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<h2>HL Paper 3</h2><div class="specification">
<p><strong>In this question you will be exploring the strategies required to solve a system of linear differential equations.</strong></p>
<p> </p>
<p>Consider the system of linear differential equations of the form:</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mi>x</mi><mo>-</mo><mi>y</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mi>a</mi><mi>x</mi><mo>+</mo><mi>y</mi></math>,</p>
<p>where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>,</mo><mo> </mo><mi>y</mi><mo>,</mo><mo> </mo><mi>t</mi><mo>∈</mo><msup><mi mathvariant="normal">ℝ</mi><mo>+</mo></msup></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> is a parameter.</p>
<p>First consider the case where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>0</mn></math>.</p>
</div>
<div class="specification">
<p>Now consider the case where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mo>-</mo><mn>1</mn></math>.</p>
</div>
<div class="specification">
<p>Now consider the case where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mo>-</mo><mn>4</mn></math>.</p>
</div>
<div class="specification">
<p>From previous cases, we might conjecture that a solution to this differential equation is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>F</mi><msup><mtext>e</mtext><mrow><mi>λ</mi><mi>t</mi></mrow></msup></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>F</mi></math> is a constant.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By solving 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>t</mi></mrow></mfrac><mo>=</mo><mi>y</mi></math>, show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>A</mi><msup><mtext>e</mtext><mi>t</mi></msup></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> is a constant.</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>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>-</mo><mi>x</mi><mo>=</mo><mo>-</mo><mi>A</mi><msup><mtext>e</mtext><mi>t</mi></msup></math>.</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>Solve the differential equation in part (a)(ii) to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> as a function of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By differentiating <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mo>-</mo><mi>x</mi><mo>+</mo><mi>y</mi></math> with respect to <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>, show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>y</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mn>2</mn><mfrac><mstyle displaystyle="true"><mo>d</mo><mi>y</mi></mstyle><mstyle displaystyle="true"><mo>d</mo><mi>t</mi></mstyle></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>By substituting <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi><mo>=</mo><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></math>, show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi><mo>=</mo><mi>B</mi><msup><mtext>e</mtext><mrow><mn>2</mn><mi>t</mi></mrow></msup></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> is a constant.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> as a function of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>.</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>Hence show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mo>-</mo><mfrac><mi>B</mi><mn>2</mn></mfrac><msup><mtext>e</mtext><mrow><mn>2</mn><mi>t</mi></mrow></msup><mo>+</mo><mi>C</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math> is a constant.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.iv.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>y</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>-</mo><mn>2</mn><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>-</mo><mn>3</mn><mi>y</mi><mo>=</mo><mn>0</mn></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the two values for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi></math> that satisfy <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>y</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>-</mo><mn>2</mn><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>-</mo><mn>3</mn><mi>y</mi><mo>=</mo><mn>0</mn></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Let the two values found in part (c)(ii) be <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>λ</mi><mn>1</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>λ</mi><mn>2</mn></msub></math>.</p>
<p>Verify that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>F</mi><msup><mtext>e</mtext><mrow><msub><mi>λ</mi><mn>1</mn></msub><mi>t</mi></mrow></msup><mo>+</mo><mi>G</mi><msup><mtext>e</mtext><mrow><msub><mi>λ</mi><mn>2</mn></msub><mi>t</mi></mrow></msup></math> is a solution to the differential equation in (c)(i),where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math> is a constant.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.iii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mtext>t</mtext></mrow></mfrac><mo>=</mo><mi>y</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mi>y</mi></mfrac><mo>=</mo><mo>∫</mo><mo>d</mo><mtext>t</mtext></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mi>y</mi><mo>=</mo><mi>t</mi><mo>+</mo><mi>c</mi></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mfenced open="|" close="|"><mi>y</mi></mfenced><mo>=</mo><mi>t</mi><mo>+</mo><mi>c</mi></math> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>A1</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mi>y</mi></math> and <em><strong>A1</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math>.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>A</mi><msup><mtext>e</mtext><mi>t</mi></msup></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>rearranging to <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mtext>t</mtext></mrow></mfrac><mo>-</mo><mi>y</mi><mo>=</mo><mn>0</mn></math> AND multiplying by integrating factor <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup><mo>=</mo><mi>A</mi></math> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>A</mi><msup><mtext>e</mtext><mi>t</mi></msup></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>substituting <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>A</mi><msup><mtext>e</mtext><mi>t</mi></msup></math> into differential equation in <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mtext>t</mtext></mrow></mfrac><mo>=</mo><mi>x</mi><mo>-</mo><mi>A</mi><msup><mtext>e</mtext><mi>t</mi></msup></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mtext>t</mtext></mrow></mfrac><mo>-</mo><mi>x</mi><mo>=</mo><mo>-</mo><mi>A</mi><msup><mtext>e</mtext><mi>t</mi></msup></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>integrating factor (IF) is <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>e</mtext><mrow><mo>∫</mo><mo>-</mo><mn>1</mn><mo>d</mo><mi>t</mi></mrow></msup></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mtext>t</mtext></mrow></mfrac><mo>-</mo><mi>x</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup><mo>=</mo><mo>-</mo><mi>A</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup><mo>=</mo><mo>-</mo><mi>A</mi><mi>t</mi><mo>+</mo><mi>D</mi></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfenced><mrow><mo>-</mo><mi>A</mi><mi>t</mi><mo>+</mo><mi>D</mi></mrow></mfenced><msup><mtext>e</mtext><mi>t</mi></msup></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> The first constant must be <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math>, and the second can be any constant for the final <em><strong>A1</strong></em> to be awarded. Accept a change of constant applied at the end.</p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>y</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mo>-</mo><mfrac><mstyle displaystyle="true"><mo>d</mo><mi>x</mi></mstyle><mstyle displaystyle="true"><mo>d</mo><mi>t</mi></mstyle></mfrac><mo>+</mo><mfrac><mstyle displaystyle="true"><mo>d</mo><mi>y</mi></mstyle><mstyle displaystyle="true"><mo>d</mo><mi>t</mi></mstyle></mfrac></math> <em><strong>A1</strong></em></p>
<p><br><strong>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo>+</mo><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>+</mo><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></math> <em><strong>A1</strong></em></p>
<p><strong><br>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo>+</mo><mfenced><mrow><mo>-</mo><mi>x</mi><mo>+</mo><mi>y</mi></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>2</mn><mfenced><mrow><mo>-</mo><mi>x</mi><mo>+</mo><mi>y</mi></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><br><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>2</mn><mfrac><mstyle displaystyle="true"><mo>d</mo><mi>y</mi></mstyle><mstyle displaystyle="true"><mo>d</mo><mi>t</mi></mstyle></mfrac></math> <em><strong>AG</strong></em></p>
<p><em><strong><br>[3 marks]</strong></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>Y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mn>2</mn><mi>Y</mi></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><mfrac><mrow><mo>d</mo><mi>Y</mi></mrow><mi>Y</mi></mfrac><mo>=</mo><mo>∫</mo><mn>2</mn><mo>d</mo><mi>t</mi></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mfenced open="|" close="|"><mi>Y</mi></mfenced><mo>=</mo><mn>2</mn><mi>t</mi><mo>+</mo><mi>c</mi></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mi>Y</mi><mo>=</mo><mn>2</mn><mi>t</mi><mo>+</mo><mi>c</mi></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi><mo>=</mo><mi>B</mi><msup><mtext>e</mtext><mrow><mn>2</mn><mi>t</mi></mrow></msup></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mi>B</mi><msup><mtext>e</mtext><mrow><mn>2</mn><mi>t</mi></mrow></msup></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>∫</mo><mi>B</mi><msup><mtext>e</mtext><mrow><mn>2</mn><mi>t</mi></mrow></msup><mtext> </mtext><mo>d</mo><mi>t</mi></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfrac><mi>B</mi><mn>2</mn></mfrac><msup><mtext>e</mtext><mrow><mn>2</mn><mi>t</mi></mrow></msup><mo>+</mo><mi>C</mi></math> <em><strong>A1</strong></em></p>
<p><strong><br>Note:</strong> The first constant must be <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math>, and the second can be any constant for the final <em><strong>A1</strong></em> to be awarded. Accept a change of constant applied at the end.</p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>substituting <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mi>B</mi><msup><mtext>e</mtext><mrow><mn>2</mn><mi>t</mi></mrow></msup></math> and their (iii) into <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mo>-</mo><mi>x</mi><mo>+</mo><mi>y</mi></math> <em><strong>M1(M1)</strong></em></p>
<p><em><strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><msup><mtext>e</mtext><mrow><mn>2</mn><mi>t</mi></mrow></msup><mo>=</mo><mo>-</mo><mi>x</mi><mo>+</mo><mfrac><mi>B</mi><mn>2</mn></mfrac><msup><mtext>e</mtext><mrow><mn>2</mn><mi>t</mi></mrow></msup><mo>+</mo><mi>C</mi></math> A1</strong></em></p>
<p><em><strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mo>-</mo><mfrac><mi>B</mi><mn>2</mn></mfrac><msup><mtext>e</mtext><mrow><mn>2</mn><mi>t</mi></mrow></msup><mo>+</mo><mi>C</mi></math> AG</strong></em></p>
<p><strong>Note:</strong> Follow through from incorrect part (iii) cannot be awarded if it does not lead to the <em><strong>AG</strong></em>.</p>
<p><br><strong>METHOD 2</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mi>x</mi><mo>-</mo><mfrac><mi>B</mi><mn>2</mn></mfrac><msup><mtext>e</mtext><mrow><mn>2</mn><mi>t</mi></mrow></msup><mo>-</mo><mi>C</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>-</mo><mi>x</mi><mo>=</mo><mo>-</mo><mfrac><mi>B</mi><mn>2</mn></mfrac><msup><mtext>e</mtext><mrow><mn>2</mn><mi>t</mi></mrow></msup><mo>-</mo><mi>C</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mfenced><mrow><mi>x</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup></mrow></mfenced></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mo>-</mo><mfrac><mi>B</mi><mn>2</mn></mfrac><msup><mtext>e</mtext><mi>t</mi></msup><mo>-</mo><mi>C</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup><mo>=</mo><mo>∫</mo><mo>-</mo><mfrac><mi>B</mi><mn>2</mn></mfrac><msup><mtext>e</mtext><mi>t</mi></msup><mo>-</mo><mi>C</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup><mtext> </mtext><mo>d</mo><mi>t</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup><mo>=</mo><mo>-</mo><mfrac><mi>B</mi><mn>2</mn></mfrac><msup><mtext>e</mtext><mi>t</mi></msup><mo>-</mo><mi>C</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup><mo>+</mo><mi>D</mi></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mo>-</mo><mfrac><mi>B</mi><mn>2</mn></mfrac><msup><mtext>e</mtext><mrow><mn>2</mn><mi>t</mi></mrow></msup><mo>+</mo><mi>C</mi><mo>+</mo><mi>D</mi><msup><mtext>e</mtext><mi>t</mi></msup></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mo>-</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo>⇒</mo><mi>B</mi><msup><mtext>e</mtext><mrow><mn>2</mn><mi>t</mi></mrow></msup><mo>=</mo><mfrac><mi>B</mi><mn>2</mn></mfrac><msup><mtext>e</mtext><mrow><mn>2</mn><mi>t</mi></mrow></msup><mo>-</mo><mi>C</mi><mo>-</mo><mi>D</mi><msup><mtext>e</mtext><mi>t</mi></msup><mo>+</mo><mfrac><mi>B</mi><mn>2</mn></mfrac><msup><mtext>e</mtext><mrow><mn>2</mn><mi>t</mi></mrow></msup><mo>+</mo><mi>C</mi><mo>⇒</mo><mi>D</mi><mo>=</mo><mn>0</mn></math> <em><strong>M1</strong></em></p>
<p><em><strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mo>-</mo><mfrac><mi>B</mi><mn>2</mn></mfrac><msup><mtext>e</mtext><mrow><mn>2</mn><mi>t</mi></mrow></msup><mo>+</mo><mi>C</mi></math> AG</strong></em></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.iv.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mo>-</mo><mn>4</mn><mi>x</mi><mo>+</mo><mi>y</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>y</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mo>-</mo><mn>4</mn><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>+</mo><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></math> seen anywhere <em><strong>M1</strong></em></p>
<p> </p>
<p><strong>METHOD 1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>y</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mo>-</mo><mn>4</mn><mfenced><mrow><mi>x</mi><mo>-</mo><mi>y</mi></mrow></mfenced><mo>+</mo><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></math></p>
<p>attempt to eliminate <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><mn>4</mn><mfenced><mrow><mfrac><mn>1</mn><mn>4</mn></mfrac><mfenced><mrow><mi>y</mi><mo>-</mo><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></mrow></mfenced><mo>-</mo><mi>y</mi></mrow></mfenced><mo>+</mo><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>2</mn><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>+</mo><mn>3</mn><mi>y</mi></math><em><strong> A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>y</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>-</mo><mn>2</mn><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>-</mo><mn>3</mn><mi>y</mi><mo>=</mo><mn>0</mn></math><em><strong> AG</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>rewriting LHS in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>y</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>-</mo><mn>2</mn><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>-</mo><mn>3</mn><mi>y</mi><mo>=</mo><mfenced><mrow><mo>-</mo><mn>8</mn><mi>x</mi><mo>+</mo><mn>5</mn><mi>y</mi></mrow></mfenced><mo>-</mo><mn>2</mn><mfenced><mrow><mo>-</mo><mn>4</mn><mi>x</mi><mo>+</mo><mi>y</mi></mrow></mfenced><mo>-</mo><mn>3</mn><mi>y</mi></math><em><strong> A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>0</mn></math><em><strong> AG</strong></em></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mi>F</mi><mi>λ</mi><msup><mtext>e</mtext><mrow><mi>λ</mi><mi>t</mi></mrow></msup><mo>,</mo><mo> </mo><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>y</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mi>F</mi><msup><mi>λ</mi><mn>2</mn></msup><msup><mtext>e</mtext><mrow><mi>λ</mi><mi>t</mi></mrow></msup></math><em><strong> (A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>F</mi><msup><mi>λ</mi><mn>2</mn></msup><msup><mtext>e</mtext><mrow><mi>λ</mi><mi>t</mi></mrow></msup><mo>-</mo><mn>2</mn><mi>F</mi><mi>λ</mi><msup><mtext>e</mtext><mrow><mi>λ</mi><mi>t</mi></mrow></msup><mo>-</mo><mn>3</mn><mi>F</mi><msup><mtext>e</mtext><mrow><mi>λ</mi><mi>t</mi></mrow></msup><mo>=</mo><mn>0</mn></math><em><strong> (M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>λ</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>λ</mi><mo>-</mo><mn>3</mn><mo>=</mo><mn>0</mn></math> (since <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>e</mtext><mrow><mi>λ</mi><mi>t</mi></mrow></msup><mo>≠</mo><mn>0</mn></math>)<em><strong> A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>λ</mi><mn>1</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>λ</mi><mn>2</mn></msub></math> are <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>1</mn></math> (either order)<em><strong> A1</strong></em></p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>F</mi><msup><mtext>e</mtext><mrow><mn>3</mn><mi>t</mi></mrow></msup><mo>+</mo><mi>G</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mn>3</mn><mi>F</mi><msup><mtext>e</mtext><mrow><mn>3</mn><mi>t</mi></mrow></msup><mo>-</mo><mi>G</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup><mo>,</mo><mo> </mo><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>y</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mn>9</mn><mi>F</mi><msup><mtext>e</mtext><mrow><mn>3</mn><mi>t</mi></mrow></msup><mo>-</mo><mi>G</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup></math> <em><strong>(A1)</strong></em><em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>y</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>-</mo><mn>2</mn><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>-</mo><mn>3</mn><mi>y</mi><mo>=</mo><mn>9</mn><mi>F</mi><msup><mtext>e</mtext><mrow><mn>3</mn><mi>t</mi></mrow></msup><mo>+</mo><mi>G</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup><mo>-</mo><mn>2</mn><mfenced><mrow><mn>3</mn><mi>F</mi><msup><mtext>e</mtext><mrow><mn>3</mn><mi>t</mi></mrow></msup><mo>-</mo><mi>G</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup></mrow></mfenced><mo>-</mo><mn>3</mn><mfenced><mrow><mi>F</mi><msup><mtext>e</mtext><mrow><mn>3</mn><mi>t</mi></mrow></msup><mo>-</mo><mi>G</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup></mrow></mfenced></math><em><strong> M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>9</mn><mi>F</mi><msup><mtext>e</mtext><mrow><mn>3</mn><mi>t</mi></mrow></msup><mo>+</mo><mi>G</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup><mo>-</mo><mn>6</mn><mi>F</mi><msup><mtext>e</mtext><mrow><mn>3</mn><mi>t</mi></mrow></msup><mo>+</mo><mn>2</mn><mi>G</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup><mo>-</mo><mn>3</mn><mi>F</mi><msup><mtext>e</mtext><mrow><mn>3</mn><mi>t</mi></mrow></msup><mo>-</mo><mn>3</mn><mi>G</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup></math><em><strong> A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>0</mn></math><em><strong> AG</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>F</mi><msup><mtext>e</mtext><mrow><msub><mi>λ</mi><mn>1</mn></msub><mi>t</mi></mrow></msup><mo>+</mo><mi>G</mi><msup><mtext>e</mtext><mrow><msub><mi>λ</mi><mn>2</mn></msub><mi>t</mi></mrow></msup></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mi>F</mi><msub><mi>λ</mi><mn>1</mn></msub><msup><mtext>e</mtext><mrow><msub><mi>λ</mi><mn>1</mn></msub><mi>t</mi></mrow></msup><mo>+</mo><mi>G</mi><msub><mi>λ</mi><mn>2</mn></msub><msup><mtext>e</mtext><mrow><msub><mi>λ</mi><mn>2</mn></msub><mi>t</mi></mrow></msup><mo>,</mo><mo> </mo><mfrac><mstyle displaystyle="true"><msup><mo>d</mo><mn>2</mn></msup><mi>y</mi></mstyle><mstyle displaystyle="true"><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mstyle></mfrac><mo>=</mo><mi>F</mi><msup><msub><mi>λ</mi><mn>1</mn></msub><mn>2</mn></msup><msup><mtext>e</mtext><mrow><msub><mi>λ</mi><mn>1</mn></msub><mi>t</mi></mrow></msup><mo>+</mo><mi>G</mi><msup><msub><mi>λ</mi><mn>2</mn></msub><mn>2</mn></msup><msup><mtext>e</mtext><mrow><msub><mi>λ</mi><mn>2</mn></msub><mi>t</mi></mrow></msup></math> <em><strong>(A1)</strong></em><em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>y</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>-</mo><mn>2</mn><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>-</mo><mn>3</mn><mi>y</mi><mo>=</mo><mi>F</mi><msup><msub><mi>λ</mi><mn>1</mn></msub><mn>2</mn></msup><msup><mtext>e</mtext><mrow><msub><mi>λ</mi><mn>1</mn></msub><mi>t</mi></mrow></msup><mo>+</mo><mi>G</mi><msup><msub><mi>λ</mi><mn>2</mn></msub><mn>2</mn></msup><msup><mtext>e</mtext><mrow><msub><mi>λ</mi><mn>2</mn></msub><mi>t</mi></mrow></msup><mo>-</mo><mn>2</mn><mfenced><mrow><mi>F</mi><msub><mi>λ</mi><mn>1</mn></msub><msup><mtext>e</mtext><mrow><msub><mi>λ</mi><mn>1</mn></msub><mi>t</mi></mrow></msup><mo>+</mo><mi>G</mi><msub><mi>λ</mi><mn>2</mn></msub><msup><mtext>e</mtext><mrow><msub><mi>λ</mi><mn>2</mn></msub><mi>t</mi></mrow></msup></mrow></mfenced><mo>-</mo><mn>3</mn><mfenced><mrow><mi>F</mi><msup><mtext>e</mtext><mrow><msub><mi>λ</mi><mn>1</mn></msub><mi>t</mi></mrow></msup><mo>+</mo><mi>G</mi><msup><mtext>e</mtext><mrow><msub><mi>λ</mi><mn>2</mn></msub><mi>t</mi></mrow></msup></mrow></mfenced></math><em><strong> M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi>F</mi><msup><mtext>e</mtext><mrow><msub><mi>λ</mi><mn>1</mn></msub><mi>t</mi></mrow></msup><mfenced><mrow><msup><mi>λ</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>λ</mi><mo>-</mo><mn>3</mn></mrow></mfenced><mo>+</mo><mi>G</mi><msup><mtext>e</mtext><mrow><msub><mi>λ</mi><mn>2</mn></msub><mi>t</mi></mrow></msup><mfenced><mrow><msup><mi>λ</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>λ</mi><mo>-</mo><mn>3</mn></mrow></mfenced></math><em><strong> A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>0</mn></math><em><strong> AG</strong></em></p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">c.iii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.iv.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.iii.</div>
</div>
<br><hr><br><div class="specification">
<p><strong>In this question you will explore some of the properties of special functions <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">f</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">g</mi></math> and their relationship with the trigonometric functions, sine and cosine.</strong></p>
<p><br>Functions <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> are defined as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>z</mi></mfenced><mo>=</mo><mfrac><mrow><msup><mtext>e</mtext><mi>z</mi></msup><mo>+</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mi>z</mi></mrow></msup></mrow><mn>2</mn></mfrac></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>z</mi></mfenced><mo>=</mo><mfrac><mrow><msup><mtext>e</mtext><mi>z</mi></msup><mo>-</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mi>z</mi></mrow></msup></mrow><mn>2</mn></mfrac></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>∈</mo><mi mathvariant="normal">ℂ</mi></math>.</p>
<p>Consider <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi></math>, such that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>,</mo><mo> </mo><mi>u</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
</div>
<div class="specification">
<p>Using <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>e</mtext><mrow><mtext>i</mtext><mi>u</mi></mrow></msup><mo>=</mo><mi>cos</mi><mo> </mo><mi>u</mi><mo>+</mo><mtext>i</mtext><mo> </mo><mi>sin</mi><mo> </mo><mi>u</mi></math>, find expressions, in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mo> </mo><mi>u</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mo> </mo><mi>u</mi></math>, for</p>
</div>
<div class="specification">
<p>The functions <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mo> </mo><mi>x</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mo> </mo><mi>x</mi></math> are known as circular functions as the general point (<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mo> </mo><mi>θ</mi><mo>,</mo><mo> </mo><mi>sin</mi><mo> </mo><mi>θ</mi></math>) defines points on the unit circle with equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>1</mn></math>.</p>
<p>The functions <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> are known as hyperbolic functions, as the general point ( <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>θ</mi><mo>)</mo><mo>,</mo><mo> </mo><mi>g</mi><mo>(</mo><mi>θ</mi><mo>)</mo></math> ) defines points on a curve known as a hyperbola with equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>1</mn></math>. This hyperbola has two asymptotes.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Verify that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi><mo>=</mo><mi>f</mi><mfenced><mi>t</mi></mfenced></math> satisfies the differential equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>u</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mi>u</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>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>f</mi><mfenced><mi>t</mi></mfenced></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mi>g</mi><mfenced><mi>t</mi></mfenced></mrow></mfenced><mn>2</mn></msup><mo>=</mo><mi>f</mi><mfenced><mrow><mn>2</mn><mi>t</mi></mrow></mfenced></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mrow><mtext>i</mtext><mi>u</mi></mrow></mfenced></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mrow><mtext>i</mtext><mi>u</mi></mrow></mfenced></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find, and simplify, an expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>f</mi><mfenced><mrow><mtext>i</mtext><mi>u</mi></mrow></mfenced></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mi>g</mi><mfenced><mrow><mtext>i</mtext><mi>u</mi></mrow></mfenced></mrow></mfenced><mn>2</mn></msup></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 <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>f</mi><mfenced><mi>t</mi></mfenced></mrow></mfenced><mn>2</mn></msup><mo>-</mo><msup><mfenced><mrow><mi>g</mi><mfenced><mi>t</mi></mfenced></mrow></mfenced><mn>2</mn></msup><mo>=</mo><msup><mfenced><mrow><mi>f</mi><mfenced><mrow><mtext>i</mtext><mi>u</mi></mrow></mfenced></mrow></mfenced><mn>2</mn></msup><mo>-</mo><msup><mfenced><mrow><mi>g</mi><mfenced><mrow><mtext>i</mtext><mi>u</mi></mrow></mfenced></mrow></mfenced><mn>2</mn></msup></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>1</mn></math>, stating the coordinates of any axis intercepts and the equation of each asymptote.</p>
<div class="marks">[4]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The hyperbola with equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>1</mn></math> can be rotated to coincide with the curve defined by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mi>y</mi><mo>=</mo><mi>k</mi><mo>,</mo><mo> </mo><mi>k</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
<p>Find the possible values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">g.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>t</mi></mfenced><mo>=</mo><mfrac><mrow><msup><mtext>e</mtext><mi>t</mi></msup><mo>-</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup></mrow><mn>2</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>''</mo><mfenced><mi>t</mi></mfenced><mo>=</mo><mfrac><mrow><msup><mtext>e</mtext><mi>t</mi></msup><mo>+</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup></mrow><mn>2</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi>f</mi><mfenced><mi>t</mi></mfenced></math> <em><strong>AG</strong></em></p>
<p><em><strong><br>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>f</mi><mfenced><mi>t</mi></mfenced></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mi>g</mi><mfenced><mi>t</mi></mfenced></mrow></mfenced><mn>2</mn></msup></math></p>
<p>substituting <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> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><msup><mfenced><mrow><msup><mtext>e</mtext><mi>t</mi></msup><mo>+</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><msup><mtext>e</mtext><mi>t</mi></msup><mo>-</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup></mrow></mfenced><mn>2</mn></msup></mrow><mn>4</mn></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><msup><mfenced><msup><mtext>e</mtext><mi>t</mi></msup></mfenced><mn>2</mn></msup><mo>+</mo><mn>2</mn><mo>+</mo><msup><mfenced><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><msup><mtext>e</mtext><mi>t</mi></msup></mfenced><mn>2</mn></msup><mo>-</mo><mn>2</mn><mo>+</mo><msup><mfenced><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup></mfenced><mn>2</mn></msup></mrow><mn>4</mn></mfrac></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><msup><mfenced><msup><mtext>e</mtext><mi>t</mi></msup></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup></mfenced><mn>2</mn></msup></mrow><mn>2</mn></mfrac><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mfrac><mrow><msup><mtext>e</mtext><mrow><mn>2</mn><mi>t</mi></mrow></msup><mo>+</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mn>2</mn><mi>t</mi></mrow></msup></mrow><mn>2</mn></mfrac></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi>f</mi><mfenced><mrow><mn>2</mn><mi>t</mi></mrow></mfenced></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mrow><mn>2</mn><mi>t</mi></mrow></mfenced><mo>=</mo><mfrac><mrow><msup><mtext>e</mtext><mrow><mn>2</mn><mi>t</mi></mrow></msup><mo>+</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mn>2</mn><mi>t</mi></mrow></msup></mrow><mn>2</mn></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><msup><mfenced><msup><mtext>e</mtext><mi>t</mi></msup></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup></mfenced><mn>2</mn></msup></mrow><mn>2</mn></mfrac><mo> </mo></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><msup><mfenced><mrow><msup><mtext>e</mtext><mi>t</mi></msup><mo>+</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><msup><mtext>e</mtext><mi>t</mi></msup><mo>-</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup></mrow></mfenced><mn>2</mn></msup></mrow><mn>4</mn></mfrac></math> <em><strong>M1A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mfenced><mrow><mi>f</mi><mfenced><mi>t</mi></mfenced></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mi>g</mi><mfenced><mi>t</mi></mfenced></mrow></mfenced><mn>2</mn></msup></math> <em><strong>AG</strong></em></p>
<p><em><strong><br></strong></em><strong>Note: </strong>Accept combinations of METHODS 1 & 2 that meet at equivalent expressions.</p>
<p><em><strong><br></strong></em><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>substituting <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>e</mtext><mrow><mtext>i</mtext><mi>u</mi></mrow></msup><mo>=</mo><mi>cos</mi><mo> </mo><mi>u</mi><mo>+</mo><mtext>i</mtext><mo> </mo><mi>sin</mi><mo> </mo><mi>u</mi></math> into the expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> <em><strong>(M1)</strong></em></p>
<p>obtaining <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>e</mtext><mrow><mtext>-i</mtext><mi>u</mi></mrow></msup><mo>=</mo><mi>cos</mi><mo> </mo><mi>u</mi><mo>-</mo><mtext>i</mtext><mo> </mo><mi>sin</mi><mo> </mo><mi>u</mi></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mrow><mtext>i</mtext><mi>u</mi></mrow></mfenced><mo>=</mo><mfrac><mrow><mi>cos</mi><mo> </mo><mi>u</mi><mo>+</mo><mtext>i</mtext><mo> </mo><mi>sin</mi><mo> </mo><mi>u</mi><mo>+</mo><mi>cos</mi><mo> </mo><mi>u</mi><mo>-</mo><mtext>i</mtext><mo> </mo><mi>sin</mi><mo> </mo><mi>u</mi></mrow><mn>2</mn></mfrac></math></p>
<p><br><strong>Note:</strong> The <em><strong>M1</strong></em> can be awarded for the use of sine and cosine being odd and even respectively.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mn>2</mn><mo> </mo><mi>cos</mi><mo> </mo><mi>u</mi></mrow><mn>2</mn></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi>cos</mi><mo> </mo><mi>u</mi></math> <em><strong>A1</strong></em></p>
<p><em><strong><br></strong></em><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mrow><mtext>i</mtext><mi>u</mi></mrow></mfenced><mo>=</mo><mfrac><mrow><mi>cos</mi><mo> </mo><mi>u</mi><mo>+</mo><mtext>i</mtext><mo> </mo><mi>sin</mi><mo> </mo><mi>u</mi><mo>-</mo><mi>cos</mi><mo> </mo><mi>u</mi><mo>+</mo><mtext>i</mtext><mo> </mo><mi>sin</mi><mo> </mo><mi>u</mi></mrow><mn>2</mn></mfrac></math></p>
<p>substituting and attempt to simplify <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mn>2</mn><mtext>i</mtext><mo> </mo><mi>sin</mi><mo> </mo><mi>u</mi></mrow><mn>2</mn></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mtext>i</mtext><mo> </mo><mi>sin</mi><mo> </mo><mi>u</mi></math> <em><strong>A1</strong></em></p>
<p><em><strong><br></strong></em><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>f</mi><mfenced><mrow><mtext>i</mtext><mi>u</mi></mrow></mfenced></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mi>g</mi><mfenced><mrow><mtext>i</mtext><mi>u</mi></mrow></mfenced></mrow></mfenced><mn>2</mn></msup></math></p>
<p>substituting expressions found in part (c) <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mi>cos</mi><mn>2</mn></msup><mo> </mo><mi>u</mi><mo>-</mo><msup><mi>sin</mi><mn>2</mn></msup><mo> </mo><mi>u</mi><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mi>cos</mi><mo> </mo><mn>2</mn><mi>u</mi></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mrow><mn>2</mn><mtext>i</mtext><mi>u</mi></mrow></mfenced><mo>=</mo><mfrac><mrow><msup><mtext>e</mtext><mrow><mn>2</mn><mtext>i</mtext><mi>u</mi></mrow></msup><mo>+</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mn>2</mn><mtext>i</mtext><mi>u</mi></mrow></msup></mrow><mn>2</mn></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mi>cos</mi><mo> </mo><mn>2</mn><mi>u</mi><mo>+</mo><mtext>i</mtext><mo> </mo><mi>sin</mi><mo> </mo><mn>2</mn><mi>u</mi><mo>+</mo><mi>cos</mi><mo> </mo><mn>2</mn><mi>u</mi><mo>-</mo><mtext>i</mtext><mo> </mo><mi>sin</mi><mo> </mo><mn>2</mn><mi>u</mi></mrow><mn>2</mn></mfrac></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi>cos</mi><mo> </mo><mn>2</mn><mi>u</mi></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Accept equivalent final answers that have been simplified removing all imaginary parts eg <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo> </mo><msup><mi>cos</mi><mn>2</mn></msup><mo> </mo><mi>u</mi><mo>−</mo><mn>1</mn></math>etc</p>
