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<h2>SL Paper 1</h2><div class="specification">
<p>Mia baked a very large apple pie that she cuts into slices to share with her friends. The smallest slice is cut first. The volume of each successive slice of pie forms a geometric sequence.</p>
<p>The second smallest slice has a volume of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>30</mn><mo> </mo><msup><mtext>cm</mtext><mn>3</mn></msup></math>. The fifth smallest slice has a volume of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>240</mn><mo> </mo><msup><mtext>cm</mtext><mn>3</mn></msup></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the common ratio of the sequence.</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 the volume of the smallest slice of pie.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The apple pie has a volume of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>61</mn><mo> </mo><mn>425</mn><mo> </mo><msup><mtext>cm</mtext><mn>3</mn></msup></math>.</p>
<p>Find the total number of slices Mia can cut from this pie.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p>Solve the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo> </mo><mi>ln</mi><mo> </mo><mi>x</mi><mo>=</mo><mo> </mo><mi>ln</mi><mo> </mo><mn>9</mn><mo>+</mo><mn>4</mn></math>. Give your answer in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mi>p</mi><msup><mi mathvariant="normal">e</mi><mi>q</mi></msup></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>,</mo><mo> </mo><mi>q</mi><mo>∈</mo><msup><mi mathvariant="normal">ℤ</mi><mo>+</mo></msup></math>.</p>
</div>
<br><hr><br><div class="specification">
<p>Consider the series <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mi>x</mi><mo>+</mo><mi>p</mi><mo> </mo><mi>ln</mi><mo> </mo><mi>x</mi><mo>+</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>ln</mi><mo> </mo><mi>x</mi><mo>+</mo><mo>…</mo></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>1</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi><mo>,</mo><mo> </mo><mi>p</mi><mo>≠</mo><mn>0</mn></math>.</p>
</div>
<div class="specification">
<p>Consider the case where the series is geometric.</p>
</div>
<div class="specification">
<p>Now consider the case where the series is arithmetic with common difference <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mo>±</mo><mfrac><mn>1</mn><msqrt><mn>3</mn></msqrt></mfrac></math>.</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>Given that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>></mo><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mo>∞</mo></msub><mo>=</mo><mn>3</mn><mo>+</mo><msqrt><mn>3</mn></msqrt></math>, find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>.</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>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math> in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo> </mo><mi>ln</mi><mo> </mo><mi>x</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>∈</mo><mi mathvariant="normal">ℚ</mi></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>The sum of the first <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> terms of the series is <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>3</mn><mo> </mo><mi>ln</mi><mo> </mo><mi>x</mi></math>.</p>
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.iii.</div>
</div>
<br><hr><br><div class="specification">
<p>The <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math><sup>th</sup> term of an arithmetic sequence is given by <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mi>n</mi></msub><mo>=</mo><mn>15</mn><mo>-</mo><mn>3</mn><mi>n</mi></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State the value of the first term, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>1</mn></msub></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>Given that the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math><sup>th</sup> term of this sequence is <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>33</mn></math>, find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the common difference, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>In an arithmetic sequence, the first term is 3 and the second term is 7.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the common difference.</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 the tenth term.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the sum of the first ten terms of the sequence.</p>
<div class="marks">[2]</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="\left( {\begin{array}{*{20}{c}} {11} \\ a \end{array}} \right) = \frac{{11{\text{!}}}}{{a{\text{!}}\,9{\text{!}}}}">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>11</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>a</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mn>11</mn>
<mrow>
<mtext>!</mtext>
</mrow>
</mrow>
<mrow>
<mi>a</mi>
<mrow>
<mtext>!</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>9</mn>
<mrow>
<mtext>!</mtext>
</mrow>
</mrow>
</mfrac>
</math></span>.</p>
</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="a"> <mi>a</mi> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence or otherwise find the coefficient of the term in <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^9}"> <mrow> <msup> <mi>x</mi> <mn>9</mn> </msup> </mrow> </math></span> in the expansion of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {x + 3} \right)^{11}}"> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>+</mo> <mn>3</mn> </mrow> <mo>)</mo> </mrow> <mrow> <mn>11</mn> </mrow> </msup> </mrow> </math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Juan buys a bicycle in a sale. He gets a discount of 30% off the original price and pays 560 US dollars (USD).</p>
</div>
<div class="specification">
<p>To buy the bicycle, Juan takes a loan of 560 USD for 6 months at a nominal annual interest rate of 75%, <strong>compounded monthly</strong>. Juan believes that the total amount he will pay will be less than the original price of the bicycle.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate the original price of the bicycle.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate the difference between the original price of the bicycle and the total amount Juan will pay.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Last year a South American candy factory sold 4.8 × 10<sup>8</sup> spherical sweets. Each sweet has a diameter of 2.5 cm.</p>
<p>The factory is producing an advertisement showing all of these sweets placed in a straight line.</p>
<p style="text-align: center;"><img 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pykJBO/p4sHqmhHD8OIin5Ms8fe5QnpXlbCC2bxWm0gUmp7BLmbGx+Tsbs7mVOU3Pn8sPZh63dRxd1p3qoV0XSttOZy1PXef65rDRVLbTzXO2cQr+BjJOK347FDnOc7+t05o3344XntjgjgnSZta1GWllVao30tXu91nl784o+R8I5w+Fwzvtfgbbqtff7hf2itUQy7rTVvXaxR1hqt5uuoGwbS7mjfNGzUiuLy1ZAtwH9FGcjfNvF6H/+UZOI3H2kVRevj8DXADDKQjv3+PZeBBhrs/pXWXuZw8oQJEzw1ojLHzKL8+/tE1Zl53yiamhV1yfzPzUbauXE7lf+6grZV1VGLjUiBeUHBuDp58mQ9zM+eCUJeDz/8sLfEZz5I+fkjoiMrmUNp1NTs+HSkGb94ESam05AQqfH+s9h30Zj8mXR/lOTpQ/eNGkvxh9g1qt/5Lu0vf5XKtlVRbcvN+DwO+BN4ZfbEiRN6Kz777DNf1mckLxM76Wb9Htq4fwv9umwrVdW2xI/sBJxfM2fO1DfLCkIzooagTASfOnVK7+jJfI/clXb0m0l/nXN3jKK/pZb3f0PlCG/VldNzi+ZRzlO/pCpLAWWY98xeRq/X8zz1TWqp/Vd6bdl8WlV7kdrr3+yez55Pa2svUmdnOx2vXEXzMH+W/RxtO+2PEGRlgjCXX8aaNUOG0GN/PYb6Wz6QKL+IKGrecxIte/1I9zw1eFlH77+2jMauqqUb7UfodX0+O5vmrf0TtXd+T+3Hd9CqedmUkTGJlm075ZsQhDL54osvdFTwypjnxpolP4j6PZZLOf2tRVJny7/Rq+WYUttP5c8tokdzFtOqqtMxsIxeC5K97E2q57U7HS1U+/5rtGzsWqq9cZHqX+9e4zPvf1Ft+/fU2V5Plavm62sSspdV0mmfDHlRmcBYwzfkpUnxZGJnC73/6jvUTu1UV/4zWvRoPj21qiqGUxSDXxQMmThx4kR9Ya3ZWxrS8I0JsRsi8Po5X0LIpo28o31ds14rqvhI+8b0vvHid1pbw++1raVFGkJ2lL9Ra/jGGMrsCl/m5/9Kq2nDFMN3WuvuF7QsWqxtbf6L9nXNGi0LeWmitrRit9agP4MQ9RMaZS3VKnZ9qDWjzDutWs2aIo2KtmrNxiqMZLn0H2F/fPDHnxCyRaO+rtFKi8xwN3veDr80TUP4Mv8JbU1Nqz7FcKd1t/ZsVpZWtLVJu4P6ssAv0rKWVmi7G9q0O1o3j3Ue/k6racYExHdaW82vtHydz7fNiHH9GngFnvkVQjZv4GWtpnSpVtFw3fy24eqdtgZtz9ZSLV8fI09Y5MN4WaDlr9mvtWFs3Dmn7X52YvdYQX3Tu8YnxtPuhq5ndB5n6TzcVdOsj/c7bfu1NfljuvhsoMOLv/xxJv5QFsaZ/ylBmXinTWvY87ZWmp+lEWVp+aayNBa/UF9wZCLkAKbYZE/WZjZbBz4