mo><mn>3</mn></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><mfrac><mn>1</mn><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>x</mi><mo>-</mo><mn>3</mn></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>3</mn></math>.</p>
</div>
<div class="specification">
<p>The inverse of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>g</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math>.</p>
</div>
<div class="specification">
<p>A function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math> is defined by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mtext>arctan</mtext><mfrac><mi>x</mi><mn>2</mn></mfrac></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></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><mo>(</mo><mi>x</mi><mo>)</mo></math>, clearly indicating any asymptotes with their equations. State the coordinates of any local maximum or minimum points and any points of intersection with the coordinate axes.</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>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>g</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>1</mn><mo>+</mo><mfrac><msqrt><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>x</mi></msqrt><mi>x</mi></mfrac></math>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.i.</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">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>h</mi><mo>∘</mo><mi>g</mi></mrow></mfenced><mfenced><mi>a</mi></mfenced><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac></math>, find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math>.</p>
<p>Give your answer in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>+</mo><mfrac><mi>q</mi><mn>2</mn></mfrac><msqrt><mi>r</mi></msqrt></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>,</mo><mo> </mo><mi>q</mi><mo>,</mo><mo> </mo><mi>r</mi><mo>∈</mo><msup><mi mathvariant="normal">ℤ</mi><mo>+</mo></msup></math>.</p>
<div class="marks">[7]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use the substitution <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u = {x^{\frac{1}{2}}}">
<mi>u</mi>
<mo>=</mo>
<mrow>
<msup>
<mi>x</mi>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
</msup>
</mrow>
</math></span> to find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {\frac{{{\text{d}}x}}{{{x^{\frac{3}{2}}} + {x^{\frac{1}{2}}}}}} ">
<mo>∫</mo>
<mrow>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mrow>
<mfrac>
<mn>3</mn>
<mn>2</mn>
</mfrac>
</mrow>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mi>x</mi>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
</msup>
</mrow>
</mrow>
</mfrac>
</mrow>
</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>Hence find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{2}\int\limits_1^9 {\frac{{{\text{d}}x}}{{{x^{\frac{3}{2}}} + {x^{\frac{1}{2}}}}}} ">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<munderover>
<mo>∫</mo>
<mn>1</mn>
<mn>9</mn>
</munderover>
<mrow>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mrow>
<mfrac>
<mn>3</mn>
<mn>2</mn>
</mfrac>
</mrow>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mi>x</mi>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
</msup>
</mrow>
</mrow>
</mfrac>
</mrow>
</math></span>, expressing your answer in the form arctan <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q">
<mi>q</mi>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q \in \mathbb{Q}">
<mi>q</mi>
<mo>∈</mo>
<mrow>
<mi mathvariant="double-struck">Q</mi>
</mrow>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>A camera at point C is 3 m from the edge of a straight section of road as shown in the following diagram. The camera detects a car travelling along the road at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t"> <mi>t</mi> </math></span> = 0. It then rotates, always pointing at the car, until the car passes O, the point on the edge of the road closest to the camera.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<p style="text-align: left;">A car travels along the road at a speed of 24 ms<sup>−1</sup>. Let the position of the car be X and let OĈX = <em>θ</em>.</p>
<p style="text-align: left;">Find <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>, the rate of rotation of the camera, in radians per second, at the instant the car passes the point O .</p>
</div>
<br><hr><br><div class="question">
<p>Use l’Hôpital’s rule to determine the value of <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><mn>2</mn><mi>x</mi><mo> </mo><mi>cos</mi><mfenced><msup><mi>x</mi><mn>2</mn></msup></mfenced></mrow><mrow><mn>5</mn><mo> </mo><mi>tan</mi><mo> </mo><mi>x</mi></mrow></mfrac></mfenced></math>.</p>
</div>
<br><hr><br><div class="specification">
<p>The acceleration, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo> </mo><msup><mtext>ms</mtext><mrow><mo>-</mo><mn>2</mn></mrow></msup></math>, of a particle moving in a horizontal line at time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> seconds, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>≥</mo><mn>0</mn></math>, is given by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mo>-</mo><mo>(</mo><mn>1</mn><mo>+</mo><mi>v</mi><mo>)</mo></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mo> </mo><msup><mtext>ms</mtext><mrow><mo>-</mo><mn>1</mn></mrow></msup></math> is the particle’s velocity and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mo>></mo><mo>-</mo><mn>1</mn></math>.</p>
<p>At <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn></math>, the particle is at a fixed origin <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math> and has initial velocity <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>v</mi><mn>0</mn></msub><mo> </mo><msup><mtext>ms</mtext><mrow><mo>-</mo><mn>1</mn></mrow></msup></math>.</p>
</div>
<div class="specification">