<p><em><strong><br></strong></em><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>f</mi><mfenced><mi>t</mi></mfenced></mrow></mfenced><mn>2</mn></msup><mo>-</mo><msup><mfenced><mrow><mi>g</mi><mfenced><mi>t</mi></mfenced></mrow></mfenced><mn>2</mn></msup><mo>=</mo><mfrac><mrow><msup><mfenced><mrow><msup><mtext>e</mtext><mi>t</mi></msup><mo>+</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup></mrow></mfenced><mn>2</mn></msup><mo>-</mo><msup><mfenced><mrow><msup><mtext>e</mtext><mi>t</mi></msup><mo>-</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup></mrow></mfenced><mn>2</mn></msup></mrow><mn>4</mn></mfrac></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mfenced><mrow><msup><mtext>e</mtext><mrow><mn>2</mn><mi>t</mi></mrow></msup><mo>+</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mn>2</mn><mi>t</mi></mrow></msup><mo>+</mo><mn>2</mn></mrow></mfenced><mo>-</mo><mfenced><mrow><msup><mtext>e</mtext><mrow><mn>2</mn><mi>t</mi></mrow></msup><mo>+</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mn>2</mn><mi>t</mi></mrow></msup><mo>-</mo><mn>2</mn></mrow></mfenced></mrow><mn>4</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>4</mn><mn>4</mn></mfrac><mo>=</mo><mn>1</mn></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>A1</strong></em> for a value of 1 obtained from either LHS or RHS of given expression.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>f</mi><mfenced><mrow><mtext>i</mtext><mi>u</mi></mrow></mfenced></mrow></mfenced><mn>2</mn></msup><mo>-</mo><msup><mfenced><mrow><mi>g</mi><mfenced><mrow><mtext>i</mtext><mi>u</mi></mrow></mfenced></mrow></mfenced><mn>2</mn></msup><mo>=</mo><msup><mi>cos</mi><mn>2</mn></msup><mo> </mo><mi>u</mi><mo>+</mo><msup><mi>sin</mi><mn>2</mn></msup><mo> </mo><mi>u</mi></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>1</mn></math> (hence <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>f</mi><mfenced><mi>t</mi></mfenced></mrow></mfenced><mn>2</mn></msup><mo>-</mo><msup><mfenced><mrow><mi>g</mi><mfenced><mi>t</mi></mfenced></mrow></mfenced><mn>2</mn></msup><mo>=</mo><msup><mfenced><mrow><mi>f</mi><mfenced><mrow><mtext>i</mtext><mi>u</mi></mrow></mfenced></mrow></mfenced><mn>2</mn></msup><mo>-</mo><msup><mfenced><mrow><mi>g</mi><mfenced><mrow><mtext>i</mtext><mi>u</mi></mrow></mfenced></mrow></mfenced><mn>2</mn></msup></math>) <em><strong>AG</strong></em></p>
<p><br><strong>Note:</strong> Award full marks for showing that <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>f</mi><mfenced><mi>z</mi></mfenced></mrow></mfenced><mn>2</mn></msup><mo>-</mo><msup><mfenced><mrow><mi>g</mi><mfenced><mi>z</mi></mfenced></mrow></mfenced><mn>2</mn></msup><mo>=</mo><mn>1</mn><mo>,</mo><mo> </mo><mo>∀</mo><mi>z</mi><mo>∈</mo><mi mathvariant="normal">ℂ</mi></math>.<br><br><em><strong><br></strong></em><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img src="data:image/png;base64,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"> <em><strong>A1</strong></em><em><strong>A1</strong></em><em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>A1</strong></em> for correct curves in the upper quadrants, <em><strong>A1</strong></em> for correct curves in the lower quadrants, <em><strong>A1</strong></em> for correct <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-intercepts of <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mo>−</mo><mn>1</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>1</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math> (condone <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mo>−</mo><mn>1</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math>), <em><strong>A1</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>x</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>−</mo><mi>x</mi></math>.</p>
<p><br><em><strong><br></strong></em><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to rotate by <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>45</mn><mo>°</mo></math> in either direction <em><strong>(M1)</strong></em></p>
<p><br><strong>Note:</strong> Evidence of an attempt to relate to a sketch of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mi>y</mi><mo>=</mo><mi>k</mi></math> would be sufficient for this <em><strong>(M1)</strong></em>.</p>
<p><br>attempting to rotate a particular point, eg <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>1</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>1</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math> rotates to <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mfrac><mn>1</mn><msqrt><mn>2</mn></msqrt></mfrac><mo>,</mo><mo> </mo><mo>±</mo><mfrac><mstyle displaystyle="true"><mn>1</mn></mstyle><mstyle displaystyle="true"><msqrt><mn>2</mn></msqrt></mstyle></mfrac></mrow></mfenced></math> (or similar) <em><strong>(A1)</strong></em></p>
<p>hence <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mo>±</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p><em><strong><br>[5 marks]</strong></em></p>
<div class="question_part_label">g.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">g.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the functions <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g">
<mi>g</mi>
</math></span> : <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mathbb{R} \times \mathbb{R} \to \mathbb{R} \times \mathbb{R}">
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
<mo>×<!-- × --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
<mo stretchy="false">→<!-- → --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
<mo>×<!-- × --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
</math></span> defined by</p>
<p style="text-align: center;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( {\left( {x{\text{,}}\,\,y} \right)} \right) = \left( {x + y,\,\,x - y} \right)">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>y</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mi>y</mi>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>−<!-- − --></mo>
<mi>y</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g\left( {\left( {x{\text{,}}\,\,y} \right)} \right) = \left( {xy,\,\,x + y} \right)">
<mi>g</mi>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>y</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mi>y</mi>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>+</mo>
<mi>y</mi>
</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="\left( {f \circ g} \right)\left( {\left( {x{\text{,}}\,\,y} \right)} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mi>f</mi>
<mo>∘</mo>
<mi>g</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>y</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</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>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {g \circ f} \right)\left( {\left( {x{\text{,}}\,\,y} \right)} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mi>g</mi>
<mo>∘</mo>
<mi>f</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>y</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</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>State with a reason whether or not <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> commute.</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 inverse of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {f \circ g} \right)\left( {\left( {x{\text{,}}\,\,y} \right)} \right) = f\left( {g\left( {\left( {x{\text{,}}\,\,y} \right)} \right)} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mi>f</mi>
<mo>∘</mo>
<mi>g</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>y</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>g</mi>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>y</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span> (<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = f\left( {\left( {xy,\,\,x + y} \right)} \right)">
<mo>=</mo>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mi>y</mi>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>+</mo>
<mi>y</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span>) <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left( {xy + x + y,\,\,xy - x - y} \right)">
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mi>y</mi>
<mo>+</mo>
<mi>x</mi>
<mo>+</mo>
<mi>y</mi>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mi>y</mi>
<mo>−</mo>
<mi>x</mi>
<mo>−</mo>
<mi>y</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>A1A1</em></strong></p>
<p> </p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {g \circ f} \right)\left( {\left( {x{\text{,}}\,\,y} \right)} \right) = g\left( {f\left( {\left( {x{\text{,}}\,\,y} \right)} \right)} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mi>g</mi>
<mo>∘</mo>
<mi>f</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>y</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mi>g</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>y</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = g\left( {\left( {x + y,\,\,x - y} \right)} \right)">
<mo>=</mo>
<mi>g</mi>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mi>y</mi>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>−</mo>
<mi>y</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left( {\left( {x + y} \right)\left( {x - y} \right),\,\,x + y + x - y} \right)">
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mi>y</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mi>y</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>+</mo>
<mi>y</mi>
<mo>+</mo>
<mi>x</mi>
<mo>−</mo>
<mi>y</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left( {{x^2} - {y^2}{\text{,}}\,\,2x} \right)">
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mrow>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
</mrow>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>A1A1</em></strong></p>
<p> </p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>no because <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f \circ g \ne g \circ f">
<mi>f</mi>
<mo>∘</mo>
<mi>g</mi>
<mo>≠</mo>
<mi>g</mi>
<mo>∘</mo>
<mi>f</mi>
</math></span> <strong><em>R1</em></strong></p>
<p><strong>Note:</strong> Accept counter example.</p>
<p> </p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( {\left( {x{\text{,}}\,\,y} \right)} \right) = \left( {a{\text{,}}\,\,b} \right) \Rightarrow \left( {x + y,\,\,x - y} \right) = \left( {a{\text{,}}\,\,b} \right)">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>y</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>a</mi>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>b</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo stretchy="false">⇒</mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mi>y</mi>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>−</mo>
<mi>y</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>a</mi>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>b</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left\{ {\begin{array}{*{20}{c}} {x = \frac{{a + b}}{2}} \\ {y = \frac{{a - b}}{2}} \end{array}} \right.">
<mrow>
<mo>{</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mi>x</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mi>a</mi>
<mo>+</mo>
<mi>b</mi>
</mrow>
<mn>2</mn>
</mfrac>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mi>y</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mi>a</mi>
<mo>−</mo>
<mi>b</mi>
</mrow>
<mn>2</mn>
</mfrac>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo fence="true" stretchy="true" symmetric="true"></mo>
</mrow>
</math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f^{ - 1}}\left( {\left( {x{\text{,}}\,\,y} \right)} \right) = \left( {\frac{{x + y}}{2},\,\,\frac{{x - y}}{2}} \right)">
<mrow>
<msup>
<mi>f</mi>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>y</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mi>y</mi>
</mrow>
<mn>2</mn>
</mfrac>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mfrac>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mi>y</mi>
</mrow>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p><strong>This question asks you to explore cubic polynomials of the form</strong> <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>x</mi><mo>-</mo><mi>r</mi></mrow></mfenced><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>a</mi><mi>x</mi><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow></mfenced></math> <strong>for</strong> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math> <strong>and corresponding cubic equations with one real root and two complex roots of the form </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>z</mi><mo>-</mo><mi>r</mi><mo>)</mo><mo>(</mo><msup><mi>z</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>a</mi><mi>z</mi><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup><mo>)</mo><mo>=</mo><mn>0</mn></math> <strong>for</strong> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>∈</mo><mi mathvariant="normal">ℂ</mi></math>.</p>
<p> </p>
</div>
<div class="specification">
<p>In parts (a), (b) and (c), let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo> </mo><mi>a</mi><mo>=</mo><mn>4</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>=</mo><mn>1</mn></math>.</p>
<p>Consider the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>z</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><msup><mi>z</mi><mn>2</mn></msup><mo>-</mo><mn>8</mn><mi>z</mi><mo>+</mo><mn>17</mn></mrow></mfenced><mo>=</mo><mn>0</mn></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>∈</mo><mi mathvariant="normal">ℂ</mi></math>.</p>
</div>
<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><mfenced><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>8</mn><mi>x</mi><mo>+</mo><mn>17</mn></mrow></mfenced></math> for <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>Consider the function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfenced><mrow><mi>x</mi><mo>-</mo><mi>r</mi></mrow></mfenced><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>a</mi><mi>x</mi><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow></mfenced></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</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>b</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi><mo>,</mo><mo> </mo><mi>b</mi><mo>></mo><mn>0</mn></math>.</p>
</div>
<div class="specification">
<p>The equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>z</mi><mo>-</mo><mi>r</mi></mrow></mfenced><mfenced><mrow><msup><mi>z</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>a</mi><mi>z</mi><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow></mfenced><mo>=</mo><mn>0</mn></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>∈</mo><mi mathvariant="normal">ℂ</mi></math> has roots <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>±</mo><mi>b</mi><mtext>i</mtext></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</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>b</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi><mo>,</mo><mo> </mo><mi>b</mi><mo>></mo><mn>0</mn></math>.</p>
</div>
<div class="specification">
<p>On the Cartesian plane, the points <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>C</mtext><mn>1</mn></msub><mfenced><mrow><mi>a</mi><mo>,</mo><mo> </mo><msqrt><mi>g</mi><mo>'</mo><mfenced><mi>a</mi></mfenced></msqrt></mrow></mfenced></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>C</mtext><mn>2</mn></msub><mfenced><mrow><mi>a</mi><mo>,</mo><mo> </mo><mo>-</mo><msqrt><mi>g</mi><mo>'</mo><mfenced><mi>a</mi></mfenced></msqrt></mrow></mfenced></math> represent the real and imaginary parts of the complex roots of the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>z</mi><mo>-</mo><mi>r</mi></mrow></mfenced><mfenced><mrow><msup><mi>z</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>a</mi><mi>z</mi><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow></mfenced><mo>=</mo><mn>0</mn></math>.</p>
<p><br>The following diagram shows a particular curve of the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfenced><mrow><mi>x</mi><mo>-</mo><mi>r</mi></mrow></mfenced><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>a</mi><mi>x</mi><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><mn>16</mn></mrow></mfenced></math> and the tangent to the curve at the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext><mfenced><mrow><mi>a</mi><mo>,</mo><mo> </mo><mn>80</mn></mrow></mfenced></math>. The curve and the tangent both intersect the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis at the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>R</mtext><mfenced><mrow><mo>-</mo><mn>2</mn><mo>,</mo><mo> </mo><mn>0</mn></mrow></mfenced></math>. The points <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>C</mtext><mn>1</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>C</mtext><mn>2</mn></msub></math> are also shown.</p>
<p style="text-align: center;"><img 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KyWzt0kXh173KqmR17W2qIy511O9KXKPzsiJcTrXP+7T8UOzRs+RpuScjQjvbNaNDlTF7W+Qk3jr1P7y85y36m2vlbpP1/TxjMGaMaj9+iHq0bptuHzg99MWfy+9rIRs0aECQCIcr4Q0bNnkv7nf/Kda9Z4atasWafcVFkbzsbLVW9r/uAO7jHMBP1k8NM6ctqlDq+SMh06fn+uUPmrz2pqcbIeGnKdW0EoU9Hal6XeHdWmpgLCCSp0uOgppaW8pp/+5lfqlzZBz63NVpfCifq1s5nyDDVPuNwbeYr1wcffHWE99vH7eksX66qE890rCBRhAgCilH+I+Otf16hZs3N18cUtQzQm3N14+ehL2l3ykdOKvfJli+adcqnjLMU1r9rwyg0O/W5Vxzi7JVkgWKgp872XAt6keVQ7X1mp7ceXMSqPeo4de722b9vrnMw4o01H9W66WwVb97p7Mb7UB1s3q6zpjbq2zTnOlRMkXq5WzbhV1oT/hwAgyviHiA0bNqhv3393jnuOGfOA0yeiLtWISpXNnaY6Gy+r7mlorptT+in+yBJNmVZQpfnTOWpz7Y1qWva+Sj5zDmp6faGDn5RL+w/qcIU1spqjJ2cVao/aqPmhjwJsHuV+fG3SiqWbnP+2orRI//v6HrW7qlVl1SOus0Zm91Tx4r/o1dIvdbh4tfIW71Gv7KHq6oQZnwMq2bZPTa9urYu4U9aI/4sAIEr4hwibqTF+/Hjt3r1b+/fv1/Ll+br99h7ue9bNsaIZurlHpv565BU9ctu/K6fou24Mx4qm6/rk6SrTEW3Pu0edWo9SfqkvODRR3PV9NLzdP/Xmu74jlxfpp6n9nb0NP71qjFboZ0rueaGONN2nHd801Q/871IVO5TX1zpb3un9nNUd2fR+/G4P6oW5qdKT/b2f+xK17pStT27JUd6vb3CXUM5Uy76ZmtfrA43o1EbtbvuzdP+fNS054cQbYvl7erPwCg1P6VAPR1ijD1NDATiYGhq5LESsWbNGOTnTnVbXAwcOdHpHzJ49ywkUoQoRofG1Stf/QcPX3qwFk7oFdaOuKJ6vXxd20mNpbevpN2JrD/57dV17q14N8muMNVQmACBC2RFP61pplYgdO3Zozpw/66677tKoUSOdHhKrV68JsyBhzlSLbun6Y+IrmrH+Y3ffQi1VlKroxTw99r+XKmtwfQUJ+zQFmrG2rZZkdiVI1BKVCQAOKhORw4LCCy+8oOzsScrMzHIChDWays3N1apVK72vH9c111zjvne4smFb87RCPTQsuW1I53MEr0KHi1dq9grpjmG9lcjGy1ojTABwECbCny1dLFmyRIWFr2vMmAx17dpV8fHxzvX09NHq1au393V6YB0sgRAgTABwECbCk+2HsPAwa9Zs5/nw4cOcfREWGGyZY9q0adq1a5eysrIioBqBaEWYAOAgTIQXCwpWhVi8eJETHmwpwz8srFu3VhMmTNCwYcOVkpJCNQKNijABwEGYCA/WF2LFihVONWLAgIFOiLClDB//asSkSZOCmKcBhB5hAoCDMNF4LCC8+uqrztHOiy5qoQceeEDXXnvtSdUGqhEIV4QJAA7CRMOzKsRrr72mmTMfd05l3H777dVWGuz0xsSJE53HDz/8cAg6WAKhRZgA4CBMNAwLBgUFBU5DqXbtrlJKSr/jGyqrYx0t09NHKTf3iRDM0wDqB2ECgIMwUX/sRMbbb7+t+fPna/v2bc5eiL59+562wkA1ApGEMAHAQZgIPev/sHr1amcZY+TI0frZz36mzp07u2+tngWPZcuWOZUL6yVBNQKRgDABwEGYCA1fd0o70mmbKa0vRMeO151wIuNUiouLnX4Rbdq00YMPPlir/wYIB4QJAA7CRPB8pzEWLlzoPLcljO7du9d6acK/GhF+g7mAmhEmADgIE4GxAFFUtNkbAp4/vg/CljAC7UJJNQLRgDABwEGYqJn/RsqVK1909kH07NkzqDbW9rEiazAXcGqMRAOA07CbvvWDmDp1qtq2TXT6Qtx9993asaNYY8eODSoE2MZMGxtubEw4QQKRjsoEAAeVie/4KhC+hlK+kxjVdaUMBNUIRCsqEwDgVV0FwgKErwJh+yHqEiSsFbZVI+Li4qhGIOpQmQDgiMXKRH1VIPwxmAuxgMoEgJhS3xUIf1aNuPPOZLVr107PPPMMQQJRi8oEAEc0VyZ8xzhffrlACxbMq5cKhD+qEYg1hAkAjmgLE/59IHzHOOszQPgwmAuxiDABwBENYaKxAoRhMBdiGWECgCNSw4R/gLBOlL169W6wAOFDNQKxjjABwBFJYaJqgAi2lXVd+bfCHjlyJNUIxCzCBABHuIeJcAkQxk6EMJgL+A5hAoAjHMNEOAUIHwZzAScjTABwhEuYsN/6CwtfdwKE/yZKCxGNiWoEcGqECQCOxgwTDdGJsi5sMFd6+mhnc2d6enpYfE1AOCFMAHA0Rpiwm7R1o8zOnqTevfs40zjDJUAYCzkM5gJqRpgA4GioMGH9GAoKCpzlgosuaqHhw4epY8frwm7vAdUIoPYIEwAc9RkmfBspZ82arU8+KdWwYcPVvXv3sDxKaV+rrxW2bbSkGgHUjDABwFEfYcKWMPz3QfTs2TOsb842mGvChAlO2ElJSaEaAdQSYQKAI1Rhwn6zX7VqlbOM0a7dVd6bcj916XJzWN+Y/asRDOYCAkeYAOCoa5iwKsSKFSucY53WD6Jv374R0RGSagRQd4QJAI5gwkTVKkS4ncY4HQZzAaHTxH0NALVmXSCnTp2qDh3a69ChQ1q4cJGefPJJp7FUJAQJG8x10003Kikpyfm6CRJA3VCZAOCoqTLhayw1Y8YM5/mgQYOcm3EkLQtQjQDqB2ECgONUYcJCxJo1a5STM93ZSHnHHXc0emvrQPm3wh4zJoMx4UCIscwBoFq2H8KWANq2TdS+ffs0Z86fNXny5IgLErYkc88992j79u1avjyfIAHUAyoTABy+yoTdfNetW6fFixc5pzLuuuuuiJyM6V+NYDAXUL8IEwAcFiZSUwc7RzttKSDS9kP4Y0w40LAIE0CMsxvv3LlztWDBPOXmPhHRIYLBXEDjYM8EEKPsZMO4ceM0dOgQderUyblm+wkiNUjYYK6ePZOcx6tXryFIAA2IygQQY2xj5ZIlS5w9Ef7LGcE0rQoHVCOAxkdlAogRdtNdsGCB02jq3HPPdX57j+RKhLFW2FaNiIuLoxoBNCLCBBADrOOj3XStW+WWLVuVmpoa0SHCqiu2RGMjze3I6ogRIyL6zwNEOsIEEMVsH0H//v21adOm4zfdSD/ZYNWIO+9MVrt27fTMM88w4RMIA+yZAKKQb6S2HfPMzp5Sq0ZT4b5nwvdnYkw4EH6oTABRxpY0bF+E/eZu+wgirWNldXx/Jjt1snTpUoIEEGYIE0CUsH4RviWNN974W8TvizB2fNWWZmw2iP2ZaIUNhCfCBBDhfKc0rF/E8OHDnPkZ0TANkzHhQOQgTAARrOoQq2iYP1G1wkI1Agh/bMAEIlB9DLFq7A2Y9fFnAtAwqEwAEcb2EURjNSLa/kxALKEyAUQQ67Hwy1/eq6efnhvyG25jVCaoRgDRgcoEEAHspmsdH5cte97ZRxANN13fYC6rtNgRVoIEELmoTABhzm62gwYN1IABA5WWllZvxz0bqjJhwYjBXEB0oTIBhDFb1rDjkdbFMhrmTzAmHIhOVCaAMGS/vefl5amgoEA5OTkN0mOhPisT/q2ws7KyCBFAlKEyAYQZu/FmZGToww8/dAZZRXqzpqqDuQgSQPShMgGEEd/+iGHDhjvtsBtSqCsTDOYCYgeVCSBMbNiw4fj+iIYOEqHGmHAgtlCZAMKAzdawXgsLFy5qtGWNUFQmrLIyceJE5/HDDz/MPA0gRlCZABqZDbEqLCxs1CARCgzmAmIXlQmgkdiJDdtoaaZPn97oxz6DrUxQjQBAZQJoBL4g0b59e+e3+EjsH2F/BluesQ2jKSn9qEYAMYwwATQw+03eGjdZkLBGVJGIwVwA/LHMATQg39HPcBxqVZtlDqtGMJgLQFVUJoAG4gsSdvQzEm/CVCMAnAqVCaABWA+JzMyHnCDRuXNn92p4OVVlwqoRDOYCcDpUJoB6ZkFiwID+YR0kToXBXABqg8oEUI98FYlI6CHhX5mgGgEgEFQmgHoSSUHCn7XCtmqEfc1UIwDUBpUJoB5EYpCwykRq6mAGcwEIGJUJIMQiMUhYNcIwmAtAMKhMACEUaUHCjqvOnDnTqUZs3LihzoO+AMQmKhNAiERakPAN5urUqZOWLl3qXgWAwFGZAELAjlD26dNbL764Muw3LJ5qMFewg74AgMoEUEd2c05PH63Fi5eGfZBgTDiA+kCYAOrAv0V2ODekslbY/fv316ZNm7Rly1YlJye7bwGAuiNMAEGKhCDhGxM+dOgQDR8+TJMnT1Z8fLz7VgAIDcIEEAS7SY8ZM8b7khG2QYLBXAAaChswgQBZkMjIyFD79u01YsQI92r4sK8vLy9PixcvCmhMOBswAQSLygQQoHAOEr7BXOXl5U4rbKoRABoCYQIIgJ2AMGlpac7rcGHViKlTpzqnSmww19ixY3X22We7bwWA+kWYAGrJgsTWrVs1ffr0sLpRMyYcQGNjzwRQC77ulraRMVxOQ5SVlWnatGkhG8zFngkAwaIyAdTAv012uAQJG8x1553JDOYCEBaoTACn4eslYfsQwmH5INTVCH9UJgAEi8oEcAq2qdHXlCocgoS1wqYaASAcUZkAqhFOvSRONZgr1KhMAAgWlQmgGrm5uTr//PMbPUgwmAtAJCBMAFXYLIvdu3c73SMbi1UjLMisWbNGb7zxNwZzAQhrhAnAj53cmD17lrOc0Bi9JGx5xcKM7dVISelHNQJARCBMAC6rBtgR0Dlz/twoN3AGcwGIVGzABLysImA3chvT3dA3cfvcy5YtcyoigQzmCjU2YAIIFpUJwGvChAnq3r17g9/IrRU21QgAkY7KBGKe7VEoLCxs0JkbVo2wEyOrVq0Mm4ZYVCYABIvKBGKab8OldZNsqCDBYC4A0YbKBGKWr1W2bbhsiG6S4ViN8EdlAkCwqEwgJtmNfcyYMd6XjAYJEjaYy6oRdkqEagSAaENlAjFp6tSpzuuxY8c6r+tLfQ7mCjUqEwCCRWUCMceqBJs3b1Z6erp7pX4wJhxArKAygZhijaFuu62706K6vhpT2V6MmTNnRkQ1wh+VCdRVRWmRVq5eppmPPKPtdqFpksZk3687Wm1W3t7empR8qfN+iD5UJhAzbJ/E0KFDtHjx0noLEr7BXJ06ddLSpUupRiBGVOjwjvka3r2vfvf6RfrtS9udYFqybZJuOZSv0Sn/rU/c90R0ojKBmDFu3Dhdeuml9TIJtKHGhNcnKhMIWvl6Zd2Yqvk//J3ynxuljs38f0/9UsV59+s3ylR+Wlt+g41SfF8RE6xicODAAaWlpblXQocx4Yht3rDw/GzNP9JUHQYkqcMJQcKcpcR+abq+/Igq3CuIPoQJRD2rGuTkTA/5JFDbf9G/f39t2rRJW7ZsZUw4YtS/tO31v3tft1b39q2qv6nEddMj6R11hvsU0Ycwgajm6ydhA7RCVTGwj2ktuG3/hQ0Gmzx5suLj4923AjHm2L/0wVtl3geJuqzlWZXXEHMIE4hq1nHyuuuuC9kALcaEA1U0aarzE5p6HxzQwUPHKq8h5hAmELVC2U/CqhG2H4JqBFBFkwvU+uoLvQ/2qHjv4cpriDmEiYhyTKX5o5xd9ye/9FFW3noVH65hi9PhYr3k/W19/o5y90JNKrz/yVrlDO303efJ36Ggf2Qc3qH8rD6VH6vnI8ov9n0d5doxP1cz3ywNySYt2ydhY8VDMcDLN5irvLzcaYVNNQLw11xd04aqg3Zr/pSFKqruZ9DhIuXlFQX/cwPhz46GIsJ8XuAZd2UrzzXTN3u+seeH3vUsH3eH59JLf+RJmr7Jc8h5p2oc+odn7rj/59l86Fv3Qi0c2uSZPiTbU7DvK4/n272egvH2eW7w3Ld8jyeAj1Lp2z2e5ffd4LnyvuWevd9+5dlXMMGTdGW6Z/le78d22LU/ecbN/cep/wy1dP/993uWL1/uPgvO0aNHPVOmTPF07fozz5YtW9yr0evSS1u5j4BAfeXZuzzdc6X379CVQ/7kWb55n/vzwftvevMCz7ghjwf2cwcRh8pEJDonTs2/7z42zdoqeexvdHfTI9r+zGvaWd2yZUWJ8h98SMU9+lU5A346FSp/a6uaj31A3VqcKTVpqW4P2uf5WKtWFOlT971qp0KHt+TrqVWJGvubnmrZ5Ey16PoLDUhcrcyZG1RZn/Be6zZYPYqna3bRQedKMGxzpKnL6QrGhAOBOFMtk/+ogvxZGntRgdKTr1drp5KZose3tlLajF8d/7lzrGi6Otjbhuar1LmCaECYiDYJ8Tr3pO+qeyN/5Vr16BjIOn8TxXUborsT/XZox12hG7oEs1fgqHa+slLb469T+8vcj9ekldp3b60jK4u063gAilfHHq00a9y86sulNbANkrNnz3KWN4Jhg7msuZX99zaa3AaBhfI4KRC9vL8MdOyltEkvVna/dF5e1KS0bkr0+wXmjI4Zeis/w/svHdGEMBEFKkrf1Pypf9L8I9cqLePnuuKk72qZ3lq2VHv63aqOcXX8llcc1cFPpQ43/1g/cC/VzgGVbNtXJeycpZaXJXq/vPdV8pkvTXgDzJUd1WX7Ui1zjpvVnq9ddnb2lKA2RzKYCwCCQ5iIYGU5fXV5wiVq3SlVzypF8156Vo92a3nyN/VYid5euVvfbx6nc9xLwap47w09//VQ/f4/EoP7y/OD86qpnFTRPEFXxf9Lf9+9P6DNmHYMtFev3urcubN7pXZ81YhZs2Y71YjU1FSqEQBiU2m+hjpLVJ10f36JKipK9WbuEP04ob2G5n/ovtPJCBMRLH7MC3p/+3KNaSdt31yuuIubuW+p4mi5PvuqqRLOb+r3Df9Q+UPbuyc0qn/pkFOkE7ZfVJTohdx/qN9jaQHsuwjWERUX76v17m/b47Bq1cqAj4FaK2yqEQDgapGsOSX/UP6YVlr19ELNe2yOtiXl6J8lWzXnNFNfCRORrtkN+vXTU9Vrz0yNm735NDffIyo54N8b/1Ilz9nqt7Z58suWMf7tbw+q6LE/qSglQ4PbxrnXAnG+Eq66WHrrfX18PKF8qY8/KPamosuV0Dz4RrtWWUhPH+1UFWpbUbCjozbwa82aNVq4cBHVCAA47jx1uOPf1WHL89p42eBa/cwnTESBJi176ZHsntqT86j+sP7jk5cGqp7+CNjX+jj/Ka24dqx+X90ySq2cozbX3qimZZu19YMvKy9V7NXWgt1q2ruj2pyUJZoqMfFinaLWcoJp06Zp2LDhta4qMJgLaESl+bo/ebrK/jpKPatWPxE2mlzWXt3jz9QF551Tq5/5hIlI5CxbSF99Vq6jzoUz1bLvb5Td6xPlDb5P4+e/qVL/RHFGgq7t3Vpl20r0mXup9r5W6fonNPNgH2UcDxJ2LVc56wP5aE0U13Wo92vco8XzC71fX7mKX3hWi4t7KntkZ52Qez8r0bayC3V16wtq/AtqmyZ37dqllJQU98qp+Vcj3njjbwzmAhqDU0avrvqJ8OH9BXLFKu24vqmeX/sP9+h+Ddx+E4gI33j2LR/pNBf67mWkZ/k+p3WV59u9yz33Xem7/nPP9M2+1k/feg5tfsyTdOV4T8HngTSO+dyzc/l4T9IJn8996TPXs9P/Q337rmdunx9535bs/bwH3Isn+3ZfoeexITdUfoyk8Z7lOz933+LzrefzgvGeK5Meq7HJzf79+52GUjt37nSvVM+aT82fP99537Vr/+peRVX2PQGqY//WrAmcvdhjRDPv/eLdBZ5xj2307LWfxdf8ybP5qz2eF+YVeu8Ip0aYiBVO98p7POMK/uVeqB/f7pznGTX33cC7Yx73L0/BuORafZ2ZmZlOSDgdCxq/+MUvnPflh+DpESZQnY8++sgJ4r5fJOyxXUP0+XbnXE8f66Q8brlnp/0y53ZbvnLIk55N1gX5NL5n/+MWKRDtDm9TXvYatR49qrKjZShVlKpo5Wq98kFrDRzVTS2CWkCrXFJ5fHeSMtOuOu1+CVvesKOcdgKjuo2T1nNi2bJlTgMrGz/OPI2a2QkeKz0D/mxfUXb2iU3gJk+e4mxaBnzYMxFLml2ltMzeOrholt+ArRBp0kId+6RpTHqwQaJcxfmztOhg7xqDhJ3eON0QLzsmyphwIDRswF1Vhw4dch8BlahMIOJYgynrC1H1NyOrRljjKus3kZv7OPM0AkRlAtWxcN6nT2/3WaUXX1zJvy+cgMoEIsqpTm8wmAuoH/ZvycJDaupg54UggepQmUDEsOUN61Zpzal8PSWoRoQOlQkAwaIygYhRtTmVVSmsGmFNp6hGAEDt2C9h9stZKBEmEBE2bNhwfHmDwVwAELydO3eqQ4f2TjfgUGGZA2HLKg8vv1ygY8eOaf36Aj377ELt2bPbOclhFQoLFoSI0GGZA4gdxcXFysrKch7bybi6DjkkTCAsWSViwID+7rNKd9zRR599tj8kf/FxMsIEEHusOpGePsrpHVKXX9COhwn7QQIAAGKXzS0KZvghlQmEpalTp2rmzMfdZ5Vyc59gOFc9ojIBxCYbgjhx4kTt379fOTk5QYUJNmAiLF144YXuo0qXXXa5unbt6j4DANSVnepYsGCBbrrpRnXp0sUZTxBMkDCECYQV2xTUv39/bdmyxXlu63hPPz3XaYsdHx/vXAMA1I39rLWxA4WFhc7SRl1PxREmEBZ8CXno0CEaPnyYvvnmG2dZw/6C22wNggQAhIZtcLeftYMGDXIGuQVbjfBHmECj8yVk32AuY2t37I8AgNC79tprnZ+1ofwZywZMNBqrRuTl5Wnx4kXHx4TbNetq6d8yGw2DDZgAgkVlAo3CN5jLxhtbK2zfmHCbszFgwECCBBBGbAnSwqa9WPdZC/2APyoTaFD2Q+hUg7lsucPW8Sxc0Nmy4VGZQHWqayDHMW1URWUCDaamMeHW2tWWOwgSQPj49NNP3Uff2bRpk/sIqESYQL3zDeayNti2F2Ls2LEnBQZr6XrBBRccX+4AEB6aNWvqPvpOu3bt3EdAJcIE6pUN67rzzmTnh481RKluL4SFjZyc6crIyHCvAAgXXbrcrNTUwe4zqXfvPs4MB8AfeyZQLywgTJs2zRkbXtNgLjvnbEaMGOG8RuNgzwROx1ouHz16lM3RqBaVCYScLVnUVI3wsU2XdjQ0LS3NvQIgHFljI4IEToXKBELGNyzGPPzww7XqqmbViJSUfuyVCANUJgAEi8oEQsKqETYsJikpqdbtWW0/hXW6JEgAQGSjMoE6CaYaYeh0GX6oTAAIFpUJBMXCgHXFGzRooLNMEeiwmGXLlqlXr94ECQCIAoQJBKzqYK5Alykq+048pGHDhrlXAESDitIivZiXpZ4Jla23E348RDn5b6r4zaeUlf+h+16IRoQJ1JqvGuEbEz558uSgRoPbkVFrx8tYcSBaVOjwjvka3r2vfvf6RfrtS9udJbOSbZN0y6F8jU75b33ivieiE2ECtWKtsOtSjfCxqkZh4evORk0AUaL8VWUnZ+qvP/ydnp0xSrclxlVeb9JCHe8er8cf7ahPDx71Rg5EK8IETsuqEVOnTlV6+mhndkaw1Qgf+xjZ2VOYvwFEjS9V/PxszT/SVB0GJKlDs6q3lbOU2C9N15cfIUxEMcIETqmmwVyBsumDpnPnzs5rANHgX9r2+t+9r1ure/tW1d9U4rrpkfSOOsN9iuhDmMBJ/KsRNia8usFcgbKPmZn5kNNaG0AUOfYvffBWmfdBoi5reVblNcQcwgROYI2krBphxzxDUY3wWbNmjTMwiKOgQJRp0lTnJ9hk0QM6eOhY5TXEHMIEHL4x4bNmzXYaSaWmpoZsX4NvKujIkSPdKwCiRpML1PrqC70P9qh47+HKa4g5hAnUakx4XSxZskQDBgwMqKkVgEjRXF3ThqqDdmv+lIUqOlzNNsvDRcrLKxJRI3oRJmKYtcKur2qEj1UlsrMn6a677nKvAIg2TRIH6ancFDXd/gf95wMzlF9U6p7c+FqlRc8q64ENav8fHdTMnq+fWNnU6sdDlPum7/0Q6QgTUc42PtqpjKp8g7k6deqkpUuX1tteBhpUAbHgTLVM/qMK8mdp7EUFSk++Xq2dLpgpenxrK6XN+JU62pHR8o1a9OpNytu9XWvH/h89OWG13iNNRAUGfUU5Cw22X8EaTdkNPdjBXMGwBlXWLdM2ctJXIvwx6AsNqjRfQ4cf1Nj8NCXya23E41sYxawqYUHigw/edyoEwYwJr4u5c+dqzJgMggSAKipU/u67OvuXt+oK7kJRgW9jFLPjmBYkzIIF8/Tcc3/Rli1blZyc7FyrT1aV2LVrV4N8LgCRpaL0VeVt/amy+iZwE4oSfB+jlK8q4e/DDxtuap+1zX7ggQfcZwDgOrxNzy79Uv8xqpt+8Ok2bSn92n0DIhlhIkr5VyV87Pns2bPdZ/WHttkAqlNRukG5D9yrrGlD9dPWl6h19yU6eA5NtqMBGzCjlLXD/vzzz3XppZfq4osvdq9Wqu+lh/79+ztVCcJEZGEDJoBgESYQUlaVmDFjhnPcFJGFMAEgWCxzIKQsSNh+CQBA7CBMIGSsKtGmTZuQDQcDAEQGwgRCxqoS9957r/sMABArCBMICV9VghHjABB7CBMICaoSABC7CBOoM6oSABDbCBOoM6oSABDbCBOoE1+3S6oSABC7CBOoE6tKMIMDAGIbYQJB81UlaJsNALGNMIGgrVixgqoEAIAwgeAUFxdr165dVCUAAIQJBGfu3LkaNGiQ+wwAEMsIEwhYWVmZCgtfV1JSknsFABDLCBMI2JIlSzRs2HCdffbZ7hUAQCz7nsfLfQzU6IsvvlDbtonasmWr4uPj3auIBgkJl6ik5CP3GQDUHpUJBMSWN0aOHE2QAAAcR5hAQGbNmq2ePXu6zwAAIEwgAHYc1FxzzTXOawAADGECtbZu3TqOgwIATkKYQK0tXrxIXbt2dZ8BAFCJMIFasd4S48ePZ+MlAOAkHA0F4OBoKIBgUZkAAAB1QpgAAAB1QpgAAAB1QpgAAAB1QpgAAAB1IP1/YnvqxiD2k50AAAAASUVORK5CYII="></p>
</div>
<div class="specification">
<p>Consider the curve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>(</mo><mi>x</mi><mo>-</mo><mi>r</mi><mo>)</mo><mo>(</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>a</mi><mi>x</mi><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup><mo>)</mo></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>≠</mo><mi>r</mi><mo>,</mo><mo> </mo><mi>b</mi><mo>></mo><mn>0</mn></math>. The points <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext><mo>(</mo><mi>a</mi><mo>,</mo><mo> </mo><mi>g</mi><mo>(</mo><mi>a</mi><mo>)</mo><mo>)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>R</mtext><mo>(</mo><mi>r</mi><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math> are as defined in part (d)(ii). The curve has a point of inflexion at point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math>.</p>
</div>
<div class="specification">
<p>Consider the special case where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mi>r</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>></mo><mn>0</mn></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>+</mo><mtext>i</mtext></math> are roots of the equation, write down the third root.</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>Verify that the mean of the two complex roots is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math>.</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>Show that the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>x</mi><mo>-</mo><mn>1</mn></math> is 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> at the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext><mfenced><mrow><mn>4</mn><mo>,</mo><mo> </mo><mn>3</mn></mrow></mfenced></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>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> and the tangent to the curve at point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math>, clearly showing where the tangent crosses the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis.</p>
<div class="marks">[2]</div>
<div class="question_part_label">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>g</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>2</mn><mfenced><mrow><mi>x</mi><mo>-</mo><mi>r</mi></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>-</mo><mi>a</mi></mrow></mfenced><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>a</mi><mi>x</mi><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, or otherwise, prove that the tangent to the curve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>g</mi><mfenced><mi>x</mi></mfenced></math> at the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext><mfenced><mrow><mi>a</mi><mo>,</mo><mo> </mo><mi>g</mi><mfenced><mi>a</mi></mfenced></mrow></mfenced></math> intersects the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis at the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>R</mtext><mfenced><mrow><mi>r</mi><mo>,</mo><mo> </mo><mn>0</mn></mrow></mfenced></math>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">d.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Deduce from part (d)(i) that the complex roots of the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>z</mi><mo>-</mo><mi>r</mi></mrow></mfenced><mfenced><mrow><msup><mi>z</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>a</mi><mi>z</mi><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow></mfenced><mo>=</mo><mn>0</mn></math> can be expressed as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>±</mo><mtext>i</mtext><msqrt><mi>g</mi><mo>'</mo><mfenced><mi>a</mi></mfenced></msqrt></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use this diagram to determine the roots of the corresponding equation of the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>z</mi><mo>-</mo><mi>r</mi></mrow></mfenced><mfenced><mrow><msup><mi>z</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>a</mi><mi>z</mi><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><mn>16</mn></mrow></mfenced><mo>=</mo><mn>0</mn></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>∈</mo><mi mathvariant="normal">ℂ</mi></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">f.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State the coordinates of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>C</mtext><mn>2</mn></msub></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">f.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-coordinate of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>3</mn></mfrac><mfenced><mrow><mn>2</mn><mi>a</mi><mo>+</mo><mi>r</mi></mrow></mfenced></math>.</p>
<p>You are <strong>not</strong> required to demonstrate a change in concavity.</p>
<div class="marks">[2]</div>
<div class="question_part_label">g.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence describe numerically the horizontal position of point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math> relative to the horizontal positions of the points <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>R</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">g.ii.</div>