d4ZV4u3DMoqEdDfR25VDa8PPcGF6kmLcPZU1/kp599ffUVvMryq97l3bWG3dWQ/iyhvLWv0AFWZhi6ENZjyyhkqX/la598z3dXbCBWpu3UhH9kO6b9t9puv7M3fRX0yZTVvttGjx5Jo2Dh5Q5hIaPvYfo08+p1SfPBFuMYtEfPH/sTOZ/ukHH3/43Grbhf9L0GF5kD512+NUVvvxN3ku0uuB+fYohM+sR+qeSx4iu3aRbdxfQq61NtLUoi+i+aVQ4PYsyKZP6/9UUysu6Sl8NnkL54xA16kP3Dh9BfegMNbfGXdrWQ6KDZ6NHj9anar788ku9VP9f8eukjuPvUeWwF+jn0wfaamlm1nQqfvZVqmnbT2vyG2njzkbqFfvClM5vJtD61Y9SFqRcZhY98k8/p6X0NX1zazAVvHqYmrdi4dz9NK3w4a5n+o+maXnZ1P7VYJqcP04f75n3jqCxfW7Rp81tMSIMtshO6qERI0bo+T799FP97ZpRo0YlVY6jmRKViZlZNL14Bb1ac4xq1kyiuo17qP6mYfF0TH5RoGQi3sY4c+aMo5C7UZiUyh9zyHiNzP+OfoOOb95Hg9f8o01FIrKoD2UVvEDrS4l2VH8SLZxufkLVO74SH6bMrFn0i8p36WWbAjAqs49/YKBh57jz588TQpT+JiiS39HOwcttK5IeemPwi65RQ/V+utXzsK5M8n6xlapfzrNpFIqZ/TsfOXKkXjnmjzEF4HvqaKDNOwfRGtvGdQ/FmVmP0ur1/0i04wA1RCmTTrrZcIB2RDGsD2XlvUCV1b9IYiz31On1Gd4fx1s1vM+//w5RCjIx834qWL2KSmk/VTeIDlF4+IX+wQ6R130l0fqkVP74QAISW72JNsqZ5/EK2e/o1GO/oGfHx5rrj1Vbfxqek0OTcrJNFEQb1Tee8cWbiEVxovdgoMFQw5a+8Cr9TPorZKdm0/pnJ5rgbYeyWPwiaq//mP7TpwiLHertPMNraMAzNgTs5HPlGf0VsrP02Pq/p/G2ojRGKjKp//CxNGnSWBpulr/9E2r8z14xAWMh0v+Hwv/44491I8BfYh2Qif2zKWdSDuUMN1mNExJ+ZWdnR9ZA+cuv2LVLqfwvX76sU+3flr7fU3vtNto54Cn6aUTx36TTlTuoLsrDiA0uUQe1fj6JVv3NuOjV6PoAIKorWU+/jlqwdINO1H4cHSWIV4XP90UDDV/78yt1ttfSv+y8i/7up6z4O6nj9Pu0re5KAiRZ8ItgFIwmqqugl3+9R1iw1EkdJw7R8YT6RALkuPQowpJ//vOfCULKt9R5kWr/ZR8N+Lv/0aP4O05R5bYjCfT/Tupo/TNNX/U0jYuSZN1GAf2RSl4uj15023GKao+LXqdvCNiuGBG1q1ev+jwN6pBM7Gijz6c/R38zTnzPP1z84kV/thns04NRQ8YnGnpVe+nSJR9DkjeppepX9Myj/0wvPTqcfhDZqe8emvDe/6Psbg+j82IVPZf9JL3GG8lgBf7ef+/ZCbCznepf30h1hT+l/LsNMGeOo59sKKF8rFpeNIEGROr4B6q55wFKNs7QC0gPLogGmj8hSay0r6I1z/wDvfTSo5T9g+5d3zJ+QAMm7CTK7vYwOr+kqudyad5r3RvJJMIvuovG/eRlqsjHixyLKGfAD7p3phtOT9X8kHKM/PUA91SqGDRokL4i2bdITcdpqlrzLD360j/To9k/7MYygzIGFNF7NKQramPkF+GNiX+nvZGdG7+n9vq3qaxuOq3MH9oLjsxxC2lDxRNEdf+bFuXc01PHU3+ie3LsrS3oVaiPF65fv05QKv4kOzLxe7pY9SJlz3s9shlaZ/tx2rv3eGTnxk68BVPWSIUrZ/eScWHjF/gk/Yp/GVckYvMYbJYQK2HVMt4IcDbxqvuuldv6an19RTH+d6/s7q6wa7W3sNKYV95HVunzCm8rCrHKfHv3Cli8FVCqbdVXGHdveBJVb6PWhE1QItew4dD/7VmxrF+3swmRFS3W14uLi+NutLR4cRdt1qVoGlYtoyynUxcfzPhF0W9B6Ku9p/esNE6YX5p2p+0j7R1+i4OKtNKtNfpbF12bQAk0FG3VmpqwMVTPtazS/dr5yIrlruvxNyFKHC2MCYwNrA63SriPvgSeWCVsBIQxGOsZq7wxr/OqewGbnn6Nt12634Iw8ivypkQ3dksrNF6Rb1kfVpm/w28FZGn5pVu73rro3gQqqt6mRn1jqMg1bDh0fn/kLQ79etxNiCwpsbwBXmBTGN54yexB8CAvL0/nq9l9XAO/ITdRnrPJrkzsfk54u6nrTQms8Kfot5SsCLTilyaPTAQv7IwLtNnxsWOFW5LXKcl8rmYDuOjIZgmAQokA3HgGgll+dS0xBPSB24211c5i4Ie4059YAysRLke8p86dRwBjAljHMtrefvtt/RkzhcNKhPkluwBzHkFvS2RDDDst4tzMaAMPHnzwQUsjnF+zBc+cV/7e4iF7beAFj41YRhuekX3sGOLR/gSVzGrt3z96QQhCKKWlpZSTk6OHLM3yqGvuIYAFfZjff+ONN6L2GucaEUoWE97YwD7kQ4YM0RcDivfUufsI4DO9s2fP1scM9vAX03fffaf/5cV/fO/IkSOEj5MsW7aML6mjRwhgAeaiRYtoyZIlBD4Y09mzZ2nYsGFRl/F67YoVK2j+/Pn6otuom+qP6wisXLmSfvzjHxM+jsW7nHKl+CIjr13ja7IdpVX+DJRSIoyEHEd0eCgIdHhORqGklAgj4//RzGgTF2mCQlYiMBaghFTyDwEzo824lgaO0IYNG3RHCB/+Uck/BESjDZ9b5oQ9T7B2TeYktfJXSkTOrsMdHl9chOJgr18pETn5BapEo40XaSolIi+/zIy2b7/9Vje6EU3DdzRUkgcBGG3YKhvRaWOkTR4qDZTIOMeycuVKrbCwMDK3wnMs6tizgEwmLLAYac6cOYpfpovY5OPZI488ongVEF5hnGM9AI6TJk1SfAsI36C/ZF9/IaXnf9ddd9G8efMI8yaxEnYo0zRN/VzEIBb+uIfdx/B1P6zFwPvj8ZLil7v9Nd6ufRhTTz31lM6msrKyeOwifLRJ8cw9nu3evTsuD/jLiz/72c/iPovyFL/c4xfGQ7yEMTh27Nh4j/l+X0rlD1TGjx9PBw8eJDuDw3cU05AAKJHq6mrCNxgQ9sdv8+bNdPjw4bhGWxrCJUWTN23apI+p6dOn6/SsXbtWV+52jDYpGpBmRECJYLfTF198UW95UVGRvtmPHaMtzaCSorlwhPD55fLychowYIAUNMUiQlrlD6IxN7lw4UK9w8fzaGI1Ut1zFgFWIhBGSJjr5zRr1ixltDEYkhxFJYIxdfHixQhlWEymjLYIHFKciEoEH14SV5LjDQ1ltEnBpggRcITgpMIR4s+cY1Mm2ZPUyp/BQ4eHNYWQCwaGSv4gYFQiVlQoo80KGW+vG5UI1/6Xv/yFTyNHZbRFoPDthJXIb3/724gSATH8rRORMGW0iWj4dw5H6IMPPtCdVJEKvIXh346MIiXW51Iq/4EDB1JbW1svqtHhYV0h3IyBopI3CFgpEa79Rz/6EdXW1vLfyFEZbREoPD2xUiJMxNChXdvhihEb3FNGGyPk/VFUIuCDMeXm5tKpU6eMl4mNtsrKyl731AX3EIAjBGcUUzLG/TLcq9XZkv+Ls8U5Uxrm+/GNeKuEcPOcOXN08K2eUdedQQCGFrA2E0hcA741jp9VYqPNbPMSqzzqenIIwFBbvXp1TIF05Ursjx2x0Ya1AGwoJEeNyhUPAXiHUCIYI1YJm8VkZlr7aRibWBT45JNPUjzeWtWhrttDAOMB65pgdFklcZrG6hkZrkup/AFMvDkTdHieX5EByLDSwPP6sdrHih/vjceygmMNmFjlq3v2EbCD8fnz5/WQJJRKLKUT6559itSTsRCwgzE2i8nKyqLTp0/HKkofe7HGX8zM6qYtBIBvvDHG0zR2eGurUpcesjYnXarQTrGwhtXOVXaQ8v8ZcUML5XX4zw87FCDcP23aNOl3ILPTlnR4hqdAb9y4kQ7NDXwbZd/WlwGWUvnzd+FFxcIEq6NcCLDCxzzzuXPn5CJOUWOKAHYjw+d843mSppnVRc8RQKQmLy9PfSPDc+STq9DfT9Lbp1lK5Y/XW5BYsdhvjnrSawTOnDmjb+6Dvay/+uorr6tX9SWIAC/ye/DBB6m+vj7B3OpxPxDAVr8w1pAwtaaS3AhgvdrIkSPlJpKIpFT+QA2LjaBYVJIbAXiPU6ZMoZkzZ5quRpab+vSjDtEZRGnAL0QAgrI4Kf041dViVvb46BISogAqyY0ADGzZ5/uBoLTKHwol1op/udmfPtTBe0RHh2cCD0UluRGAYEKUhr/sx4uT5KY6falramrSG49oqHKI5O8HMNZgVI8aNUp6YqVV/vhCklImcvcfsaNzmIvDynJTnr7UHThwQPf68bYMXgs0e3c8fdGRr+XYLpZ3N8Urt8ohko9HIkVsrCnPX0QlwfMJEyboOZQySRA4Dx9HR0cIGWC0xNYAAA24SURBVB0dr8AoZeIh+ElUxcYa7zxWWFiolEkSOHqZhY011DlmzBjlEHkJfhJ1icZaEtk9zZKg5/8ttWz7CWVkZNP8baepM4rUK1S7KtfiHufLoOxVtXQzKp/5H6VMzHGR6So+vLRgwYIISVAm+/bti/xXJ3IhIBproAzzyIiuqXl/ufjE1BiNNbyeiXTy5El+RB0lQ0A01uyRxrrRfZ1qpCdB5W/M7u5/fNRHKRN3MU6ldGy1jG1HOUGZYH8GpUwYEbmORmMNn2FGamxslItQRY2OQENDQySyhguYqsG8P3aYU0k+BPBqOub78VpmEFKGho8/S5pg4WLu/+rVqzF3jpOU/FCTZcWbjIyMuNtfhhoYSRsHgwz7Zxi3Jt2wYQPdfffdkc/GSkp+WpK1YsUK/U0a/qQvQNi/fz/98pe/VK9pStgjtm/fTlVVVfr3ZyQkrxdJUnv+2L4Xc8qwgFWSCwEoEXghxu1E8a1xeJgqyYUAe/ccOmbq5s6dSytXrlTRGgZEkiNC/oii8St+TBYibceOHVOhfwZEoiMUP6LVQUlSK3+AiA9WvPXWW0HBMy3ohBcJK3fx4sW92vv444/TunXr1GYkvZDx9wLCkTDMEDoWExsDbByI99S5fwhwyN/4/RIY2yr07x9frGrmkH9BQYHVI9Jdl175Y0EZBJda9S9P34GigPeBV4+MiaM1dXV1xlvqv08IwIvEwj4YZsYEYwBGAcaYSvIgAIcHjo9ZgtEN41utrTFDx59rWP+Et514d1p/qEisVumVP8AEqJjrUkkOBN555x3C98eNXiRThzlKCCeV5EDgD3/4gz59ZvQimToYBTAOYCSo5D8CcHRgjIlv0ohUweiG8a2iNSIq/p1zJPT555/3j4gkapZe+aNN2OQC85JKOCXBYYezQDBhLjLWp37xXXEIryNHjjhcuyouUQQgmN544w1avXq1ZVYYBTCw33vvPctn1A3vEHj//fd1mWflRcLohvGtNkHzjiexajp06JB+2ywSGiuf3/ekXu3P4ECAYWESBFiQFlQw/WE6YnX42bNnafPmzTGbhefwCdLy8vKYz6mb7iIAAwyLxuK9MWP3OXepVaXDwRkyZEivtzKMyNh9zphP/XceAURosMeJ+FaG87U4X2IglD+ajZWUr7zyir6S3Crc7Dw8qkQRgUQEDiIEeI+8ubk5EB+5ENsZpnO7gkkZ2HJwHVEabBSDOeR4qbS0VH9EGdjxkHLvfpCN5sAofyWc3OvAdkuGN48P+dgRTChTCSe7yLrzXKKCiQ3sDz74oNcrnO5QqEoVEUjEuEY+NrCxpazVeg6xfHXuPAJ2jWvna3agRGzyE5S0e/dubcaMGdrVq1eDQnJo6GxubsZmUNqJEydst4nzHD582HYe9aAzCNy6dUsfK5s2bbJdYDJ5bBeuHoyLQElJiVZcXBz3OfGBZPKI+dV58ghUV1frMjGo+igwnj/sHHj/S5YsCeT8igN2mq9FJOvFJxot8LWRIaqcvXhsuJTINBlHC/CpX6sFZyGCSZqmJOvFc7Sguro65iJcaRoaEkI4Eo15fqtXMqVvavJ2jz854UXCA4VXqZI3CKSCOaxi8AtRG5W8QSBVzJcvX67Bo1TJOwTg8SeLeWVlpR7lQeRGJW8QCAPm5A1UztaCQQIBpZL7CHAoGJ092QTFDwMgqOGxZNvtV75UQ8E8XYOwpkruI5Dq+OAxWlZW5j6xqgbd8YQ8C/p0ZiCV/4ULF5Q36dEghEDBOotUvYpUPBuPmhqKangeMtXIGHs2ymBzt1s4JcuwFgcKKZE1Oe62LJylQw6GRZYFUvmjW7G1jMGjkjsIOClQlDfpDo/EUqGoYailEqXh8sIk5LhNMh4RwXQqiumUoS4jTrLQFCajOLDKH52Bw5upeqWydCyZ6GBFkshq8Xj0K4MtHkKp3WdF4tR4UAZbavyIlxuKBN66Uw4M+A7jT4X/4yGf3H12hsIyHRZo5c8KSnX25DqzVS4IEacVCdeFchE2c0pBcbnpfnRakTCeymBjJJw9uqVIlMHmLJ+4tDDqmkArfzDGrUHETE/HI7x9eBBOeSQihuydJLuyWSxLnXchwG9juLUACbxyYt2H4lcXAhhXwNPJqJqILRtsqa77EMtM53M3nSE/cQ288gd43NndEn5+MsjrunnBmJtYQvgh3OmW8PMaMz/r8wJLCD9EaxC1URGb1LjtlSLh+X83DPjUEAhebjedIT/RCIXyB4BhZZCXnYM9SBhTbieuy00jw+02+F0+e5BeRFG4LjXFljzXWfHD60cY2c3EdakpttRQhl4J61sUoVH+YDE8E7fC1al1IflzszL20htXEZvk+wWEO/q6l944zyd72UeSR0i+nF574+gjKmKTfD/wIgqaPHWp5wyV8mdrVxkAiXUM9ur8EOpsWasIgH2ecT/3UvEzdX4YiVx3kI/o537IJR7bfvSVIPOL+3mY5VKolD86GwtGPwZaEDs7d3I/FD/jpQwARiL+EcIcgtxPYS5Dn4mPlBxPQB75pfgZAWUAMBL2juzxezH9aY8id54KnfIHTMoAsNdZZBLiygCIzzOZhLjYdzDeVOqNgExySOw7bq836I1EcK6kkxwKpfJHV8PAw0IoLNaAoFIpGgGeb0dnlyXJSJMs2GC+3es5/nhtx2u2stEUj2av7kPBIjojUwRSRpq84ke8eqAv0knxA4/QKn9mNjPUiS1PucwgH8VOLuNOVexRwnADrSppGochsWBMNkzYo8TCMpyrpOkffmGjSDZM0H/YKFFOUVdvFY0iGLTpkkKv/MFIFp7prlAgiHjgy9zJlULpEj8Q1FD4iF7JPP+oFEqPugiKrFFOURfPOKKWjsZrWih/sJmZDIsc5+mWIJRk9UbMeAGFwtM2Mis+M9qduIY+CoEEnslsqIlt5S2GZYxQiHS6cR7E/grPP0gywWm+pXN/BZZpo/zRWNGTAuPxP+wJIS1WokFsM68DQBvQlrAn9Mkgt5nXAQTJaEm1T3GbYawFzbHgaKDs0aVUeSTm5zajj8IpSteUVsqfmcxecNgFFLcziEKJeYUjD1YWUGE12kRvP8hCSTSyEQUIq9EmGtYIowe5X7LBiWnBoBkwoqyIdQ7+sLcPZwJyJZ1TWip/MFwUUOgIYerwrESgLIPo7VsNSAgoGGwwZsK0WElUImGKcMAj5qkL8C7IylHsk2gHK8ugG9Ziu6AM0f8gN2DMhMlo4ymOdPf2RX6nrfJnEKAoYe2GocOjLTx4w6REmFc4QiDxYqWgeyliW8KkRER+saKE0MUvyEabsS0wAMKYwCP0R3YegmwEiG0JenTG6b6W9sqfARU7CUKVQYoEwMMSlX6QaGf8Ez2Khg4EVZCUCjwsNmCCRnuifOLnRUMHRkCQIgGgHfSyARMk2hn/RI+ioRM0xwi0i/I8bFGMRHlp9bxS/gZk0Gk4EoAj/qMzyZYgkDAvzBY6lH86KH0jH9BmUZFCMMvoqbBAYiMtXZS+kV/gDaaioEhZqcg69yr2raAZLEbck/2PfmuUM3A2ZExmfUtGWSALdkr5W3BCHPgQUogGoNP7aQiwAgEtoAkCSVm1XQw0DnwYbhBafg9+9BnwiJVd0KJKFsMj5ctGpQJjCEaB34YA6gcdbFSzA5Byg0NQABwhNl5Z9vhtCGB8Y5yzwwa+pUNkxonulIFCSCVLBG7fvk2NjY108OBBWrdunf5cWVkZ5ebm0kMPPUTDhw+3zOvEjZaWFjp16hTt27ePtmzZEql/7ty5NGvWLCeqCF0ZJ0+epMOHD9P27dvp2LFjtHz5cpozZw7NnDmTxo0b52p7W1tb6bPPPqOGhoZIf0H9ixcv1mno27evq/UHsXBgVltbS1VVVbR3714qLi6mwsJCmj17NuXk5JCbmF27do2amproxIkTdODAgUj9CxcupIKCAtfHdxD5BczQv3ft2qXLpBkzZtCCBQsIMmnChAk0ePBg15oFedzc3EyffPIJHTp0KFL/0qVL9f4ydepU1+oOW8FK+SfAUTYEIChYsSB7SUkJTZw4ke677z4aNWoUDR06NOEBgAF15coVOnfuHEHhnz9/nioqKnTqvBSGCcARiEdhCEBQsGIB0VDGU6ZMoezsbJ1v/fr1S1jIoy9cuHCBLl++TJcuXaKjR49SXV2dbmx4KQwDwYQEiIQhUF9fr+Mp9v+8vDwaP348jR49msCvZIw4jKtbt27RmTNn6PTp03o9MDaQMIZhcHhh0CcAh/SPsiEAY2DPnj2R/p+fn68b2/feey8NGzaMRowYkbARh74AfsH5aWtro48//jjiAHlp0EvPhCQJVMo/SeCQjb2GL774Qu+gLPzFIiFUYiUWcPwMFD0EG7xUCLqRI0cmbEhwWeoYjQB7DRD+UNZQBiz8+UkIlUGDBvHfXkcjj6HoIehg/I0ZM4YeeOCBhA2JXpWoCxEEOPIFZX327NmI8OcHeLzwf+PRiscw/jDOYKwnY0gY61H/uxDgyBdwFx0YxofHC/83Hq9fv27KYzb+MM6SMSSM9aj/REr5u9AL0PGR2CuMVQV7Msl4n7HKVffsI8AeBnuFsXKyJ4NnlNKIhZR79zhKhhrgFcZK/fv31xU8nlFKIxZS7t3jKBlqQGSzo6MjZmVQ8EjJRFBjFqxuRiGglH8UHOqPQkAhoBBQCCgEwo/A/wcLnlDKUwi3uAAAAABJRU5ErkJggg=="></p>
