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<h2>HL Paper 3</h2><div class="specification">
<p><strong>This question asks you to examine various polygons for which the numerical value of the area is the same as the numerical value of the perimeter. For example, a <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math> by <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn></math> rectangle has an area of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>18</mn></math> and a perimeter of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>18</mn></math>.</strong></p>
<p> </p>
<p>For each polygon in this question, let the numerical value of its area be <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> and let the numerical value of its perimeter be <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math>.</p>
</div>
<div class="specification">
<p>An <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>-sided regular polygon can be divided into <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> congruent isosceles triangles. Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> be the length of each of the two equal sides of one such isosceles triangle and let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> be the length of the third side. The included angle between the two equal sides has magnitude <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>2</mn><mi mathvariant="normal">π</mi></mrow><mi>n</mi></mfrac></math>.</p>
<p>Part of such an <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>-sided regular polygon is shown in the following diagram.</p>
<p style="text-align: center;"><img 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"></p>
</div>
<div class="specification">
<p>Consider a <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>-sided regular polygon such that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mi>P</mi></math>.</p>
</div>
<div class="specification">
<p>The Maclaurin series for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tan</mi><mo> </mo><mi>x</mi></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>+</mo><mfrac><msup><mi>x</mi><mn>3</mn></msup><mn>3</mn></mfrac><mo>+</mo><mfrac><mrow><mn>2</mn><msup><mi>x</mi><mn>5</mn></msup></mrow><mn>15</mn></mfrac><mo>+</mo><mo>…</mo></math></p>
</div>
<div class="specification">
<p>Consider a right-angled triangle with side lengths <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>,</mo><mo> </mo><mi>b</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></msqrt></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>≥</mo><mi>b</mi></math>, such that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mi>P</mi></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the side length, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mo>></mo><mn>0</mn></math>, of a square such that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mi>P</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>Write down, 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>n</mi></math>, an expression for the area, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>A</mi><mi>T</mi></msub></math>, of one of these isosceles triangles.</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>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>2</mn><mi>x</mi><mo> </mo><mi>sin</mi><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use the results from parts (b) and (c) to show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mi>P</mi><mo>=</mo><mn>4</mn><mi>n</mi><mo> </mo><mi>tan</mi><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac></math>.</p>
<div class="marks">[7]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use the Maclaurin series for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tan</mi><mo> </mo><mi>x</mi></math> to find <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi>lim</mi><mrow><mi>n</mi><mo>→</mo><mo>∞</mo></mrow></munder><mfenced><mrow><mn>4</mn><mi>n</mi><mo> </mo><mi>tan</mi><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac></mrow></mfenced></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">e.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Interpret your answer to part (e)(i) geometrically.</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>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mfrac><mn>8</mn><mrow><mi>b</mi><mo>-</mo><mn>4</mn></mrow></mfrac><mo>+</mo><mn>4</mn></math>.</p>
<div class="marks">[7]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By using the result of part (f) or otherwise, determine the three side lengths of the only two right-angled triangles for which <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>,</mo><mo> </mo><mi>b</mi><mo>,</mo><mo> </mo><mi>A</mi><mo>,</mo><mo> </mo><mi>P</mi><mo>∈</mo><mi mathvariant="normal">ℤ</mi></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>Determine the area and perimeter of these two right-angled triangles.</p>