</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><mfenced><mrow><mi>x</mi><mo>-</mo><mi>r</mi></mrow></mfenced><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>a</mi><mi>x</mi><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow></mfenced></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mi>r</mi><mo>=</mo><mn>1</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>=</mo><mn>2</mn></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">h.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>For <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mi>r</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>></mo><mn>0</mn></math>, state in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math>, the coordinates of points <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">h.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>-</mo><mtext>i</mtext></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>mean<math xmlns="http://www.w3.org/1998/Math/MathML"><mo> </mo><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mfenced><mrow><mn>4</mn><mo>+</mo><mtext>i</mtext><mo>+</mo><mn>4</mn><mo>-</mo><mtext>i</mtext></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>4</mn></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>attempts product rule differentiation <em><strong>(M1)</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for attempting to express <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced></math> as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mn>9</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>25</mn><mi>x</mi><mo>-</mo><mn>17</mn></math></p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mfenced><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mn>2</mn><mi>x</mi><mo>-</mo><mn>8</mn></mrow></mfenced><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>8</mn><mi>x</mi><mo>+</mo><mn>17</mn><mo> </mo><mo> </mo><mfenced><mrow><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>18</mn><mi>x</mi><mo>+</mo><mn>25</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mn>4</mn></mfenced><mo>=</mo><mn>1</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced></math> is correct, award <em><strong>A1</strong></em> for solving <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>1</mn></math> and obtaining <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>4</mn></math>.</p>
<p><strong><br>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>-</mo><mn>3</mn><mo>=</mo><mn>1</mn><mfenced><mrow><mi>x</mi><mo>-</mo><mn>4</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><strong><br>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>x</mi><mo>+</mo><mi>c</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>=</mo><mn>4</mn><mo>+</mo><mi>c</mi><mo>⇒</mo><mi>c</mi><mo>=</mo><mo>-</mo><mn>1</mn></math> <em><strong>A1</strong></em></p>
<p><strong><br>OR</strong></p>
<p>states the gradient of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>x</mi><mo>-</mo><mn>1</mn></math> is also <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math> and verifies that <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>4</mn><mo>,</mo><mo> </mo><mn>3</mn></mrow></mfenced></math> lies on the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>x</mi><mo>-</mo><mn>1</mn></math> <em><strong>A1</strong></em></p>
<p><strong><br>THEN</strong></p>
<p>so <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>x</mi><mo>-</mo><mn>1</mn></math> is the tangent to the curve at <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext><mfenced><mrow><mn>4</mn><mo>,</mo><mo> </mo><mn>3</mn></mrow></mfenced></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award a maximum of <em><strong>(M0)A0A1A1</strong></em> to a candidate who does not attempt to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced></math>.</p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>sets <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>x</mi><mo>-</mo><mn>1</mn></math> to form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>-</mo><mn>1</mn><mo>=</mo><mfenced><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>8</mn><mi>x</mi><mo>+</mo><mn>17</mn></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><strong><br>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>8</mn><mi>x</mi><mo>+</mo><mn>16</mn></mrow></mfenced><mo>=</mo><mn>0</mn><mo> </mo><mo> </mo><mfenced><mrow><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mn>9</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>24</mn><mi>x</mi><mo>-</mo><mn>16</mn><mo>=</mo><mn>0</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p>attempts to solve a correct cubic equation <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfenced><msup><mfenced><mrow><mi>x</mi><mo>-</mo><mn>4</mn></mrow></mfenced><mn>2</mn></msup><mo>=</mo><mn>0</mn><mo>⇒</mo><mi>x</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo> </mo><mn>4</mn></math></p>
<p><strong><br>OR</strong></p>
<p>recognises that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>≠</mo><mn>1</mn></math> and forms <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>8</mn><mi>x</mi><mo>+</mo><mn>17</mn><mo>=</mo><mn>1</mn><mo> </mo><mo> </mo><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>8</mn><mi>x</mi><mo>+</mo><mn>16</mn><mo>=</mo><mn>0</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p>attempts to solve a correct quadratic equation <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>x</mi><mo>-</mo><mn>4</mn></mrow></mfenced><mn>2</mn></msup><mo>=</mo><mn>0</mn><mo>⇒</mo><mi>x</mi><mo>=</mo><mn>4</mn></math></p>
<p><strong><br>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>4</mn></math> is a double root <em><strong>R1</strong></em></p>
<p>so <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>x</mi><mo>-</mo><mn>1</mn></math> is the tangent to the curve at <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext><mfenced><mrow><mn>4</mn><mo>,</mo><mo> </mo><mn>3</mn></mrow></mfenced></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Candidates using this method are not required to verify that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>3</mn></math>.</p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="padding-left:60px;"><img src="data:image/png;base64,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"></p>
<p>a positive cubic with an <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-intercept <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>x</mi><mo>=</mo><mn>1</mn></mrow></mfenced></math>, and a local maximum and local minimum in the first quadrant both positioned to the left of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> As the local minimum and point A are very close to each other, condone graphs that seem to show these points coinciding.<br>For the point of tangency, accept labels such as <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext><mo>,</mo><mo> </mo><mfenced><mrow><mn>4</mn><mo>,</mo><mn>3</mn></mrow></mfenced></math> or the point labelled from both axes. Coordinates are not required.</p>
<p> </p>
<p>a correct sketch of the tangent passing through <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> and crossing the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis at the same point <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>x</mi><mo>=</mo><mn>1</mn></mrow></mfenced></math> as the curve <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A1A0</strong></em> if both graphs cross the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis at distinctly different points.</p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mfenced><mrow><mi>x</mi><mo>-</mo><mi>r</mi></mrow></mfenced><mfenced><mrow><mn>2</mn><mi>x</mi><mo>-</mo><mn>2</mn><mi>a</mi></mrow></mfenced><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>a</mi><mi>x</mi><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></math> <em><strong>(M1)A1</strong></em></p>
<p><br><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mfenced><mrow><mn>2</mn><mi>a</mi><mo>+</mo><mi>r</mi></mrow></mfenced><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mfenced><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>a</mi><mi>r</mi></mrow></mfenced><mi>x</mi><mo>-</mo><mfenced><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow></mfenced><mi>r</mi></math></p>
<p>attempts to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>'</mo><mfenced><mi>x</mi></mfenced></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mfenced><mrow><mn>2</mn><mi>a</mi><mo>+</mo><mi>r</mi></mrow></mfenced><mi>x</mi><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>a</mi><mi>r</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mfenced><mrow><mi>a</mi><mo>+</mo><mi>r</mi></mrow></mfenced><mi>x</mi><mo>+</mo><mn>2</mn><mi>a</mi><mi>r</mi><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>a</mi><mi>x</mi><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>=</mo><mn>2</mn><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mi>a</mi><mi>x</mi><mo>-</mo><mi>r</mi><mi>x</mi><mo>+</mo><mi>a</mi><mi>r</mi></mrow></mfenced><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>a</mi><mi>x</mi><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow></mfenced></math></p>
<p><br><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>2</mn><mfenced><mrow><mi>x</mi><mo>-</mo><mi>r</mi></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>-</mo><mi>a</mi></mrow></mfenced><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>a</mi><mi>x</mi><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>a</mi></mfenced><mo>=</mo><msup><mi>b</mi><mn>2</mn></msup><mfenced><mrow><mi>a</mi><mo>-</mo><mi>r</mi></mrow></mfenced></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>'</mo><mfenced><mi>a</mi></mfenced><mo>=</mo><msup><mi>b</mi><mn>2</mn></msup></math> <em><strong>(A1)</strong></em></p>
<p>attempts to substitute their <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>a</mi></mfenced></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>'</mo><mfenced><mi>a</mi></mfenced></math> into <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>-</mo><mi>g</mi><mfenced><mi>a</mi></mfenced><mo>=</mo><mi>g</mi><mo>'</mo><mfenced><mi>a</mi></mfenced><mfenced><mrow><mi>x</mi><mo>-</mo><mi>a</mi></mrow></mfenced></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>-</mo><msup><mi>b</mi><mn>2</mn></msup><mfenced><mrow><mi>a</mi><mo>-</mo><mi>r</mi></mrow></mfenced><mo>=</mo><msup><mi>b</mi><mn>2</mn></msup><mfenced><mrow><mi>x</mi><mo>-</mo><mi>a</mi></mrow></mfenced></math></p>
<p><br><strong>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><msup><mi>b</mi><mn>2</mn></msup><mfenced><mrow><mi>x</mi><mo>-</mo><mi>r</mi></mrow></mfenced><mo> </mo><mfenced><mrow><mi>y</mi><mo>=</mo><msup><mi>b</mi><mn>2</mn></msup><mi>x</mi><mo>-</mo><msup><mi>b</mi><mn>2</mn></msup><mi>r</mi></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p>sets <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>0</mn></math> so <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>b</mi><mn>2</mn></msup><mfenced><mrow><mi>x</mi><mo>-</mo><mi>r</mi></mrow></mfenced><mo>=</mo><mn>0</mn></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>></mo><mn>0</mn><mo>⇒</mo><mi>x</mi><mo>=</mo><mi>r</mi></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>≠</mo><mn>0</mn><mo>⇒</mo><mi>x</mi><mo>=</mo><mi>r</mi></math> <em><strong>R1</strong></em></p>
<p><br><strong>OR </strong></p>
<p>sets <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>0</mn></math> so <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><msup><mi>b</mi><mn>2</mn></msup><mfenced><mrow><mi>a</mi><mo>-</mo><mi>r</mi></mrow></mfenced><mo>=</mo><msup><mi>b</mi><mn>2</mn></msup><mfenced><mrow><mi>x</mi><mo>-</mo><mi>a</mi></mrow></mfenced></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>></mo><mn>0</mn></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>≠</mo><mn>0</mn><mo>⇒</mo><mo>-</mo><mfenced><mrow><mi>a</mi><mo>-</mo><mi>r</mi></mrow></mfenced><mo>=</mo><mi>x</mi><mo>-</mo><mi>a</mi></math> <em><strong>R1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mi>r</mi></math> <em><strong>A1</strong></em><br><strong><br>THEN</strong></p>
<p>so the tangent intersects the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis at the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>R</mtext><mfenced><mrow><mi>r</mi><mo>,</mo><mo> </mo><mn>0</mn></mrow></mfenced></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>'</mo><mfenced><mi>a</mi></mfenced><mo>=</mo><msup><mi>b</mi><mn>2</mn></msup></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>a</mi></mfenced><mo>=</mo><msup><mi>b</mi><mn>2</mn></msup><mfenced><mrow><mi>a</mi><mo>-</mo><mi>r</mi></mrow></mfenced></math> <em><strong>(A1)</strong></em></p>
<p>attempts to substitute their <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>a</mi></mfenced></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>'</mo><mfenced><mi>a</mi></mfenced></math> into <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>g</mi><mo>'</mo><mfenced><mi>a</mi></mfenced><mi>x</mi><mo>+</mo><mi>c</mi></math> and attempts to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>=</mo><mo>-</mo><msup><mi>b</mi><mn>2</mn></msup><mi>r</mi></math></p>
<p><br><strong>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><msup><mi>b</mi><mn>2</mn></msup><mfenced><mrow><mi>x</mi><mo>-</mo><mi>r</mi></mrow></mfenced><mo> </mo><mfenced><mrow><mi>y</mi><mo>=</mo><msup><mi>b</mi><mn>2</mn></msup><mi>x</mi><mo>-</mo><msup><mi>b</mi><mn>2</mn></msup><mi>r</mi></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p>sets <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>0</mn></math> so <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>b</mi><mn>2</mn></msup><mfenced><mrow><mi>x</mi><mo>-</mo><mi>r</mi></mrow></mfenced><mo>=</mo><mn>0</mn></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>></mo><mn>0</mn><mo>⇒</mo><mi>x</mi><mo>=</mo><mi>r</mi></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>≠</mo><mn>0</mn><mo>⇒</mo><mi>x</mi><mo>=</mo><mi>r</mi></math> <em><strong>R1</strong></em></p>
<p><br><strong>OR</strong></p>
<p>sets <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>0</mn></math> so <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>b</mi><mn>2</mn></msup><mfenced><mrow><mi>x</mi><mo>-</mo><mi>r</mi></mrow></mfenced><mo>=</mo><mn>0</mn></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>></mo><mn>0</mn></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>≠</mo><mn>0</mn><mo>⇒</mo><mi>x</mi><mo>-</mo><mi>r</mi><mo>=</mo><mn>0</mn></math> <em><strong>R1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mi>r</mi></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>METHOD 3</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>'</mo><mfenced><mi>a</mi></mfenced><mo>=</mo><msup><mi>b</mi><mn>2</mn></msup></math> <em><strong>(A1)</strong></em></p>
<p>the line through <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mfenced><mrow><mi>r</mi><mo>,</mo><mo> </mo><mn>0</mn></mrow></mfenced></math> parallel to the tangent at <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> has equation<br><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><msup><mi>b</mi><mn>2</mn></msup><mfenced><mrow><mi>x</mi><mo>-</mo><mi>r</mi></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p>sets <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><msup><mi>b</mi><mn>2</mn></msup><mfenced><mrow><mi>x</mi><mo>-</mo><mi>r</mi></mrow></mfenced></math> to form <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>b</mi><mn>2</mn></msup><mfenced><mrow><mi>x</mi><mo>-</mo><mi>r</mi></mrow></mfenced><mo>=</mo><mfenced><mrow><mi>x</mi><mo>-</mo><mi>r</mi></mrow></mfenced><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>a</mi><mi>x</mi><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow></mfenced></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>b</mi><mn>2</mn></msup><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>a</mi><mi>x</mi><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup><mo>,</mo><mo> </mo><mfenced><mrow><mi>x</mi><mo>≠</mo><mi>r</mi></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>x</mi><mo>-</mo><mi>a</mi></mrow></mfenced><mn>2</mn></msup><mo>=</mo><mn>0</mn></math> <em><strong>A1</strong></em></p>
<p>since there is a double root <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>x</mi><mo>=</mo><mi>a</mi></mrow></mfenced></math>, this parallel line through <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>R</mi><mfenced><mrow><mi>r</mi><mo>,</mo><mo> </mo><mn>0</mn></mrow></mfenced></math> is the required tangent at <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> <em><strong>R1</strong></em></p>
<p> </p>
<p><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">d.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>'</mo><mfenced><mi>a</mi></mfenced><mo>=</mo><msup><mi>b</mi><mn>2</mn></msup><mo>⇒</mo><mi>b</mi><mo>=</mo><msqrt><mi>g</mi><mo>'</mo><mfenced><mi>a</mi></mfenced></msqrt></math> (since <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>></mo><mn>0</mn></math>) <em><strong>R1</strong></em><br><br><br><strong>Note:</strong> Accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>=</mo><mo>±</mo><msqrt><mi>g</mi><mo>'</mo><mfenced><mi>a</mi></mfenced></msqrt></math>.</p>
<p><br><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>a</mi><mo>±</mo><mi>b</mi><mtext>i=</mtext></mrow></mfenced><mi>a</mi><mo>±</mo><mtext>i</mtext><msqrt><msup><mi>b</mi><mn>2</mn></msup></msqrt></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>'</mo><mfenced><mi>a</mi></mfenced><mo>=</mo><msup><mi>b</mi><mn>2</mn></msup></math> <em><strong>R1</strong></em></p>
<p><br><strong>THEN</strong></p>
<p>hence the complex roots can be expressed as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>±</mo><mtext>i</mtext><msqrt><mi>g</mi><mo>'</mo><mfenced><mi>a</mi></mfenced></msqrt></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>=</mo><mn>4</mn></math> (seen anywhere) <em><strong>A1</strong></em></p>
<p><strong><br>EITHER</strong></p>
<p>attempts to find the gradient of the tangent in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> and equates to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>16</mn></math> <em><strong>(M1)</strong></em><br><br><br><strong>OR</strong></p>
<p>substitutes <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mo>-</mo><mn>2</mn><mo>,</mo><mo> </mo><mi>x</mi><mo>=</mo><mi>a</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>80</mn></math> to form <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>80</mn><mo>=</mo><mfenced><mrow><mi>a</mi><mo>-</mo><mfenced><mrow><mo>-</mo><mn>2</mn></mrow></mfenced></mrow></mfenced><mfenced><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><mn>16</mn></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><br><strong>OR</strong></p>
<p>substitutes <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mo>-</mo><mn>2</mn><mo>,</mo><mo> </mo><mi>x</mi><mo>=</mo><mi>a</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>80</mn></math> into <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>16</mn><mfenced><mrow><mi>x</mi><mo>-</mo><mi>r</mi></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><br><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>80</mn><mrow><mi>a</mi><mo>+</mo><mn>2</mn></mrow></mfrac><mo>=</mo><mn>16</mn><mo>⇒</mo><mi>a</mi><mo>=</mo><mn>3</mn></math></p>
<p>roots are <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>2</mn></math> (seen anywhere) and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>±</mo><mn>4</mn><mtext>i</mtext></math> <em><strong>A1A1</strong></em></p>
<p> </p>
<p><strong>Note: </strong>Award <em><strong>A1</strong> </em>for <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>2</mn></math> and <em><strong>A1</strong> </em>for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>±</mo><mn>4</mn><mtext>i</mtext></math>. Do not accept coordinates.</p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">f.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>3</mn><mo>,</mo><mo> </mo><mo>-</mo><mn>4</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note: </strong>Accept “<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>3</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>−</mo><mn>4</mn></math>”.<br>Do not award <em><strong>A1FT</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>a</mi><mo>,</mo><mo> </mo><mo>−</mo><mn>4</mn><mo>)</mo></math>. </p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">f.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>2</mn><mfenced><mrow><mi>x</mi><mo>-</mo><mi>r</mi></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>-</mo><mi>a</mi></mrow></mfenced><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>a</mi><mi>x</mi><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></math></p>
<p>attempts to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>''</mo><mfenced><mi>x</mi></mfenced></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>''</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>2</mn><mfenced><mrow><mi>x</mi><mo>-</mo><mi>a</mi></mrow></mfenced><mo>+</mo><mn>2</mn><mfenced><mrow><mi>x</mi><mo>-</mo><mi>r</mi></mrow></mfenced><mo>+</mo><mn>2</mn><mi>x</mi><mo>-</mo><mn>2</mn><mi>a</mi><mo> </mo><mfenced><mrow><mo>=</mo><mn>6</mn><mi>x</mi><mo>-</mo><mn>2</mn><mi>r</mi><mo>-</mo><mn>4</mn><mi>a</mi></mrow></mfenced></math></p>
<p>sets <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>''</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>0</mn></math> and correctly solves for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> <em><strong>A1</strong></em></p>
<p>for example, obtaining <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>-</mo><mi>r</mi><mo>+</mo><mn>2</mn><mfenced><mrow><mi>x</mi><mo>-</mo><mi>a</mi></mrow></mfenced><mo>=</mo><mn>0</mn></math> leading to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mi>x</mi><mo>=</mo><mn>2</mn><mi>a</mi><mo>+</mo><mi>r</mi></math></p>
<p>so <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mfenced><mrow><mn>2</mn><mi>a</mi><mo>+</mo><mi>r</mi></mrow></mfenced></math> <em><strong>AG</strong></em></p>
<p><br><strong>Note:</strong> Do not award <em><strong>A1</strong></em> if the answer does not lead to the <em><strong>A</strong><strong>G</strong></em>.</p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">g.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>2</mn><mn>3</mn></mfrac></math> of the horizontal distance (way) from point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>R</mtext></math> to point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Accept equivalent numerical statements or a clearly labelled diagram displaying the numerical relationship.<br>Award <em><strong>A0</strong></em> for non-numerical statements such as “<math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math> is between <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>R</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math>, closer to <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math>”.</p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">g.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfenced><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>5</mn></mrow></mfenced></math> <em><strong>(A1)</strong></em></p>
<p><img 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"></p>
<p>a positive cubic with no stationary points and a non-stationary point of inflexion at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>1</mn></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Graphs may appear approximately linear. Award this <em><strong>A1</strong> </em>if a change of concavity either side of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>1</mn></math> is apparent.<br>Coordinates are not required and the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-intercept need not be indicated.</p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">h.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>r</mi><mo>,</mo><mo> </mo><mn>0</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">h.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>Part (a) (i) was generally well done with a significant majority of candidates using the conjugate root theorem to state <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>-</mo><mtext>i</mtext></math> as the third root. A number of candidates, however, wasted considerable time attempting an algebraic method to determine the third root. Part (a) (ii) was reasonably well done. A few candidates however attempted to calculate the product of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>+</mo><mtext>i</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>-</mo><mtext>i</mtext></math>.</p>
<p>Part (b) was reasonably well done by a significant number of candidates. Most were able to find a correct expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mo>(</mo><mi>x</mi><mo>)</mo></math> and a good number of those candidates were able to determine that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mo>(</mo><mn>4</mn><mo>)</mo><mo>=</mo><mn>1</mn></math>. Candidates that did not determine the equation of the tangent had to state that the gradient of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>x</mi><mo>-</mo><mn>1</mn></math> is also 1 and verify that the point (4,3) lies on the line. A few candidates only met one of those requirements. Weaker candidates tended to only verify that the point (4,3) lies on the curve and the tangent line without attempting to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mo>(</mo><mi>x</mi><mo>)</mo></math>.</p>
<p>Part (c) was not answered as well as anticipated. A number of sketches were inaccurate and carelessly drawn with many showing both graphs crossing the <em>x-</em>axis at distinctly different points.</p>
<p>Part (d) (i) was reasonably well done by a good number of candidates. Most successful responses involved use of the product rule. A few candidates obtained full marks by firstly expanding <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo></math>, then differentiating to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>'</mo><mo>(</mo><mi>x</mi><mo>)</mo></math>and finally simplifying to obtain the desired result. A number of candidates made elementary mistakes when differentiating. In general, the better candidates offered reasonable attempts at showing the general result in part (d) (ii). A good number gained partial credit by determining that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>'</mo><mo>(</mo><mi>a</mi><mo>)</mo><mo>=</mo><msup><mi>b</mi><mn>2</mn></msup></math> and/or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>(</mo><mi>a</mi><mo>)</mo><mo>=</mo><msup><mi>b</mi><mn>2</mn></msup><mo>(</mo><mi>a</mi><mo>-</mo><mi>r</mi><mo>)</mo></math>. Only the very best candidates obtained full marks by concluding that as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>></mo><mn>0</mn></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>≠</mo><mn>0</mn></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mi>r</mi></math> when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>0</mn></math>.</p>
<p>In general, only the best candidates were able to use the result <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>'</mo><mo>(</mo><mi>a</mi><mo>)</mo><mo>=</mo><msup><mi>b</mi><mn>2</mn></msup></math> to deduce that the complex roots of the equation can be expressed as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>±</mo><mtext>i</mtext><msqrt><mi>g</mi><mo>'</mo><mo>(</mo><mi>a</mi><mo>)</mo></msqrt></math>. Although given the complex roots <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>±</mo><mi>b</mi><mtext>i</mtext></math>, a significant number of candidates attempted, with mixed success, to use the quadratic formula to solve the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>z</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>a</mi><mi>z</mi><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup><mo>=</mo><mn>0</mn></math>.</p>
<p>In part (f) (i), only a small number of candidates were able to determine all the roots of the equation. Disappointingly, a large number did not state <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>2</mn></math> as a root. Some candidates determined that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>=</mo><mn>4</mn></math> but were unable to use the diagram to determine that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>3</mn></math>. Of the candidates who determined all the roots in part (f) (i), very few gave the correct coordinates for C<sub>2</sub> . The most frequent error was to give the <em>y-</em>coordinate as <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>-</mo><mn>4</mn><mtext>i</mtext></math>.</p>
<p>Of the candidates who attempted part (g) (i), most were able to find an expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>''</mo><mo>(</mo><mi>x</mi><mo>)</mo></math> and a reasonable number of these were then able to convincingly show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mo>(</mo><mn>2</mn><mi>a</mi><mo>+</mo><mi>r</mi><mo>)</mo></math>. It was very rare to see a correct response to part (g) (ii). A few candidates stated that P is between R and A with some stating that P was closer to A. A small number restated <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mo>(</mo><mn>2</mn><mi>a</mi><mo>+</mo><mi>r</mi><mo>)</mo></math> in words.</p>
<p>Of the candidates who attempted part (h) (i), most were able to determine that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>(</mo><mi>x</mi><mo>-</mo><mn>1</mn><mo>)</mo><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>5</mn></mrow></mfenced></math>. However, most graphs were poorly drawn with many showing a change in concavity at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn></math> rather than at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>1</mn></math>. In part (h) (ii), only a very small number of candidates determined that A and P coincide at (<em>r</em>,0).</p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">f.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">f.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">g.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">g.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">h.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">h.ii.</div>
</div>
<br><hr><br><div class="specification">
<p><strong>This question asks you to explore the behaviour and some key features of the function</strong> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mi>n</mi></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><msup><mi>x</mi><mi>n</mi></msup><mo>(</mo><mi>a</mi><mo>-</mo><mi>x</mi><msup><mo>)</mo><mi>n</mi></msup><mo> </mo></math><strong>, where</strong> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>∈</mo><msup><mi mathvariant="normal">ℝ</mi><mo>+</mo></msup></math> <strong>and</strong> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>∈</mo><msup><mi mathvariant="normal">ℤ</mi><mo>+</mo></msup></math><strong>.</strong></p>
<p>In parts (a) and (b), <strong>only</strong> consider the case where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>2</mn></math>.</p>
<p>Consider <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mn>1</mn></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>x</mi><mo>(</mo><mn>2</mn><mo>-</mo><mi>x</mi><mo>)</mo></math>.</p>
</div>
<div class="specification">
<p>Consider <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mi>n</mi></msub><mfenced><mi>x</mi></mfenced><mo>=</mo><msup><mi>x</mi><mi>n</mi></msup><msup><mfenced><mrow><mn>2</mn><mo>-</mo><mi>x</mi></mrow></mfenced><mi>n</mi></msup></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>∈</mo><msup><mi mathvariant="normal">ℤ</mi><mo>+</mo></msup><mo>,</mo><mo> </mo><mi>n</mi><mo>></mo><mn>1</mn></math>.</p>
</div>
<div class="specification">
<p>Now consider <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mi>n</mi></msub><mfenced><mi>x</mi></mfenced><mo>=</mo><msup><mi>x</mi><mi>n</mi></msup><msup><mfenced><mrow><mi>a</mi><mo>-</mo><mi>x</mi></mrow></mfenced><mi>n</mi></msup></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>∈</mo><msup><mi mathvariant="normal">ℝ</mi><mo>+</mo></msup></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>∈</mo><msup><mi mathvariant="normal">ℤ</mi><mo>+</mo></msup><mo>,</mo><mo> </mo><mi>n</mi><mo>></mo><mn>1</mn></math>.</p>
</div>
<div class="specification">
<p>By using the result from part (f) and considering the sign of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><msub><mi>f</mi><mi>n</mi></msub><mo>'</mo></msup><mfenced><mrow><mo>-</mo><mn>1</mn></mrow></mfenced></math>, show that the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>0</mn><mo>,</mo><mo> </mo><mn>0</mn></mrow></mfenced></math> on the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><msub><mi>f</mi><mi>n</mi></msub><mfenced><mi>x</mi></mfenced></math> is</p>
</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><msub><mi>f</mi><mn>1</mn></msub><mo>(</mo><mi>x</mi><mo>)</mo></math>, stating the values of any axes intercepts and the coordinates of any local maximum or minimum points.</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 your graphic display calculator to explore the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><msub><mi>f</mi><mi>n</mi></msub><mo>(</mo><mi>x</mi><mo>)</mo></math> for</p>
<p>• the odd values <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>3</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>5</mn></math>;</p>
<p>• the even values <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>2</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>4</mn></math>.</p>
<p>Hence, copy and complete the following table.</p>