</div>
<div class="specification">
<p>The advertisement claims that the length of this line is <em>x</em> times the length of the Amazon River. The length of the Amazon River is 6400 km.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the length, in cm, of this line. Give your answer in the form <em>a</em> × 10<sup><em>k</em></sup> , where 1 ≤ <em>a</em> < 10 and k ∈ <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mathbb{Z}">
<mrow>
<mi mathvariant="double-struck">Z</mi>
</mrow>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the length of the Amazon River in cm.</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>Find the value of <em>x</em>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>A hydraulic hammer drives a metal post vertically into the ground by striking the top of the post. The distance that the post is driven into the ground, by the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n{\text{th}}">
<mi>n</mi>
<mrow>
<mtext>th</mtext>
</mrow>
</math></span> strike of the hammer, is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{d_n}">
<mrow>
<msub>
<mi>d</mi>
<mi>n</mi>
</msub>
</mrow>
</math></span>.</p>
<p>The distances <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{d_1},{\text{ }}{d_2},{\text{ }}{d_3}{\text{ }} \ldots ,{\text{ }}{d_n}">
<mrow>
<msub>
<mi>d</mi>
<mn>1</mn>
</msub>
</mrow>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<msub>
<mi>d</mi>
<mn>2</mn>
</msub>
</mrow>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<msub>
<mi>d</mi>
<mn>3</mn>
</msub>
</mrow>
<mrow>
<mtext> </mtext>
</mrow>
<mo>…<!-- … --></mo>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<msub>
<mi>d</mi>
<mi>n</mi>
</msub>
</mrow>
</math></span> form a geometric sequence.</p>
<p>The distance that the post is driven into the ground by the first strike of the hammer, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{d_1}">
<mrow>
<msub>
<mi>d</mi>
<mn>1</mn>
</msub>
</mrow>
</math></span>, is 64 cm.</p>
<p>The distance that the post is driven into the ground by the second strike of the hammer, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{d_2}">
<mrow>
<msub>
<mi>d</mi>
<mn>2</mn>
</msub>
</mrow>
</math></span>, is 48 cm.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of the common ratio for this sequence.</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 the distance that the post is driven into the ground by the eighth strike of the hammer.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the <strong>total depth </strong>that the post has been driven into the ground after 10 strikes of the hammer.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><msup><mi>a</mi><mi>x</mi></msup></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>,</mo><mo> </mo><mi>a</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>></mo><mn>0</mn><mo>,</mo><mo> </mo><mi>a</mi><mo>></mo><mn>1</mn></math>.</p>
<p>The graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> contains the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mfrac><mn>2</mn><mn>3</mn></mfrac><mo>,</mo><mo> </mo><mn>4</mn></mrow></mfenced></math>.</p>
</div>
<div class="specification">
<p>Consider the arithmetic sequence <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>log</mi><mn>8</mn></msub><mo> </mo><mn>27</mn><mo> </mo><mo>,</mo><mo> </mo><msub><mi>log</mi><mn>8</mn></msub><mo> </mo><mi>p</mi><mo> </mo><mo>,</mo><mo> </mo><msub><mi>log</mi><mn>8</mn></msub><mo> </mo><mi>q</mi><mo> </mo><mo>,</mo><mo> </mo><msub><mi>log</mi><mn>8</mn></msub><mo> </mo><mn>125</mn><mo> </mo><mo>,</mo></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>></mo><mn>1</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi><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>a</mi><mo>=</mo><mn>8</mn></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down an expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>f</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mi>x</mi></mfenced></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>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>f</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><msqrt><mn>32</mn></msqrt></mfenced></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>27</mn><mo>,</mo><mo> </mo><mi>p</mi><mo>,</mo><mo> </mo><mi>q</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>125</mn></math> are four consecutive terms in a geometric sequence.</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math> and the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">d.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>Iron in the asteroid <em>16 Psyche</em> is said to be valued at <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8973</mn></math> quadrillion euros <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtext>EUR</mtext></mfenced></math>, where one quadrillion <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mn>10</mn><mn>15</mn></msup></math>.</p>
</div>
<div class="specification">
<p>James believes the asteroid is approximately spherical with radius <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>113</mn><mo> </mo><mtext>km</mtext></math>. He uses this information to estimate its volume.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the value of the iron in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>×</mo><msup><mn>10</mn><mi>k</mi></msup></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>≤</mo><mi>a</mi><mo><</mo><mn>10</mn><mo> </mo><mo>,</mo><mo> </mo><mi>k</mi><mo>∈</mo><mi mathvariant="normal">ℤ</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate James’s estimate of its volume, in <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>km</mtext><mn>3</mn></msup></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The actual volume of the asteroid is found to be <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mo>.</mo><mn>074</mn><mo>×</mo><msup><mn>10</mn><mn>6</mn></msup><mo> </mo><msup><mtext>km</mtext><mn>3</mn></msup></math>.</p>
<p>Find the percentage error in James’s estimate of the volume.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</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( {2n - 1} \right)^2} + {\left( {2n + 1} \right)^2} = 8{n^2} + 2">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mi>n</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>8</mn>
<mrow>
<msup>
<mi>n</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>2</mn>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n \in \mathbb{Z}">
<mi>n</mi>
<mo>∈</mo>
<mrow>