<div class="marks">[1]</div>
<div class="question_part_label">g.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><msup><mi>s</mi><mn>2</mn></msup></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>=</mo><mn>4</mn><mi>s</mi></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mi>P</mi><mo>⇒</mo><msup><mi>s</mi><mn>2</mn></msup><mo>=</mo><mn>4</mn><mi>s</mi></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mfenced><mrow><mi>s</mi><mo>-</mo><mn>4</mn></mrow></mfenced><mo>=</mo><mn>0</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>s</mi><mo>=</mo><mn>4</mn><mfenced><mrow><mi>s</mi><mo>></mo><mn>0</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note: </strong>Award <em><strong>A1M1A0</strong></em> if both <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mo>=</mo><mn>4</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mo>=</mo><mn>0</mn></math> are stated as final answers.</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"><msub><mi>A</mi><mi>T</mi></msub><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>x</mi><mrow><mn>2</mn><mo> </mo></mrow></msup><mi>sin</mi><mfrac><mrow><mn>2</mn><mi mathvariant="normal">π</mi></mrow><mi>n</mi></mfrac></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note: </strong>Award <em><strong>A1 </strong></em>for a correct alternative form expressed 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>n</mi></math> only.</p>
<p> For example, using Pythagoras’ theorem, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>A</mi><mi>T</mi></msub><mo>=</mo><mi>x</mi><mo> </mo><mi>sin</mi><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac><msqrt><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><msup><mi>x</mi><mrow><mn>2</mn><mo> </mo></mrow></msup><msup><mi>sin</mi><mn>2</mn></msup><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac></msqrt></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>A</mi><mi>T</mi></msub><mo>=</mo><mn>2</mn><mfenced><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mfenced><mrow><mi>x</mi><mo> </mo><mi>sin</mi><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac></mrow></mfenced><mfenced><mrow><mi>x</mi><mo> </mo><mi>cos</mi><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac></mrow></mfenced></mrow></mfenced></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>A</mi><mi>T</mi></msub><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mo> </mo><mi>sin</mi><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac><mi>cos</mi><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac></math>.</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><strong>METHOD 1</strong></p>
<p>uses <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mo> </mo><mi>θ</mi><mo>=</mo><mfrac><mtext>opp</mtext><mtext>hyp</mtext></mfrac></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mstyle displaystyle="true"><mfrac><mi>y</mi><mn>2</mn></mfrac></mstyle><mi>x</mi></mfrac><mo>=</mo><mi>sin</mi><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>2</mn><mi>x</mi><mo> </mo><mi>sin</mi><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>uses Pythagoras’ theorem <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mfrac><mi>y</mi><mn>2</mn></mfrac></mfenced><mn>2</mn></msup><mo>+</mo><msup><mi>h</mi><mn>2</mn></msup><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>=</mo><mi>x</mi><mo> </mo><mi>cos</mi><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mfrac><mi>y</mi><mn>2</mn></mfrac></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mi>x</mi><mo> </mo><mi>cos</mi><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac></mrow></mfenced><mn>2</mn></msup><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mo> </mo><mo> </mo><mfenced><mrow><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mfenced><mrow><mn>1</mn><mo>-</mo><msup><mi>cos</mi><mn>2</mn></msup><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac></mrow></mfenced></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo> </mo><msup><mi>sin</mi><mn>2</mn></msup><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>2</mn><mi>x</mi><mo> </mo><mi>sin</mi><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>METHOD 3</strong></p>
<p>uses the cosine rule <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo> </mo><mi>cos</mi><mfrac><mrow><mn>2</mn><mi mathvariant="normal">π</mi></mrow><mi>n</mi></mfrac><mo> </mo><mfenced><mrow><mo>=</mo><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mfenced><mrow><mn>1</mn><mo>-</mo><mi>cos</mi><mfrac><mrow><mn>2</mn><mi mathvariant="normal">π</mi></mrow><mi>n</mi></mfrac></mrow></mfenced></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup><mo> </mo><msup><mi>sin</mi><mn>2</mn></msup><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>2</mn><mi>x</mi><mo> </mo><mi>sin</mi><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>METHOD 4</strong></p>