<p>Initially at <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math>, the particle moves in the positive direction until it reaches its maximum displacement from <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math>. The particle then returns to <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math>.</p>
<p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi></math> metres represent the particle’s displacement from <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>s</mi><mtext>max</mtext></msub></math> its maximum displacement from <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math>.</p>
</div>
<div class="specification">
<p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mo>(</mo><mi>T</mi><mo>-</mo><mi>k</mi><mo>)</mo></math> represent the particle’s velocity <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> seconds before it reaches <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>s</mi><mtext>max</mtext></msub></math>, where</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mo>(</mo><mi>T</mi><mo>-</mo><mi>k</mi><mo>)</mo><mo>=</mo><mfenced><mrow><mn>1</mn><mo>+</mo><msub><mi>v</mi><mn>0</mn></msub></mrow></mfenced><msup><mtext>e</mtext><mrow><mo>-</mo><mo>(</mo><mi>T</mi><mo>-</mo><mi>k</mi><mo>)</mo></mrow></msup><mo>-</mo><mn>1</mn></math>.</p>
</div>
<div class="specification">
<p>Similarly, let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mo>(</mo><mi>T</mi><mo>+</mo><mi>k</mi><mo>)</mo></math> represent the particle’s velocity <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> seconds after it reaches <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>s</mi><mtext>max</mtext></msub></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By solving an appropriate differential equation, show that the particle’s velocity at time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> is given by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mo>(</mo><mn>1</mn><mo>+</mo><msub><mi>v</mi><mn>0</mn></msub><mo>)</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup><mo>-</mo><mn>1</mn></math>.</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>Show that the time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math> taken for the particle to reach <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>s</mi><mtext>max</mtext></msub></math> satisfies the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>e</mtext><mi>T</mi></msup><mo>=</mo><mn>1</mn><mo>+</mo><msub><mi>v</mi><mn>0</mn></msub></math>.</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>By solving an appropriate differential equation and using the result from part (b) (i), find an expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>s</mi><mtext>max</mtext></msub></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>v</mi><mn>0</mn></msub></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By using the result to part (b) (i), show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mfenced><mrow><mi>T</mi><mo>-</mo><mi>k</mi></mrow></mfenced><mo>=</mo><msup><mtext>e</mtext><mi>k</mi></msup><mo>-</mo><mn>1</mn></math>.</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>Deduce a similar expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mo>(</mo><mi>T</mi><mo>+</mo><mi>k</mi><mo>)</mo></math> in terms 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">d.</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"><mi>v</mi><mfenced><mrow><mi>T</mi><mo>-</mo><mi>k</mi></mrow></mfenced><mo>+</mo><mi>v</mi><mfenced><mrow><mi>T</mi><mo>+</mo><mi>k</mi></mrow></mfenced><mo>≥</mo><mn>0</mn></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="question">
<p>Use l’Hôpital’s rule to 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><mtext>arctan</mtext><mo> </mo><mn>2</mn><mi>x</mi></mrow><mrow><mi>tan</mi><mo> </mo><mn>3</mn><mi>x</mi></mrow></mfrac></mfenced></math>.</p>
</div>
<br><hr><br><div class="specification">
<p>Consider the curves <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{C_1}">
<mrow>
<msub>
<mi>C</mi>
<mn>1</mn>
</msub>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{C_2}">
<mrow>
<msub>
<mi>C</mi>
<mn>2</mn>
</msub>
</mrow>
</math></span> defined as follows</p>
<p style="text-align: center;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{C_1}\,{\text{:}}\,xy = 4">
<mrow>
<msub>
<mi>C</mi>
<mn>1</mn>
</msub>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>:</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mi>y</mi>
<mo>=</mo>
<mn>4</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></p>
<p style="text-align: center;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{C_2}\,{\text{:}}\,{y^2} - {x^2} = 2">
<mrow>
<msub>
<mi>C</mi>
<mn>2</mn>
</msub>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>:</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−<!-- − --></mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>2</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></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Using implicit differentiation, or otherwise, 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> for each curve 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>Let P(<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>) be the unique point where the curves <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{C_1}"> <mrow> <msub> <mi>C</mi> <mn>1</mn> </msub> </mrow> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{C_2}"> <mrow> <msub> <mi>C</mi> <mn>2</mn> </msub> </mrow> </math></span> intersect.</p>
<p>Show that the tangent to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{C_1}"> <mrow> <msub> <mi>C</mi> <mn>1</mn> </msub> </mrow> </math></span> at P is perpendicular to the tangent to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{C_2}"> <mrow> <msub> <mi>C</mi> <mn>2</mn> </msub> </mrow> </math></span> at P.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>Solve 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>ln</mi><mstyle displaystyle="true"><mo> </mo></mstyle><mstyle displaystyle="true"><mn>2</mn></mstyle><mstyle displaystyle="true"><mi>x</mi></mstyle></mrow><msup><mi>x</mi><mn>2</mn></msup></mfrac><mo>-</mo><mfrac><mrow><mn>2</mn><mi>y</mi></mrow><mi>x</mi></mfrac><mo>,</mo><mo> </mo><mi>x</mi><mo>></mo><mn>0</mn></math>, given that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>4</mn></math> at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math>.</p>