<mi mathvariant="double-struck">Z</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, or otherwise, prove that the sum of the squares of any two consecutive odd integers is even.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>4</mn><mo> </mo><mi>cos</mi><mo> </mo><mi>x</mi><mfenced><mrow><mn>1</mn><mo>-</mo><mn>3</mn><mo> </mo><mi>cos</mi><mo> </mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>3</mn><mo> </mo><msup><mi>cos</mi><mn>2</mn></msup><mo> </mo><mn>2</mn><mi>x</mi><mo>-</mo><msup><mi>cos</mi><mn>3</mn></msup><mo> </mo><mn>2</mn><mi>x</mi></mrow></mfenced></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Expand and simplify <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow><mo>(</mo><mn>1</mn><mo>-</mo><mi>a</mi><mo>)</mo></mrow><mn>3</mn></msup></math> in ascending powers of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math>.</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>By using a suitable substitution for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math>, show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>-</mo><mn>3</mn><mo> </mo><mi>cos</mi><mo> </mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>3</mn><mo> </mo><msup><mi>cos</mi><mn>2</mn></msup><mo> </mo><mn>2</mn><mi>x</mi><mo>-</mo><mo> </mo><msup><mi>cos</mi><mn>3</mn></msup><mo> </mo><mn>2</mn><mi>x</mi><mo>=</mo><mn>8</mn><mo> </mo><msup><mi>sin</mi><mn>6</mn></msup><mo> </mo><mi>x</mi></math>.</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>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mn>0</mn><mi>m</mi></msubsup><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>d</mo><mi>x</mi><mo>=</mo><mfrac><mn>32</mn><mn>7</mn></mfrac><msup><mi>sin</mi><mn>7</mn></msup><mo> </mo><mi>m</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math> is a positive real constant.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>It is given that <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mi>m</mi><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></msubsup><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>d</mo><mi>x</mi><mo>=</mo><mfrac><mn>127</mn><mn>28</mn></mfrac></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>≤</mo><mi>m</mi><mo>≤</mo><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></math>. Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the binomial expansion <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow><mo>(</mo><mi>x</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mn>7</mn></msup><mo>=</mo><msup><mi>x</mi><mrow><mn>7</mn></mrow></msup><mo>+</mo><mi>a</mi><msup><mi>x</mi><mn>6</mn></msup><mo>+</mo><mi>b</mi><msup><mi>x</mi><mn>5</mn></msup><mo>+</mo><mn>35</mn><msup><mi>x</mi><mn>4</mn></msup><mo>+</mo><mo>…</mo><mo>+</mo><mn>1</mn></math> where <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>a</mi><mo>,</mo><mo> </mo><mi>b</mi><mo>∈</mo><msup><mi mathvariant="normal">ℤ</mi><mo>+</mo></msup></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>=</mo><mn>21</mn></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The third term in the expansion is the mean of the second term and the fourth term in the expansion.</p>
<p>Find the possible values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The first three terms of a geometric sequence are <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_1} = 486,{\text{ }}{u_2} = 162,{\text{ }}{u_3} = 54">
<mrow>
<msub>
<mi>u</mi>
<mn>1</mn>
</msub>
</mrow>
<mo>=</mo>
<mn>486</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<msub>
<mi>u</mi>
<mn>2</mn>
</msub>
</mrow>
<mo>=</mo>
<mn>162</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<msub>
<mi>u</mi>
<mn>3</mn>
</msub>
</mrow>
<mo>=</mo>
<mn>54</mn>
</math></span>.</p>
</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="r">
<mi>r</mi>
</math></span>, the common ratio of the sequence.</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 the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span> for which <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_n} = 2">
<mrow>
<msub>
<mi>u</mi>
<mi>n</mi>
</msub>
</mrow>
<mo>=</mo>
<mn>2</mn>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the sum of the first 30 terms of the sequence.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>x</mi><mo>-</mo><mn>3</mn><mo>-</mo><mfrac><mn>6</mn><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>5</mn><mi>x</mi><mo>-</mo><mn>3</mn></mrow><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfrac><mo>,</mo><mo> </mo><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi><mo>,</mo><mo> </mo><mi>x</mi><mo>≠</mo><mn>1</mn></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence or otherwise, solve the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo> </mo><mi>sin</mi><mo> </mo><mn>2</mn><mi>θ</mi><mo>-</mo><mn>3</mn><mo>-</mo><mfrac><mn>6</mn><mrow><mi>sin</mi><mo> </mo><mn>2</mn><mi>θ</mi><mo>-</mo><mn>1</mn></mrow></mfrac><mo>=</mo><mn>0</mn></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>≤</mo><mi>θ</mi><mo>≤</mo><mi mathvariant="normal">π</mi><mo>,</mo><mo> </mo><mi>θ</mi><mo>≠</mo><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The volume of a hemisphere, <em>V</em>, is given by the formula</p>
<p style="text-align: center;"><em>V</em> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sqrt {\frac{{4\,{S^3}}}{{243\,\pi }}} ">
<msqrt>
<mfrac>
<mrow>
<mn>4</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<msup>
<mi>S</mi>
<mn>3</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mn>243</mn>
<mspace width="thinmathspace"></mspace>
<mi>π<!-- π --></mi>
</mrow>
</mfrac>
</msqrt>
</math></span>,</p>
<p>where <em>S</em> is the total surface area.</p>
<p>The total surface area of a given hemisphere is 350 cm<sup>2</sup>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate the volume of this hemisphere in cm<sup>3</sup>.</p>
<p>Give your answer correct to <strong>one decimal place</strong>.</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>Write down your answer to part (a) correct to the nearest integer.</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>Write down your answer to <strong>part (b)</strong> in the form <em>a</em> × 10<sup><em>k</em></sup> , where 1 ≤ <em>a</em> < 10 and <em>k </em>∈<em> </em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mathbb{Z}">
<mrow>
<mi mathvariant="double-struck">Z</mi>
</mrow>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>In an arithmetic sequence, the first term is 8 and the second term is 5.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the common difference.</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 the tenth term.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Sergei is training to be a weightlifter. Each day he trains at the local gym by lifting a metal bar that has heavy weights attached. He carries out successive lifts. After each lift, the same amount of weight is <strong>added</strong> to the bar to increase the weight to be lifted.</p>
<p>The weights of each of Sergei’s lifts form an arithmetic sequence.</p>
<p>Sergei’s friend, Yuri, records the weight of each lift. Unfortunately, last Monday, Yuri misplaced all but two of the recordings of Sergei’s lifts.</p>
<p>On that day, Sergei lifted 21 kg on the third lift and 46 kg on the eighth lift.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>For that day find how much weight was added after each lift.</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>For that day find the weight of Sergei’s first lift.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>On that day, Sergei made 12 successive lifts. Find the total combined weight of these lifts.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider any three consecutive integers, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>-</mo><mn>1</mn></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>+</mo><mn>1</mn></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Prove that the sum of these three integers is always divisible by <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Prove that the sum of the squares of these three integers is never divisible by <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>In the Canadian city of Ottawa:</p>
<p><span class="mjpage mjpage__block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" alttext="\begin{array}{*{20}{l}} {{\text{97% of the population speak English,}}} \\ {{\text{38% of the population speak French,}}} \\ {{\text{36% of the population speak both English and French.}}} \end{array}">
<mtable columnalign="left" rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mrow>