<p>uses the sine rule <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>y</mi><mrow><mi>sin</mi><mstyle displaystyle="true"><mfrac><mrow><mn>2</mn><mi mathvariant="normal">π</mi></mrow><mi>n</mi></mfrac></mstyle></mrow></mfrac><mo>=</mo><mfrac><mi>x</mi><mrow><mi>sin</mi><mstyle displaystyle="true"><mfenced><mrow><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac><mo>-</mo><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac></mrow></mfenced></mstyle></mrow></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo> </mo><mi>cos</mi><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac><mo>=</mo><mn>2</mn><mi>x</mi><mo> </mo><mi>sin</mi><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac><mi>cos</mi><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>2</mn><mi>x</mi><mo> </mo><mi>sin</mi><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac></math> <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"><mi>A</mi><mo>=</mo><mi>P</mi><mo>⇒</mo><mi>n</mi><msub><mi>A</mi><mi>T</mi></msub><mo>=</mo><mi>n</mi><mi>y</mi></math> <em><strong>(M1)</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>M1</strong></em> for equating correct expressions for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math>.</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>n</mi><msup><mi>x</mi><mn>2</mn></msup><mo> </mo><mi>sin</mi><mfrac><mrow><mn>2</mn><mi mathvariant="normal">π</mi></mrow><mi>n</mi></mfrac><mo>=</mo><mn>2</mn><mi>n</mi><mi>x</mi><mo> </mo><mi>sin</mi><mfrac><mstyle displaystyle="true"><mi mathvariant="normal">π</mi></mstyle><mstyle displaystyle="true"><mi>n</mi></mstyle></mfrac><mo> </mo><mfenced><mrow><mi>n</mi><msup><mi>x</mi><mn>2</mn></msup><mo> </mo><mi>sin</mi><mfrac><mstyle displaystyle="true"><mi mathvariant="normal">π</mi></mstyle><mstyle displaystyle="true"><mi>n</mi></mstyle></mfrac><mi>cos</mi><mfrac><mstyle displaystyle="true"><mi mathvariant="normal">π</mi></mstyle><mstyle displaystyle="true"><mi>n</mi></mstyle></mfrac><mo>=</mo><mn>2</mn><mi>n</mi><mi>x</mi><mo> </mo><mi>sin</mi><mfrac><mstyle displaystyle="true"><mi mathvariant="normal">π</mi></mstyle><mstyle displaystyle="true"><mi>n</mi></mstyle></mfrac></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>x</mi><mn>2</mn></msup><mo> </mo><mi>sin</mi><mfrac><mrow><mn>2</mn><mi mathvariant="normal">π</mi></mrow><mi>n</mi></mfrac><mo>=</mo><mn>2</mn><mi>x</mi><mo> </mo><mi>sin</mi><mfrac><mstyle displaystyle="true"><mi mathvariant="normal">π</mi></mstyle><mstyle displaystyle="true"><mi>n</mi></mstyle></mfrac><mo> </mo><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo> </mo><mi>sin</mi><mfrac><mstyle displaystyle="true"><mi mathvariant="normal">π</mi></mstyle><mstyle displaystyle="true"><mi>n</mi></mstyle></mfrac><mi>cos</mi><mfrac><mstyle displaystyle="true"><mi mathvariant="normal">π</mi></mstyle><mstyle displaystyle="true"><mi>n</mi></mstyle></mfrac><mo>=</mo><mn>2</mn><mi>x</mi><mo> </mo><mi>sin</mi><mfrac><mstyle displaystyle="true"><mi mathvariant="normal">π</mi></mstyle><mstyle displaystyle="true"><mi>n</mi></mstyle></mfrac></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p>uses <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mfrac><mrow><mn>2</mn><mi mathvariant="normal">π</mi></mrow><mi>n</mi></mfrac><mo>=</mo><mn>2</mn><mo> </mo><mi>sin</mi><mfrac><mstyle displaystyle="true"><mi mathvariant="normal">π</mi></mstyle><mstyle displaystyle="true"><mi>n</mi></mstyle></mfrac><mi>cos</mi><mfrac><mstyle displaystyle="true"><mi mathvariant="normal">π</mi></mstyle><mstyle displaystyle="true"><mi>n</mi></mstyle></mfrac></math> (seen anywhere in part (d) or in part (b)) <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo> </mo><mi>sin</mi><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac><mi>cos</mi><mfrac><mstyle displaystyle="true"><mi mathvariant="normal">π</mi></mstyle><mstyle displaystyle="true"><mi>n</mi></mstyle></mfrac><mo>=</mo><mn>2</mn><mi>x</mi><mo> </mo><mi>sin</mi><mfrac><mstyle displaystyle="true"><mi mathvariant="normal">π</mi></mstyle><mstyle displaystyle="true"><mi>n</mi></mstyle></mfrac></math></p>