<p>Give your answer in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mfenced><mi>x</mi></mfenced></math>.</p>
</div>
<br><hr><br><div class="question">
<p>A particle moves in a straight 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 \geqslant 0)">
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo>⩾</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</math></span>, its velocity <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v">
<mi>v</mi>
</math></span>, in <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{m}}{{\text{s}}^{ - 1}}">
<mrow>
<mtext>m</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>s</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</math></span>, is given by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v = 10t{{\text{e}}^{ - 2t}}">
<mi>v</mi>
<mo>=</mo>
<mn>10</mn>
<mi>t</mi>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mn>2</mn>
<mi>t</mi>
</mrow>
</msup>
</mrow>
</math></span>. Find the exact distance travelled by the particle in the first half-second.</p>
</div>
<br><hr><br><div class="specification">
<p>Consider the function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q\left( x \right) = {x^5} - 10{x^2} + 15x - 6,{\text{ }}x \in \mathbb{R}">
<mi>q</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>5</mn>
</msup>
</mrow>
<mo>−<!-- − --></mo>
<mn>10</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>15</mn>
<mi>x</mi>
<mo>−<!-- − --></mo>
<mn>6</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>x</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = q(x)"> <mi>y</mi> <mo>=</mo> <mi>q</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></span> is concave up for <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>.</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>Sketch the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = q(x)"> <mi>y</mi> <mo>=</mo> <mi>q</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></span> showing clearly any intercepts with the axes.</p>
<div class="marks">[3]</div>
<div class="question_part_label">e.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the curve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math> defined by <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mi>sin</mi><mo> </mo><mfenced><mrow><mi>x</mi><mi>y</mi></mrow></mfenced><mo> </mo><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><mtext>d</mtext><mi>y</mi></mrow><mrow><mtext>d</mtext><mi>x</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>y</mi><mo> </mo><mi>cos</mi><mo> </mo><mfenced><mrow><mi>x</mi><mi>y</mi></mrow></mfenced></mrow><mrow><mn>2</mn><mi>y</mi><mo>-</mo><mi>x</mi><mo> </mo><mi>cos</mi><mo> </mo><mfenced><mrow><mi>x</mi><mi>y</mi></mrow></mfenced></mrow></mfrac></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>Prove that, when <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><mn>0</mn><mo> </mo><mo>,</mo><mo> </mo><mi>y</mi><mo>=</mo><mo>±</mo><mn>1</mn></math>.</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>Hence find the coordinates of all points on <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math>, for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo><</mo><mi>x</mi><mo><</mo><mn>4</mn><mi mathvariant="normal">π</mi></math>, where <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><mn>0</mn></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p>Find the coordinates of the points on the curve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{y^3} + 3x{y^2} - {x^3} = 27">
<mrow>
<msup>
<mi>y</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>3</mn>
<mi>x</mi>
<mrow>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>27</mn>
</math></span> at which <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>.</p>
</div>
<br><hr><br><div class="question">
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int {\arcsin x\,{\text{d}}x} ">
<mo>∫</mo>
<mrow>
<mi>arcsin</mi>
<mo></mo>
<mi>x</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</math></span></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{e}}^{{\text{sin}}\,x}}">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
</msup>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the first two derivatives of <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> and hence find the Maclaurin series for <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> up to and including the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^2}">
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</math></span> term.</p>
<div class="marks">[8]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the coefficient of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^3}">
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
</math></span> in the Maclaurin series for <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> is zero.</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>Using the Maclaurin series for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{arctan}}\,x">
<mrow>
<mtext>arctan</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="{{\text{e}}^{3x}} - 1">
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mn>3</mn>
<mi>x</mi>
</mrow>
</msup>
</mrow>
<mo>−</mo>
<mn>1</mn>
</math></span>, find the Maclaurin series for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{arctan}}\left( {{{\text{e}}^{3x}} - 1} \right)">
<mrow>
<mtext>arctan</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mn>3</mn>
<mi>x</mi>
</mrow>
</msup>
</mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span> up to and including the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^3}">