<mtext>97% of the population speak English,</mtext>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mrow>
<mtext>38% of the population speak French,</mtext>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mrow>
<mtext>36% of the population speak both English and French.</mtext>
</mrow>
</mrow>
</mtd>
</mtr>
</mtable>
</math></span></p>
</div>
<div class="specification">
<p>The total population of Ottawa is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="985\,000">
<mn>985</mn>
<mspace width="thinmathspace"></mspace>
<mn>000</mn>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate the percentage of the population of Ottawa that speak English but not French.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate the number of people in Ottawa that speak both English and French.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down your answer to part (b) in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a \times {10^k}">
<mi>a</mi>
<mo>×</mo>
<mrow>
<msup>
<mn>10</mn>
<mi>k</mi>
</msup>
</mrow>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1 \leqslant a < 10">
<mn>1</mn>
<mo>⩽</mo>
<mi>a</mi>
<mo><</mo>
<mn>10</mn>
</math></span> and <em>k </em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \in \mathbb{Z}">
<mo>∈</mo>
<mrow>
<mi mathvariant="double-struck">Z</mi>
</mrow>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p = \frac{{\cos x + \sin y}}{{\sqrt {{w^2} - z} }}">
<mi>p</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mi>cos</mi>
<mo><!-- --></mo>
<mi>x</mi>
<mo>+</mo>
<mi>sin</mi>
<mo><!-- --></mo>
<mi>y</mi>
</mrow>
<mrow>
<msqrt>
<mrow>
<msup>
<mi>w</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−<!-- − --></mo>
<mi>z</mi>
</msqrt>
</mrow>
</mfrac>
</math></span>,</p>
<p>where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 36^\circ ,{\text{ }}y = 18^\circ ,{\text{ }}w = 29">
<mi>x</mi>
<mo>=</mo>
<msup>
<mn>36</mn>
<mo>∘<!-- ∘ --></mo>
</msup>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>y</mi>
<mo>=</mo>
<msup>
<mn>18</mn>
<mo>∘<!-- ∘ --></mo>
</msup>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>w</mi>
<mo>=</mo>
<mn>29</mn>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z = 21.8">
<mi>z</mi>
<mo>=</mo>
<mn>21.8</mn>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
<mi>p</mi>
</math></span>. Write down your full calculator display.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write your answer to part (a)</p>
<p>(i) correct to two decimal places;</p>
<p>(ii) correct to three significant figures.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write your answer to <strong>part (b)(ii) </strong>in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a \times {10^k}">
<mi>a</mi>
<mo>×</mo>
<mrow>
<msup>
<mn>10</mn>
<mi>k</mi>
</msup>
</mrow>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1 \leqslant a < 10,{\text{ }}k \in \mathbb{Z}">
<mn>1</mn>
<mo>⩽</mo>
<mi>a</mi>
<mo><</mo>
<mn>10</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>k</mi>
<mo>∈</mo>
<mrow>
<mi mathvariant="double-struck">Z</mi>
</mrow>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the functions <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mn>1</mn><mrow><mi>x</mi><mo>-</mo><mn>4</mn></mrow></mfrac><mo>+</mo><mn>1</mn></math>, for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>≠</mo><mn>4</mn></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>x</mi><mo>-</mo><mn>3</mn></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
<p>The following diagram shows the graphs of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</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>The graphs of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math> intersect at points <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math>. The coordinates of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> are <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>3</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math>.</p>
</div>
<div class="specification">
<p>In the following diagram, the shaded region is enclosed by the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math>, the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math>, the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis, and the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mi>k</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>∈</mo><mi mathvariant="normal">ℤ</mi></math>.</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" 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"></p>
<p>The area of the shaded region can be written as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo>(</mo><mi>p</mi><mo>)</mo><mo>+</mo><mn>8</mn></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>∈</mo><mi mathvariant="normal">ℤ</mi></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the coordinates of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> and the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math>.</p>
<div class="marks">[10]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The expression <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>3</mn><msqrt><mi>x</mi></msqrt><mo>-</mo><mn>5</mn></mrow><msqrt><mi>x</mi></msqrt></mfrac></math> can be written as <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>-</mo><mn>5</mn><msup><mi>x</mi><mi>p</mi></msup></math>. Write down the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, 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 class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p><strong>In this question, give all answers to two decimal places.</strong></p>
<p>Karl invests 1000 US dollars (USD) in an account that pays a nominal annual interest of 3.5%, <strong>compounded quarterly</strong>. He leaves the money in the account for 5 years.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate the amount of money he has in the account after 5 years.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the amount of <strong>interest</strong> he earned after 5 years.</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>Karl decides to donate this <strong>interest</strong> to a charity in France. The charity receives 170 euros (EUR). The exchange rate is 1 USD = <em>t</em> EUR.</p>
<p>Calculate the value of <em>t</em>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The first two terms of an infinite geometric sequence are <em>u</em><sub>1</sub> = 18 and <em>u</em><sub>2</sub> = 12sin<sup>2</sup> <em>θ</em> , where 0 < <em>θ</em> < 2<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\pi ">
<mi>π<!-- π --></mi>
</math></span> , and <em>θ</em> ≠ <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\pi ">
<mi>π<!-- π --></mi>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an expression for <em>r</em> in terms of <em>θ</em>.</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>Find the values of <em>θ</em> which give the greatest value of the sum.</p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>The company Snakezen’s Ladders makes ladders of different lengths. All the ladders that the company makes have the same design such that:</p>
<p style="padding-left: 90px;">the first rung is 30 cm from the base of the ladder,</p>
<p style="padding-left: 90px;">the second rung is 57 cm from the base of the ladder,</p>
<p style="padding-left: 90px;">the distance between the first and second rung is equal to the distance between all adjacent rungs on the ladder.</p>
<p>The ladder in the diagram was made by this company and has eleven equally spaced rungs.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-15_om_10.59.54.png" alt="M17/5/MATSD/SP1/ENG/TZ1/05"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the distance from the base of this ladder to the top rung.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The company also makes a ladder that is 1050 cm long.</p>
<p>Find the maximum number of rungs in this 1050 cm long ladder.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the geometric sequence <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_1} = 18,{\text{ }}{u_2} = 9,{\text{ }}{u_3} = 4.5,{\text{ }} \ldots ">
<mrow>