<p><img src="data:image/png;base64,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"></p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><msub><mi>f</mi><mi>n</mi></msub><mo>'</mo></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>n</mi><msup><mi>x</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mrow><mi>a</mi><mo>-</mo><mn>2</mn><mi>x</mi></mrow></mfenced><msup><mfenced><mrow><mi>a</mi><mo>-</mo><mi>x</mi></mrow></mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State the three solutions to the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><msub><mi>f</mi><mi>n</mi></msub><mo>'</mo></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>0</mn></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 point <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mfrac><mi>a</mi><mn>2</mn></mfrac><mo>,</mo><mo> </mo><msub><mi>f</mi><mi>n</mi></msub><mfenced><mfrac><mi>a</mi><mn>2</mn></mfrac></mfenced></mrow></mfenced></math> on the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><msub><mi>f</mi><mi>n</mi></msub><mfenced><mi>x</mi></mfenced></math> is always above the horizontal axis.</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>Hence, or otherwise, show that <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><msub><mi>f</mi><mi>n</mi></msub><mo>'</mo></msup><mfenced><mfrac><mi>a</mi><mn>4</mn></mfrac></mfenced><mo>></mo><mn>0</mn></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">[2]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>a local minimum point for even values 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><mo>></mo><mn>1</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>∈</mo><msup><mi mathvariant="normal">ℝ</mi><mo>+</mo></msup></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">g.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>a point of inflexion with zero gradient for odd values 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><mo>></mo><mn>1</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>∈</mo><msup><mi mathvariant="normal">ℝ</mi><mo>+</mo></msup></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">g.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Consider the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><msup><mi>x</mi><mi>n</mi></msup><msup><mfenced><mrow><mi>a</mi><mo>-</mo><mi>x</mi></mrow></mfenced><mi>n</mi></msup><mo>-</mo><mi>k</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>∈</mo><msup><mi mathvariant="normal">ℤ</mi><mo>+</mo></msup></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>∈</mo><msup><mi mathvariant="normal">ℝ</mi><mo>+</mo></msup></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
<p>State the conditions on <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> such that the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mi>n</mi></msup><msup><mfenced><mrow><mi>a</mi><mo>-</mo><mi>x</mi></mrow></mfenced><mi>n</mi></msup><mo>=</mo><mi>k</mi></math> has four solutions for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">h.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><img 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GCIehpKpOz2YTKaTJR7LZcKi/KpraVV1CqmvNyvX7+mWYeDRkdH6eHDfur68UeytdjIUldPly5dChKpLy9nkaX+LjW1NVF7ayt19/bS0NAQOWed9Ma1nfLy44EgoBUCECkNWDKUSHmq5h6TQkvKwyPw0/Xbb7Sxukqj9hEylhvpwoUL0mxJ9ob+5+w/U7nRSDablf7j55+leOwxwrcVxec8W5DzYAe1f/97P9211tO/596grMuXpbw4z5qaGimPeYeTdlyuwGLgGgRAQIYAREoGimhBkUQK66ROLcqLdJeXlmjg8WMym+uosLCQCkoKqLKymto6OmhiYozWV9eJZ06+fRu8GJdzGhoc8hOpM2fP0tjY+OlDfM4O3x7R4d4+LS6vkX3YTu3t7VR35w59910xlZeXU1NzE42PjdHLzS1pAbFPUpyCAAgQEURKA68BREreiG/fH9LK8hr19D6Qts5gQfpfN29SZ1eXtOnh/uGu34Jd+VzkQ3nNVKPNStXV1TT1YlI+UoRQFsGt11s0bLdLvvxuGo1kKL1Jbe3NUjdhqD2vImSL2yCgKQIQKQ2YM7JI2UPuRKuB6nursL+/T8uLy/R4aJAqjBWUf6OUmpobaW5+XmoZRfK7580ojSfsmunVq9f0oOcBlZWXkaG8TGp9TTmmieuHAwT0RgAiJbDF+UuXv4B53IPHPJ4/fU7cegg82GlqqO3SA+OKdr29s0PDI8NUaTJRcWEx2ZqbaWpyWtZvnmh14/Ky/7/NjZfU2/0PKiwpodraGhodfUK7u/EtKhaRAcqsbwIQKYHtz14MeDty3z+56mhp00NuaXBX2+DgEJUaSslkqpC+tOWcucqxED2MW1OPHz+m0hPBYg8Wm+vBznJFryfKDwIeAhApDwkNf4o+cYJbjL+srlNLcwsVFBRJ40sT4+N0/O6dhq0WuWrcmrI/G6HqmjoqKSmSWpToEozMDTHEIgCREstecZVWZAez3Eo015jpTGYGjdgHpVZjXBA0nohnI5aUG6RuXx6XwwECWiEAkdKKJcPUw92Smg8TQ3232FN4o7WRygwG6unro+3tHfUVUoUlmpmeIbO5hngtV1tbW9wOcVVYNRRJpwQgUjowvChjUuxhvLWllf5qKKHmvzeTY3ZWB9ZRropPnz0jU0Ul1dbV0Movq8o9CDmDgIIEIFIKwlVL1m4Hs+rdT4o9kdfU1lBVdRVNPp302xFXLQxFLsfq6irdtTSQ0WgkbmnhAAGRCECkRLJWnGVVo1sknqXX2dEpdUvdbrBKXiDirB6SRUngzZsd6mhrp5KCErJYG9AVGCU3REsvAYhUevmn5OlqakkdHR5Sc1MLfVdWRGNjE5gIkZI3IPghjpkZMhrKIVbBaBCiMgIQKZUZRIniuMek0utg1tNyKjCU0IuJZ0pUE3nGQYDFqqLcSE02G/EPCBwgoDYCECm1WUSB8rBnbocjfWNS/zM/Tzk5OdJU8g86X9ukgHmTkqWt2UZXs6+SY9KB1m1SiCKTZBGASCWLpIrzSddi3q2tLbptsVBOXg7x3lYi+M5TsRkVLRrbZmx8nNgPZO3f/kY8mQUHCKiBAERKDVZQuAypHpM62D+gjvY2KiwqpeWVFYVrh+yTTWBzc5MqjEZqbm7C5Ipkw0V+MROASMWMTLwEqVonxb/G2V1RXn4ePX3xQjxQKLEfAR6vys+7ToP2IcmLvN9NXIBAighApFIEOp2PcbekHIoVgcWJx7yMFbwXUju69RQjnfqM2Ynxg54eys3JkZYMwDdg6m2g9ydCpHTwBrjXSSVfpH4nopWlJcrPz5ecvr7Z2tYBTf1Wcfxfz+g7QyG9mIxvk0f9kkPNEyEAkUqEniBplWhJ8XRlbjVVVJhof29XEBIoZqIE3h+9lXwq1tbWwp9iojCRPioCEKmoMIkdye0FPXnrpNgzebmhnGqqasQGg9LHTaCvr5+uXLlCWFIQN0IkjJIARCpKUCJHS+YUdN5wsNxYRve672HsSeSXIglln/jXOOX+5TqxYB2/P05CjsgCBIIJQKSCmWguJBmz+3haOTso5RbU8gr2K9LcSxJnhbirt7+/n/Ly8sg5J9Z2MHFWGclSTAAilWLg6XhcouukHE4nFZaU0sLCYjqKj2cKQIDFylxXR1ZrI/GMQBwgkCwCEKlkkUxTPtzt1tzcTv0PH9Kjh49kSxGvF3T+sulo76CaWjMmR8iSRWAggdH/NyrN9lzb2Ai8hWsQiIsARCoubOpI9OjRAFVWVXoLU1lZTY8GBr3XnpN4WlK8AWFZ+U16LJOfJ198goAcge3tN2Q03aQHvQ/kbiMMBGIiAJGKCZe6In9+8XPimXueg885LPCItSXl+u03KjDk0sDAQGBWuAaBqAjwDFDewp59Abp2dqJKg0ggIEcAIiVHRYCw9dV1ysjIoOWl00kMKyvLUtgvq+t+NYhlndTU1BRlX82mB/d6/fLABQjESoDX0tlarVRWVkara9i+PlZ+iO8mAJES9E0YGxuXBIldEnkO3rOJhYvv+R7RTkHv/vEfZKquhp82X3g4T5jA1tZrKi0spZEnownnhQz0RwAiJajNH/b1UWZmZtBaJRYp3y5Arl6kManDoyMy36qjrs5OQWmg2CIQaLI1ksXSgM0VRTCWisoIkVKRMWIpyqOH/SFF6vHg6TgV5xlunZQ0QaLEQM+f+be+YikL4oJAtATGJyao7GaZ5JDYtxcg2vSIpz8CEClBbT4xMSF17fmu9H/1alMKm5qe8atVqDGpXzdfUX5RHi36jGv5JcQFCChAYG1tjYwmE9WZv0erSgG+WssSIiWoRXdcLvroozO0tHi6yn98bIzOnD0btKYp0As6/4IdtNupoKiQ4Llc0BdAA8V+PjVJhtJS4h9LOEAgFAGIVCgyAoTzFN/u3tNZeLbmZukXamDRAx3Msq+1jzMyiP3w4QCBdBLgfcjYUS1PWccBAnIEIFJyVAQJczpmKT8vX/oHP373jq5fv+43Jd1TDe7um511e0Gf4l+vhjKsXfHAwWfaCYwMD1NxaSm6/tJgiUg/DhYXFiXfjPX19cT+O/mYdTrprqWB9o5Ss/5NVSLFs8xwxEaAnb1yS8lutwd1m7x9fyi1tLKzs6nEUEh99/vpptFIPFUdR3II4J1NDselpSW6lptL/zU9KW2gWVJUSnWWO94vxuQ8RV+5HL49Cpr960uA3Z6VlBhoJmAM2zcO78S8vLhMX399ldpaW6Xxa3OdmbKyvqLVl6npiVGVSOXmXvflg/MECRTkF0sTKXhaOv9J41WH2KAwQax+yasrT91S+d3ARcwEPAvUPe8rf+bk5sScDxK4CYyOjkrLT8LxcL1x0bfXsqn+liXsj9d7vQ/oypWvqKe3J1x2itxTlUh9+acvqeGHu2S5W0936uvJUneXLDYz3THXk+X2Sbjn3GzxjxdNmrv1ZDFbTvNONM3deulXX1DZwj0nTBqpnhaLu96+ZfNJw1w8jKRPi4VstyzU0HibrHf4082p0dJAH511i5PvP72hqJQaGhq8aSy3b5PnT2J++65Upztch9t3JVa+5xaLhXyvPee3zGbZcCnPEGn43q275pieEy6NVBbmJ1PuUM8Jl0a6Vx/M4DTNbfoq6yuy1LvfRX5vJZvYzNK76bGTN/zkvfC7TiQN52cze98Hv3xP/j9kyxAqzUn5kpbmbj1Zm93vo2/ZpPxPytdovSvxszbUS1vB+L6rnvPqmkqJp+f7QPr8of40jM85v8Awrs/Jd4hsGr5ff1I+juebnm0YGOa5HypNuHLEkyZc2SPU11p/V+o9KSrOd39Pmt3vJv+fSvxP6sLca6pr6ExGBl3Nzg65L9jrrV+lH7k7W9uKCFG4TFUlUtdyrtHW5pb0t721TfzH155z38/AcN9r3/PANDs7O978fOP5nkebhuPJpeOwUM8Jl4bvedIF5it3zWGeP0++vtfnz3/q15Lif3r+deWJw5+cLvDPE+759OTNn1w+uXBPmOczmjSeOMlK48mPP+XO5Z4jF883fSR7lJeVeflx/p4/T76eZ3rCPfwCwwOvfdOHSuMbJzB9PGl88wt1HvicUPF8wwPT8HVgGMefePGfQe8rv7PcwvKkwefpOxaJRd+jfnpwr8/LevPk+8qTzmOjHzu7KfPfMmkgxC4KLCCvftmms2fPhu0aDCc0idxTlUg1N7ckUhek9SHA41Fff/0NfZyZ6f3HN5nQNeWDKCmn3ffuJSUfZOImwJ78PS0o7p7+4osvaA77mMX1esw558gx479mMjCjhwOP6TuDgTY3Xgbe8l7z5Ip/3L9H/IPM2mCVwlM5rq0qkfJSwUnCBKxWqzShYmtri25ZLNTeCpdHCUNFBikhMD8/Tz3d3eSYdkizUHmK+sGBe2ZZSgqgo4fk/HtuxJm+ra02afJEe3s75eTkSLOJuUcmVQdEKlWkU/icyRcvyGQyeWf2ROtgNoVFxKNAIGoCE08nyHjTiJl+UROLPiIvXZE7eNbqN998Q82tLdR/sh3Q9IyDzp07R60trWEnWcjll0gYRCoReipM+/TFCyq/6T/NPJKDWRVWA0UCAT8CTqeTSv9ajAXoflSUvRgdfULswd73GBsfIx5KSOUBkUolbYWfxYvsKipMQb9y0JJSGDyyTwkB9jHJXlb+0dtHv6fkiXiIGghApNRghSSUgQWq5LsS2S4RiFQSACML1RCw2Rqpq7sTQqUaiyhbEIiUsnwVz93jLNZQZqDDQ/lmOERKcTPgASkm0NXeRbYmW4qfiselg4CqRYr3OpqcnEwHF2GeaWtqoiqTkZhVqAMiFYpM4uEL86de6BPPDTm4dl3SWr6RER4P2QoLpKuzG0IVghB7kmBXaex7T/RDlSLFG6I75+fpfGYm/Tn7z6IzVqz87EH6xg1DxPzdXtCnI8ZDhOgJuH77jTo7u6U1PdGnQsxwBFw7O3ThwgXvOin+/+dFyeEOfv+fTj4LF0V393gKv+/6yGmH2P/7qhQp3itpeHhEcsORfTVbdy9ZNBXmX0oF+QW0vf0mYnS0pCIiiikCz24afzZO9XduQ6RiIhc+coPVSj09PeR0LtLwyLAkWAUFBWETvdnZoevX87HT7wml5ZUVMplqyDE7K7Wirnz9NWVnXwvLUO03VSlSHmifnj+PlpQHhs/n7v4BFRYX0lqU+0Fxs9/hcPjkgNNkEJienoJIJQMkEe3t7Uo/TH2zGx4ZkfhG8hfHQsUtqpaWZu/aQN989HS+vr7mV13ewfviHy/6hYl2oXqRyslBS8r3pTo6PJQWNi4uRd/XjHVSvgSTd87drezCB0fiBHgCUODBvuWYb6gJQYHxeYyKZ/3hOCUwZLdLi29PQ8Q7U/V/2CfnzxO6+05fquP3x1RV9b/p+cyL08AoztDdFwWkOKJApOKAFkOSgcFB4r2LYjl4enr3vdRvJxFLGVMRl0V/cXme8vLzyOVypeKRij0jpSLF4NhZYai/wF9TPIiKiRNu2zOzH9gf3+BgzC+D7868MSdGgpAE0N0XEk3CN/i7oNRgIO7Ki/VgoXp84son1rRaib+5uSltUvjp+U/p0qVLQu96nFKRam1uJnYW6fuXdfmy97qvt9fvHWGR4rg4iNjJY7mhPC4UdvugNLAcV2IkCknA4XCiuy8kncRusAPTvr7+uDPh743ll8txp9dKQp6Ewl2m/f3xs0w3i5SK1PzcPPH6h2H7sN8ff4ny4P7KyqofD+7uQ0uK6OHDPrqafTWqmXx+AE8u0N0nRyXxMHT3Jc5QLgfe76jRag1y7yUXN1TY3MI83Swvj3o8K1Q+WgjPzs6mRpu4C59TKlKxGpxn9+l9TOrJyBMy37oTKzq/+Jg44YcjaRcQqaSh9GY0Mz1F3ONysH+6NQfP/IvnmHE4qKLCqPttPligBh4/jgehKtKoWqS4JaXn2X0zsw5pyw2eMJHI4e7uwxT0RBjKpcWYlByV+MMmnz+n4sLvpK6pgUcD0mdHewf19z+KO9PxiQnKukUbEFIAABgZSURBVJpF7NtSDwcvS+EuPs/BYl9YVES/y8ye9MRR+6cqRer90VsaGhmiy198SZ//6XMaHBzSXbN95ZdV+muZwe8XZbwvk7slBZGKl19gOh7U51ZUQ0MjXfzDRXrQ95B4cTWO+AmsrCzTZ5/9kS5dukiXL38ucWW2n1/8nFZXEhtbermxQQZDKa2t+a8hir+06k1523JLGoPK+upravjhLnV0dMY1+URNNVSlSKkJUDrKwl94vDgxnplNcuXFmJQcFYTpiQB7ZiksyA/r41IrPHgm8MG+SzMLmyFSKnszeZvswtJSWvvFfxJJIsWESCVCD2m1QoBbUoVFpSnftE8r/NJVD4hUusjLPJcFiqfOdgdMxZeJGlMQHMzGhAuRNUygt6eXulvbKdi/hYYrLXjVIFIqMSCPc/Bi3Vtmc9IX3sF3n0qMjGKkncDx8TGVlhqId/nFIQYBiJQK7MRbYfd036PuLmXcuWAKugqMjCKohgDvvZafX5C0MV/VVEyjBYFIqcCwz8bHqcZco1hJMCalGFpkLCiBXzdf0Y0bRcR7WOFQNwGIVJrts7C0RN+VGhSdiQPffWk2Mh6vSgIO5wyd/7d/I+ccdldWpYFOCgWRSqN1eKp53o38pKyFClcNbknxuh4cIAAC/gSOXHuUl3+dNl9v+9/AlWoIQKTSZAreF4qnw7K3YqUPdPcpTRj5i0xga+s15efnJ33CkshM1FR2iFSarNHX30+dKdqgzW4fQUsqTXbGY8UgwMs+kr30Q4yaq7+UEKk02Ojp5DMqKCkhdv+UigO++1JBGc8QmYA0Nb2wlNgdGQ51EYBIpdgeg48HpAW77KYlVQd896WKNJ4jMgEeIy4sKSDX7p7I1dBc2SFSKTSpwzFLFRUmaWfiZDyWx7VstibKyckJmx3GpMLiwU0Q8BJYWFgkY3l50v5HvRnjJG4CEKm40cWWcGtri0rKSpI2k29/f59Gn4xSxpmMiLvDQqRisxVi65vAiH1E2mSUJ1TgSD8BiFQKbMD93XlFf1FkJt+Xf/oyCpGyY+JECuyMR2iHAC/ZyM8rUHT9onZoKVsTiJSyfKXca2tqyTE9o8iTLl/OiihS8N2nCHpkqnECk//5nGprazVeS/VXDyKlsI14Jp/RZFLsKdGIFLygK4YfGWucQLmhnBbn4ZEinWaGSClInyc2lJffpPVX64o9JSvrq5AtKR632trcop6eezQ2Nk6uPfgpU8wQyFiTBHg7dkNpGXGXPY70EIBIKcSdd8fkroLJ58+jfgKLGq9p4okOgX/cGuKwhQX/X3XhWlIjw8NkNpsp91ouFZcUU0tbW9RlQUQQAAE3gdGxJ1Rffws40kQAIqUA+IP9AyopN1Bn972YcneL1AjxGFLwnzt8PsAZZriWlOfhmN3nIYFPEIiPQHNTC/GPPhypJwCRSjJz7haorKyg3u7eJOcsn124lpQnBUTKQwKfIBA/geLSUuLuPxypJQCRSiJv3l23q7OTHj8aSGKu4bOKpiUFt0jhGeIuCERDgL3EFBpKsQdVNLCSGAcilUSYBwcHdOVqdhJzjJzV+cxMaeJEuM3b3A5mnZEzQwwQAIGwBPp6eyW3Zvy/jiM1BCBSSeRcWVlFc/OzScwxdFbH74/JOes8GbtiL+cOevv+UDYBuvtksSAQBGImwOPGpr+ZqLW1Pea0SBAfAYhUfNyCUt3v6aHue91B4WoIcDuYxaaHarAFyqANAubvzTQ1NaWNyqi8FhCpJBiIW0+VpkrVulBBSyoJRkYWIOBDgFtUBUWFxJ7TcShLACKVIF/eWTcvPy9pjmMTLI5scoiULBYEgkBCBFbXVqmsrEy1P04TqpyKEkOkEjDGzPQMZZ6/QKur6t4oDd19CRgZSUEgDIHBwUFpRm+YKLiVIAGIVJwAuZlfXCLGdFQ4mI3TyEgGAhEI/P7hA1VX19BsiiZMRSiOJm9DpOI0q9VqpdHR0ThTpzYZHMymljeepi8C7B+zIO+GviqdwtpCpOKAvfLLKvHqc1EOjEmJYimUU1QC/KP16cRTUYuv6nJDpGI0D/vlKyszEK8+F+Vwj0k5RCkuygkCwhGQNja9nk/O//Z3AC1cRVRYYIhUjEapNX9Pk9MTMaZKb3S4RUovfzxdHwTWf12jCxcukHMmNQv69UGVCCIVg6XtI0PU2tIaQwp1REVLSh12QCm0T2DvaIcKSkrI5cL6qWRZGyIVBckP795J3o9Li4uJZ/OIdmAKumgWQ3lFJsDb6dRUV4tcBVWVHSIVhTnYeSsv2hNpHMq3Wpg44UsD5yCgPIHWtnZpk1Lln6T9J0CkItiYd9i9ccNAza3idfN5qub2gg7ffR4e+ASBVBDgHRH4+wNHYgQgUhH4sePYYkMJsa8uUQ+0pES1HMotMoGe7nv06FG/yFVQRdkhUmHMsLS8RFXVlcL/GmKR4q08cIAACKSOAI9f3yy/SasvX6buoRp8EkQqhFHfHh3RtdxrtLsv/uZmcIsUwsgIBgGFCWxtvabiUoPwP3QVxhQ2e4hUCDxV1dXkdGhjvQO6+0IYGcEgkAICQ/ZBys6+imnpcbKGSMmAGx4epra2Npk7YgZhCrqYdtN7qdmLw/jYqX/M9x8+EPvJGxkZoaGhobTs5TT2ZCyuVtGg3U6Vlerdc07N7xpEKsA6W1tbVFxUSvwPopUDY1JasaR+6rG7t0eWhga/fdqqq6vo48xMuviHi3Tu3Dn6JPMTGkuxk+e1jQ0ylhtp8vnzmI3R0d5B9uHhmNPpPQFEKuANMJpM5HQuBoSKfel2i+QUuxIova4I1FRV+f0fLi8uk6GkhPhHJB9PJ59RRkYGXc2+mnIurt9+oz9n/5kOjmIbrz44OKDsq9nEzgFwRE8AIuXDanllhYwVNzW30ybcIvkYGaeqJsAz4rhr7Pr1XL9ydnffC+res1is9NFHZ2h/f98vbiouurq6qLU19iGB7q5OLPKN0UAQqRNg/Csn/3p+0D9CjDxVGR1jUqo0CwolQ2Bhfp7OZGTQT48e+d1lry+Bx8zMrNSa2t3dC7yl+PXe7i5lZmbStky5wj2chxFKSkppezu4PuHS6fkeRIqINl9v0+XLWTT57F+afBcwu0+TZo2pUhMvXlCDxUo5174l7q7yHHVmM7V1dHguk/J5/O4dTYxPUFFJCeXk5EhddDzpob7+lnQd7iHm77+XhMc5G7l7+vnzp/TF5cvhsovpHpdxbHhMKndeXi65XG+ksema2loqKQzeP+6Pn12m9s7YPdH8uvmKysvKNddjExPsGCLrWqS4a4FfzJvGMhp/Nh4DNrGicktqNop/erFqhdLGSoAH/XkcZ9Ru9yZ9MjxMFaZK77Xn5PDoiAyGcsklGLsFk/trtLZ4ost+jo2NS8/r7PqR+vseEq/X6+r4P964P/30E337bTaZzfWSGBy/P6ZPP70gpXG9iexFvNHaSA96erz5JetkcGBAKkNXdzfdf/CA+P/nfn/wc4oLC+mbb76N67HsyWZwcCCutHpLpGuRYmPf675Hjx9p+2Vxz+6D7z69/XMH1pcH7D85f57a2zu9t3ixaU+IL/q9PRd5/4523Oe7u7TrctH+GxftH+1685E7Odzfp88uX6ar2dk0PTXpF+X/joxL40ksmvzX3t4hdbV/dHLNPx7DHbx3U15evmwUTruysko82SLU36/rv8qm5UD2t8f7Qn399VWamZ4JGa+6upo+PX+eWNBjPbiMhlJ2Wo1uv0jsdC1SS0tLdNNojMRI+PuY3Se8CZNWgT9+9jkZTWZvfve6e73nSpw025rpzNmzQV1bPd3dkjh5RKrWbJbWQHmuw5WFx3WqqqvC7kpguX2bzDW1ZDabZf94Oni4o77eTBf+cCFcFOJn8LgUj2fHc3C3Xym8UUREp2uRMplM9HJjIyIk0SPAC7roFkxe+a9cuSJNn+YceWFsd6+8SHE75n/m58P+bayuRixYa6tNEiNu0fgePJWc1zx5RGl0dJR2dna8175xA8959h/HV/LwlFtuwobnuRaLRapDvCLF+VSZjDTYP+jJEp8yBHQrUjyLyFQZ3Bcvw0iVQbzY0WZronJDOVls5rAunDBxQpUmTEuheCIDd8Hxcb/vIfE4kNzBrZXC4lIqKCiQutVyc/OkT+5i47Dc4gJqbGiUS+oNW19fo+7ObsrKyqIhn3EwT4QnP/9M+fkF1NzSIrW0eCr5mXNnJaE6OpDfdcDhmCbL7QZPFtInd50lc+0RfzfwuNT585/S6OgTv2f5XnB3Hy8sjtQ16Zsm8Hx1dZUuXr5Ac7NzgbdwfUJAlyLF227wP4eo22+8P3pLubnXqLOrizo6OqSZibxe5PkL/35/z1uOKegeEvgsKSqVRGBiYpwWFpVZtM7Ts1nkPF1q3JVns1ql7rnlZf8WVaBFvsr6Uirf+up64C36wdpA3HrZ3tqiNzs70t/y0hK1tcQ+wy4ocyLicr99f0hdPz6QbtdU19BdSwNxtxyPcQUe16/nSzMBA8NjveYWLc+E5P9rHMEEdClSDVYbPZ16GkxDkJDp6Sl6vfnar7SFBYVkvFnuF+a5gFskDwl8VlZWSSLwqL9PERjcPcaTMyqrqrwujQYeD0gujLjlFungSRzcBTjyxL87r7GpWQqXugjPuCdbZJx8Pp1I/H95e2ubMjM/oeqqGu+P14HHj6UJFP39/mu2uA7ceuJW1PBIctwc2Z8MU1t7eyQ8uryvO5GyWm1krjsdOBbR6vwrNfCwNlqpwmgKDJausVWHLBZdBlru1tPozz8rVvffiWh+bs77Rc8P4i/05cWlqJ7J3difX/ycmptsfvEP9l3EAri3tye1eLjVw7MMd/d2Eupu833IrMPh57OTy72wuOAbxXv+am2NLn35mV987804T6oqTLSwpEzrNs4iqSKZrkSKXZnwLruJ9CGrwmoyhaisqAjqP+eV+DwYzdPsR5+MkWsP011l0Okm6MX4U5qdV//2M3tHO3Tt29yws/fSbbSav92imamppBaD14aVlZUlVfiSWsA0ZaYbkeK1DzyzSasHD4ZzHX0Pq9VKGedPukY+zoi42t83Lc61Q+DZs0np3fix80dhKsVOntvbkzPWlOxK86xHo0m+az3RZ90ym4knh+A4JaAbkeIJBtztpeaDJ3Lw1Npwf+wE1/fgNOwNYGhwyDdYOudBYJ4e2//Pfnr6dIJCzZgKSogAzRCYmpwmHruxWm8Lt8v0wMAgTSZhvCmZxuQeiR972hXz8bm9/YbKDGWa7O2J1w66EKnlpWWqrq6Jl1HK0h0eHlJfXz/1POgL+uP1LBw+43D4laeltY0mJ+Vn9XkiuidORPaF5omPT+0Q4PHLqekZYb/01OaRYev1tuIvx8N/3qeB/seKP0eUB+hCpHKvX1d1/3a8L8tPP92npeXIA9Lw3RcvYaQDgfQQKCop8u6dlZ4SqOepmheppqZmmhjTnvPY7q5u6u8foPm5BWlzuDnnLHV0tNPc/HzQ24Up6EFIEAACqiawtrYmbTev6kKmqHCaFqmFhUUy16q/my9WW3d1+vs987iW+eKLz2S7dbCYN1bCiA8C6SfQ1NRE5hqz7P90+kuXuhJoWqSu/0Wbmxiy1+X9w11pUgRPjOApu/wZyhsz3CKl7h8KTyLiLXBwJE6AXVYZyk3Eznj1fGhWpNgJZV8I55l6MzgczOrN4qivVgjwBpXZV7ODlpdopX7R1EOTIsXrhQrytdmKisaogXEwJhVIBNdKEdBTKypVde0fuK/5Pe/CvY+aEymPW5WRJ2Ph6q2re+ju05W5UVmNEWAPOcUFhcTfbXo8NCVS/Mum/nsz9fbI75GjRwNznd0TJ/zXV+mVBeoNAiIS4O1DqqqrdTnepymRejE5STZr+D1uRHxBEy0zuvsSJYj00RJIVRdYtOVRMl6q62ptsJHDqb9F+ZoRqYP9A7qWe0330zXl/inR3SdHBWEgIBYB/o7Lu54vbVSZzE0e1U5BMyLFG6I5dfgrI5oXDC2paCghDgjERiDVLSkuHTsmYFdoejo0IVK8s2VeXq6e7BZTXQcHBsnh0F83QUyQEDlhAnr6dZ8wrAQyuJql3d0c5LBoQqQaG21BjlflKqvXMHic0KvlUW8tEhgaGpR+lPMOCHo4hBepOeccVddW6cFWcdcRY1Jxo0NCEAhJIB3dfVwYnpJef8tCbW362G5eaJHiXxL5RXnEA4o4QhPAFPTQbHAnOQTS9YWdnNKLmYux/Catr6+JWfgYSi20SPG26COjozFUV59R0ZLSp91Ra20TkDZILDFou5JEJKxIjYyMUPlN7RsoGW8gZvclgyLyAAF/AmpoPba0ttBTle1e7E8p8SshRerpixeUlXVFkxsZJm7S4BwwcSKYCUKSS0ANX9jJrZEYuTH3638poP39fTEKHEcphRQpQ0kJTf78H3FUV59J7HZMQden5VFrPRDo/PvfiZeZaPUQTqTmeCPDOrNW7aFIvTAmpQhWZKpzAmppPfK+U0UlJaTVKelCiRS/FMWlxbr1BhzvdwLGpOIlh3TREMAi3mgoKRtnaHCI+n/6SdmHpCl3oUSqo62d7PahNKES97GYgi6u7VBy9RJQS0vKQ6ggv0iT4/TCiNTm6226WV7usQc+YyCA7r4YYCFq1ATU9iUddcE1GnFxaZFu36nXXO2EEakKUzmtrq5qzgCpqBBm96WCMp4BAuknwOP17a2d6S9IEksghEixA9maqpokVltfWcHBrL7sjdqmhoAax+JcOzuUkZFB/J2plUMIkaqpqaKFhUWtME95PdDdl3LkeCAIpI1Af/8/yWSqSNvzk/1gVYsUO1Ls7Owka6Mt2fXWVX6Y3acrc6esshiTShnqmB/U0GDTzP56qhapsZ9/pitXrpBr1xWzkZDglADGpE5Z4AwEkkVAzSK9u7dH+dfzk1XVtOajWpFiz+bX83Lp4AAezhN9Q9zdffOJZoP0IOBHQM1f0n4F1elF/+NHNPDwIf0ueP1VK1LNTS3kmJkRHK9yxeeu0JWlJckdisMxTcfHxyEf5u7umw55HzdAAARiJ6B2kebvhAINeKJQpUjxHik3TTdjf2t0koL/Odo7ush4s5yuXcuVZvNkZ38TUqjcIoXt43XyeqCaIOAlMOtw0u07d7zXIp6oUqQsVjM5nZjNF+qF4mmmfX293ttGo1ESqlDM7PYR4tYWDhBIBgE1Tr1ORr20mkfJjRt09O6dsNVTnUjtuFxUUlgqbZEsLFWFC84TSbi7z3PwC3ju3Dman5vzBPl9usek0JLyg4ILEEiQgNq7+zzVGxp+Ql1dXZ5L4T5VJ1J1t77HmqgYX6OXGxtUWHQj5AApZvfFCBTRQUBjBAxlxeR6I+YsaVWJ1OLiAtXU1mrs9VCuOuyif+ixXerqGxt5EvJB7jEpR8j7uAECsRAQpQURS520Hnd5YZGqaqqErKaqRKqtpZVcLjHVPh3Wn5p8QY22Rvr84ueSUD1/MelXjLEno2SxmaX1Ejxu1dHR4XcfFyAAAvETEE2sf7A2Ul9fP4lWblWJVPyvizZSulxvKC8vl/LzC4L+btwwSGH37/fIVraoIF+a7Sd7E4EgAAK6J7C9vUNWq1W4zREhUhp5dfsePqJSQ5lGaoNqqJmAaL/ElWIJDkqR9c8XIuXPQ9ir/v5+6uxtF7b8KDgIgAAIyBGASMlRUXnY/t4uLS4seqeh8xT0CqOR1jY2VF5yFA8EQAAEYiMAkYqNlypiT09N0yfnz1NOzrdUV28mq9VGS0tLqigbCqFdAuje8rctePjzUOoKIqUUWYXzZb9c7HzXd1Gvwo9E9iAAAiCQcgIQqZQjxwNBAARAAASiJQCRipYU4oGAzgmge8v/BQAPfx5KXUGklCKLfEEABEAABBImAJFKGCEyAAEQAAEQUIoAREopssgXBEAABEAgYQIQqYQRIgMQAAEQAAGlCECklCKLfEEABEAABBImAJFKGCEyAAEQAAEQUIoAREopssgXBEAABEAgYQIQqYQRIgMQAAEQAAGlCECklCKLfEEABEAABBImAJFKGCEyAAEQAAEQUIoAREopssgXBEAABEAgYQIQqYQRIgMQAAEQAAGlCECklCKLfEEABEAABBImAJFKGCEyAAEQAAEQUIoAREopssgXBEAABEAgYQIQqYQRIgMQAAEQAAGlCECklCKLfEEABEAABBImAJFKGCEyAAEQAAEQUIoAREopssgXBEAABEAgYQL/HzyGj4UTtCVeAAAAAElFTkSuQmCC"></p>
<p>inverted parabola extended below the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis intercept values <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn><mo>,</mo><mo> </mo><mn>2</mn></math> <em><strong>A1</strong></em><br><br><br><strong>Note:</strong> Accept a graph passing through the origin as an indication of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn></math>.<br><br></p>
<p>local maximum at <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>1</mn><mo>,</mo><mo> </mo><mn>1</mn></mrow></mfenced></math> <em><strong> A1</strong></em></p>
<p><br><strong>Note:</strong> Coordinates must be stated to gain the final <em><strong>A1</strong></em>.<br> Do not accept decimal approximations.</p>
<p><br><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img 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"> <em><strong>A1</strong></em><em><strong>A1A1A1A1A1</strong></em></p>
<p><strong><br>Note:</strong> Award <em><strong>A1</strong></em> for each correct value.</p>
<p style="padding-left:30px;">For a table not sufficiently or clearly labelled, assume that their values are in the same order as the table in the question paper and award marks accordingly.</p>
<p><br><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>attempts to use the product rule <em><strong>(M</strong></em><em><strong>1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><msub><mi>f</mi><mi>n</mi></msub><mo>'</mo></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><mo>-</mo><mi>n</mi><msup><mi>x</mi><mi>n</mi></msup><msup><mfenced><mrow><mi>a</mi><mo>-</mo><mi>x</mi></mrow></mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><mo>+</mo><mi>n</mi><msup><mi>x</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><msup><mfenced><mrow><mi>a</mi><mo>-</mo><mi>x</mi></mrow></mfenced><mi>n</mi></msup></math> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>A1</strong> </em>for a correct <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi><mfrac><mrow><mo>d</mo><mi>v</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac></math> and <em><strong>A1</strong></em> for a correct <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mfrac><mrow><mo>d</mo><mi>u</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac></math>.</p>
<p><br><strong>EITHER</strong></p>
<p>attempts to factorise <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><msub><mi>f</mi><mi>n</mi></msub><mo>'</mo></msup><mfenced><mi>x</mi></mfenced></math> (involving at least one of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><msup><mi>x</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>a</mi><mo>-</mo><mi>x</mi></mrow></mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup></math>) <em><strong>(M</strong></em><em><strong>1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi>n</mi><msup><mi>x</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><msup><mfenced><mrow><mi>a</mi><mo>-</mo><mi>x</mi></mrow></mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mrow><mfenced><mrow><mi>a</mi><mo>-</mo><mi>x</mi></mrow></mfenced><mo>-</mo><mi>x</mi></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><br><strong>OR</strong></p>
<p>attempts to express <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><msub><mi>f</mi><mi>n</mi></msub><mo>'</mo></msup><mfenced><mi>x</mi></mfenced></math> as the difference of two products with each product containing at least one of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><msup><mi>x</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>a</mi><mo>-</mo><mi>x</mi></mrow></mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup></math> <em><strong>(M</strong></em><em><strong>1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced><mrow><mo>-</mo><mi>x</mi></mrow></mfenced><mfenced><mrow><mi>n</mi><msup><mi>x</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup></mrow></mfenced><msup><mfenced><mrow><mi>a</mi><mo>-</mo><mi>x</mi></mrow></mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><mo>+</mo><mfenced><mrow><mi>a</mi><mo>-</mo><mi>x</mi></mrow></mfenced><mfenced><mrow><mi>n</mi><msup><mi>x</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup></mrow></mfenced><msup><mfenced><mrow><mi>a</mi><mo>-</mo><mi>x</mi></mrow></mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup></math> <em><strong>A1</strong></em></p>
<p><br><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><msub><mi>f</mi><mi>n</mi></msub><mo>'</mo></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>n</mi><msup><mi>x</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mrow><mi>a</mi><mo>-</mo><mn>2</mn><mi>x</mi></mrow></mfenced><msup><mfenced><mrow><mi>a</mi><mo>-</mo><mi>x</mi></mrow></mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup></math> <em><strong>AG</strong></em></p>
<p><br><strong>Note:</strong> Award the final <em><strong>(M1)A1</strong></em> for obtaining any of the following forms: </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><msub><mi>f</mi><mi>n</mi></msub><mo>'</mo></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>n</mi><msup><mi>x</mi><mi>n</mi></msup><msup><mfenced><mrow><mi>a</mi><mo>-</mo><mi>x</mi></mrow></mfenced><mi>n</mi></msup><mfenced><mfrac><mrow><mi>a</mi><mo>-</mo><mi>x</mi><mo>-</mo><mi>x</mi></mrow><mrow><mi>x</mi><mfenced><mrow><mi>a</mi><mo>-</mo><mi>x</mi></mrow></mfenced></mrow></mfrac></mfenced><mo>;</mo><mo> </mo><mo> </mo><mo> </mo><msup><msub><mi>f</mi><mi>n</mi></msub><mo>'</mo></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mrow><mi>n</mi><msup><mi>x</mi><mi>n</mi></msup><msup><mfenced><mrow><mi>a</mi><mo>-</mo><mi>x</mi></mrow></mfenced><mi>n</mi></msup></mrow><mrow><mi>x</mi><mfenced><mrow><mi>a</mi><mo>-</mo><mi>x</mi></mrow></mfenced></mrow></mfrac><mfenced><mrow><mi>a</mi><mo>-</mo><mi>x</mi><mo>-</mo><mi>x</mi></mrow></mfenced><mo>;</mo></math></p>
<p> <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><msub><mi>f</mi><mi>n</mi></msub><mo>'</mo></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>n</mi><msup><mi>x</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mrow><msup><mfenced><mrow><mi>a</mi><mo>-</mo><mi>x</mi></mrow></mfenced><mi>n</mi></msup><mo>-</mo><mi>x</mi><msup><mfenced><mrow><mi>a</mi><mo>-</mo><mi>x</mi></mrow></mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup></mrow></mfenced><mo>;</mo></math></p>
<p> <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><msub><mi>f</mi><mi>n</mi></msub><mo>'</mo></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><msup><mfenced><mrow><mi>a</mi><mo>-</mo><mi>x</mi></mrow></mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mrow><mi>n</mi><msup><mi>x</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><msup><mfenced><mrow><mi>a</mi><mo>-</mo><mi>x</mi></mrow></mfenced><mi>n</mi></msup><mo>-</mo><mi>n</mi><msup><mi>x</mi><mi>n</mi></msup></mrow></mfenced></math></p>
<p> </p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mi>n</mi></msub><mfenced><mi>x</mi></mfenced><mo>=</mo><msup><mfenced><mrow><mi>x</mi><mfenced><mrow><mi>a</mi><mo>-</mo><mi>x</mi></mrow></mfenced></mrow></mfenced><mi>n</mi></msup></math> <em><strong>(M</strong></em><em><strong>1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mfenced><mrow><mi>a</mi><mi>x</mi><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced><mi>n</mi></msup></math> <em><strong>A1</strong></em></p>
<p>attempts to use the chain rule <em><strong>(M</strong></em><em><strong>1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><msub><mi>f</mi><mi>n</mi></msub><mo>'</mo></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>n</mi><mfenced><mrow><mi>a</mi><mo>-</mo><mn>2</mn><mi>x</mi></mrow></mfenced><msup><mfenced><mrow><mi>a</mi><mi>x</mi><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup></math> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p><strong><br>Note:</strong> Award <em><strong>A1</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mfenced><mrow><mi>a</mi><mo>-</mo><mn>2</mn><mi>x</mi></mrow></mfenced></math> and <em><strong>A1</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>a</mi><mi>x</mi><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup></math>.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><msub><mi>f</mi><mi>n</mi></msub><mo>'</mo></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>n</mi><msup><mi>x</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mrow><mi>a</mi><mo>-</mo><mn>2</mn><mi>x</mi></mrow></mfenced><msup><mfenced><mrow><mi>a</mi><mo>-</mo><mi>x</mi></mrow></mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn><mo>,</mo><mo> </mo><mi>x</mi><mo>=</mo><mfrac><mi>a</mi><mn>2</mn></mfrac><mo>,</mo><mo> </mo><mi>x</mi><mo>=</mo><mi>a</mi></math> <em>A2</em></strong></p>
<p><strong>Note: </strong>Award <em><strong>A1</strong></em> for either two correct solutions or for obtaining <strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn><mo>,</mo><mo> </mo><mi>x</mi><mo>=</mo><mo>-</mo><mi>a</mi><mo>,</mo><mo> </mo><mi>x</mi><mo>=</mo><mo>-</mo><mfrac><mi>a</mi><mn>2</mn></mfrac></math><br> </strong> Award<em><strong> A0 </strong></em>otherwise.</p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempts to find an expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mi>n</mi></msub><mfenced><mfrac><mi>a</mi><mn>2</mn></mfrac></mfenced></math><strong> <em>(M1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mi>n</mi></msub><mfenced><mfrac><mi>a</mi><mn>2</mn></mfrac></mfenced><mo>=</mo><msup><mfenced><mfrac><mi>a</mi><mn>2</mn></mfrac></mfenced><mi>n</mi></msup><msup><mfenced><mrow><mi>a</mi><mo>-</mo><mfrac><mi>a</mi><mn>2</mn></mfrac></mrow></mfenced><mi>n</mi></msup></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mfenced><mfrac><mi>a</mi><mn>2</mn></mfrac></mfenced><mi>n</mi></msup><msup><mfenced><mfrac><mi>a</mi><mn>2</mn></mfrac></mfenced><mi>n</mi></msup><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><msup><mfenced><mfrac><mi>a</mi><mn>2</mn></mfrac></mfenced><mrow><mn>2</mn><mi>n</mi></mrow></msup></mrow></mfenced><mo>,</mo><mo> </mo><mfenced><mrow><mo>=</mo><msup><mfenced><msup><mfenced><mfrac><mi>a</mi><mn>2</mn></mfrac></mfenced><mi>n</mi></msup></mfenced><mn>2</mn></msup></mrow></mfenced></math><strong> <em>A1</em></strong></p>
<p><br><strong>EITHER</strong></p>
<p>since <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>∈</mo><msup><mi mathvariant="normal">ℝ</mi><mo>+</mo></msup><mo>,</mo><mo> </mo><msup><mfenced><mfrac><mi>a</mi><mn>2</mn></mfrac></mfenced><mrow><mn>2</mn><mi>n</mi></mrow></msup><mo>></mo><mn>0</mn></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><mo>,</mo><mo> </mo><mi>n</mi><mo>></mo><mn>1</mn></math> and so <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mi>n</mi></msub><mfenced><mfrac><mi>a</mi><mn>2</mn></mfrac></mfenced><mo>></mo><mn>0</mn></math>) <em><strong>R1</strong></em></p>
<p><br><strong>Note:</strong> Accept any logically equivalent conditions/statements on <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>.<br> Award <em><strong>R0</strong></em> if any conditions/statements specified involving <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> or both are incorrect.</p>
<p> </p>
<p><strong>OR</strong></p>
<p>(since <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>∈</mo><msup><mi mathvariant="normal">ℝ</mi><mo>+</mo></msup></math>), <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>a</mi><mn>2</mn></mfrac></math> raised to an even power (<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>n</mi></math>) (or equivalent reasoning) is always positive (and so <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mi>n</mi></msub><mfenced><mfrac><mi>a</mi><mn>2</mn></mfrac></mfenced><mo>></mo><mn>0</mn></math>) <em><strong>R1</strong></em></p>
<p><br><strong>Note:</strong> The condition <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>∈</mo><msup><mi mathvariant="normal">ℝ</mi><mo>+</mo></msup></math> is given in the question. Hence some candidates will assume <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>∈</mo><msup><mi mathvariant="normal">ℝ</mi><mo>+</mo></msup></math> and not state it. In these instances, award <em><strong>R1</strong></em> for a convincing argument.<br> Accept any logically equivalent conditions/statements on on <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>.<br> Award <em><strong>R0</strong></em> if any conditions/statements specified involving <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> or both are incorrect.</p>
<p><br><strong>THEN</strong></p>
<p>so <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mfrac><mi>a</mi><mn>2</mn></mfrac><mo>,</mo><mo> </mo><msub><mi>f</mi><mi>n</mi></msub><mfenced><mfrac><mi>a</mi><mn>2</mn></mfrac></mfenced></mrow></mfenced></math> is always above the horizontal axis<strong> <em>AG</em></strong></p>
<p><br><strong>Note:</strong> Do not award <em><strong>(M1)A0R1</strong></em>.</p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><msub><mi>f</mi><mi>n</mi></msub><mo>'</mo></msup><mfenced><mfrac><mi>a</mi><mn>4</mn></mfrac></mfenced><mo>=</mo><mi>n</mi><msup><mfenced><mfrac><mi>a</mi><mn>4</mn></mfrac></mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mrow><mi>a</mi><mo>-</mo><mfrac><mi>a</mi><mn>2</mn></mfrac></mrow></mfenced><msup><mfenced><mrow><mi>a</mi><mo>-</mo><mfrac><mi>a</mi><mn>4</mn></mfrac></mrow></mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mi>n</mi><msup><mfenced><mfrac><mi>a</mi><mn>4</mn></mfrac></mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mfrac><mi>a</mi><mn>2</mn></mfrac></mfenced><msup><mfenced><mfrac><mrow><mn>3</mn><mi>a</mi></mrow><mn>4</mn></mfrac></mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup></mrow></mfenced></math><strong> <em>A1</em></strong></p>
<p><br><strong>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><msup><mfenced><mfrac><mi>a</mi><mn>4</mn></mfrac></mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mfrac><mi>a</mi><mn>2</mn></mfrac></mfenced><msup><mfenced><mfrac><mrow><mn>3</mn><mi>a</mi></mrow><mn>4</mn></mfrac></mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><mo>></mo><mn>0</mn></math> as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>∈</mo><msup><mi mathvariant="normal">ℝ</mi><mo>+</mo></msup></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>∈</mo><msup><mi mathvariant="normal">ℤ</mi><mo>+</mo></msup></math> <em><strong>R1</strong></em></p>
<p><br><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><msup><mfenced><mfrac><mi>a</mi><mn>4</mn></mfrac></mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><mo>,</mo><mo> </mo><mfenced><mrow><mi>a</mi><mo>-</mo><mfrac><mi>a</mi><mn>2</mn></mfrac></mrow></mfenced></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mo> </mo><msup><mfenced><mrow><mi>a</mi><mo>-</mo><mfrac><mi>a</mi><mn>4</mn></mfrac></mrow></mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup></math> are all <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>></mo><mn>0</mn></math> <em><strong>R1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Do not award <em><strong>A0R1</strong></em>.<br> Accept equivalent reasoning on correct alternative expressions for <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><msub><mi>f</mi><mi>n</mi></msub><mo>'</mo></msup><mfenced><mfrac><mi>a</mi><mn>4</mn></mfrac></mfenced></math> and accept any logically equivalent conditions/statements on <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>.</p>
<p> Exceptions to the above are condone <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>></mo><mn>1</mn></math> and condone <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>></mo><mn>0</mn></math>.</p>
<p> An alternative form for <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><msub><mi>f</mi><mi>n</mi></msub><mo>'</mo></msup><mfenced><mfrac><mi>a</mi><mn>4</mn></mfrac></mfenced></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>2</mn><mi>n</mi></mrow></mfenced><msup><mfenced><mn>3</mn></mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><msup><mfenced><mfrac><mi>a</mi><mn>4</mn></mfrac></mfenced><mrow><mn>2</mn><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup></math>.</p>
<p><br><strong>THEN</strong></p>
<p>hence <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><msub><mi>f</mi><mi>n</mi></msub><mo>'</mo></msup><mfenced><mfrac><mi>a</mi><mn>4</mn></mfrac></mfenced><mo>></mo><mn>0</mn></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mi>n</mi></msub><mfenced><mn>0</mn></mfenced><mo>=</mo><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mi>n</mi></msub><mfenced><mfrac><mi>a</mi><mn>2</mn></mfrac></mfenced><mo>></mo><mn>0</mn></math><strong> <em>A1</em></strong></p>
<p>(since <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mi>n</mi></msub></math> is continuous and there are no stationary points between <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><mfrac><mi>a</mi><mn>2</mn></mfrac></math>)</p>
<p>the gradient (of the curve) must be positive between <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><mfrac><mi>a</mi><mn>2</mn></mfrac></math> <em><strong>R1</strong></em></p>
<p><br><strong>Note:</strong> Do not award <em><strong>A0R1</strong></em>.</p>