<p>attempts to either factorise or divide their expression <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo> </mo><mi>sin</mi><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac><mfenced><mrow><mi>x</mi><mo> </mo><mi>cos</mi><mfrac><mstyle displaystyle="true"><mi mathvariant="normal">π</mi></mstyle><mstyle displaystyle="true"><mi>n</mi></mstyle></mfrac><mo>-</mo><mn>2</mn></mrow></mfenced><mo>=</mo><mn>0</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mn>2</mn><mrow><mi>cos</mi><mfrac><mstyle displaystyle="true"><mi mathvariant="normal">π</mi></mstyle><mstyle displaystyle="true"><mi>n</mi></mstyle></mfrac></mrow></mfrac><mo>,</mo><mo> </mo><mfenced><mrow><mi>x</mi><mo> </mo><mi>sin</mi><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac><mo>≠</mo><mn>0</mn></mrow></mfenced></math> (or equivalent) <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>EITHER</strong></p>
<p>substitutes <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mn>2</mn><mrow><mi>cos</mi><mfrac><mstyle displaystyle="true"><mi mathvariant="normal">π</mi></mstyle><mstyle displaystyle="true"><mi>n</mi></mstyle></mfrac></mrow></mfrac></math> (or equivalent) into <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>=</mo><mi>n</mi><mi>y</mi></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>=</mo><mn>2</mn><mi>n</mi><mfenced><mfrac><mn>2</mn><mrow><mi>cos</mi><mfrac><mstyle displaystyle="true"><mi mathvariant="normal">π</mi></mstyle><mstyle displaystyle="true"><mi>n</mi></mstyle></mfrac></mrow></mfrac></mfenced><mfenced><mrow><mi>sin</mi><mfrac><mstyle displaystyle="true"><mi mathvariant="normal">π</mi></mstyle><mstyle displaystyle="true"><mi>n</mi></mstyle></mfrac></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><strong><br>Note:</strong> Other approaches are possible. For example, award<em><strong> A1</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>=</mo><mn>2</mn><mi>n</mi><mi>x</mi><mo> </mo><mi>cos</mi><mfrac><mstyle displaystyle="true"><mi mathvariant="normal">π</mi></mstyle><mstyle displaystyle="true"><mi>n</mi></mstyle></mfrac><mi>tan</mi><mfrac><mstyle displaystyle="true"><mi mathvariant="normal">π</mi></mstyle><mstyle displaystyle="true"><mi>n</mi></mstyle></mfrac></math> and <em><strong>M1</strong></em> for substituting <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mn>2</mn><mrow><mi>cos</mi><mfrac><mstyle displaystyle="true"><mi mathvariant="normal">π</mi></mstyle><mstyle displaystyle="true"><mi>n</mi></mstyle></mfrac></mrow></mfrac></math> into <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math>.</p>
<p><strong><br>OR</strong></p>
<p>substitutes <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mn>2</mn><mrow><mi>cos</mi><mfrac><mstyle displaystyle="true"><mi mathvariant="normal">π</mi></mstyle><mstyle displaystyle="true"><mi>n</mi></mstyle></mfrac></mrow></mfrac></math> (or equivalent) into <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mi>n</mi><msub><mi>A</mi><mi>T</mi></msub></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>n</mi><msup><mfenced><mfrac><mn>2</mn><mrow><mi>cos</mi><mfrac><mstyle displaystyle="true"><mi mathvariant="normal">π</mi></mstyle><mstyle displaystyle="true"><mi>n</mi></mstyle></mfrac></mrow></mfrac></mfenced><mn>2</mn></msup><mfenced><mrow><mi>sin</mi><mfrac><mstyle displaystyle="true"><mn>2</mn><mi mathvariant="normal">π</mi></mstyle><mstyle displaystyle="true"><mi>n</mi></mstyle></mfrac></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>n</mi><msup><mfenced><mfrac><mn>2</mn><mrow><mi>cos</mi><mfrac><mstyle displaystyle="true"><mi mathvariant="normal">π</mi></mstyle><mstyle displaystyle="true"><mi>n</mi></mstyle></mfrac></mrow></mfrac></mfenced><mn>2</mn></msup><mfenced><mrow><mn>2</mn><mo> </mo><mi>sin</mi><mfrac><mstyle displaystyle="true"><mi mathvariant="normal">π</mi></mstyle><mstyle displaystyle="true"><mi>n</mi></mstyle></mfrac><mi>cos</mi><mfrac><mstyle displaystyle="true"><mi mathvariant="normal">π</mi></mstyle><mstyle displaystyle="true"><mi>n</mi></mstyle></mfrac></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"><mi>A</mi><mo>=</mo><mi>P</mi><mo>=</mo><mn>4</mn><mi>n</mi><mo> </mo><mi>tan</mi><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[7 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempts to use the Maclaurin series for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tan</mi><mo> </mo><mi>x</mi></math> with <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tan</mi><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac><mo>=</mo><mfrac><mstyle displaystyle="true"><mi mathvariant="normal">π</mi></mstyle><mstyle displaystyle="true"><mi>n</mi></mstyle></mfrac><mo>+</mo><mfrac><mstyle displaystyle="true"><msup><mfenced><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac></mfenced><mn>3</mn></msup></mstyle><mn>3</mn></mfrac><mo>+</mo><mfrac><mstyle displaystyle="true"><mn>2</mn><msup><mfenced><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac></mfenced><mn>5</mn></msup></mstyle><mn>15</mn></mfrac><mfenced><mrow><mo>+</mo><mo>…</mo></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mi>n</mi><mo> </mo><mi>tan</mi><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac><mo>=</mo><mn>4</mn><mi>n</mi><mfenced><mrow><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac><mo>+</mo><mfrac><msup><mi