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
</math></span> term.</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>Hence, or otherwise, find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mathop {{\text{lim}}}\limits_{x \to 0} \frac{{f\left( x \right) - 1}}{{{\text{arctan}}\left( {{{\text{e}}^{3x}} - 1} \right)}}">
<munder>
<mrow>
<mrow>
<mtext>lim</mtext>
</mrow>
</mrow>
<mrow>
<mi>x</mi>
<mo stretchy="false">→</mo>
<mn>0</mn>
</mrow>
</munder>
<mo></mo>
<mfrac>
<mrow>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mrow>
<mrow>
<mtext>arctan</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mn>3</mn>
<mi>x</mi>
</mrow>
</msup>
</mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
</mfrac>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="question">
<p>Find the equation of the tangent to the curve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><msup><mtext>e</mtext><mrow><mn>2</mn><mi>x</mi></mrow></msup><mo>–</mo><mn>3</mn><mi>x</mi></math> at the point where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn></math>.</p>
</div>
<br><hr><br><div class="question">
<p>Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_0^{{\text{ln}}\,k} {{{\text{e}}^{2x}}} {\text{d}}x = 12"> <msubsup> <mo>∫</mo> <mn>0</mn> <mrow> <mrow> <mtext>ln</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>k</mi> </mrow> </msubsup> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mn>2</mn> <mi>x</mi> </mrow> </msup> </mrow> </mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> <mo>=</mo> <mn>12</mn> </math></span>, find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k"> <mi>k</mi> </math></span>.</p>
</div>
<br><hr><br><div class="specification">
<p>The lines <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>l</mi><mn>1</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>l</mi><mn>2</mn></msub></math> have the following vector equations where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi><mo>,</mo><mo> </mo><mi>μ</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>l</mi><mn>1</mn></msub><mo>:</mo><msub><mi mathvariant="bold-italic">r</mi><mn>1</mn></msub><mo>=</mo><mfenced><mtable><mtr><mtd><mn>3</mn></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mi>λ</mi><mfenced><mtable><mtr><mtd><mn>2</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable></mfenced></math></p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>l</mi><mn>2</mn></msub><mo>:</mo><msub><mi mathvariant="bold-italic">r</mi><mn>2</mn></msub><mo>=</mo><mfenced><mtable><mtr><mtd><mn>2</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>4</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mi>μ</mi><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr></mtable></mfenced></math></p>
</div>
<div class="question">
<p>By using the substitution <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi><mo>=</mo><mi>sin</mi><mo> </mo><mi>x</mi></math>, find <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><mfrac><mrow><mi>sin</mi><mo> </mo><mi>x</mi><mo> </mo><mi>cos</mi><mo> </mo><mi>x</mi></mrow><mrow><msup><mi>sin</mi><mn>2</mn></msup><mo> </mo><mi>x</mi><mo>-</mo><mi>sin</mi><mo> </mo><mi>x</mi><mo>-</mo><mn>2</mn></mrow></mfrac><mo>d</mo><mi>x</mi></math>.</p>
</div>
<br><hr><br><div class="specification">
<p>A right circular cone of radius <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r">
<mi>r</mi>
</math></span> is inscribed in a sphere with centre O and radius <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="R">
<mi>R</mi>
</math></span> as shown in the following diagram. The perpendicular height of the cone is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="h">
<mi>h</mi>
</math></span>, X denotes the centre of its base and B a point where the cone touches the sphere.</p>
<p style="text-align: center;"><img 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"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the volume of the cone may be expressed by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="V = \frac{\pi }{3}\left( {2R{h^2} - {h^3}} \right)">
<mi>V</mi>
<mo>=</mo>
<mfrac>
<mi>π</mi>
<mn>3</mn>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mi>R</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>
</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>Given that there is one inscribed cone having a maximum volume, show that the volume of this cone is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{32\pi {R^3}}}{{81}}">
<mfrac>
<mrow>
<mn>32</mn>
<mi>π</mi>
<mrow>
<msup>
<mi>R</mi>
<mn>3</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mn>81</mn>
</mrow>
</mfrac>
</math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the functions <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f,\,\,g,">
<mi>f</mi>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>g</mi>
<mo>,</mo>
</math></span> defined for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x \in \mathbb{R}">
<mi>x</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
</math></span>, given by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = {{\text{e}}^{ - x}}\,{\text{sin}}\,x">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−<!-- − --></mo>
<mi>x</mi>
</mrow>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</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="g\left( x \right) = {{\text{e}}^{ - x}}\,{\text{cos}}\,x">
<mi>g</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−<!-- − --></mo>
<mi>x</mi>
</mrow>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</math></span>.</p>
</div>
<div class="question">
<p>Hence, or otherwise, find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int\limits_0^\pi {{{\text{e}}^{ - x}}\,{\text{sin}}\,x\,{\text{d}}x} ">
<munderover>
<mo>∫</mo>
<mn>0</mn>
<mi>π</mi>
</munderover>
<mrow>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mi>x</mi>
</mrow>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</math></span>.</p>
</div>
<br><hr><br>