<p><br>hence <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><msub><mi>f</mi><mi>n</mi></msub><mo>'</mo></msup><mfenced><mfrac><mi>a</mi><mn>4</mn></mfrac></mfenced><mo>></mo><mn>0</mn></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><msub><mi>f</mi><mi>n</mi></msub><mo>'</mo></msup><mfenced><mrow><mo>-</mo><mn>1</mn></mrow></mfenced><mo>=</mo><mi>n</mi><msup><mfenced><mrow><mo>-</mo><mn>1</mn></mrow></mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mrow><mi>a</mi><mo>+</mo><mn>2</mn></mrow></mfenced><msup><mfenced><mrow><mi>a</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup></math></p>
<p>for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> even:</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><msup><mfenced><mrow><mo>-</mo><mn>1</mn></mrow></mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mrow><mo>=</mo><mo>-</mo><mi>n</mi></mrow></mfenced><mo><</mo><mn>0</mn></math> (and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>a</mi><mo>+</mo><mn>2</mn></mrow></mfenced><mo>,</mo><mo> </mo><msup><mfenced><mrow><mi>a</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup></math> are both <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>></mo><mn>0</mn></math>) <em><strong>R1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><msub><mi>f</mi><mi>n</mi></msub><mo>'</mo></msup><mfenced><mrow><mo>-</mo><mn>1</mn></mrow></mfenced><mo><</mo><mn>0</mn></math><strong> <em>A1</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><msub><mi>f</mi><mi>n</mi></msub><mo>'</mo></msup><mfenced><mn>0</mn></mfenced><mo>=</mo><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><msub><mi>f</mi><mi>n</mi></msub><mo>'</mo></msup><mfenced><mfrac><mi>a</mi><mn>4</mn></mfrac></mfenced><mo>></mo><mn>0</mn></math> (seen anywhere)<strong> <em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Candidates can give arguments based on the sign of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mo>-</mo><mn>1</mn></mrow></mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup></math> to obtain the <em><strong>R</strong></em> mark.<br> For example, award<em><strong> R1</strong></em> for the following:<br> If <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> is even, then <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>-</mo><mn>1</mn></math> is odd and hence <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mo>-</mo><mn>1</mn></mrow></mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><mo><</mo><mn>0</mn><mo> </mo><mfenced><mrow><mo>=</mo><mo>-</mo><mn>1</mn></mrow></mfenced></math>.<br> Do not award <em><strong>R0A1</strong></em>.<br> The second <strong><em>A1</em></strong> is independent of the other two marks.<br> The<em><strong> A</strong></em> marks can be awarded for correct descriptions expressed in words.<br> Candidates can state <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>0</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math> as a point of zero gradient from part (d) or show, state or explain (words or diagram) that <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><msub><mi>f</mi><mi>n</mi></msub><mo>'</mo></msup><mfenced><mn>0</mn></mfenced><mo>=</mo><mn>0</mn></math>. The last <em><strong>A </strong></em>mark can be awarded for a clearly labelled diagram showing changes in the sign of the gradient.<br> The last <em><strong>A1</strong></em> can be awarded for use of a specific case (e.g. <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>2</mn></math>).</p>
<p><br>hence <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>0</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math> is a local minimum point<strong> <em>AG</em></strong></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">g.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> odd:</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><msup><mfenced><mrow><mo>-</mo><mn>1</mn></mrow></mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mrow><mo>=</mo><mi>n</mi></mrow></mfenced><mo><</mo><mn>0</mn></math>, (and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>a</mi><mo>+</mo><mn>2</mn></mrow></mfenced><mo>,</mo><mo> </mo><msup><mfenced><mrow><mi>a</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup></math> are both <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>></mo><mn>0</mn></math>) so <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><msub><mi>f</mi><mi>n</mi></msub><mo>'</mo></msup><mfenced><mrow><mo>-</mo><mn>1</mn></mrow></mfenced><mo>></mo><mn>0</mn></math> <em><strong>R1</strong></em></p>
<p><br><strong>Note:</strong> Candidates can give arguments based on the sign of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mo>-</mo><mn>1</mn></mrow></mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup></math> to obtain the <em><strong>R</strong></em> mark.<br> For example, award<em><strong> R1</strong></em> for the following:<br> If <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> is odd, then <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>-</mo><mn>1</mn></math> is even and hence <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mo>-</mo><mn>1</mn></mrow></mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><mo>></mo><mn>0</mn><mo> </mo><mfenced><mrow><mo>=</mo><mn>1</mn></mrow></mfenced></math>.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><msub><mi>f</mi><mi>n</mi></msub><mo>'</mo></msup><mfenced><mn>0</mn></mfenced><mo>=</mo><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><msub><mi>f</mi><mi>n</mi></msub><mo>'</mo></msup><mfenced><mfrac><mi>a</mi><mn>4</mn></mfrac></mfenced><mo>></mo><mn>0</mn></math> (seen anywhere)<strong> <em>A1</em></strong></p>
<p><br><strong>Note:</strong> The <em><strong>A1</strong></em> is independent of the <em><strong>R1</strong></em>.<br> Candidates can state <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>0</mn><mo>,</mo><mo> </mo><mn>0</mn></mrow></mfenced></math> as a point of zero gradient from part (d) or show, state or explain (words or diagram) that <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><msub><mi>f</mi><mi>n</mi></msub><mo>'</mo></msup><mfenced><mn>0</mn></mfenced><mo>=</mo><mn>0</mn></math>. The last <em><strong>A</strong></em> mark can be awarded for a clearly labelled diagram showing changes in the sign of the gradient.<br> The last <em><strong>A1</strong></em> can be awarded for use of a specific case (e.g. <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>3</mn></math>).</p>
<p> </p>
<p>hence <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>0</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math> is a point of inflexion with zero gradient<strong> <em>AG</em></strong></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">g.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>considers the parity of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> <em><strong> (M1)</strong></em></p>
<p><br><strong>Note:</strong> Award<em><strong> M1</strong> </em>for stating at least one specific even value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> must be even (for four solutions) <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> The above 2 marks are independent of the 3 marks below.</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo><</mo><mi>k</mi><mo><</mo><msup><mfenced><mfrac><mi>a</mi><mn>2</mn></mfrac></mfenced><mrow><mn>2</mn><mi>n</mi></mrow></msup></math> <em><strong>A1A1A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for the correct lower endpoint, <em><strong>A1</strong></em> for the correct upper endpoint and <em><strong>A1</strong></em> for strict inequality signs.</p>
<p> The third <em><strong>A1</strong></em> (strict inequality signs) can only be awarded if <em><strong>A1</strong></em><em><strong>A1</strong></em> has been awarded.<br> For example, award <em><strong>A1A1A0</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><msup><mfenced><mfrac><mi>a</mi><mn>2</mn></mfrac></mfenced><mrow><mn>2</mn><mi>n</mi></mrow></msup></math>. Award <em><strong>A1A0A0</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>></mo><mn>0</mn></math>.</p>
<p> Award <em><strong>A1A0A0</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo><</mo><mi>k</mi><mo><</mo><msub><mi>f</mi><mi>n</mi></msub><mfenced><mfrac><mi>a</mi><mn>2</mn></mfrac></mfenced></math>.</p>
<p> </p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">h.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">g.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">g.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">h.</div>
</div>
<br><hr><br><div class="specification">
<p><strong>This question asks you to explore the behaviour and key features of cubic polynomials of the form</strong> <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mn>3</mn><mi>c</mi><mi>x</mi><mo>+</mo><mi>d</mi></math>.</p>
<p> </p>
<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>x</mi><mn>3</mn></msup><mo>-</mo><mn>3</mn><mi>c</mi><mi>x</mi><mo>+</mo><mn>2</mn></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math> and where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math> is a parameter, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
<p>The graphs of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>=</mo><mo>-</mo><mn>1</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>=</mo><mn>0</mn></math> are shown in the following diagrams.</p>
<p style="text-align: left;"><br> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>=</mo><mo>-</mo><mn>1</mn></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>=</mo><mn>0</mn></math></p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" 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"></p>
</div>
<div class="specification">
<p>On separate axes, 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> showing the value of the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-intercept and the coordinates of any points with zero gradient, for</p>
</div>
<div class="specification">
<p>Hence, or otherwise, find the set of values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math> such that the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> has</p>
</div>
<div class="specification">
<p>Given that the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> has one local maximum point and one local minimum point, show that</p>
</div>
<div class="specification">
<p>Hence, for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>></mo><mn>0</mn></math>, find the set of values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math> such that the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> has</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>=</mo><mn>1</mn></math>.</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><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>=</mo><mn>2</mn></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>Write down an expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><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>a point of inflexion with zero gradient.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>one local maximum point and one local minimum point.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>no points where the gradient is equal to zero.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-coordinate of the local maximum point is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><msup><mi>c</mi><mfrac><mn>3</mn><mn>2</mn></mfrac></msup><mo>+</mo><mn>2</mn></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-coordinate of the local minimum point is <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>2</mn><msup><mi>c</mi><mfrac><mn>3</mn><mn>2</mn></mfrac></msup><mo>+</mo><mn>2</mn></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">d.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>exactly one <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis intercept.</p>
<div class="marks">[2]</div>
<div class="question_part_label">e.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>exactly two <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis intercepts.</p>
<div class="marks">[2]</div>
<div class="question_part_label">e.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>exactly three <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis intercepts.</p>
<div class="marks">[2]</div>
<div class="question_part_label">e.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Consider the function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mn>3</mn><mi>c</mi><mi>x</mi><mo>+</mo><mi>d</mi></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math> and where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo> </mo><mo>,</mo><mo> </mo><mi>d</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
<p>Find all conditions on <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math> such that the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> has exactly one <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis intercept, explaining your reasoning.</p>
<div class="marks">[6]</div>
<div class="question_part_label">f.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><img style="display:block;margin-left:auto;margin-right:auto;" src="data:image/png;base64,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"></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>=</mo><mn>1</mn></math>: positive cubic with correct <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-intercept labelled <em><strong>A1</strong></em></p>
<p>local maximum point correctly labelled <em><strong>A1</strong></em></p>
<p>local minimum point correctly labelled <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img style="display:block;margin-left:auto;margin-right:auto;" 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"></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>=</mo><mn>2</mn></math>: positive cubic with correct <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-intercept labelled <em><strong>A1</strong></em></p>
<p>local maximum point correctly labelled <em><strong>A1</strong></em></p>
<p>local minimum point correctly labelled <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Accept the following exact answers:<br> Local maximum point coordinates <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>-</mo><msqrt><mn>2</mn></msqrt><mo>,</mo><mo> </mo><mn>2</mn><mo>+</mo><mn>4</mn><msqrt><mn>2</mn></msqrt></mrow></mfenced></math>.<br> Local minimum point coordinates <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><msqrt><mn>2</mn></msqrt><mo>,</mo><mo> </mo><mn>2</mn><mo>-</mo><mn>4</mn><msqrt><mn>2</mn></msqrt></mrow></mfenced></math>.</p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mo>(</mo><mi>x</mi><mo>)</mo><mo> </mo><mo>=</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>3</mn><mi>c</mi></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>3</mn><mi>c</mi></math> (an expression).</p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>=</mo><mn>0</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>considers the number of solutions to their <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mn>0</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>3</mn><mi>c</mi><mo>=</mo><mn>0</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>></mo><mn>0</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo><</mo><mn>0</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> The <em><strong>(M1)</strong></em> in part (c)(ii) can be awarded for work shown in either (ii) or (iii). </p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">c.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempts to solve their <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mn>0</mn></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>±</mo><msqrt><mi>c</mi></msqrt></math> <em><strong>(A1)</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>(A1)</strong></em> if either <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mo>-</mo><msqrt><mi>c</mi></msqrt></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><msqrt><mi>c</mi></msqrt></math> is subsequently considered.<br> Award the above <em><strong>(M1)(A1)</strong></em> if this work is seen in part (c).</p>
<p> </p>
<p>correctly evaluates <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mrow><mo>-</mo><msqrt><mi>c</mi></msqrt></mrow></mfenced></math> <em><strong>A1 </strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mrow><mo>-</mo><msqrt><mi>c</mi></msqrt></mrow></mfenced><mo>=</mo><mo>-</mo><msup><mi>c</mi><mfrac><mn>3</mn><mn>2</mn></mfrac></msup><mo>+</mo><mn>3</mn><msup><mi>c</mi><mfrac><mn>3</mn><mn>2</mn></mfrac></msup><mo>+</mo><mn>2</mn><mo> </mo><mfenced><mrow><mo>=</mo><mo>-</mo><mi>c</mi><msqrt><mi>c</mi></msqrt><mo>+</mo><mn>3</mn><mi>c</mi><msqrt><mi>c</mi></msqrt><mo>+</mo><mn>2</mn></mrow></mfenced></math></p>
<p>the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-coordinate of the local maximum point is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><msup><mi>c</mi><mfrac><mn>3</mn><mn>2</mn></mfrac></msup><mo>+</mo><mn>2</mn></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p> </p>
<p>correctly evaluates <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><msqrt><mi>c</mi></msqrt></mfenced></math> <em><strong>A1 </strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><msqrt><mi>c</mi></msqrt></mfenced><mo>=</mo><msup><mi>c</mi><mfrac><mn>3</mn><mn>2</mn></mfrac></msup><mo>-</mo><mn>3</mn><msup><mi>c</mi><mfrac><mn>3</mn><mn>2</mn></mfrac></msup><mo>+</mo><mn>2</mn><mo> </mo><mfenced><mrow><mo>=</mo><mi>c</mi><msqrt><mi>c</mi></msqrt><mo>-</mo><mn>3</mn><mi>c</mi><msqrt><mi>c</mi></msqrt><mo>+</mo><mn>2</mn></mrow></mfenced></math></p>
<p>the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-coordinate of the local minimum point is <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>2</mn><msup><mi>c</mi><mfrac><mn>3</mn><mn>2</mn></mfrac></msup><mo>+</mo><mn>2</mn></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">d.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>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> will have one <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis intercept if</p>
<p><strong>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>2</mn><msup><mi>c</mi><mfrac><mn>3</mn><mn>2</mn></mfrac></msup><mo>+</mo><mn>2</mn><mo>></mo><mn>0</mn></math> (or equivalent reasoning) <em><strong>R1</strong></em></p>
<p> </p>
<p><strong>OR</strong></p>
<p>the minimum point is above the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis <em><strong>R1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award<em><strong> R1</strong></em> for a rigorous approach that does not (only) refer to sketched graphs.</p>
<p> </p>
<p><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo><</mo><mi>c</mi><mo><</mo><mn>1</mn></math> <em><strong>A1 </strong></em></p>
<p> </p>
<p><strong>Note:</strong> Condone <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo><</mo><mn>1</mn></math>. The <em><strong>A1</strong></em> is independent of the <em><strong>R1</strong></em>.</p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">e.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>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> will have two <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis intercepts if</p>
<p><strong>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>2</mn><msup><mi>c</mi><mfrac><mn>3</mn><mn>2</mn></mfrac></msup><mo>+</mo><mn>2</mn><mo>=</mo><mn>0</mn></math> (or equivalent reasoning) <em><strong>(M1)</strong></em></p>
<p> </p>
<p><strong>OR</strong></p>
<p>evidence from the graph in part(a)(i) <em><strong>(M1)</strong></em></p>
<p> </p>
<p><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>=</mo><mn>1</mn></math> <em><strong>A1 </strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">e.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>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> will have three <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis intercepts if</p>
<p><strong>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>2</mn><msup><mi>c</mi><mfrac><mn>3</mn><mn>2</mn></mfrac></msup><mo>+</mo><mn>2</mn><mo><</mo><mn>0</mn></math> (or equivalent reasoning) <em><strong>(M1)</strong></em></p>
<p> </p>
<p><strong>OR</strong></p>
<p>reasoning from the results in both parts (e)(i) and (e)(ii) <em><strong>(M1)</strong></em></p>
<p> </p>
<p><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>></mo><mn>1</mn></math> <em><strong>A1 </strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">e.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>case 1:</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>≤</mo><mn>0</mn></math> (independent of the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math>) <em><strong>A1 </strong></em></p>
<p><strong>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>'</mo><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mn>0</mn></math> does not have two solutions (has no solutions or <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math> solution) <em><strong> R1</strong></em></p>
<p><strong><br>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>g</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>≥</mo><mn>0</mn></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><msup><mo>∈</mo><mo>~</mo></msup></math> <em><strong> R1</strong></em></p>
<p><strong><br>OR</strong></p>
<p>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> has no local maximum or local minimum points, hence any vertical translation of this graph (<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>g</mi><mfenced><mi>x</mi></mfenced></math>) will also have no local maximum or local minimum points <em><strong> R1</strong></em></p>
<p><strong><br>THEN</strong></p>
<p>therefore there is only one <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis intercept <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award at most <em><strong>A0R1</strong></em> if only <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo><</mo><mn>0</mn></math> is considered.</p>
<p> </p>
<p><br>case 2</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>></mo><mn>0</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>-</mo><msqrt><mi>c</mi></msqrt><mo>,</mo><mo> </mo><mn>2</mn><msup><mi>c</mi><mfrac><mn>3</mn><mn>2</mn></mfrac></msup><mo>+</mo><mi>d</mi></mrow></mfenced></math> is a local maximum point and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><msqrt><mi>c</mi></msqrt><mo>,</mo><mo> </mo><mo>-</mo><mn>2</mn><msup><mi>c</mi><mfrac><mn>3</mn><mn>2</mn></mfrac></msup><mo>+</mo><mi>d</mi></mrow></mfenced></math> is a local minimum point <em><strong>(A1)</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>(A1)</strong></em> for a correct <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-coordinate seen for either the maximum or the minimum.</p>
<p> </p>
<p>considers the positions of the local maximum point and/or the local minimum point <em><strong>(M1)</strong></em></p>
<p> </p>
<p><strong>EITHER</strong><br>considers both points above the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis or both points below the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis<br><br><br><strong>OR</strong></p>
<p>considers either the local minimum point only above the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis OR the local maximum point only below the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis<br><br><br><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo>></mo><mn>2</mn><msup><mi>c</mi><mfrac><mn>3</mn><mn>2</mn></mfrac></msup></math> (both points above the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis) <em><strong>A1 </strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo><</mo><mo>-</mo><mn>2</mn><msup><mi>c</mi><mfrac><mn>3</mn><mn>2</mn></mfrac></msup></math> (both points above the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis) <em><strong>A1 </strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award at most <em><strong>(A1)(M1)A0A0</strong></em> for case 2 if <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>></mo><mn>0</mn></math> is not clearly stated.</p>
<p> </p>
<p><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">f.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">f.</div>
</div>
<br><hr><br><div class="specification">
<p><strong>This question asks you to explore properties of a family of curves of the type</strong> <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi></math> <strong>for various values of</strong> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> <strong>and</strong> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math>, <strong>where</strong> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>,</mo><mo> </mo><mi>b</mi><mo>∈</mo><mi mathvariant="normal">ℕ</mi></math>.</p>
</div>
<div class="specification">
<p>On the same set of axes, sketch the following curves for <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>2</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mn>2</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>2</mn><mo>≤</mo><mi>y</mi><mo>≤</mo><mn>2</mn></math>, clearly indicating any points of intersection with the coordinate axes.</p>
</div>
<div class="specification">
<p>Now, consider curves of the form <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mi>b</mi></math>, for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>≥</mo><mo>-</mo><mroot><mi>b</mi><mn>3</mn></mroot></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>∈</mo><msup><mi mathvariant="normal">ℤ</mi><mo>+</mo></msup></math>.</p>
</div>
<div class="specification">
<p>Next, consider the curve <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mi>x</mi><mo>,</mo><mo> </mo><mi>x</mi><mo>≥</mo><mn>0</mn></math>.</p>
</div>
<div class="specification">
<p>The curve <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mi>x</mi></math> has two points of inflexion. Due to the symmetry of the curve these points have the same <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-coordinate.</p>
</div>
<div class="specification">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mo>(</mo><mi>x</mi><mo>,</mo><mo> </mo><mi>y</mi><mo>)</mo></math> is defined to be a rational point on a curve if <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> are rational numbers.</p>
<p>The tangent to the curve <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi></math> at a rational point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math> intersects the curve at another rational point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Q</mtext></math>.</p>
<p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math> be the curve <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>2</mn></math>, for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>≥</mo><mo>-</mo><mroot><mn>2</mn><mn>3</mn></mroot></math>. The rational point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mo>(</mo><mo>-</mo><mn>1</mn><mo>,</mo><mo> </mo><mo>-</mo><mn>1</mn><mo>)</mo></math> lies on <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup><mo>,</mo><mo> </mo><mi>x</mi><mo>≥</mo><mn>0</mn></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><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>1</mn><mo>,</mo><mo> </mo><mi>x</mi><mo>≥</mo><mo>-</mo><mn>1</mn></math></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>Write down the coordinates of the two points of inflexion on the curve <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>1</mn></math>.</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>By considering each curve from part (a), identify two key features that would distinguish one curve from the other.</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>By varying the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math>, suggest two key features common to these curves.</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>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mo>±</mo><mfrac><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow><mrow><mn>2</mn><msqrt><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mi>x</mi></msqrt></mrow></mfrac></math>, for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>></mo><mn>0</mn></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence deduce that the curve <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mi>x</mi><mo> </mo></math>has no local minimum or maximum points.</p>
<div class="marks">[1]</div>
<div class="question_part_label">d.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of this <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-coordinate, giving your answer in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><msqrt><mfrac><mrow><mi>p</mi><msqrt><mn>3</mn></msqrt><mo>+</mo><mi>q</mi></mrow><mi>r</mi></mfrac></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><mi mathvariant="normal">ℤ</mi></math>.</p>
<div class="marks">[7]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the equation of the tangent to <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math> at <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">f.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, find the coordinates of the rational point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Q</mtext></math> where this tangent intersects <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math>, expressing each coordinate as a fraction.</p>
<div class="marks">[2]</div>
<div class="question_part_label">f.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>S</mtext><mo>(</mo><mo>-</mo><mn>1</mn><mo> </mo><mo>,</mo><mo> </mo><mn>1</mn><mo>)</mo></math> also lies on <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math>. The line <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>[QS]</mtext></math> intersects <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math> at a further point. Determine the coordinates of this point.</p>
<div class="marks">[5]</div>
<div class="question_part_label">g.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><img src="data:image/png;base64,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"></p>
<p>approximately symmetric about the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup></math> <em><strong>A1</strong></em></p>
<p>including cusp/sharp point at <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>0</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Final <em><strong>A1</strong> </em>can be awarded if intersections are in approximate correct place with respect to the axes shown. Award <em><strong>A1A1A1A0</strong></em> if graphs ‘merge’ or ‘cross’ or are discontinuous at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis but are otherwise correct. Award <em><strong>A1A0A0A0</strong></em> if only one correct branch of both curves are seen.</p>
<p><strong>Note:</strong> If they sketch graphs on separate axes, award a maximum of 2 marks for the ‘best’ response seen. This is likely to be <em><strong>A1A1A0A0</strong></em>.</p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>approximately symmetric about the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>1</mn></math> with approximately correct gradient at axes intercepts <em><strong>A1</strong></em><br>some indication of position of intersections at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mo>−</mo><mn>1</mn></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>±</mo><mn>1</mn></math> <em><strong>A1</strong></em> </p>
<p><em><strong>[2 marks]</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Final <em><strong>A1</strong> </em>can be awarded if intersections are in approximate correct place with respect to the axes shown. Award <em><strong>A1A1A1A0</strong> </em>if graphs ‘merge’ or ‘cross’ or are discontinuous at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis but are otherwise correct. Award <em><strong>A1A0A0A0</strong></em> if only one correct branch of both curves are seen.</p>
<p><strong>Note:</strong> If they sketch graphs on separate axes, award a maximum of 2 marks for the ‘best’ response seen. This is likely to be <em><strong>A1A1A0A0</strong></em>.</p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>0</mn><mo>,</mo><mo> </mo><mn>1</mn></mrow></mfenced></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>0</mn><mo>,</mo><mo> </mo><mo>-</mo><mn>1</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Any <strong>two</strong> from:</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup></math> has a cusp/sharp point, (the other does not)</p>
<p>graphs have different domains</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>1</mn></math> has points of inflexion, (the other does not)</p>
<p>graphs have different <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis intercepts (one goes through the origin, and the other does not)</p>
<p>graphs have different <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-axis intercepts <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Follow through from their sketch in part (a)(i). In accordance with marking rules, mark their first two responses and ignore any subsequent.</p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Any <strong>two</strong> from:</p>
<p>as , <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>→</mo><mo>∞</mo><mo>,</mo><mo> </mo><mi>y</mi><mo>→</mo><mo>±</mo><mo>∞</mo></math></p>
<p>as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>→</mo><mo>∞</mo><mo>,</mo><mo> </mo><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mi>b</mi></math> is approximated by <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup></math> (or similar)</p>
<p>they have <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> intercepts at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mo>-</mo><mroot><mi>b</mi><mn>3</mn></mroot></math></p>
<p>they have <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> intercepts at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfenced><mo>±</mo></mfenced><msqrt><mi>b</mi></msqrt></math></p>
<p>they all have the same range</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>0</mn></math> (or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis) is a line of symmetry</p>
<p>they all have the same line of symmetry <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>y</mi><mo>=</mo><mn>0</mn></mrow></mfenced></math></p>