<msub>
<mi>u</mi>
<mn>1</mn>
</msub>
</mrow>
<mo>=</mo>
<mn>18</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<msub>
<mi>u</mi>
<mn>2</mn>
</msub>
</mrow>
<mo>=</mo>
<mn>9</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<msub>
<mi>u</mi>
<mn>3</mn>
</msub>
</mrow>
<mo>=</mo>
<mn>4.5</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mo>…<!-- … --></mo>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the common ratio of the sequence.</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="{u_5}">
<mrow>
<msub>
<mi>u</mi>
<mn>5</mn>
</msub>
</mrow>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the smallest value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span> for which <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_n}">
<mrow>
<msub>
<mi>u</mi>
<mi>n</mi>
</msub>
</mrow>
</math></span> is less than <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{10^{ - 3}}">
<mrow>
<msup>
<mn>10</mn>
<mrow>
<mo>−</mo>
<mn>3</mn>
</mrow>
</msup>
</mrow>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g\left( x \right) = {p^x} + q">
<mi>g</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mi>p</mi>
<mi>x</mi>
</msup>
</mrow>
<mo>+</mo>
<mi>q</mi>
</math></span>, for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x{\text{, }}p{\text{, }}q \in \mathbb{R}{\text{, }}p > 1">
<mi>x</mi>
<mrow>
<mtext>, </mtext>
</mrow>
<mi>p</mi>
<mrow>
<mtext>, </mtext>
</mrow>
<mi>q</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
<mrow>
<mtext>, </mtext>
</mrow>
<mi>p</mi>
<mo>></mo>
<mn>1</mn>
</math></span>. The point <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{A}}\left( {0{\text{, }}a} \right)">
<mrow>
<mtext>A</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>0</mn>
<mrow>
<mtext>, </mtext>
</mrow>
<mi>a</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span> lies on the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g">
<mi>g</mi>
</math></span>.</p>
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = {g^{ - 1}}\left( x \right)">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mi>g</mi>
<mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span>. The point <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{B}}">
<mrow>
<mtext>B</mtext>
</mrow>
</math></span> lies on the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> and is the reflection of point <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{A}}">
<mrow>
<mtext>A</mtext>
</mrow>
</math></span> in the line <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = x">
<mi>y</mi>
<mo>=</mo>
<mi>x</mi>
</math></span>.</p>
</div>
<div class="specification">
<p>The line <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{L_1}">
<mrow>
<msub>
<mi>L</mi>
<mn>1</mn>
</msub>
</mrow>
</math></span> is tangent to the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{B}}">
<mrow>
<mtext>B</mtext>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the coordinates of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{B}}"> <mrow> <mtext>B</mtext> </mrow> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f'\left( a \right) = \frac{1}{{{\text{ln}}\,p}}"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mi>a</mi> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <mrow> <mtext>ln</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>p</mi> </mrow> </mfrac> </math></span>, find the equation of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{L_1}"> <mrow> <msub> <mi>L</mi> <mn>1</mn> </msub> </mrow> </math></span> <strong>in terms of</strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p"> <mi>p</mi> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q"> <mi>q</mi> </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>The line <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{L_2}"> <mrow> <msub> <mi>L</mi> <mn>2</mn> </msub> </mrow> </math></span> is tangent to the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g"> <mi>g</mi> </math></span> at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{A}}"> <mrow> <mtext>A</mtext> </mrow> </math></span> and has equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = \left( {{\text{ln}}\,p} \right)x + q + 1"> <mi>y</mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>ln</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>p</mi> </mrow> <mo>)</mo> </mrow> <mi>x</mi> <mo>+</mo> <mi>q</mi> <mo>+</mo> <mn>1</mn> </math></span>.</p>
<p>The line <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{L_2}"> <mrow> <msub> <mi>L</mi> <mn>2</mn> </msub> </mrow> </math></span> passes through the point <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( { - 2{\text{, }} - 2} \right)"> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>2</mn> <mrow> <mtext>, </mtext> </mrow> <mo>−</mo> <mn>2</mn> </mrow> <mo>)</mo> </mrow> </math></span>.</p>
<p>The gradient of the normal to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g"> <mi>g</mi> </math></span> at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{A}}"> <mrow> <mtext>A</mtext> </mrow> </math></span> is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{{{\text{ln}}\left( {\frac{1}{3}} \right)}}"> <mfrac> <mn>1</mn> <mrow> <mrow> <mtext>ln</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </math></span>.</p>
<p> </p>
<p>Find the equation of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{L_1}"> <mrow> <msub> <mi>L</mi> <mn>1</mn> </msub> </mrow> </math></span> in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>.</p>
<div class="marks">[7]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>The speed of light is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{300}}\,{\text{000}}">
<mrow>
<mtext>300</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>000</mtext>
</mrow>
</math></span> kilometres per second. The average distance from the Sun to the Earth is 149.6 million km.</p>
</div>
<div class="specification">
<p>A light-year is the distance light travels in one year and is equal to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{9}}\,{\text{467}}\,{\text{280}}">
<mrow>
<mtext>9</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>467</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>280</mtext>
</mrow>
</math></span> million km. Polaris is a bright star, visible from the Northern Hemisphere. The distance from the Earth to Polaris is 323 light-years.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate the time, <strong>in minutes</strong>, it takes for light from the Sun to reach the Earth.</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>Find the distance from the Earth to Polaris in millions of km. Give your answer in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a \times {10^k}">
<mi>a</mi>
<mo>×</mo>
<mrow>
<msup>
<mn>10</mn>
<mi>k</mi>
</msup>
</mrow>
</math></span> with <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1 \leqslant a < 10">
<mn>1</mn>
<mo>⩽</mo>
<mi>a</mi>
<mo><</mo>
<mn>10</mn>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k \in \mathbb{Z}">
<mi>k</mi>
<mo>∈</mo>
<mrow>
<mi mathvariant="double-struck">Z</mi>
</mrow>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>A sphere with diameter 3 474 000 metres can model the shape of the Moon.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use this model to calculate the circumference of the Moon in <strong>kilometres</strong>. Give your full calculator display.</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>Give your answer to part (a) correct to three significant figures.</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>Write your answer to<strong> part (b)</strong> in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a \times {10^k}">
<mi>a</mi>
<mo>×</mo>
<mrow>
<msup>
<mn>10</mn>
<mi>k</mi>
</msup>
</mrow>
</math></span>, where 1 ≤ <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span> < 10 , <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k \in \mathbb{Z}">
<mi>k</mi>
<mo>∈</mo>
<mrow>
<mi mathvariant="double-struck">Z</mi>