mathvariant="normal">π</mi><mn>3</mn></msup><mrow><mn>3</mn><msup><mi>n</mi><mn>3</mn></msup></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>2</mn><msup><mi mathvariant="normal">π</mi><mn>5</mn></msup></mrow><mrow><mn>15</mn><msup><mi>n</mi><mn>5</mn></msup></mrow></mfrac><mfenced><mrow><mo>+</mo><mo>…</mo></mrow></mfenced></mrow></mfenced></math> (or equivalent) <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>4</mn><mfenced><mrow><mi mathvariant="normal">π</mi><mo>+</mo><mfrac><msup><mi mathvariant="normal">π</mi><mn>3</mn></msup><mrow><mn>3</mn><msup><mi>n</mi><mn>2</mn></msup></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>2</mn><msup><mi mathvariant="normal">π</mi><mn>5</mn></msup></mrow><mrow><mn>15</mn><msup><mi>n</mi><mn>4</mn></msup></mrow></mfrac><mo>+</mo><mo>…</mo></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><munder><mi>lim</mi><mrow><mi>n</mi><mo>→</mo><mo>∞</mo></mrow></munder><mfenced><mrow><mn>4</mn><mi>n</mi><mo> </mo><mi>tan</mi><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac></mrow></mfenced><mo>=</mo><mn>4</mn><mi mathvariant="normal">π</mi></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award a maximum of <em><strong>M1A1A0</strong></em> if <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi>lim</mi><mrow><mi>n</mi><mo>→</mo><mo>∞</mo></mrow></munder></math> is not stated anywhere.</p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">e.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>(as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>→</mo><mo>∞</mo><mo>,</mo><mo> </mo><mi>P</mi><mo>→</mo><mn>4</mn><mi mathvariant="normal">π</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>→</mo><mn>4</mn><mi mathvariant="normal">π</mi></math>)</p>
<p>the polygon becomes a circle of radius <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math> <em><strong>R1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>R1</strong></em> for alternative responses such as:<br>the polygon becomes a circle of area <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mi mathvariant="normal">π</mi></math> OR<br>the polygon becomes a circle of perimeter <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mi mathvariant="normal">π</mi></math> OR<br>the polygon becomes a circle with <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mi>P</mi><mo>=</mo><mn>4</mn><mi mathvariant="normal">π</mi></math>.<br>Award <em><strong>R0</strong></em> for polygon becomes a circle.</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><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>a</mi><mi>b</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>=</mo><mi>a</mi><mo>+</mo><mi>b</mi><mo>+</mo><msqrt><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></msqrt></math> <em><strong>(A1)(A1)</strong></em></p>
<p>equates their expressions for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mi>P</mi><mo>⇒</mo><mi>a</mi><mo>+</mo><mi>b</mi><mo>+</mo><msqrt><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></msqrt><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>a</mi><mi>b</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></msqrt><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>a</mi><mi>b</mi><mo>-</mo><mfenced><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow></mfenced></math> <em><strong>M1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>M1</strong></em> for isolating <math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></msqrt></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>±</mo><mn>2</mn><msqrt><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></msqrt></math>. This step may be seen later.</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup><mo>=</mo><msup><mfenced><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>a</mi><mi>b</mi><mo>-</mo><mfenced><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow></mfenced></mrow></mfenced><mn>2</mn></msup></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup><mo>=</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><msup><mi>a</mi><mn>2</mn></msup><msup><mi>b</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mfenced><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>a</mi><mi>b</mi></mrow></mfenced><mfenced><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow></mfenced><mo>+</mo><msup><mfenced><mrow><mi>a</mi><mo>+</mo><mi>b</mi></mrow></mfenced><mn>2</mn></msup></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>=</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><msup><mi>a</mi><mn>2</mn></msup><msup><mi>b</mi><mn>2</mn></msup><mo>-</mo><msup><mi>a</mi><mn>2</mn></msup><mi>b</mi><mo>-</mo><mi>a</mi><msup><mi>b</mi><mn>2</mn></msup><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>a</mi><mi>b</mi><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow></mfenced></math></p>