<p>they have one <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis intercept</p>
<p>they have two <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-axis intercepts</p>
<p>they have two points of inflexion</p>
<p>at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis intercepts, curve is vertical/infinite gradient</p>
<p>there is no cusp/sharp point at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis intercepts <em><strong>A1A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> The last example is the only valid answer for things “not” present. Do not credit an answer of “they are all symmetrical” without some reference to the line of symmetry.</p>
<p><strong>Note:</strong> Do not allow same/ similar shape or equivalent.</p>
<p><strong>Note:</strong> In accordance with marking rules, mark their first two responses and ignore any subsequent.</p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>attempt to differentiate implicitly <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>y</mi><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow><mrow><mn>2</mn><mi>y</mi></mrow></mfrac></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mo>±</mo></mfenced><mn>2</mn><msqrt><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mi>x</mi></msqrt><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mo>±</mo><mfrac><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow><mrow><mn>2</mn><msqrt><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mi>x</mi></msqrt></mrow></mfrac></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>attempt to use chain rule <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfenced><mo>±</mo></mfenced><msqrt><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mi>x</mi></msqrt></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mfenced><mo>±</mo></mfenced><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mfenced><mrow><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mi>x</mi></mrow></mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup><mfenced><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfenced></math> <em><strong>A1A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A1</strong> </em>for <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mo>±</mo></mfenced><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mfenced><mrow><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mi>x</mi></mrow></mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup></math>, <em><strong>A1</strong> </em>for <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfenced></math></p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mo>±</mo><mfrac><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow><mrow><mn>2</mn><msqrt><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mi>x</mi></msqrt></mrow></mfrac></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p>local minima/maxima occur when<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mn>0</mn></math><br><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>+</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>=</mo><mn>0</mn></math> has no (real) solutions (or equivalent) <em><strong>R1</strong></em></p>
<p><br><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>≥</mo><mn>0</mn><mo>⇒</mo></mrow></mfenced><mo> </mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn><mo>></mo><mn>0</mn></math>, so <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>≠</mo><mn>0</mn></math> <em><strong>R1</strong></em></p>
<p><br><strong>THEN</strong></p>
<p>so, no local minima/maxima exist <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">d.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p>attempt to use quotient rule to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>y</mi></mrow><mrow><mo>d</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>y</mi></mrow><mrow><mo>d</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mfenced><mo>±</mo></mfenced><mfrac><mrow><mn>12</mn><mi>x</mi><msqrt><mi>x</mi><mo>+</mo><msup><mi>x</mi><mn>3</mn></msup></msqrt><mo>-</mo><mfenced><mrow><mn>1</mn><mo>+</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced><msup><mfenced><mrow><mi>x</mi><mo>+</mo><msup><mi>x</mi><mn>3</mn></msup></mrow></mfenced><mrow><mo>-</mo><mstyle displaystyle="true"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle></mrow></msup><mfenced><mrow><mn>1</mn><mo>+</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced></mrow><mrow><mn>4</mn><mfenced><mrow><mi>x</mi><mo>+</mo><msup><mi>x</mi><mn>3</mn></msup></mrow></mfenced></mrow></mfrac></math> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p><strong><br>Note:</strong> Award <em><strong>A1</strong> </em>for correct <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>12</mn><mi>x</mi><msqrt><mi>x</mi><mo>+</mo><msup><mi>x</mi><mn>3</mn></msup></msqrt></math> and correct denominator, <em><strong>A1</strong> </em>for correct <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfenced><mrow><mn>1</mn><mo>+</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced><msup><mfenced><mrow><mi>x</mi><mo>+</mo><msup><mi>x</mi><mn>3</mn></msup></mrow></mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup><mfenced><mrow><mn>1</mn><mo>+</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced></math>.</p>
<p><strong>Note:</strong> Future <em><strong>A</strong></em> marks may be awarded if the denominator is missing or incorrect.</p>
<p><br>stating or using <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>y</mi></mrow><mrow><mo>d</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mn>0</mn></math> (may be seen anywhere) <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>12</mn><mi>x</mi><msqrt><mi>x</mi><mo>+</mo><msup><mi>x</mi><mn>3</mn></msup></msqrt><mo>=</mo><mfenced><mrow><mn>1</mn><mo>+</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced><msup><mfenced><mrow><mi>x</mi><mo>+</mo><msup><mi>x</mi><mn>3</mn></msup></mrow></mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup><mfenced><mrow><mn>1</mn><mo>+</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced></math></p>
<p><br><strong>OR</strong></p>
<p>attempt to use product rule to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>y</mi></mrow><mrow><mo>d</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>y</mi></mrow><mrow><mo>d</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mfenced><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mfenced><mfenced><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfenced><msup><mfenced><mrow><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mi>x</mi></mrow></mfenced><mrow><mo>-</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow></msup><mo>+</mo><mn>3</mn><mi>x</mi><msup><mfenced><mrow><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mi>x</mi></mrow></mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup></math> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p><strong><br>Note:</strong> Award <em><strong>A1</strong></em> for correct first term, <em><strong>A1 </strong></em>for correct second term.</p>
<p><br>setting <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>y</mi></mrow><mrow><mo>d</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mn>0</mn></math> <em><strong>(M1)</strong></em></p>
<p><br><strong>OR</strong></p>
<p>attempts implicit differentiation on <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>y</mi><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><msup><mfenced><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac></mfenced><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>y</mi><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>y</mi></mrow><mrow><mo>d</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mn>6</mn><mi>x</mi></math> <em><strong>A1</strong></em></p>
<p>recognizes that <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>y</mi></mrow><mrow><mo>d</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mn>0</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mo>±</mo><msqrt><mn>3</mn><mi>x</mi></msqrt></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mo>±</mo></mfenced><mfrac><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow><mrow><mn>2</mn><msqrt><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mi>x</mi></msqrt></mrow></mfrac><mo>=</mo><mfenced><mo>±</mo></mfenced><msqrt><mn>3</mn><mi>x</mi></msqrt></math> <em><strong>(A1)</strong></em></p>
<p><br><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>12</mn><mi>x</mi><mfenced><mrow><mi>x</mi><mo>+</mo><msup><mi>x</mi><mn>3</mn></msup></mrow></mfenced><mo>=</mo><msup><mfenced><mrow><mn>1</mn><mo>+</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced><mn>2</mn></msup></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>12</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>12</mn><msup><mi>x</mi><mn>4</mn></msup><mo>=</mo><mn>9</mn><msup><mi>x</mi><mn>4</mn></msup><mo>+</mo><mn>6</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><msup><mi>x</mi><mn>4</mn></msup><mo>+</mo><mn>6</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn><mo>=</mo><mn>0</mn></math> <em><strong>A1</strong></em></p>
<p>attempt to use quadratic formula or equivalent <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>=</mo><mfrac><mrow><mo>-</mo><mn>6</mn><mo>±</mo><msqrt><mn>48</mn></msqrt></mrow><mn>6</mn></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>x</mi><mo>></mo><mn>0</mn><mo>⇒</mo></mrow></mfenced><mi>x</mi><mo>=</mo><msqrt><mfrac><mrow><mn>2</mn><msqrt><mn>3</mn></msqrt><mo>-</mo><mn>3</mn></mrow><mn>3</mn></mfrac></msqrt><mo> </mo><mfenced><mrow><mi>p</mi><mo>=</mo><mn>2</mn><mo>,</mo><mo> </mo><mi>q</mi><mo>=</mo><mo>-</mo><mn>3</mn><mo>,</mo><mo> </mo><mi>r</mi><mo>=</mo><mn>3</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Accept any integer multiple of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>,</mo><mo> </mo><mi>q</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math> (e.g. <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>,</mo><mo> </mo><mo>-</mo><mn>6</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn></math>).</p>
<p> </p>
<p><em><strong>[7 marks]</strong></em></p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to find tangent line through <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>-</mo><mn>1</mn><mo>,</mo><mo> </mo><mo>-</mo><mn>1</mn></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>+</mo><mn>1</mn><mo>=</mo><mo>-</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mfenced><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfenced></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>-</mo><mn>1</mn><mo>.</mo><mn>5</mn><mi>x</mi><mo>-</mo><mn>2</mn><mo>.</mo><mn>5</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">f.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to solve simultaneously with <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>2</mn></math> <em><strong>(M1)</strong></em></p>
<p><br><strong>Note:</strong> The <em><strong>M1</strong></em> mark can be awarded for an unsupported correct answer in an incorrect format (e.g. <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>4</mn><mo>.</mo><mn>25</mn><mo>,</mo><mo> </mo><mo>-</mo><mn>8</mn><mo>.</mo><mn>875</mn><mo>)</mo></math>).</p>
<p><br>obtain <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mfrac><mn>17</mn><mn>4</mn></mfrac><mo>,</mo><mo> </mo><mo>-</mo><mfrac><mn>71</mn><mn>8</mn></mfrac></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">f.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to find equation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>[QS]</mtext></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>y</mi><mo>-</mo><mn>1</mn></mrow><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mo>=</mo><mo>-</mo><mfrac><mn>79</mn><mn>42</mn></mfrac><mfenced><mrow><mo>=</mo><mo>-</mo><mn>1</mn><mo>.</mo><mn>88095</mn><mo>…</mo></mrow></mfenced></math> <em><strong>(A1)</strong></em></p>
<p>solve simultaneously with <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>2</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>28798</mn><mo>…</mo><mfenced><mrow><mo>=</mo><mfrac><mn>127</mn><mn>441</mn></mfrac></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>-</mo><mn>1</mn><mo>.</mo><mn>4226</mn><mo>…</mo><mfenced><mrow><mo>=</mo><mfrac><mn>13175</mn><mn>9261</mn></mfrac></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>0</mn><mo>.</mo><mn>228</mn><mo>,</mo><mo> </mo><mo>-</mo><mn>1</mn><mo>.</mo><mn>42</mn></mrow></mfenced></math></p>
<p> </p>
<p><strong>OR</strong></p>
<p>attempt to find vector equation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>[QS]</mtext></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mi>x</mi></mtd></mtr><mtr><mtd><mi>y</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mtable><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mi>λ</mi><mfenced><mtable><mtr><mtd><mfrac><mn>21</mn><mn>4</mn></mfrac></mtd></mtr><mtr><mtd><mo>-</mo><mfrac><mn>79</mn><mn>8</mn></mfrac></mtd></mtr></mtable></mfenced></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mo>-</mo><mn>1</mn><mo>+</mo><mfrac><mn>21</mn><mn>4</mn></mfrac><mi>λ</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>1</mn><mo>-</mo><mfrac><mn>79</mn><mn>8</mn></mfrac><mi>λ</mi></math></p>
<p>attempt to solve <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mn>1</mn><mo>-</mo><mfrac><mn>79</mn><mn>8</mn></mfrac><mi>λ</mi></mrow></mfenced><mn>2</mn></msup><mo>=</mo><msup><mfenced><mrow><mo>-</mo><mn>1</mn><mo>+</mo><mfrac><mn>21</mn><mn>4</mn></mfrac><mi>λ</mi></mrow></mfenced><mn>3</mn></msup><mo>+</mo><mn>2</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>2453</mn><mo>…</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>28798</mn><mo>…</mo><mfenced><mrow><mo>=</mo><mfrac><mn>127</mn><mn>441</mn></mfrac></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>-</mo><mn>1</mn><mo>.</mo><mn>4226</mn><mo>…</mo><mfenced><mrow><mo>=</mo><mfrac><mn>13175</mn><mn>9261</mn></mfrac></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>0</mn><mo>.</mo><mn>228</mn><mo>,</mo><mo> </mo><mo>-</mo><mn>1</mn><mo>.</mo><mn>42</mn></mrow></mfenced></math></p>
<p> </p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">g.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>This was a relatively straightforward start, though it was disappointing to see so many candidates sketch their graphs on two separate axes, despite the question stating they should be sketched on the same axes.</p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Of those candidates producing clear sketches, the vast majority were able to recognise the points of inflexion and write down their coordinates. A small number embarked on a mostly fruitless algebraic approach rather than use their graphs as intended. The distinguishing features between curves tended to focus on points of intersection with the axes, which was accepted. Only a small number offered ideas such as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>→</mo><mo>∞</mo></math> on both curves. A number of (incorrect) suggestions were seen, stating that both curves tended towards a linear asymptote.</p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>A majority of candidates' suggestions related to the number of intersection points with the coordinate axes, while the idea of the <em>x</em>-axis acting as a line of symmetry was also often seen.</p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>The required differentiation was straightforward for the majority of candidates.</p>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>The majority employed the quotient rule here, often doing so successfully to find a correct expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mtext>d</mtext><mn>2</mn></msup><mi>y</mi></mrow><mrow><mtext>d</mtext><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></math>. Despite realising that <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mtext>d</mtext><mn>2</mn></msup><mi>y</mi></mrow><mrow><mtext>d</mtext><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mn>0</mn></math>, the resulting algebra to find the required solution proved a step too far for most. A number of slips were seen in candidates' working, though better candidates were able to answer the question confidently.</p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Mistakes proved to be increasingly common by this stage of the paper. Various equations of lines were suggested, with the incorrect <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>1</mn><mo>.</mo><mn>5</mn><mi>x</mi><mo>+</mo><mn>2</mn><mo>.</mo><mn>5</mn></math> appearing more than once. Only the better candidates were able to tackle the final part of the question with any success; it was pleasing to see a number of clear algebraic (only) approaches, though this was not necessary to obtain full marks.</p>
<div class="question_part_label">f.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">f.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Significant work on this question part was rarely seen, and it may have been the case that many candidates chose to spend their remaining time on the second question, especially if they felt they were making little progress with part f. Having said that, correct final answers were seen from better candidates, though these were few and far between.</p>
<div class="question_part_label">g.</div>
</div>
<br><hr><br><div class="specification">
<p><strong>This question asks you to explore some properties of polygonal numbers and to determine and prove interesting results involving these numbers.</strong></p>
<p><br>A polygonal number is an integer which can be represented as a series of dots arranged in the shape of a regular polygon. Triangular numbers, square numbers and pentagonal numbers are examples of polygonal numbers.</p>
<p>For example, a triangular number is a number that can be arranged in the shape of an equilateral triangle. The first five triangular numbers are <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>,</mo><mo> </mo><mn>3</mn><mo>,</mo><mo> </mo><mn>6</mn><mo>,</mo><mo> </mo><mn>10</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>15</mn></math>.</p>
<p>The following table illustrates the first five triangular, square and pentagonal numbers respectively. In each case the first polygonal number is one represented by a single dot.</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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"></p>
<p>For an <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math>-sided regular polygon, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>∈</mo><msup><mi mathvariant="normal">ℤ</mi><mo>+</mo></msup><mo>,</mo><mo> </mo><mi>r</mi><mo>≥</mo><mn>3</mn></math>, the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>th polygonal number <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mi>r</mi></msub><mfenced><mi>n</mi></mfenced></math> is given by</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mi>r</mi></msub><mfenced><mi>n</mi></mfenced><mo>=</mo><mfrac><mrow><mfenced><mrow><mi>r</mi><mo>-</mo><mn>2</mn></mrow></mfenced><msup><mi>n</mi><mn>2</mn></msup><mo>-</mo><mfenced><mrow><mi>r</mi><mo>-</mo><mn>4</mn></mrow></mfenced><mi>n</mi></mrow><mn>2</mn></mfrac></math>, where <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>
<p style="text-align: left;">Hence, for square numbers, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>4</mn></msub><mfenced><mi>n</mi></mfenced><mo>=</mo><mfrac><mrow><mfenced><mrow><mn>4</mn><mo>-</mo><mn>2</mn></mrow></mfenced><msup><mi>n</mi><mn>2</mn></msup><mo>-</mo><mfenced><mrow><mn>4</mn><mo>-</mo><mn>4</mn></mrow></mfenced><mi>n</mi></mrow><mn>2</mn></mfrac><mo>=</mo><msup><mi>n</mi><mn>2</mn></msup></math>.</p>
</div>
<div class="specification">
<p>The <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>th pentagonal number can be represented by the arithmetic series</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>5</mn></msub><mfenced><mi>n</mi></mfenced><mo>=</mo><mn>1</mn><mo>+</mo><mn>4</mn><mo>+</mo><mn>7</mn><mo>+</mo><mo>…</mo><mo>+</mo><mfenced><mrow><mn>3</mn><mi>n</mi><mo>-</mo><mn>2</mn></mrow></mfenced></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>For triangular numbers, verify that <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>3</mn></msub><mfenced><mi>n</mi></mfenced><mo>=</mo><mfrac><mrow><mi>n</mi><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mrow><mn>2</mn></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>The number <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>351</mn></math> is a triangular number. Determine which one it is.</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>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>3</mn></msub><mfenced><mi>n</mi></mfenced><mo>+</mo><msub><mi>P</mi><mn>3</mn></msub><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>≡</mo><msup><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup></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>State, in words, what the identity given in part (b)(i) shows for two consecutive triangular numbers.</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>For <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>4</mn></math>, sketch a diagram clearly showing your answer to part (b)(ii).</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.iii.</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>8</mn><msub><mi>P</mi><mn>3</mn></msub><mfenced><mi>n</mi></mfenced><mo>+</mo><mn>1</mn></math> is the square of an odd number for all <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">[3]</div>
<div class="question_part_label">c.</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"><msub><mi>P</mi><mn>5</mn></msub><mfenced><mi>n</mi></mfenced><mo>=</mo><mfrac><mrow><mi>n</mi><mfenced><mrow><mn>3</mn><mi>n</mi><mo>-</mo><mn>1</mn></mrow></mfenced></mrow><mn>2</mn></mfrac></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">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By using a suitable table of values or otherwise, determine the smallest positive integer, greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math>, that is both a triangular number and a pentagonal number.</p>
<div class="marks">[5]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>A polygonal number, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mi>r</mi></msub><mfenced><mi>n</mi></mfenced></math>, can be represented by the series</p>
<p style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mtext>Σ</mtext><mrow><mi>m</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mfenced><mrow><mn>1</mn><mo>+</mo><mfenced><mrow><mi>m</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mi>r</mi><mo>-</mo><mn>2</mn></mrow></mfenced></mrow></mfenced></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>∈</mo><msup><mi mathvariant="normal">ℤ</mi><mo>+</mo></msup><mo>,</mo><mo> </mo><mi>r</mi><mo>≥</mo><mn>3</mn></math>.</p>
<p>Use mathematical induction to prove that <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mi>r</mi></msub><mfenced><mi>n</mi></mfenced><mo>=</mo><mfrac><mrow><mfenced><mrow><mi>r</mi><mo>-</mo><mn>2</mn></mrow></mfenced><msup><mi>n</mi><mn>2</mn></msup><mo>-</mo><mfenced><mrow><mi>r</mi><mo>-</mo><mn>4</mn></mrow></mfenced><mi>n</mi></mrow><mn>2</mn></mfrac></math> where <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">[8]</div>
<div class="question_part_label">f.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>3</mn></msub><mfenced><mi>n</mi></mfenced><mo>=</mo><mfrac><mrow><mfenced><mrow><mn>3</mn><mo>-</mo><mn>2</mn></mrow></mfenced><msup><mi>n</mi><mn>2</mn></msup><mo>-</mo><mfenced><mrow><mn>3</mn><mo>-</mo><mn>4</mn></mrow></mfenced><mi>n</mi></mrow><mn>2</mn></mfrac></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>3</mn></msub><mfenced><mi>n</mi></mfenced><mo>=</mo><mfrac><mrow><msup><mi>n</mi><mn>2</mn></msup><mo>-</mo><mfenced><mrow><mo>-</mo><mi>n</mi></mrow></mfenced></mrow><mn>2</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>3</mn></msub><mfenced><mi>n</mi></mfenced><mo>=</mo><mfrac><mrow><msup><mi>n</mi><mn>2</mn></msup><mo>+</mo><mi>n</mi></mrow><mn>2</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A0A1</strong></em> if <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>3</mn></msub><mfenced><mi>n</mi></mfenced><mo>=</mo><mfrac><mrow><msup><mi>n</mi><mn>2</mn></msup><mo>+</mo><mi>n</mi></mrow><mn>2</mn></mfrac></math> only is seen.</p>
<p>Do not award any marks for numerical verification.</p>
<p> </p>
<p>so for triangular numbers, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>3</mn></msub><mfenced><mi>n</mi></mfenced><mo>=</mo><mfrac><mrow><mi>n</mi><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mrow><mn>2</mn></mfrac></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>uses a table of values to find a positive integer that satisfies <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>3</mn></msub><mfenced><mi>n</mi></mfenced><mo>=</mo><mn>351</mn></math> <em><strong>(M1)</strong></em></p>
<p>for example, a list showing at least <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math> consecutive terms <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>…</mo><mn>325</mn><mo>,</mo><mo> </mo><mn>351</mn><mo>,</mo><mo> </mo><mn>378</mn><mo>…</mo></mrow></mfenced></math></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for use of a GDC’s numerical solve or graph feature.</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>26</mn></math> (<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>26</mn></math>th triangular number) <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A0</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mo>−</mo><mn>27</mn><mo>,</mo><mn>26</mn></math>. Award <em><strong>A0</strong></em> if additional solutions besides <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>26</mn></math> are given.</p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>attempts to solve <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>n</mi><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mrow><mn>2</mn></mfrac><mo>=</mo><mn>351</mn><mo> </mo><mfenced><mrow><msup><mi>n</mi><mn>2</mn></msup><mo>+</mo><mi>n</mi><mo>-</mo><mn>702</mn><mo>=</mo><mn>0</mn></mrow></mfenced></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mfrac><mrow><mo>-</mo><mn>1</mn><mo>±</mo><msqrt><msup><mn>1</mn><mn>2</mn></msup><mo>-</mo><mn>4</mn><mfenced><mn>1</mn></mfenced><mfenced><mrow><mo>-</mo><mn>702</mn></mrow></mfenced></msqrt></mrow><mn>2</mn></mfrac></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>n</mi><mo>-</mo><mn>26</mn></mrow></mfenced><mfenced><mrow><mi>n</mi><mo>+</mo><mn>27</mn></mrow></mfenced><mo>=</mo><mn>0</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>26</mn></math> (<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>26</mn></math>th triangular number) <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A0</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mo>−</mo><mn>27</mn><mo>,</mo><mn>26</mn></math>. Award <em><strong>A0</strong></em> if additional solutions besides <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>26</mn></math> are given.</p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempts to form an expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>3</mn></msub><mfenced><mi>n</mi></mfenced><mo>+</mo><msub><mi>P</mi><mn>3</mn></msub><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> <em><strong>M1</strong></em></p>
<p> </p>
<p><strong>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>3</mn></msub><mfenced><mi>n</mi></mfenced><mo>+</mo><msub><mi>P</mi><mn>3</mn></msub><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>≡</mo><mfrac><mrow><mi>n</mi><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mrow><mn>2</mn></mfrac><mo>+</mo><mfrac><mrow><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow></mfenced></mrow><mn>2</mn></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>≡</mo><mfrac><mrow><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>2</mn></mrow></mfenced></mrow><mn>2</mn></mfrac><mo> </mo><mfenced><mrow><mo>≡</mo><mfrac><mrow><mn>2</mn><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mrow><mn>2</mn></mfrac></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>3</mn></msub><mfenced><mi>n</mi></mfenced><mo>+</mo><msub><mi>P</mi><mn>3</mn></msub><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>≡</mo><mfenced><mrow><mfrac><msup><mi>n</mi><mn>2</mn></msup><mn>2</mn></mfrac><mo>+</mo><mfrac><mi>n</mi><mn>2</mn></mfrac></mrow></mfenced><mo>+</mo><mfenced><mrow><mfrac><msup><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup><mn>2</mn></mfrac><mo>+</mo><mfrac><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow><mn>2</mn></mfrac></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>≡</mo><mfenced><mfrac><mrow><msup><mi>n</mi><mn>2</mn></msup><mo>+</mo><mi>n</mi></mrow><mn>2</mn></mfrac></mfenced><mo>+</mo><mfenced><mfrac><mrow><msup><mi>n</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn><mo>+</mo><mi>n</mi><mo>+</mo><mn>1</mn></mrow><mn>2</mn></mfrac></mfenced><mo> </mo><mo> </mo><mfenced><mrow><mo>≡</mo><msup><mi>n</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>≡</mo><msup><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>the sum of the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>th and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced></math>th triangular numbers</p>
<p>is the <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced></math>th square number <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAJAAAACGCAYAAAAyyQM+AAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAADsMAAA7DAcdvqGQAAAw7SURBVHhe7Z1tbFRVGsefrl1ZZNTMRhK7S7dArSmuxJiGbIqS0A9uikTGlw8uxC3FXdN2yYayxGI3UFDEXRijqFnphk5WCC9FO1ZtlK6hUcDQ1Y5ssdRNazPQlk7pC9OWaZnSKfe/57a3dDovgHPunbHj8/syM+cO99x7zv++nDP3RxMgIIaJkJ9orwwTERwgRgoOECMFB4iRggPESMEBYqTgADFScIAYKUJPJF7+lt7f/Qk5pyyZQfMfXU1P3jeTOk8coANf9mrlYiXzH6WCJ++j27TP3x+IKj+k3Z+0iHd+JNxDjxZY6L6ZnXRizyH68tLE0oltuUP7HAmX6Nv399Inziva53Em9mVm5wnac+BL8S2NiW25LUEriIQY1Gl0X6oBCoXi7ULjR1uQZVL7dDFKjl+Eoi0DfOiqWgdz2h9RWnse3skFElyFt6cBH21eBpOaqLlbcXzwqrZMoJxHVf4DSMv9J2pdQ37bIoHiRU/jB9iclaw2L+aWHMeg34qVrirkmx9EbulJuLx+2yJDDOo0si/DBmgcL1rt65BGZiwq+RxubeWKpx6lq/JQ2tCvT0f643PCnp8hdjRL7Gi3tv5ReBr2YNWqPWjwjI6V6IcCX6sd+Wkm0KJtOO7W1q/0o6E0D6tK6+HRfSdjUacxfXmDAAkCO1TpwcntOSiwO0V2jSC4cRX3CWy3bIC91at9R28CG3cU7pNWWArsaPXp3pMaMajTgL68cYCmdGgx9pYVwrLlM3Qb1a5j+DXupj0oK3gGW451RnSE3DR+jbtp7z9QYHkFx7qNOUSuEfU69e/LmwiQyjA6qp5HunpvkrkDtQN6X0ZC4GtFVWGmaFwzMrefxICh6VERjdtRhcJ00bj0W2yv9b9PMIpY1KlvX97kMH4GJS3JJkuyeFtbTUdOu9W7MWNJ/CUteSyLkqmPaj+oodP9V7UFRpFAiUm/occsC8T7OvrgSAP1G76TsahT577UgnR91LPB+jXYdvDt4Bs/Qxg/Mtc//hIOlq0NuvEzBvXILMbj2/6FsqCbeKOIQZ069+VNBMiDpn1/xso3HWJkEPpOXne8jdi35jm8Wd8HJeSoTG8UeJv2Y83Kt1EvRnkhR0i6E4s69e/LGwQoxMjA14Ly3HTjOjRoZKDe+B1GrlmcaQ1q3OBRnmjc8jyYDTxQol+nMX15nQCNwnPmHeQufgHVXf4jAwUjTTZY1Emp9OdR1TGsleuA0o8ztjwsXn8EXVP2ZghNtlUwkQnphVXo0HGYq3hOw5a7AuurO6Y24EgjbJa5onEzUVjVquuURfTrNK4vQwZI6TmFwzufxSJ1xWm5sFbWo2diT71OHLMVabOaBNPStdj17qnJ5RHhQ099BXbmLh4LSVqOFZX1E0eEONU7P4etKHt8hpqSsTT/Nbx7bXmEKN2oP7wDuYuSxDofRI61EvU9E407COcxG4q02WIyZSF/l91veYTEoE6j+zL0b2Gjg9Tb4yGf9pF+ejvNvstEiep7DFN/Vx95/f+V//KIgKjSTT2eEe1zgljlXXSXaXyNGO6nrj7v2F6OM3V5ZIzSYG8veXwTa72Vbp/9czIlqr87KTTc30t9Xv+Rn//ySIlBnQb3JVsZjBT8OAcjBQeIkYIDxEjBAWKk4AAxUnCAGCk4QIwUHCBGCg4QIwUHiJGCA8RIcePfwnCZOk99ShXv/ZscnZdFwW1098KHKfupx2jpvDtI5qfFSVgsjDuxcAxfO46WLIM5qwT2+jZ4fKPwup2oKy9GlnkZSo626/jMCouFKvEjFo49GbgcJnMeyoN8LO3pOdMy/XWbkI+wslgoT+hHWA0SCxUMOV5FpjhCkopqMKCVTmGgBkVJItGZr8IxpOfeBjcui4U6ET2xcAAO6yOiolTk2Nu0skDaYM9JFd9Zipdr+7QyvfBrXBYLdcT/4DRULJwMh9Xh0coC8YiQLb1ByCRgsdAgYiIWxgAWCw1CX7EwTIBupVlmMRYSnXe+d2i8KIgRunxJHY6ayDzr1vEi3QCNuo7Qi2/9jHaUraW0utdow64vqM/Qxr1Cro9fp7dMf6Ky/PlU99KLtOuLHqnGvTExqHO0jT5+cT+ZdrxN+Wl19NKGf9IXfRIHp3YmCuCKGGU9K4bTZjy8u0GMgUKgfId9FjEUNT0rRmlXtEKdYLFQ+47eRFEsVC4exUb12vxIKRpHgteutJZjpUmu8pCwWGhQnVEXC33oPrYdWaY0PB0wL6F4W1BZuBhpue/gjJ7zMiwWxodYOMkwuh0HsHH5w1he+Dr2VVSg3LYT68TnFSWVaNItPCwWxpdYGMjoJWpv/o7OtXeT785f0JyUVJqfJCMSBsJiIYuFzI8SfpyDkYIDxEjBAWKk4AAxUnCAGCk4QIwUHCBGCg4QIwUHiJGCA8RIwQFipOAAMVL8QMzUqWC4g05Vv0/vVZ+iTq9CNDOJFj6UTU89sYTmmW7RviULm6lxZqaqKPC5PkVJVjqyNlegvm0APsULt/M/KN+YLbZjG4669Pyf8dlMVZ/FihszdewxT9Gw5tzDwXKd9vilKWu7zu6UvysVLUs0FnWGfgY6jszUfjis2eIIyUBRTY9W5s9VUWUxkkQDZFq/wpBWqg+Bjctm6s0SJkAxMFOH62B9QByRlAe7K8zuuOzIEaGm+19Bra46tYDN1IgIE6AYmKkT4Ui1whGuDX0OWFPV6/h1QhYxonHZTP3e8DD+GmymRkKYAMXATJ0xi8xm8eq+QL2XwpiSVy7TJfW5e7Fts2boPYHAZmpEaGeiAGJgpipOlK9UL5sW7G4MfYusOPfBIjbZtLIcrbqe69lMjQMzdRQXazaJa3MSHhGhHdFKJ5kItf6KM5upcWOmduLYlmUwmZ8JmJe4Cm9rpbjZzECu7bSucyRspsaFmeqH7wIc+4uxPMOCwjf2osJ+CDbrOvH5KZRU/k+/8LCZKk56cWOmBqKaqh3U3HSW2i/46M45yZSSOo+SpEzUQNhMZTOViTk8D8RIwQFipOAAMVJwgBgpOECMFBwgRgoOECMFB4iRggPESMEBYqTgADFScIAYKX7EZqo/Cg13/peqK+xU7eggrziuZt79a3oo+wl6YmkqmXTbSTZT48tMHWMYrqPbxvZns/1rtHl8ULwX4aw7hI3qdpR8CpeenhabqSra45fT3kxVH/PcKRo2HbnlLQEHg+pQqX/kJRlZku5UMP5+Fpup09dMHfoK1kwzKKkYNQOhjvYe1BSp4l82rI5+rUwvAjuUzVQ5om6mKhh2WPGAur4cO1xa6VR8oso8sU0m3P/ySZ11akEcmKlhRmED1NrgFK/JtDBFlbVCYaaUhaqd1k7NLs94kQzuNmo4PSgyew+lzA7zQOXsFFqYKl7PtJJrQMJlGuMquVtb6LR4l7owhWaPFwaQKKq8h1JpkM40d4pW0ZnEubRi81YqTK+jl1d/SHNeyKMl4fZdFxIo8VfLafNrBZRe9zdabUuiF/6yhGZL3KPzMD6msJkqXqermZogqhTbLt65z/dODp2nAFHloNhTUaV5lmh6vZn+ZmqYAJlpQWaGiEYbfX22R5zsQwA3nfvmvMhPBmUuCHeZ+x6Y76XMbHF96muhsxemzpNMgO5z9E27qDJ7ES0wy84H3ULmBYsoWxwnfV+fpQshd3KEus+1iIt0KmVn3jsWNv0ADTdX0KaDybS18Pe0uviv+qjG12WQmg+9SgfnraXC362h4r//Qf4PGmv3QkH8KMxUpQs1GxeLpnsSuxsHtUI/JnRrA3RjNlPjxUzt/gxb1Lmnp/egwX9flEG0Vm5AetpzsJ2JbI4kHGymxoOZeg0xtO2uw/6NjyNj+Xq8se892MvLYF1nQcaKrahs0jE8bKbGk5kayCgNtjdT07nzdMF3B82Zk0Kp8++WNFEDYTOVYabA80CMFBwgRgoOECMFB4iRggPESMEBYqTgADFScIAYKThAjBQcIEYKDhAjBQeIkYLN1DGiZab6E6U62UyNMzN1jOjWyWaqStyYqbGyYdlMjQ8zNZY2LJup091MjbUNy2bqNDdTY23DspnKSMNm6jQ2U38ANiybqdPZTI21Dctm6vQ3U2Nmw7KZGj9matTrZDM1PszUa0SvTjZT48pMDSQKdbKZyvyQ4XkgRgoOECMFB4iRggPESMEBYqTgADFScIAYKThAjARE/we1V9KZR0mcngAAAABJRU5ErkJggg=="> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Accept equivalent single diagrams, such as the one above, where the <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math>th and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn></math>th triangular numbers and the <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn></math>th square number are clearly shown.<br>Award <em><strong>A1</strong> </em>for a diagram that show <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>3</mn></msub><mfenced><mn>4</mn></mfenced></math> (a triangle with <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn></math> dots) and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>3</mn></msub><mfenced><mn>5</mn></mfenced></math> (a triangle with <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>15</mn></math> dots) and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>4</mn></msub><mfenced><mn>5</mn></mfenced></math> (a square with <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>25</mn><mo> </mo></math>dots).</p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">b.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn><msub><mi>P</mi><mn>3</mn></msub><mfenced><mi>n</mi></mfenced><mo>+</mo><mn>1</mn><mo>=</mo><mn>8</mn><mfenced><mfrac><mrow><mi>n</mi><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mrow><mn>2</mn></mfrac></mfenced><mo>+</mo><mn>1</mn><mo> </mo><mfenced><mrow><mo>=</mo><mn>4</mn><mi>n</mi><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>+</mo><mn>1</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p>attempts to expand their expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn><msub><mi>P</mi><mn>3</mn></msub><mfenced><mi>n</mi></mfenced><mo>+</mo><mn>1</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>4</mn><msup><mi>n</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mi>n</mi><mo>+</mo><mn>1</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mfenced><mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup></math> <em><strong>A1</strong></em></p>