</mrow>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>The first three terms of an arithmetic sequence are <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>1</mn></msub><mo>,</mo><mo> </mo><mn>5</mn><msub><mi>u</mi><mn>1</mn></msub><mo>-</mo><mn>8</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><msub><mi>u</mi><mn>1</mn></msub><mo>+</mo><mn>8</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"><msub><mi>u</mi><mn>1</mn></msub><mo>=</mo><mn>4</mn></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Prove that the sum of the first <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> terms of this arithmetic sequence is a square number.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>Consider an arithmetic sequence where <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>8</mn></msub><mo>=</mo><msub><mi>S</mi><mn>8</mn></msub><mo>=</mo><mn>8</mn></math>. Find the value of the first term, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>1</mn></msub></math>, and the value of the common difference, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math>.</p>
</div>
<br><hr><br><div class="specification">
<p>Consider the numbers <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p = 2.78 \times {10^{11}}">
<mi>p</mi>
<mo>=</mo>
<mn>2.78</mn>
<mo>×<!-- × --></mo>
<mrow>
<msup>
<mn>10</mn>
<mrow>
<mn>11</mn>
</mrow>
</msup>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q = 3.12 \times {10^{ - 3}}">
<mi>q</mi>
<mo>=</mo>
<mn>3.12</mn>
<mo>×<!-- × --></mo>
<mrow>
<msup>
<mn>10</mn>
<mrow>
<mo>−<!-- − --></mo>
<mn>3</mn>
</mrow>
</msup>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sqrt[3]{{\frac{p}{q}}}"> <mroot> <mrow> <mfrac> <mi>p</mi> <mi>q</mi> </mfrac> </mrow> <mn>3</mn> </mroot> </math></span>. Give your full calculator display.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down your answer to part (a) correct to two decimal places;</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>Write down your answer to part (a) correct to three significant figures.</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>Write your answer to <strong>part (b)(ii) </strong>in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a \times {10^k}"> <mi>a</mi> <mo>×</mo> <mrow> <msup> <mn>10</mn> <mi>k</mi> </msup> </mrow> </math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1 \leqslant a < 10,{\text{ }}k \in \mathbb{Z}"> <mn>1</mn> <mo>⩽</mo> <mi>a</mi> <mo><</mo> <mn>10</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mi>k</mi> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p><strong>Give your answers in this question correct to the nearest whole number.</strong></p>
<p>Imon invested <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>25</mn><mo> </mo><mn>000</mn></math> Singapore dollars (<math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>SGD</mtext></math>) in a fixed deposit account with a nominal annual interest rate of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>.</mo><mn>6</mn><mo>%</mo></math>, compounded <strong>monthly</strong>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate the value of Imon’s investment after <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn></math> years.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>At the end of the <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn></math> years, Imon withdrew <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo> </mo><mtext>SGD</mtext></math> from the fixed deposit account and reinvested this into a super-savings account with a nominal annual interest rate of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mo>.</mo><mn>7</mn><mo>%</mo></math>, compounded <strong>half-yearly</strong>.</p>
<p>The value of the super-savings account increased to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>20</mn><mo> </mo><mn>000</mn><mo> </mo><mtext>SGD</mtext></math> after <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>18</mn></math> months.</p>
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = \sqrt x \,{\text{sin}}\left( {\frac{\pi }{4}x} \right)">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msqrt>
<mi>x</mi>
</msqrt>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mi>π<!-- π --></mi>
<mn>4</mn>
</mfrac>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g(x) = \sqrt x ">
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msqrt>
<mi>x</mi>
</msqrt>
</math></span> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> ≥ 0. The first time the graphs of <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> intersect is at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 0">
<mi>x</mi>
<mo>=</mo>
<mn>0</mn>
</math></span>.</p>
</div>
<div class="specification">
<p>The set of all non-zero values that satisfy <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = g(x)">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span> can be described as an arithmetic sequence, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_n} = a + bn">
<mrow>
<msub>
<mi>u</mi>
<mi>n</mi>
</msub>
</mrow>
<mo>=</mo>
<mi>a</mi>
<mo>+</mo>
<mi>b</mi>
<mi>n</mi>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span> ≥ 1.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the <strong>two</strong> smallest non-zero values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> for which <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = g(x)"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </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>At point P, the graphs of <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> intersect for the 21st time. Find the coordinates of P.</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>The following diagram shows part of the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g"> <mi>g</mi> </math></span> <strong>reflected</strong> in the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>-axis. It also shows part of the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f"> <mi>f</mi> </math></span> and the point P.</p>
<p><img src="data:image/png;base64,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"></p>
<p>Find an expression for the area of the shaded region. Do not calculate the value of the expression.</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>In an arithmetic sequence, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_2} = 5">
<mrow>
<msub>
<mi>u</mi>
<mn>2</mn>
</msub>
</mrow>
<mo>=</mo>
<mn>5</mn>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_3} = 11">
<mrow>
<msub>
<mi>u</mi>
<mn>3</mn>
</msub>
</mrow>
<mo>=</mo>
<mn>11</mn>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the common difference.</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 the first term.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the sum of the first <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="20"> <mn>20</mn> </math></span> terms.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>In an arithmetic sequence, <em>u</em><sub>1</sub> = −5 and <em>d</em> = 3.</p>
</div>
<div class="question">
<p>Find <em>u</em><sub>8</sub>.</p>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Explain why any integer can be written in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="4k">
<mn>4</mn>
<mi>k</mi>
</math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="4k + 1">
<mn>4</mn>
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
</math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="4k + 2">
<mn>4</mn>
<mi>k</mi>
<mo>+</mo>
<mn>2</mn>
</math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="4k + 3">
<mn>4</mn>
<mi>k</mi>
<mo>+</mo>
<mn>3</mn>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k \in \mathbb{Z}">
<mi>k</mi>
<mo>∈</mo>
<mrow>
<mi mathvariant="double-struck">Z</mi>
</mrow>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence prove that the square of any integer can be written in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="4t">
<mn>4</mn>
<mi>t</mi>
</math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="4t + 1">
<mn>4</mn>
<mi>t</mi>
<mo>+</mo>
<mn>1</mn>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t \in {\mathbb{Z}^ + }">
<mi>t</mi>
<mo>∈</mo>
<mrow>
<msup>
<mrow>
<mi mathvariant="double-struck">Z</mi>
</mrow>
<mo>+</mo>
</msup>
</mrow>