<p> </p>
<p><strong>Note:</strong> Award <strong>M1</strong> for attempting to expand their RHS of either <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup><mo>=</mo><mo>…</mo></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mfenced><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow></mfenced><mo>=</mo><mo>…</mo></math>.</p>
<p> </p>
<p><strong>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>b</mi><mfenced><mrow><mfrac><mn>1</mn><mn>4</mn></mfrac><mi>a</mi><mi>b</mi><mo>-</mo><mi>a</mi><mo>-</mo><mi>b</mi><mo>+</mo><mn>2</mn></mrow></mfenced><mo>=</mo><mn>0</mn><mo> </mo><mo> </mo><mfenced><mrow><mi>a</mi><mi>b</mi><mo>≠</mo><mn>0</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>4</mn></mfrac><mi>a</mi><mi>b</mi><mo>-</mo><mi>a</mi><mo>-</mo><mi>b</mi><mo>+</mo><mn>2</mn><mo>=</mo><mn>0</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>b</mi><mo>-</mo><mn>4</mn><mi>a</mi><mo>=</mo><mn>4</mn><mi>b</mi><mo>-</mo><mn>8</mn></math></p>
<p> </p>
<p><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>4</mn></mfrac><msup><mi>a</mi><mn>2</mn></msup><msup><mi>b</mi><mn>2</mn></msup><mo>-</mo><msup><mi>a</mi><mn>2</mn></msup><mi>b</mi><mo>-</mo><mi>a</mi><msup><mi>b</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>a</mi><mi>b</mi><mo>=</mo><mn>0</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mfenced><mrow><mfrac><mn>1</mn><mn>4</mn></mfrac><msup><mi>b</mi><mn>2</mn></msup><mo>-</mo><mi>b</mi></mrow></mfenced><mo>+</mo><mfenced><mrow><mn>2</mn><mi>b</mi><mo>-</mo><msup><mi>b</mi><mn>2</mn></msup></mrow></mfenced><mo>=</mo><mn>0</mn><mo> </mo><mo> </mo><mfenced><mrow><mi>a</mi><mfenced><mrow><msup><mi>b</mi><mn>2</mn></msup><mo>-</mo><mn>4</mn><mi>b</mi></mrow></mfenced><mo>+</mo><mfenced><mrow><mn>8</mn><mi>b</mi><mo>-</mo><mn>4</mn><msup><mi>b</mi><mn>2</mn></msup></mrow></mfenced><mo>=</mo><mn>0</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mfrac><mrow><mn>4</mn><msup><mi>b</mi><mn>2</mn></msup><mo>-</mo><mn>8</mn><mi>b</mi></mrow><mrow><msup><mi>b</mi><mn>2</mn></msup><mo>-</mo><mn>4</mn><mi>b</mi></mrow></mfrac></math></p>
<p> </p>
<p><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>a</mi><mo>=</mo><mfrac><mrow><mn>4</mn><mi>b</mi><mo>-</mo><mn>8</mn></mrow><mrow><mi>b</mi><mo>-</mo><mn>4</mn></mrow></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mfrac><mrow><mn>4</mn><mi>b</mi><mo>-</mo><mn>16</mn><mo>+</mo><mn>8</mn></mrow><mrow><mi>b</mi><mo>-</mo><mn>4</mn></mrow></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mfrac><mn>8</mn><mrow><mi>b</mi><mo>-</mo><mn>4</mn></mrow></mfrac><mo>+</mo><mn>4</mn></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award a maximum of <em><strong>A1A1M1M1M0A0A0</strong></em> for attempting to verify.<br>For example, verifying that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mi>P</mi><mo>=</mo><mfrac><mn>16</mn><mrow><mi>b</mi><mo>-</mo><mn>4</mn></mrow></mfrac><mo>+</mo><mn>2</mn><mi>b</mi><mo>+</mo><mn>4</mn></math> gains <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math> of the <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn></math> marks.</p>
<p> </p>
<p><em><strong>[7 marks]</strong></em></p>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>using an appropriate method <em><strong>(M1)</strong></em></p>
<p><em>eg</em> substituting values for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math> or using divisibility properties</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>5</mn><mo>,</mo><mo> </mo><mn>12</mn><mo>,</mo><mo> </mo><mn>13</mn></mrow></mfenced></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>6</mn><mo>,</mo><mo> </mo><mn>8</mn><mo>,</mo><mo> </mo><mn>10</mn></mrow></mfenced></math> <em><strong>A1A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A1A0</strong></em> for either one set of three correct side lengths or two sets of two correct side lengths.</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><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mi>P</mi><mo>=</mo><mn>30</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mi>P</mi><mo>=</mo><mn>24</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Do not award <em><strong>A1FT</strong></em>.</p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">g.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.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">f.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">g.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">g.ii.</div>
</div>
<br><hr><br>