<p>and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></math> is odd <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn><msub><mi>P</mi><mn>3</mn></msub><mfenced><mi>n</mi></mfenced><mo>+</mo><mn>1</mn><mo>=</mo><mn>8</mn><mfenced><mrow><msup><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup><mo>-</mo><msub><mi>P</mi><mn>3</mn></msub><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mrow></mfenced><mo>+</mo><mn>1</mn><mfenced><mrow><mo>=</mo><mn>8</mn><mfenced><mrow><msup><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup><mo>-</mo><mfrac><mrow><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow></mfenced></mrow><mn>2</mn></mfrac></mrow></mfenced><mo>+</mo><mn>1</mn></mrow></mfenced><mspace linebreak="newline"></mspace></math> <em><strong>A1</strong></em></p>
<p>attempts to expand their expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn><msub><mi>P</mi><mn>3</mn></msub><mfenced><mi>n</mi></mfenced><mo>+</mo><mn>1</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn><mfenced><mrow><msup><mi>n</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>-</mo><mn>4</mn><mfenced><mrow><msup><mi>n</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn><mi>n</mi><mo>+</mo><mn>2</mn></mrow></mfenced><mo>+</mo><mn>1</mn><mo> </mo><mfenced><mrow><mo>=</mo><mn>4</mn><msup><mi>n</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mfenced><mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup></math> <em><strong>A1</strong></em></p>
<p>and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></math> is odd <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>Method 3</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn><msub><mi>P</mi><mn>3</mn></msub><mfenced><mi>n</mi></mfenced><mo>+</mo><mn>1</mn><mo>=</mo><mn>8</mn><mfenced><mfrac><mrow><mi>n</mi><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mrow><mn>2</mn></mfrac></mfenced><mo>+</mo><mn>1</mn><mo> </mo><mfenced><mrow><mo>=</mo><msup><mfenced><mrow><mi>A</mi><mi>n</mi><mo>+</mo><mi>B</mi></mrow></mfenced><mn>2</mn></msup></mrow></mfenced></math> (where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>,</mo><mi>B</mi><mo>∈</mo><msup><mi mathvariant="normal">ℤ</mi><mo>+</mo></msup></math>) <em><strong>A1</strong></em></p>
<p>attempts to expand their expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn><msub><mi>P</mi><mn>3</mn></msub><mfenced><mi>n</mi></mfenced><mo>+</mo><mn>1</mn></math> <em><strong>(M1)</strong></em></p>
<p><em><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><msup><mi>n</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mi>n</mi><mo>+</mo><mn>1</mn><mo> </mo><mfenced><mrow><mo>=</mo><msup><mi>A</mi><mn>2</mn></msup><msup><mi>n</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>A</mi><mi>B</mi><mi>n</mi><mo>+</mo><msup><mi>B</mi><mn>2</mn></msup></mrow></mfenced></math></em></p>
<p>now equates coefficients and obtains <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mo>=</mo><mn>1</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mn>2</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mfenced><mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup></math> <em><strong>A1</strong></em></p>
<p>and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></math> is odd <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>1</mn></msub><mo>=</mo><mn>1</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo>=</mo><mn>3</mn></math> <em><strong>(A1)</strong></em></p>
<p>substitutes their <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>1</mn></msub></math> and their <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math> into <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>5</mn></msub><mfenced><mi>n</mi></mfenced><mo>=</mo><mfrac><mi>n</mi><mn>2</mn></mfrac><mfenced><mrow><mn>2</mn><msub><mi>u</mi><mn>1</mn></msub><mo>+</mo><mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mi>d</mi></mrow></mfenced></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>5</mn></msub><mfenced><mi>n</mi></mfenced><mo>=</mo><mfrac><mi>n</mi><mn>2</mn></mfrac><mfenced><mrow><mn>2</mn><mo>+</mo><mn>3</mn><mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></mfenced></mrow></mfenced><mo> </mo><mfenced><mrow><mo>=</mo><mfrac><mi>n</mi><mn>2</mn></mfrac><mfenced><mrow><mn>2</mn><mo>+</mo><mn>3</mn><mi>n</mi><mo>-</mo><mn>3</mn></mrow></mfenced></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>1</mn></msub><mo>=</mo><mn>1</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mi>n</mi></msub><mo>=</mo><mn>3</mn><mi>n</mi><mo>-</mo><mn>2</mn></math> <em><strong>(A1)</strong></em></p>
<p>substitutes their <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>1</mn></msub></math> and their <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mi>n</mi></msub></math> into <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>5</mn></msub><mfenced><mi>n</mi></mfenced><mo>=</mo><mfrac><mi>n</mi><mn>2</mn></mfrac><mfenced><mrow><msub><mi>u</mi><mn>1</mn></msub><mo>+</mo><msub><mi>u</mi><mi>n</mi></msub></mrow></mfenced></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>5</mn></msub><mfenced><mi>n</mi></mfenced><mo>=</mo><mfrac><mi>n</mi><mn>2</mn></mfrac><mfenced><mrow><mn>1</mn><mo>+</mo><mn>3</mn><mi>n</mi><mo>-</mo><mn>2</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>5</mn></msub><mfenced><mi>n</mi></mfenced><mo>=</mo><mfenced><mrow><mn>3</mn><mfenced><mn>1</mn></mfenced><mo>-</mo><mn>2</mn></mrow></mfenced><mo>+</mo><mfenced><mrow><mn>3</mn><mfenced><mn>2</mn></mfenced><mo>-</mo><mn>2</mn></mrow></mfenced><mo>+</mo><mfenced><mrow><mn>3</mn><mfenced><mn>3</mn></mfenced><mo>-</mo><mn>2</mn></mrow></mfenced><mo>+</mo><mo>…</mo><mn>3</mn><mi>n</mi><mo>-</mo><mn>2</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>5</mn></msub><mfenced><mi>n</mi></mfenced><mo>=</mo><mfenced><mrow><mn>3</mn><mfenced><mn>1</mn></mfenced><mo>+</mo><mn>3</mn><mfenced><mn>2</mn></mfenced><mo>+</mo><mn>3</mn><mfenced><mn>3</mn></mfenced><mo>+</mo><mo>…</mo><mo>+</mo><mn>3</mn><mi>n</mi></mrow></mfenced><mo>-</mo><mn>2</mn><mi>n</mi><mo> </mo><mfenced><mrow><mo>=</mo><mn>3</mn><mfenced><mrow><mn>1</mn><mo>+</mo><mn>2</mn><mo>+</mo><mn>3</mn><mo>+</mo><mo>…</mo><mo>+</mo><mi>n</mi></mrow></mfenced><mo>-</mo><mn>2</mn><mi>n</mi></mrow></mfenced></math> <em><strong>(A1)</strong></em></p>
<p>substitutes <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>n</mi><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mrow><mn>2</mn></mfrac></math> into their expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>5</mn></msub><mfenced><mi>n</mi></mfenced></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>5</mn></msub><mfenced><mi>n</mi></mfenced><mo>=</mo><mn>3</mn><mfenced><mfrac><mrow><mi>n</mi><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mrow><mn>2</mn></mfrac></mfenced><mo>-</mo><mn>2</mn><mi>n</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>5</mn></msub><mfenced><mi>n</mi></mfenced><mo>=</mo><mfrac><mi>n</mi><mn>2</mn></mfrac><mfenced><mrow><mn>3</mn><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>-</mo><mn>4</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>OR</strong></p>
<p>attempts to find the arithmetic mean of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> terms <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mn>1</mn><mo>+</mo><mfenced><mrow><mn>3</mn><mi>n</mi><mo>-</mo><mn>2</mn></mrow></mfenced></mrow><mn>2</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p>multiplies the above expression by the number of terms <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>5</mn></msub><mfenced><mi>n</mi></mfenced><mo>=</mo><mfrac><mi>n</mi><mn>2</mn></mfrac><mfenced><mrow><mn>1</mn><mo>+</mo><mn>3</mn><mi>n</mi><mo>-</mo><mn>2</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>THEN</strong></p>
<p>so <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>5</mn></msub><mfenced><mi>n</mi></mfenced><mo>=</mo><mfrac><mrow><mi>n</mi><mfenced><mrow><mn>3</mn><mi>n</mi><mo>-</mo><mn>1</mn></mrow></mfenced></mrow><mn>2</mn></mfrac></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>forms a table of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>3</mn></msub><mfenced><mi>n</mi></mfenced></math> values that includes some values for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>></mo><mn>5</mn></math> <em><strong>(M1)</strong></em></p>
<p>forms a table of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>5</mn></msub><mfenced><mi>m</mi></mfenced></math> values that includes some values for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>></mo><mn>5</mn></math> <em><strong>(M1)</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>(M1)</em></strong> if at least one <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>3</mn></msub><mfenced><mi>n</mi></mfenced></math> value is correct. Award <strong><em>(M1)</em></strong> if at least one <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>5</mn></msub><mfenced><mi>m</mi></mfenced></math> value is correct. Accept as above for <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><msup><mi>n</mi><mn>2</mn></msup><mo>+</mo><mi>n</mi></mrow></mfenced></math> values and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>3</mn><msup><mi>m</mi><mn>2</mn></msup><mo>-</mo><mi>m</mi></mrow></mfenced></math> values.</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>20</mn></math> for triangular numbers <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mn>12</mn></math> for pentagonal numbers <em><strong>(A1)</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>(A1)</strong></em> if <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>20</mn></math> is seen in or out of a table. Award <em><strong>(A1)</strong></em> if <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mn>12</mn></math> is seen in or out of a table. Condone the use of the same parameter for triangular numbers and pentagonal numbers, for example, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>20</mn></math> for triangular numbers and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>12</mn></math> for pentagonal numbers.</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>210</mn></math> (is a triangular number and a pentagonal number) <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award all five marks for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>210</mn></math> seen anywhere with or without working shown.</p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><strong>EITHER</strong></p>
<p>attempts to express <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>3</mn></msub><mfenced><mi>n</mi></mfenced><mo>=</mo><msub><mi>P</mi><mn>5</mn></msub><mfenced><mi>m</mi></mfenced></math> as a quadratic in <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>n</mi><mn>2</mn></msup><mo>+</mo><mi>n</mi><mo>+</mo><mfenced><mrow><mi>m</mi><mo>-</mo><mn>3</mn><msup><mi>m</mi><mn>2</mn></msup></mrow></mfenced><mfenced><mrow><mo>=</mo><mn>0</mn></mrow></mfenced></math> (or equivalent)</p>
<p>attempts to solve their quadratic in <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mfrac><mrow><mo>-</mo><mn>1</mn><mo>±</mo><msqrt><mn>12</mn><msup><mi>m</mi><mn>2</mn></msup><mo>-</mo><mn>4</mn><mi>m</mi><mo>+</mo><mn>1</mn></msqrt></mrow><mn>2</mn></mfrac><mfenced><mrow><mo>=</mo><mfrac><mrow><mo>-</mo><mn>1</mn><mo>±</mo><msqrt><msup><mn>1</mn><mn>2</mn></msup><mo>-</mo><mn>4</mn><mfenced><mrow><mi>m</mi><mo>-</mo><mn>3</mn><msup><mi>m</mi><mn>2</mn></msup></mrow></mfenced></msqrt></mrow><mn>2</mn></mfrac></mrow></mfenced></math></p>
<p> </p>
<p style="text-align:left;"><strong>OR</strong></p>
<p>attempts to express <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>3</mn></msub><mfenced><mi>n</mi></mfenced><mo>=</mo><msub><mi>P</mi><mn>5</mn></msub><mfenced><mi>m</mi></mfenced></math> as a quadratic in <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><msup><mi>m</mi><mn>2</mn></msup><mo>-</mo><mi>m</mi><mo>-</mo><mfenced><mrow><msup><mi>n</mi><mn>2</mn></msup><mo>+</mo><mi>n</mi></mrow></mfenced><mfenced><mrow><mo>=</mo><mn>0</mn></mrow></mfenced></math> (or equivalent)</p>
<p>attempts to solve their quadratic in <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mfrac><mrow><mn>1</mn><mo>±</mo><msqrt><mn>12</mn><msup><mi>n</mi><mn>2</mn></msup><mo>-</mo><mn>12</mn><mi>n</mi><mo>+</mo><mn>1</mn></msqrt></mrow><mn>6</mn></mfrac><mfenced><mrow><mo>=</mo><mfrac><mrow><mn>1</mn><mo>±</mo><msqrt><msup><mfenced><mrow><mo>-</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup><mo>+</mo><mn>12</mn><mfenced><mrow><msup><mi>n</mi><mn>2</mn></msup><mo>+</mo><mi>n</mi></mrow></mfenced></msqrt></mrow><mn>6</mn></mfrac></mrow></mfenced></math></p>
<p> </p>
<p><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>20</mn></math> for triangular numbers <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mn>12</mn></math> for pentagonal numbers <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>210</mn></math> (is a triangular number and a pentagonal number) <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>METHOD 3</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>n</mi><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mrow><mn>2</mn></mfrac><mo>=</mo><mfrac><mrow><mi>m</mi><mfenced><mrow><mn>3</mn><mi>m</mi><mo>-</mo><mn>1</mn></mrow></mfenced></mrow><mn>2</mn></mfrac></math></p>
<p>let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mi>m</mi><mo>+</mo><mi>k</mi><mo> </mo><mfenced><mrow><mi>n</mi><mo>></mo><mi>m</mi></mrow></mfenced></math> and so <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><msup><mi>m</mi><mn>2</mn></msup><mo>-</mo><mi>m</mi><mo>=</mo><mfenced><mrow><mi>m</mi><mo>+</mo><mi>k</mi></mrow></mfenced><mfenced><mrow><mi>m</mi><mo>+</mo><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><msup><mi>m</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mfenced><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mi>m</mi><mo>-</mo><mfenced><mrow><msup><mi>k</mi><mn>2</mn></msup><mo>+</mo><mi>k</mi></mrow></mfenced><mo>=</mo><mn>0</mn></math> <em><strong>A1</strong></em></p>
<p>attempts to find the discriminant of their quadratic</p>
<p>and recognises that this must be a perfect square <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Δ</mtext><mo>=</mo><mn>4</mn><msup><mfenced><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup><mo>+</mo><mn>8</mn><mfenced><mrow><msup><mi>k</mi><mn>2</mn></msup><mo>+</mo><mi>k</mi></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>N</mi><mn>2</mn></msup><mo>=</mo><mn>4</mn><msup><mfenced><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup><mo>+</mo><mn>8</mn><mfenced><mrow><msup><mi>k</mi><mn>2</mn></msup><mo>+</mo><mi>k</mi></mrow></mfenced><mo> </mo><mfenced><mrow><mo>=</mo><mn>4</mn><mfenced><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mn>3</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mrow></mfenced></math></p>
<p>determines that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mn>8</mn></math> leading to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><msup><mi>m</mi><mn>2</mn></msup><mo>-</mo><mn>18</mn><mi>m</mi><mo>-</mo><mn>72</mn><mo>=</mo><mn>0</mn><mo>⇒</mo><mi>m</mi><mo>=</mo><mo>-</mo><mn>3</mn><mo>,</mo><mn>12</mn></math> and so <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mn>12</mn></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>210</mn></math> (is a triangular number and a pentagonal number) <em><strong>A1</strong></em></p>
<p> </p>
<p> </p>
<p><strong>METHOD 4</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi>n</mi><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mrow><mn>2</mn></mfrac><mo>=</mo><mfrac><mrow><mi>m</mi><mfenced><mrow><mn>3</mn><mi>m</mi><mo>-</mo><mn>1</mn></mrow></mfenced></mrow><mn>2</mn></mfrac></math></p>
<p>let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mi>n</mi><mo>-</mo><mi>k</mi><mo> </mo><mfenced><mrow><mi>m</mi><mo><</mo><mi>n</mi></mrow></mfenced></math> and so <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>n</mi><mn>2</mn></msup><mo>+</mo><mi>n</mi><mo>=</mo><mfenced><mrow><mi>n</mi><mo>-</mo><mi>k</mi></mrow></mfenced><mfenced><mrow><mn>3</mn><mfenced><mrow><mi>n</mi><mo>-</mo><mi>k</mi></mrow></mfenced><mo>-</mo><mn>1</mn></mrow></mfenced></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><msup><mi>n</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mfenced><mrow><mn>3</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mi>n</mi><mo>+</mo><mfenced><mrow><mn>3</mn><msup><mi>k</mi><mn>2</mn></msup><mo>+</mo><mi>k</mi></mrow></mfenced><mo>=</mo><mn>0</mn></math> <em><strong>A1</strong></em></p>
<p>attempts to find the discriminant of their quadratic</p>
<p>and recognises that this must be a perfect square <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Δ</mtext><mo>=</mo><mn>4</mn><msup><mfenced><mrow><mn>3</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup><mo>-</mo><mn>8</mn><mfenced><mrow><mn>3</mn><msup><mi>k</mi><mn>2</mn></msup><mo>+</mo><mi>k</mi></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>N</mi><mn>2</mn></msup><mo>=</mo><mn>4</mn><msup><mfenced><mrow><mn>3</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup><mo>-</mo><mn>8</mn><mfenced><mrow><mn>3</mn><msup><mi>k</mi><mn>2</mn></msup><mo>+</mo><mi>k</mi></mrow></mfenced><mo> </mo><mfenced><mrow><mo>=</mo><mn>4</mn><mfenced><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mn>3</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mrow></mfenced></math></p>
<p>determines that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mn>8</mn></math> leading to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><msup><mi>n</mi><mn>2</mn></msup><mo>-</mo><mn>50</mn><mi>n</mi><mo>+</mo><mn>200</mn><mo>=</mo><mn>0</mn><mo>⇒</mo><mi>n</mi><mo>=</mo><mn>5</mn><mo>,</mo><mn>20</mn></math> and so <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>20</mn></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>210</mn></math> (is a triangular number and a pentagonal number) <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>Note:</strong> Award a maximum of <em><strong>R1M0M0A1M1A1A1R0</strong></em> for a ‘correct’ proof using <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> </p>
<p>consider <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>1</mn><mo>:</mo><mo> </mo><msub><mi>P</mi><mi>r</mi></msub><mfenced><mn>1</mn></mfenced><mo>=</mo><mn>1</mn><mo>+</mo><mfenced><mrow><mn>1</mn><mo>-</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mi>r</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mo>=</mo><mn>1</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mi>r</mi></msub><mfenced><mn>1</mn></mfenced><mo>=</mo><mfrac><mrow><mfenced><mrow><mi>r</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mfenced><msup><mn>1</mn><mn>2</mn></msup></mfenced><mo>-</mo><mfenced><mrow><mi>r</mi><mo>-</mo><mn>4</mn></mrow></mfenced><mfenced><mn>1</mn></mfenced></mrow><mn>2</mn></mfrac><mo>=</mo><mn>1</mn></math></p>
<p>so true for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>1</mn></math> <em><strong>R1</strong> </em></p>
<p> </p>
<p><strong>Note:</strong> Accept <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mi>r</mi></msub><mfenced><mn>1</mn></mfenced><mo>=</mo><mn>1</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mi>r</mi></msub><mfenced><mn>1</mn></mfenced><mo>=</mo><mfrac><mrow><mfenced><mrow><mi>r</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mfenced><msup><mn>1</mn><mn>2</mn></msup></mfenced><mo>-</mo><mfenced><mrow><mi>r</mi><mo>-</mo><mn>4</mn></mrow></mfenced><mfenced><mn>1</mn></mfenced></mrow><mn>2</mn></mfrac><mo>=</mo><mn>1</mn></math>.<br>Do not accept one-sided considerations such as '<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mi>r</mi></msub><mfenced><mn>1</mn></mfenced><mo>=</mo><mn>1</mn></math> and so true for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>1</mn></math>'.<br>Subsequent marks after this <em><strong>R1</strong> </em>are independent of this mark can be awarded.</p>
<p> </p>
<p>Assume true for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mi>k</mi></math>, <em>ie.</em> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mi>r</mi></msub><mfenced><mi>k</mi></mfenced><mo>=</mo><mfrac><mrow><mfenced><mrow><mi>r</mi><mo>-</mo><mn>2</mn></mrow></mfenced><msup><mi>k</mi><mn>2</mn></msup><mo>-</mo><mfenced><mrow><mi>r</mi><mo>-</mo><mn>4</mn></mrow></mfenced><mi>k</mi></mrow><mn>2</mn></mfrac></math> <em><strong>M1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>M0</strong> </em>for statements such as “let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mi>k</mi></math> ”. The assumption of truth must be clear.<br>Subsequent marks after this <em><strong>M1</strong> </em>are independent of this mark and can be awarded.</p>
<p> </p>
<p>Consider <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>:</mo></math></p>
<p>(<math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mi>r</mi></msub><mfenced><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced></math> can be represented by the sum</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mtext>Σ</mtext><mrow><mi>m</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></munderover><mfenced><mrow><mn>1</mn><mo>+</mo><mfenced><mrow><mi>m</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mi>r</mi><mo>-</mo><mn>2</mn></mrow></mfenced></mrow></mfenced><mo>=</mo><munderover><mtext>Σ</mtext><mrow><mi>m</mi><mo>=</mo><mn>1</mn></mrow><mi>k</mi></munderover><mfenced><mrow><mn>1</mn><mo>+</mo><mfenced><mrow><mi>m</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mi>r</mi><mo>-</mo><mn>2</mn></mrow></mfenced></mrow></mfenced><mo>+</mo><mfenced><mrow><mn>1</mn><mo>+</mo><mi>k</mi><mfenced><mrow><mi>r</mi><mo>-</mo><mn>2</mn></mrow></mfenced></mrow></mfenced></math> and so</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mi>r</mi></msub><mfenced><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>=</mo><mfrac><mrow><mfenced><mrow><mi>r</mi><mo>-</mo><mn>2</mn></mrow></mfenced><msup><mi>k</mi><mn>2</mn></msup><mo>-</mo><mfenced><mrow><mi>r</mi><mo>-</mo><mn>4</mn></mrow></mfenced><mi>k</mi></mrow><mn>2</mn></mfrac><mo>+</mo><mfenced><mrow><mn>1</mn><mo>+</mo><mi>k</mi><mfenced><mrow><mi>r</mi><mo>-</mo><mn>2</mn></mrow></mfenced></mrow></mfenced><mo> </mo><mo> </mo><mfenced><mrow><msub><mi>P</mi><mi>r</mi></msub><mfenced><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>=</mo><msub><mi>P</mi><mi>r</mi></msub><mfenced><mi>k</mi></mfenced><mo>+</mo><mfenced><mrow><mn>1</mn><mo>+</mo><mi>k</mi><mfenced><mrow><mi>r</mi><mo>-</mo><mn>2</mn></mrow></mfenced></mrow></mfenced></mrow></mfenced></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mfenced><mrow><mi>r</mi><mo>-</mo><mn>2</mn></mrow></mfenced><msup><mi>k</mi><mn>2</mn></msup><mo>-</mo><mfenced><mrow><mi>r</mi><mo>-</mo><mn>4</mn></mrow></mfenced><mi>k</mi><mo>+</mo><mn>2</mn><mo>+</mo><mn>2</mn><mi>k</mi><mfenced><mrow><mi>r</mi><mo>-</mo><mn>2</mn></mrow></mfenced></mrow><mn>2</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p> <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mfenced><mrow><mi>r</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mfenced><mrow><msup><mi>k</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>k</mi></mrow></mfenced><mo>-</mo><mfenced><mrow><mi>r</mi><mo>-</mo><mn>4</mn></mrow></mfenced><mi>k</mi><mo>+</mo><mn>2</mn></mrow><mn>2</mn></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mfenced><mrow><mi>r</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mfenced><mrow><msup><mi>k</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>-</mo><mfenced><mrow><mi>r</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mo>-</mo><mfenced><mrow><mi>r</mi><mo>-</mo><mn>4</mn></mrow></mfenced><mi>k</mi><mo>+</mo><mn>2</mn></mrow><mn>2</mn></mfrac></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mfenced><mrow><mi>r</mi><mo>-</mo><mn>2</mn></mrow></mfenced><msup><mfenced><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup><mo>-</mo><mfenced><mrow><mi>r</mi><mo>-</mo><mn>4</mn></mrow></mfenced><mi>k</mi><mo>-</mo><mfenced><mrow><mi>r</mi><mo>-</mo><mn>4</mn></mrow></mfenced></mrow><mn>2</mn></mfrac></math> <em><strong>(A1)</strong></em></p>
<p> <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mfenced><mrow><mi>r</mi><mo>-</mo><mn>2</mn></mrow></mfenced><msup><mfenced><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup><mo>-</mo><mfenced><mrow><mi>r</mi><mo>-</mo><mn>4</mn></mrow></mfenced><mfenced><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mrow><mn>2</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p>hence true for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>1</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mi>k</mi></math> true <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>n</mi><mo>=</mo><mi>k</mi><mo>+</mo><mn>1</mn></math> true <em><strong>R1</strong></em></p>
<p>therefore true for all <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>
<p> </p>
<p><strong>Note:</strong> Only award the final <em><strong>R1</strong> </em>if the first five marks have been awarded. Award marks as appropriate for solutions that expand both the LHS and (given) RHS of the equation.</p>
<p> </p>
<p><em><strong>[8 marks]</strong></em></p>
<div class="question_part_label">f.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>Part (a) (i) was generally well done. Unfortunately, some candidates adopted numerical verification. Part (a) (ii) was generally well done with the majority of successful candidates using their GDC judiciously and disregarding <em>n </em>= −27 as a possible solution. A few candidates interpreted the question as needing to deal with P<sub>3</sub>(351).</p>
<p>Although part (b) (i) was generally well done, a significant number of candidates laboured unnecessarily to show the required result. Many candidates set their LHS to equal the RHS throughout the solution. Part (b) (ii) was generally not well done with many candidates unable to articulate clearly in words and symbols what the given identity shows for the sum of two consecutive triangular numbers. In part (b) (iii), most candidates were unable to produce a clear diagram illustrating the identity stated in part (b) (i). </p>
<p>Part (c) was reasonably well done. Most candidates were able to show algebraically that <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn><msub><mi>P</mi><mn>3</mn></msub><mo>(</mo><mi>n</mi><mo>)</mo><mo>+</mo><mn>1</mn><mo>=</mo><mn>4</mn><msup><mi>n</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mi>n</mi><mo>+</mo><mn>1</mn></math>. A good number of candidates were then able to express <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><msup><mi>n</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mi>n</mi><mo>+</mo><mn>1</mn></math> as <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mrow><mo>(</mo><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mn>2</mn></msup></math> and conclude that <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo></math> is odd. Rather than making the connection that <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><msup><mi>n</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mi>n</mi><mo>+</mo><mn>1</mn></math> is a perfect square, many candidates attempted instead to analyse the parity of either <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mi>n</mi><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>+</mo><mn>1</mn></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><msup><mi>n</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mi>n</mi><mo>+</mo><mn>1</mn></math>. As with part (b) (i), many candidates set their LHS to equal the RHS throughout the solution. A number of candidates unfortunately adopted numerical verification.</p>
<p>Part (d) was not answered as well as anticipated with many candidates not understanding what was<br>required. Instead of using the given arithmetic series to show that <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>5</mn></msub><mo>(</mo><mi>n</mi><mo>)</mo><mo>=</mo><mfrac><mrow><mi>n</mi><mo>(</mo><mn>3</mn><mi>n</mi><mo>-</mo><mn>1</mn><mo>)</mo></mrow><mn>2</mn></mfrac></math>, a large number of<br>candidates used <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>5</mn></msub><mo>(</mo><mi>n</mi><mo>)</mo><mo>=</mo><mfrac><mrow><mo>(</mo><mn>5</mn><mo>-</mo><mn>2</mn><mo>)</mo><msup><mi>n</mi><mn>2</mn></msup><mo>-</mo><mo>(</mo><mn>5</mn><mo>-</mo><mn>4</mn><mo>)</mo><mi>n</mi></mrow><mn>2</mn></mfrac></math> . Unfortunately, a number of candidates adopted numerical verification.</p>
<p>In part (e), the overwhelming majority of candidates who successfully determined that 210 is the smallest positive integer greater than 1 that is both triangular and pentagonal used a table of values. Unfortunately, a large proportion of these candidates seemingly spent quite a few minutes listing the first 20 triangular numbers and the first 12 pentagonal numbers. And it can be surmised that a number of these candidates constructed their table of values either without the use of a GDC or with the arithmetic functionality of a GDC rather than with a GDC's table of values facility. Candidates should be aware that a relevant excerpt from a table of values is sufficient evidence of correct working. A number of candidates started constructing a table of values but stopped before identifying 210. Disappointingly, a significant number of candidates attempted to solve <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>3</mn></msub><mo>(</mo><mi>n</mi><mo>)</mo><mo>=</mo><msub><mi>P</mi><mn>5</mn></msub><mo>(</mo><mi>n</mi><mo>)</mo></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>.</p>
<p>Part (f) proved beyond the reach of most with only a small number of candidates successfully proving the given result. A significant number of candidates were unable to show that the result is true for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>1</mn></math>. A number of candidates established the validity of the base case for the RHS only while a number of other candidates attempted to prove the base case for <em>r</em> = 3. A large number of candidates did not state the inductive step correctly with the assumption of truth not clear. A number of candidates then either attempted to work backwards from the given result or misinterpreted the question and attempted to prove the result stated in the question stem rather than the result stated in the question. Some candidates who were awarded the first answer mark when considering the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mi>k</mi><mo>+</mo><mn>1</mn></math> case were unable to complete the square or equivalent simplification correctly. Disappointingly, a significant number listed the steps involved in an induction proof without engaging in the actual proof.</p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">f.</div>
</div>
<br><hr><br><div class="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A">
<mi>A</mi>
</math></span> be the set <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\{ x|x \in \mathbb{R},{\text{ }}x \ne 0\} ">
<mo fence="false" stretchy="false">{</mo>
<mi>x</mi>
<mrow>
<mo stretchy="false">|</mo>
</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>
<mn>0</mn>
<mo fence="false" stretchy="false">}</mo>
</math></span>. Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="B">
<mi>B</mi>
</math></span> be the set <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\{ x|x \in ] - 1,{\text{ }} + 1[,{\text{ }}x \ne 0\} ">
<mo fence="false" stretchy="false">{</mo>
<mi>x</mi>
<mrow>
<mo stretchy="false">|</mo>
</mrow>
<mi>x</mi>
<mo>∈<!-- ∈ --></mo>
<mo stretchy="false">]</mo>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mo>+</mo>
<mn>1</mn>
<mo stretchy="false">[</mo>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>x</mi>
<mo>≠<!-- ≠ --></mo>
<mn>0</mn>
<mo fence="false" stretchy="false">}</mo>
</math></span>.</p>
<p>A function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f:A \to B">
<mi>f</mi>
<mo>:</mo>
<mi>A</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi>B</mi>
</math></span> is defined by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = \frac{2}{\pi }\arctan (x)">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mn>2</mn>