</math></span>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>The following diagram shows [CD], with length <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b{\text{ cm}}"> <mi>b</mi> <mrow> <mtext> cm</mtext> </mrow> </math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b > 1"> <mi>b</mi> <mo>></mo> <mn>1</mn> </math></span>. Squares with side lengths <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k{\text{ cm}},{\text{ }}{k^2}{\text{ cm}},{\text{ }}{k^3}{\text{ cm}},{\text{ }} \ldots "> <mi>k</mi> <mrow> <mtext> cm</mtext> </mrow> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mrow> <msup> <mi>k</mi> <mn>2</mn> </msup> </mrow> <mrow> <mtext> cm</mtext> </mrow> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mrow> <msup> <mi>k</mi> <mn>3</mn> </msup> </mrow> <mrow> <mtext> cm</mtext> </mrow> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mo>…</mo> </math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0 < k < 1"> <mn>0</mn> <mo><</mo> <mi>k</mi> <mo><</mo> <mn>1</mn> </math></span>, are drawn along [CD]. This process is carried on indefinitely. The diagram shows the first three squares.</p>
<p><img src="images/Schermafbeelding_2018-02-12_om_09.48.48.png" alt="N17/5/MATME/SP1/ENG/TZ0/10.b"></p>
<p>The <strong>total</strong> sum of the areas of all the squares is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{9}{{16}}"> <mfrac> <mn>9</mn> <mrow> <mn>16</mn> </mrow> </mfrac> </math></span>. Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b"> <mi>b</mi> </math></span>.</p>
</div>
<br><hr><br><div class="specification">
<p>A comet orbits the Sun and is seen from Earth every 37 years. The comet was first seen from Earth in the year 1064.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the year in which the comet was seen from Earth for the fifth time.</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>Determine how many times the comet has been seen from Earth up to the year 2014.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>In the expansion of <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>x</mi><mo>+</mo><mi>k</mi><msup><mo>)</mo><mn>7</mn></msup></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>, the coefficient of the term in <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>5</mn></msup></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>63</mn></math>.</p>
<p>Find the possible values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math>.</p>
</div>
<br><hr><br><div class="specification">
<p>The first three terms of a geometric sequence are <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\ln {x^{16}}">
<mi>ln</mi>
<mo><!-- --></mo>
<mrow>
<msup>
<mi>x</mi>
<mrow>
<mn>16</mn>
</mrow>
</msup>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\ln {x^8}">
<mi>ln</mi>
<mo><!-- --></mo>
<mrow>
<msup>
<mi>x</mi>
<mn>8</mn>
</msup>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\ln {x^4}">
<mi>ln</mi>
<mo><!-- --></mo>
<mrow>
<msup>
<mi>x</mi>
<mn>4</mn>
</msup>
</mrow>
</math></span>, for <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">
<p>Find the common ratio.</p>
</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="{\text{lo}}{{\text{g}}_9}\left( {{\text{cos}}\,2x + 2} \right) = {\text{lo}}{{\text{g}}_3}\sqrt {{\text{cos}}\,2x + 2} ">
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>9</mn>
</msub>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>3</mn>
</msub>
</mrow>
<msqrt>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
</msqrt>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence or otherwise solve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{lo}}{{\text{g}}_3}\left( {{\text{2}}\,{\text{sin}}\,x} \right) = {\text{lo}}{{\text{g}}_9}\left( {{\text{cos}}\,2x + 2} \right)">
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>3</mn>
</msub>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>2</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>9</mn>
</msub>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0 < x < \frac{\pi }{2}">
<mn>0</mn>
<mo><</mo>
<mi>x</mi>
<mo><</mo>
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
</math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>Consider the curve with equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>(</mo><mn>2</mn><mi>x</mi><mo>-</mo><mn>1</mn><mo>)</mo><msup><mtext>e</mtext><mrow><mi>k</mi><mi>x</mi></mrow></msup></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>∈</mo><mi mathvariant="normal">ℚ</mi></math>.</p>
<p>The tangent to the curve at the point where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>1</mn></math> is parallel to the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>5</mn><msup><mtext>e</mtext><mi>k</mi></msup><mi>x</mi></math>.</p>
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math>.</p>
</div>
<br><hr><br><div class="specification">
<p>The diameter of a spherical planet is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mo>×</mo><msup><mn>10</mn><mn>4</mn></msup><mo> </mo><mtext>km</mtext></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the radius of the planet.</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>The volume of the planet can be expressed in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">π</mi><mfenced><mrow><mi>a</mi><mo>×</mo><msup><mn>10</mn><mi>k</mi></msup></mrow></mfenced><mo> </mo><msup><mtext>km</mtext><mn>3</mn></msup></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>≤</mo><mi>a</mi><mo><</mo><mn>10</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>∈</mo><mi mathvariant="normal">ℤ</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>k</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>Consider two consecutive positive integers, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>+</mo><mn>1</mn></math>.</p>
<p>Show that the difference of their squares is equal to the sum of the two integers.</p>
</div>
<br><hr><br><div class="specification">
<p>Tomás is playing with sticks and he forms the first three diagrams of a pattern. These diagrams are shown below.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-15_om_17.25.44.png" alt="M17/5/MATSD/SP1/ENG/TZ2/05"></p>
<p>Tomás continues forming diagrams following this pattern.</p>
</div>
<div class="specification">
<p>Tomás forms a total of 24 diagrams.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Diagram <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span> is formed with 52 sticks. Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the total number of sticks used by Tomás for all 24 diagrams.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Jashanti is saving money to buy a car. The price of the car, in US Dollars (USD), can be modelled by the equation</p>
<p><span class="mjpage mjpage__block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" alttext="P = 8500{\text{ }}{(0.95)^t}.">
<mi>P</mi>
<mo>=</mo>
<mn>8500</mn>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<mo stretchy="false">(</mo>
<mn>0.95</mn>
<msup>
<mo stretchy="false">)</mo>
<mi>t</mi>
</msup>
</mrow>
<mo>.</mo>
</math></span></p>
<p>Jashanti’s savings, in USD, can be modelled by the equation</p>
<p><span class="mjpage mjpage__block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" alttext="S = 400t + 2000.">
<mi>S</mi>
<mo>=</mo>
<mn>400</mn>
<mi>t</mi>
<mo>+</mo>
<mn>2000.</mn>
</math></span></p>
<p>In both equations <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> is the time in months since Jashanti started saving for the car.</p>
</div>
<div class="specification">
<p>Jashanti does not want to wait too long and wants to buy the car two months after she started saving. She decides to ask her parents for the extra money that she needs.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the amount of money Jashanti saves per month.</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>Use your graphic display calculator to find how long it will take for Jashanti to have saved enough money to buy the car.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate how much extra money Jashanti needs.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br>