<mi>π<!-- π --></mi>
</mfrac>
<mi>arctan</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span>.</p>
</div>
<div class="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="D">
<mi>D</mi>
</math></span> be the set <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\{ x|x \in \mathbb{R},{\text{ }}x > 0\} ">
<mo fence="false" stretchy="false">{</mo>
<mi>x</mi>
<mrow>
<mo stretchy="false">|</mo>
</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>
<mn>0</mn>
<mo fence="false" stretchy="false">}</mo>
</math></span>.</p>
<p>A function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g:\mathbb{R} \to D">
<mi>g</mi>
<mo>:</mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
<mo stretchy="false">→<!-- → --></mo>
<mi>D</mi>
</math></span> is defined by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g(x) = {{\text{e}}^x}">
<mi>g</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>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>(i) Sketch the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f(x)">
<mi>y</mi>
<mo>=</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span> and hence justify whether or not <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> is a bijection.</p>
<p>(ii) Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A">
<mi>A</mi>
</math></span> is a group under the binary operation of multiplication.</p>
<p>(iii) Give a reason why <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="B">
<mi>B</mi>
</math></span> is not a group under the binary operation of multiplication.</p>
<p>(iv) Find an example to show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(a \times b) = f(a) \times f(b)">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>a</mi>
<mo>×</mo>
<mi>b</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>a</mi>
<mo stretchy="false">)</mo>
<mo>×</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>b</mi>
<mo stretchy="false">)</mo>
</math></span> is not satisfied for all <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a,{\text{ }}b \in A">
<mi>a</mi>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>b</mi>
<mo>∈</mo>
<mi>A</mi>
</math></span>.</p>
<div class="marks">[13]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>(i) Sketch the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = g(x)">
<mi>y</mi>
<mo>=</mo>
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span> and hence justify whether or not <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g">
<mi>g</mi>
</math></span> is a bijection.</p>
<p>(ii) Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g(a + b) = g(a) \times g(b)">
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>a</mi>
<mo>+</mo>
<mi>b</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>a</mi>
<mo stretchy="false">)</mo>
<mo>×</mo>
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>b</mi>
<mo stretchy="false">)</mo>
</math></span> for all <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a,{\text{ }}b \in \mathbb{R}">
<mi>a</mi>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>b</mi>
<mo>∈</mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
</math></span>.</p>
<p>(iii) Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\{ \mathbb{R},{\text{ }} + \} ">
<mo fence="false" stretchy="false">{</mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mo>+</mo>
<mo fence="false" stretchy="false">}</mo>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\{ D,{\text{ }} \times \} ">
<mo fence="false" stretchy="false">{</mo>
<mi>D</mi>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mo>×</mo>
<mo fence="false" stretchy="false">}</mo>
</math></span> are both groups, explain whether or not they are isomorphic.</p>
<div class="marks">[8]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p>(i) <img src="images/Schermafbeelding_2017-03-02_om_12.36.51.png" alt="N16/5/MATHL/HP3/ENG/TZ0/SG/M/02.a.i"> <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Notes: </strong>Award <strong><em>A1 </em></strong>for general shape, labelled asymptotes, and showing that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x \ne 0">
<mi>x</mi>
<mo>≠</mo>
<mn>0</mn>
</math></span>.</p>
<p> </p>
<p>graph shows that it is injective since it is increasing or by the horizontal line test <strong><em>R1</em></strong></p>
<p>graph shows that it is surjective by the horizontal line test <strong><em>R1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Allow any convincing reasoning.</p>
<p> </p>
<p>so <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> is a bijection <strong><em>A1</em></strong></p>
<p>(ii) closed since non-zero real times non-zero real equals non-zero real <strong><em>A1R1</em></strong></p>
<p>we know multiplication is associative <strong><em>R1</em></strong></p>
<p>identity is 1 <strong><em>A1</em></strong></p>
<p>inverse of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{x}(x \ne 0)">
<mfrac>
<mn>1</mn>
<mi>x</mi>
</mfrac>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>≠</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</math></span> <strong><em>A1</em></strong></p>
<p>hence it is a group <strong><em>AG</em></strong></p>
<p>(iii) <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="B">
<mi>B</mi>
</math></span> does not have an identity <strong><em>A2</em></strong></p>
<p>hence it is not a group <strong><em>AG</em></strong></p>
<p>(iv) <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(1 \times 1) = f(1) = \frac{1}{2}">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>×</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</math></span> whereas <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(1) \times f(1) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>×</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mo>×</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>4</mn>
</mfrac>
</math></span> is one counterexample <strong><em>A2</em></strong></p>
<p>hence statement is not satisfied <strong><em>AG</em></strong></p>
<p><strong><em>[13 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img src="images/Schermafbeelding_2017-03-02_om_12.52.46.png" alt="N16/5/MATHL/HP3/ENG/TZ0/SG/M/02.b"></p>
<p>award <strong><em>A1 </em></strong>for general shape going through (0, 1) and with domain <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mathbb{R}">
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
</math></span> <strong><em>A1</em></strong></p>
<p>graph shows that it is injective since it is increasing or by the horizontal line test and graph shows that it is surjective by the horizontal line test <strong><em>R1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Allow any convincing reasoning.</p>
<p> </p>
<p>so <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g">
<mi>g</mi>
</math></span> is a bijection <strong><em>A1</em></strong></p>
<p>(ii) <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g(a + b) = {{\text{e}}^{a + b}}">
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>a</mi>
<mo>+</mo>
<mi>b</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mi>a</mi>
<mo>+</mo>
<mi>b</mi>
</mrow>
</msup>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g(a) \times g(b) = {{\text{e}}^a} \times {{\text{e}}^b} = {{\text{e}}^{a + b}}">
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>a</mi>
<mo stretchy="false">)</mo>
<mo>×</mo>
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>b</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mi>a</mi>
</msup>
</mrow>
<mo>×</mo>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mi>b</mi>
</msup>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mi>a</mi>
<mo>+</mo>
<mi>b</mi>
</mrow>
</msup>
</mrow>
</math></span> <strong><em>M1A1</em></strong></p>
<p>hence <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g(a + b) = g(a) \times g(b)">
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>a</mi>
<mo>+</mo>
<mi>b</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>a</mi>
<mo stretchy="false">)</mo>
<mo>×</mo>
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>b</mi>
<mo stretchy="false">)</mo>
</math></span> <strong><em>AG</em></strong></p>
<p>(iii) since <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g">
<mi>g</mi>
</math></span> is a bijection and the homomorphism rule is obeyed <strong><em>R1R1</em></strong></p>
<p>the two groups are isomorphic <strong><em>A1</em></strong></p>
<p><strong><em>[8 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p><strong>This question asks you to investigate conditions for the existence of complex roots of polynomial equations of degree <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext mathvariant="bold">3</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext mathvariant="bold">4</mtext></math>.</strong></p>
<p> <br>The cubic equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mi>p</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>q</mi><mi>x</mi><mo>+</mo><mi>r</mi><mo>=</mo><mn>0</mn></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><mo>∈</mo><mo> </mo><mi mathvariant="normal">ℝ</mi></math>, has roots <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>,</mo><mo> </mo><mi>β</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>γ</mi></math>.</p>
</div>
<div class="specification">
<p>Consider the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mn>7</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>q</mi><mi>x</mi><mo>+</mo><mn>1</mn><mo>=</mo><mn>0</mn></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
</div>
<div class="specification">
<p>Noah believes that if <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>p</mi><mn>2</mn></msup><mo>≥</mo><mn>3</mn><mi>q</mi></math> then <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>,</mo><mo> </mo><mi>β</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>γ</mi></math> are all real.</p>
</div>
<div class="specification">
<p>Now consider polynomial equations of degree <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math>.</p>
<p>The equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>4</mn></msup><mo>+</mo><mi>p</mi><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mi>q</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>r</mi><mi>x</mi><mo>+</mo><mi>s</mi><mo>=</mo><mn>0</mn></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><mo> </mo><mi>s</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>, has roots <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>,</mo><mo> </mo><mi>β</mi><mo>,</mo><mo> </mo><mi>γ</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>δ</mi></math>.</p>
<p>In a similar way to the cubic equation, it can be shown that:</p>
<p style="padding-left: 30px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mo>-</mo><mo>(</mo><mi>α</mi><mo>+</mo><mi>β</mi><mo>+</mo><mi>γ</mi><mo>+</mo><mi>δ</mi><mo>)</mo></math></p>
<p style="padding-left: 30px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi><mo>=</mo><mi>α</mi><mi>β</mi><mo>+</mo><mi>α</mi><mi>γ</mi><mo>+</mo><mi>α</mi><mi>δ</mi><mo>+</mo><mi>β</mi><mi>γ</mi><mo>+</mo><mi>β</mi><mi>δ</mi><mo>+</mo><mi>γ</mi><mi>δ</mi></math></p>
<p style="padding-left: 30px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mo>-</mo><mo>(</mo><mi>α</mi><mi>β</mi><mi>γ</mi><mo>+</mo><mi>α</mi><mi>β</mi><mi>δ</mi><mo>+</mo><mi>α</mi><mi>γ</mi><mi>δ</mi><mo>+</mo><mi>β</mi><mi>γ</mi><mi>δ</mi><mo>)</mo></math></p>
<p style="padding-left: 30px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mo>=</mo><mi>α</mi><mi>β</mi><mi>γ</mi><mi>δ</mi></math>.</p>
</div>
<div class="specification">
<p>The equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>4</mn></msup><mo>-</mo><mn>9</mn><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>24</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>22</mn><mi>x</mi><mo>-</mo><mn>12</mn><mo>=</mo><mn>0</mn></math>, has one integer root.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By expanding <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>x</mi><mo>-</mo><mi>α</mi></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>-</mo><mi>β</mi></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>-</mo><mi>γ</mi></mrow></mfenced></math> show that:</p>
<p style="padding-left:30px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mo>-</mo><mfenced><mrow><mi>α</mi><mo>+</mo><mi>β</mi><mo>+</mo><mi>γ</mi></mrow></mfenced></math></p>
<p style="padding-left:30px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi><mo>=</mo><mi>α</mi><mi>β</mi><mo>+</mo><mi>β</mi><mi>γ</mi><mo>+</mo><mi>γ</mi><mi>α</mi></math></p>
<p style="padding-left:30px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mo>-</mo><mi>α</mi><mi>β</mi><mi>γ</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>p</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>q</mi><mo>=</mo><msup><mi>α</mi><mn>2</mn></msup><mo>+</mo><msup><mi>β</mi><mn>2</mn></msup><mo>+</mo><msup><mi>γ</mi><mn>2</mn></msup></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>Hence show that <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>α</mi><mo>-</mo><mi>β</mi></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mi>β</mi><mo>-</mo><mi>γ</mi></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mi>γ</mi><mo>-</mo><mi>α</mi></mrow></mfenced><mn>2</mn></msup><mo>=</mo><mn>2</mn><msup><mi>p</mi><mn>2</mn></msup><mo>-</mo><mn>6</mn><mi>q</mi></math>.</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>Given that <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>p</mi><mn>2</mn></msup><mo><</mo><mn>3</mn><mi>q</mi></math>, deduce that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>,</mo><mo> </mo><mi>β</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>γ</mi></math> cannot all be real.</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>Using the result from part (c), show that when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi><mo>=</mo><mn>17</mn></math>, this equation has at least one complex root.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By varying the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi></math> in the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mn>7</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>q</mi><mi>x</mi><mo>+</mo><mn>1</mn><mo>=</mo><mn>0</mn></math>, determine the smallest positive integer value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi></math> required to show that Noah is incorrect.</p>
<div class="marks">[2]</div>
<div class="question_part_label">e.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Explain why the equation will have at least one real root for all values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">e.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>α</mi><mn>2</mn></msup><mo>+</mo><msup><mi>β</mi><mn>2</mn></msup><mo>+</mo><msup><mi>γ</mi><mn>2</mn></msup><mo>+</mo><msup><mi>δ</mi><mn>2</mn></msup></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">f.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence state a condition in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi></math> that would imply <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>4</mn></msup><mo>+</mo><mi>p</mi><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mi>q</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>r</mi><mi>x</mi><mo>+</mo><mi>s</mi><mo>=</mo><mn>0</mn></math> has at least one complex root.</p>
<div class="marks">[1]</div>
<div class="question_part_label">f.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use your result from part (f)(ii) to show that the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>4</mn></msup><mo>-</mo><mn>2</mn><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>4</mn><mi>x</mi><mo>+</mo><mn>5</mn><mo>=</mo><mn>0</mn></math> has at least one complex root.</p>
<div class="marks">[1]</div>
<div class="question_part_label">g.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State what the result in part (f)(ii) tells us when considering this equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>4</mn></msup><mo>-</mo><mn>9</mn><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>24</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>22</mn><mi>x</mi><mo>-</mo><mn>12</mn><mo>=</mo><mn>0</mn></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">h.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the integer root of this equation.</p>
<div class="marks">[1]</div>
<div class="question_part_label">h.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By writing <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>4</mn></msup><mo>-</mo><mn>9</mn><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>24</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>22</mn><mi>x</mi><mo>-</mo><mn>12</mn></math> as a product of one linear and one cubic factor, prove that the equation has at least one complex root.</p>
<div class="marks">[4]</div>
<div class="question_part_label">h.iii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>attempt to expand <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>x</mi><mo>-</mo><mi>α</mi></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>-</mo><mi>β</mi></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>-</mo><mi>γ</mi></mrow></mfenced></math> <strong> </strong><em> <strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mfenced><mrow><mi>α</mi><mo>+</mo><mi>β</mi></mrow></mfenced><mi>x</mi><mo>+</mo><mi>α</mi><mi>β</mi></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>-</mo><mi>γ</mi></mrow></mfenced></math> <strong>OR </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced><mrow><mi>x</mi><mo>-</mo><mi>α</mi></mrow></mfenced><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mfenced><mrow><mi>β</mi><mo>+</mo><mi>γ</mi></mrow></mfenced><mi>x</mi><mo>+</mo><mi>β</mi><mi>γ</mi></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mi>p</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>q</mi><mi>x</mi><mo>+</mo><mi>r</mi></mrow></mfenced><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mfenced><mrow><mi>α</mi><mo>+</mo><mi>β</mi><mo>+</mo><mi>γ</mi></mrow></mfenced><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mfenced><mrow><mi>α</mi><mi>β</mi><mo>+</mo><mi>β</mi><mi>γ</mi><mo>+</mo><mi>γ</mi><mi>α</mi></mrow></mfenced><mi>x</mi><mo>-</mo><mi>α</mi><mi>β</mi><mi>γ</mi></math> <em><strong>A1</strong></em></p>
<p>comparing coefficients:</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mo>-</mo><mfenced><mrow><mi>α</mi><mo>+</mo><mi>β</mi><mo>+</mo><mi>γ</mi></mrow></mfenced></math> <em><strong>AG</strong> </em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi><mo>=</mo><mfenced><mrow><mi>α</mi><mi>β</mi><mo>+</mo><mi>β</mi><mi>γ</mi><mo>+</mo><mi>γ</mi><mi>α</mi></mrow></mfenced></math> <em><strong>AG</strong> </em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mo>-</mo><mi>α</mi><mi>β</mi><mi>γ</mi></math> <em><strong>AG</strong> </em></p>
<p> </p>
<p><strong>Note:</strong> For candidates who do not include the <em><strong>AG</strong> </em>lines award full marks.</p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>p</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>q</mi><mo>=</mo><msup><mfenced><mrow><mi>α</mi><mo>+</mo><mi>β</mi><mo>+</mo><mi>γ</mi></mrow></mfenced><mn>2</mn></msup><mo>-</mo><mn>2</mn><mfenced><mrow><mi>α</mi><mi>β</mi><mo>+</mo><mi>β</mi><mi>γ</mi><mo>+</mo><mi>γ</mi><mi>α</mi></mrow></mfenced></math> <strong> </strong><em> <strong>(A1)</strong></em></p>
<p>attempt to expand <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>α</mi><mo>+</mo><mi>β</mi><mo>+</mo><mi>γ</mi></mrow></mfenced><mn>2</mn></msup></math> <strong> </strong><em> <strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mi>α</mi><mn>2</mn></msup><mo>+</mo><msup><mi>β</mi><mn>2</mn></msup><mo>+</mo><msup><mi>γ</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mfenced><mrow><mi>α</mi><mi>β</mi><mo>+</mo><mi>β</mi><mi>γ</mi><mo>+</mo><mi>γ</mi><mi>α</mi></mrow></mfenced><mo>-</mo><mn>2</mn><mfenced><mrow><mi>α</mi><mi>β</mi><mo>+</mo><mi>β</mi><mi>γ</mi><mo>+</mo><mi>γ</mi><mi>α</mi></mrow></mfenced></math> or equivalent <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mi>α</mi><mn>2</mn></msup><mo>+</mo><msup><mi>β</mi><mn>2</mn></msup><mo>+</mo><msup><mi>γ</mi><mn>2</mn></msup></math> <em><strong>AG</strong> </em></p>
<p> </p>
<p><strong>Note:</strong> Accept equivalent working from RHS to LHS.</p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p>attempt to expand <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>α</mi><mo>-</mo><mi>β</mi></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mi>β</mi><mo>-</mo><mi>γ</mi></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mi>γ</mi><mo>-</mo><mi>α</mi></mrow></mfenced><mn>2</mn></msup></math> <strong> </strong><em> <strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced><mrow><msup><mi>α</mi><mn>2</mn></msup><mo>+</mo><msup><mi>β</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>α</mi><mi>β</mi></mrow></mfenced><mo>+</mo><mfenced><mrow><msup><mi>β</mi><mn>2</mn></msup><mo>+</mo><msup><mi>γ</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>β</mi><mi>γ</mi></mrow></mfenced><mo>+</mo><mfenced><mrow><msup><mi>γ</mi><mn>2</mn></msup><mo>+</mo><msup><mi>α</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>γ</mi><mi>α</mi></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>2</mn><mfenced><mrow><msup><mi>α</mi><mn>2</mn></msup><mo>+</mo><msup><mi>β</mi><mn>2</mn></msup><mo>+</mo><msup><mi>γ</mi><mn>2</mn></msup></mrow></mfenced><mo>-</mo><mn>2</mn><mfenced><mrow><mi>α</mi><mi>β</mi><mo>+</mo><mi>β</mi><mi>γ</mi><mo>+</mo><mi>γ</mi><mi>α</mi></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>2</mn><mfenced><mrow><msup><mi>p</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>q</mi></mrow></mfenced><mo>-</mo><mn>2</mn><mi>q</mi></math> or equivalent <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>2</mn><msup><mi>p</mi><mn>2</mn></msup><mo>-</mo><mn>6</mn><mi>q</mi></math> <em><strong>AG</strong> </em></p>
<p><br><strong>OR</strong></p>
<p>attempt to write <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><msup><mi>p</mi><mn>2</mn></msup><mo>-</mo><mn>6</mn><mi>q</mi></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>,</mo><mo> </mo><mi>β</mi><mo>,</mo><mo> </mo><mi>γ</mi></math> <strong> </strong><em> <strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>2</mn><mfenced><mrow><msup><mi>p</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>q</mi></mrow></mfenced><mo>-</mo><mn>2</mn><mi>q</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>2</mn><mfenced><mrow><msup><mi>α</mi><mn>2</mn></msup><mo>+</mo><msup><mi>β</mi><mn>2</mn></msup><mo>+</mo><msup><mi>γ</mi><mn>2</mn></msup></mrow></mfenced><mo>-</mo><mn>2</mn><mfenced><mrow><mi>α</mi><mi>β</mi><mo>+</mo><mi>β</mi><mi>γ</mi><mo>+</mo><mi>γ</mi><mi>α</mi></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced><mrow><msup><mi>α</mi><mn>2</mn></msup><mo>+</mo><msup><mi>β</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>α</mi><mi>β</mi></mrow></mfenced><mo>+</mo><mfenced><mrow><msup><mi>β</mi><mn>2</mn></msup><mo>+</mo><msup><mi>γ</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>β</mi><mi>γ</mi></mrow></mfenced><mo>+</mo><mfenced><mrow><msup><mi>γ</mi><mn>2</mn></msup><mo>+</mo><msup><mi>α</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>γ</mi><mi>α</mi></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msup><mfenced><mrow><mi>α</mi><mo>-</mo><mi>β</mi></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mi>β</mi><mo>-</mo><mi>γ</mi></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mi>γ</mi><mo>-</mo><mi>α</mi></mrow></mfenced><mn>2</mn></msup></math> <em><strong>AG</strong> </em></p>
<p> </p>
<p><strong>Note:</strong> Accept equivalent working where LHS and RHS are expanded to identical expressions.</p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>p</mi><mn>2</mn></msup><mo><</mo><mn>3</mn><mi>q</mi><mo>⇒</mo><mn>2</mn><msup><mi>p</mi><mn>2</mn></msup><mo>-</mo><mn>6</mn><mi>q</mi><mo><</mo><mn>0</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msup><mfenced><mrow><mi>α</mi><mo>-</mo><mi>β</mi></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mi>β</mi><mo>-</mo><mi>γ</mi></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mi>γ</mi><mo>-</mo><mi>α</mi></mrow></mfenced><mn>2</mn></msup><mo><</mo><mn>0</mn></math> <em><strong>A1</strong></em></p>
<p>if all roots were real <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>α</mi><mo>-</mo><mi>β</mi></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mi>β</mi><mo>-</mo><mi>γ</mi></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mi>γ</mi><mo>-</mo><mi>α</mi></mrow></mfenced><mn>2</mn></msup><mo>≥</mo><mn>0</mn></math> <em><strong>R1</strong></em></p>
<p><strong><br>Note:</strong> Condone strict inequality in the <em><strong>R1</strong> </em>line.<br><strong>Note:</strong> Do not award <em><strong>A0R1</strong></em>.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo></math>roots cannot all be real <em><strong>AG</strong> </em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>p</mi><mn>2</mn></msup><mo>=</mo><msup><mfenced><mrow><mo>-</mo><mn>7</mn></mrow></mfenced><mn>2</mn></msup><mo>=</mo><mn>49</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mi>q</mi><mo>=</mo><mn>51</mn></math> <em><strong>A1</strong></em></p>
<p>so <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>p</mi><mn>2</mn></msup><mo><</mo><mn>3</mn><mi>q</mi><mo>⇒</mo></math> the equation has at least one complex root <em><strong>R1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Allow equivalent comparisons; e.g. checking <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>p</mi><mn>2</mn></msup><mo><</mo><mn>6</mn><mi>q</mi></math></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>use of GDC (<em>eg</em> graphs or tables) <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi><mo>=</mo><mn>12</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">e.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>complex roots appear in conjugate pairs (so if complex roots occur the other root will be real OR all 3 roots will be real).</p>
<p>OR</p>
<p>a cubic curve always crosses the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis at at least one point. <em><strong>R1</strong></em></p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">e.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to expand <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>α</mi><mo>+</mo><mi>β</mi><mo>+</mo><mi>γ</mi><mo>+</mo><mi>δ</mi></mrow></mfenced><mn>2</mn></msup></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>α</mi><mo>+</mo><mi>β</mi><mo>+</mo><mi>γ</mi><mo>+</mo><mi>δ</mi></mrow></mfenced><mn>2</mn></msup><mo>=</mo><msup><mi>α</mi><mn>2</mn></msup><mo>+</mo><msup><mi>β</mi><mn>2</mn></msup><mo>+</mo><msup><mi>γ</mi><mn>2</mn></msup><mo>+</mo><msup><mi>δ</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mfenced><mrow><mi>α</mi><mi>β</mi><mo>+</mo><mi>α</mi><mi>γ</mi><mo>+</mo><mi>α</mi><mi>δ</mi><mo>+</mo><mi>β</mi><mi>γ</mi><mo>+</mo><mi>β</mi><mi>δ</mi><mo>+</mo><mi>γ</mi><mi>δ</mi></mrow></mfenced></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msup><mi>α</mi><mn>2</mn></msup><mo>+</mo><msup><mi>β</mi><mn>2</mn></msup><mo>+</mo><msup><mi>γ</mi><mn>2</mn></msup><mo>+</mo><msup><mi>δ</mi><mn>2</mn></msup><mo>=</mo><msup><mfenced><mrow><mi>α</mi><mo>+</mo><mi>β</mi><mo>+</mo><mi>γ</mi><mo>+</mo><mi>δ</mi></mrow></mfenced><mn>2</mn></msup><mo>-</mo><mn>2</mn><mfenced><mrow><mi>α</mi><mi>β</mi><mo>+</mo><mi>α</mi><mi>γ</mi><mo>+</mo><mi>α</mi><mi>δ</mi><mo>+</mo><mi>β</mi><mi>γ</mi><mo>+</mo><mi>β</mi><mi>δ</mi><mo>+</mo><mi>γ</mi><mi>δ</mi></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>⇒</mo><msup><mi>α</mi><mn>2</mn></msup><mo>+</mo><msup><mi>β</mi><mn>2</mn></msup><mo>+</mo><msup><mi>γ</mi><mn>2</mn></msup><mo>+</mo><msup><mi>δ</mi><mn>2</mn></msup><mo>=</mo></mrow></mfenced><msup><mi>p</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>q</mi></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">f.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>p</mi><mn>2</mn></msup><mo><</mo><mn>2</mn><mi>q</mi></math> <strong>OR </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>p</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>q</mi><mo><</mo><mn>0</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Allow <em><strong>FT</strong> </em>on their result from part (f)(i).</p>
<p> </p>
<p><em><strong>[1 mark</strong></em><em><strong>]</strong></em></p>
<div class="question_part_label">f.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo><</mo><mn>6</mn></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>2</mn><mn>2</mn></msup><mo>-</mo><mn>2</mn><mo>×</mo><mn>3</mn><mo><</mo><mn>0</mn></math> <em><strong>R1</strong></em></p>
<p>hence there is at least one complex root. <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Allow <em><strong>FT</strong> </em>from part (f)(ii) for the <em><strong>R</strong></em> mark provided numerical reasoning is seen.</p>
<p> </p>
<p><em><strong>[1 mark</strong></em><em><strong>]</strong></em></p>
<div class="question_part_label">g.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><msup><mi>p</mi><mn>2</mn></msup><mo>></mo><mn>2</mn><mi>q</mi></mrow></mfenced><mo> </mo><mfenced><mrow><mn>81</mn><mo>></mo><mn>2</mn><mo>×</mo><mn>24</mn></mrow></mfenced></math> (so) nothing can be deduced <em><strong>R1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Do not allow <em><strong>FT</strong> </em>for the <em><strong>R</strong></em> mark.</p>
<p> </p>
<p><em><strong>[1 mark</strong></em><em><strong>]</strong></em></p>
<div class="question_part_label">h.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>1</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[1 mark</strong></em><em><strong>]</strong></em></p>
<div class="question_part_label">h.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to express as a product of a linear and cubic factor <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mn>10</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>34</mn><mi>x</mi><mo>-</mo><mn>12</mn></mrow></mfenced></math> <em><strong>A1A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for each factor. Award at most <em><strong>A1A0</strong></em> if not written as a product.</p>
<p> </p>
<p>since for the cubic, <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>p</mi><mn>2</mn></msup><mo><</mo><mn>3</mn><mi>q</mi><mo> </mo><mfenced><mrow><mn>100</mn><mo><</mo><mn>102</mn></mrow></mfenced></math> <em><strong>R1</strong></em></p>
<p>there is at least one complex root <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[4 marks</strong></em><em><strong>]</strong></em></p>
<div class="question_part_label">h.iii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>The first part of this question proved to be very accessible, with the majority of candidates expanding their brackets as required, to find the coefficients <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>,</mo><mo> </mo><mi>q</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math>.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>The first part of this question was usually answered well, though presentation in the second part sometimes left a lot to be desired. The expression <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mfenced><mrow><msup><mi>p</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>q</mi></mrow></mfenced><mo>-</mo><mn>2</mn><mi>q</mi></math> was expected to be seen more often, as a 'pivot' to reaching the required result. Algebra was often lengthy, but untidily so, sometimes leaving examiners to do some mental tidying up on behalf of the candidate.</p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>A good number of candidates recognised the reasoning required in this part of the question and were able to score both marks.</p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Most candidates found applying this specific case to be very straightforward.</p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Most candidates offered incorrect answers in the first part; despite their working suggested utilisation of the GDC, it was clear that many did not appreciate what the question was asking. The second part was usually answered well, with the idea of complex roots occurring in conjugate pairs being put to good use.</p>
<div class="question_part_label">e.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Some very dubious algebra was seen here, and often no algebra at all. Despite this, a good number of candidates seemed to make the 'leap' to the correct expression <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>p</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>q</mi></math>, perhaps fortuitously so in a number of cases.</p>
<div class="question_part_label">f.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">f.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Of those finding <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>p</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>q</mi></math> in part f, a surprising number of answers seen employed the test of checking whether <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>p</mi><mn>2</mn></msup><mo><</mo><mn>3</mn><mi>q</mi></math>.</p>
<div class="question_part_label">g.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Part i was usually not answered successfully, which may have been due to shortage of time. However, it was pleasing to see a number of candidates reach the end of the paper and successfully factorise the given quartic using a variety of methods. The final part required the <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>p</mi><mn>2</mn></msup><mo><</mo><mn>3</mn><mi>q</mi></math> test. Though correct reasoning was sometimes seen, it was rare for this final mark to be gained.</p>
<div class="question_part_label">h.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">h.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">h.iii.</div>
</div>
<br><hr><br>