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<h2>HL Paper 2</h2><div class="specification">
<p>A function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> is defined by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mtext>arcsin</mtext><mfenced><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfrac></mfenced><mo>,</mo><mo> </mo><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
</div>
<div class="specification">
<p>A function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math> is defined by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mtext>arcsin</mtext><mfenced><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfrac></mfenced><mo>,</mo><mo> </mo><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi><mo>,</mo><mo> </mo><mi>x</mi><mo>≥</mo><mn>0</mn></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> is an even function.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By considering limits, show that the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> has a horizontal asymptote and state its equation.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mrow><mn>2</mn><mi>x</mi></mrow><mrow><msqrt><msup><mi>x</mi><mn>2</mn></msup></msqrt><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfenced></mrow></mfrac></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi><mo>,</mo><mo> </mo><mi>x</mi><mo>≠</mo><mn>0</mn></math>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By using the expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced></math> and the result <math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><msup><mi>x</mi><mn>2</mn></msup></msqrt><mo>=</mo><mfenced open="|" close="|"><mi>x</mi></mfenced></math>, show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> is decreasing for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo><</mo><mn>0</mn></math>.</p>
<p> </p>
<div class="marks">[3]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>g</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo>(</mo><mi>x</mi><mo>)</mo></math>, justifying your answer.</p>
<div class="marks">[5]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State the domain of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>g</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><msup><mi>g</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo>(</mo><mi>x</mi><mo>)</mo></math>, clearly indicating any asymptotes with their equations and stating the values of any axes intercepts.</p>
<div class="marks">[3]</div>
<div class="question_part_label">f.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mrow><mo>-</mo><mi>x</mi></mrow></mfenced><mo>=</mo><mtext>arcsin</mtext><mfenced><mfrac><mrow><msup><mfenced><mrow><mo>-</mo><mi>x</mi></mrow></mfenced><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow><mrow><msup><mfenced><mrow><mo>-</mo><mi>x</mi></mrow></mfenced><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfrac></mfenced><mo>=</mo><mtext>arcsin</mtext><mfenced><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfrac></mfenced><mo>=</mo><mi>f</mi><mfenced><mi>x</mi></mfenced></math> <em><strong>R1</strong></em></p>
<p><br><strong>OR</strong></p>
<p>a sketch graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mfenced><mi>x</mi></mfenced></math> with line symmetry in the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-axis indicated <em><strong>R1</strong></em></p>
<p><br><strong>THEN</strong></p>
<p>so <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced></math> is an even function. <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>→</mo><mo>±</mo><mo>∞</mo><mo>,</mo><mo> </mo><mo> </mo><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>→</mo><mtext>arcsin</mtext><mo> </mo><mn>1</mn><mfenced><mrow><mo>→</mo><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p>so the horizontal asymptote is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></math> <em><strong>A1</strong></em> </p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempting to use the quotient rule to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mtext>d</mtext><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mfenced><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfrac></mfenced></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mtext>d</mtext><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mfenced><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfrac></mfenced><mo>=</mo><mfrac><mrow><mn>2</mn><mi>x</mi><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfenced><mo>-</mo><mn>2</mn><mi>x</mi><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow></mfenced></mrow><msup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup></mfrac><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mfrac><mrow><mn>4</mn><mi>x</mi></mrow><msup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup></mfrac></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p>attempting to use the chain rule to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mtext>d</mtext><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mfenced><mrow><mtext>arcsin</mtext><mfenced><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfrac></mfenced></mrow></mfenced></math> <em><strong>M1</strong></em></p>
<p>let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi><mo>=</mo><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfrac></math> and so <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mtext>arcsin</mtext><mo> </mo><mi>u</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>u</mi></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><msqrt><mn>1</mn><mo>-</mo><msup><mi>u</mi><mn>2</mn></msup></msqrt></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mn>1</mn><msqrt><mn>1</mn><mo>-</mo><msup><mfenced><mstyle displaystyle="true"><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfrac></mstyle></mfenced><mn>2</mn></msup></msqrt></mfrac><mo>×</mo><mfrac><mrow><mn>4</mn><mi>x</mi></mrow><msup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup></mfrac></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mn>4</mn><mi>x</mi></mrow><msqrt><msup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup><mo>-</mo><msup><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup></msqrt></mfrac><mo>×</mo><mfrac><mn>1</mn><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfenced></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mn>4</mn><mi>x</mi></mrow><msqrt><mn>4</mn><msup><mi>x</mi><mn>2</mn></msup></msqrt></mfrac><mo>×</mo><mfrac><mn>1</mn><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfenced></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mn>2</mn><mi>x</mi></mrow><mrow><msqrt><msup><mi>x</mi><mn>2</mn></msup></msqrt><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfenced></mrow></mfrac></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mrow><mn>2</mn><mi>x</mi></mrow><mrow><mfenced open="|" close="|"><mi>x</mi></mfenced><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfenced></mrow></mfrac></math></p>
<p><br><strong>EITHER</strong></p>
<p>for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo><</mo><mn>0</mn><mo>,</mo><mo> </mo><mfenced open="|" close="|"><mi>x</mi></mfenced><mo>=</mo><mo>-</mo><mi>x</mi></math> <em><strong>(A1)</strong></em></p>
<p>so <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mo>-</mo><mfrac><mrow><mn>2</mn><mi>x</mi></mrow><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfrac></math> <em><strong>A1</strong></em></p>
<p><br><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mi>x</mi></mfenced><mo>></mo><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn><mo>></mo><mn>0</mn></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>x</mi><mo><</mo><mn>0</mn><mo>,</mo><mo> </mo><mi>x</mi><mo><</mo><mn>0</mn></math> <em><strong>A1</strong></em></p>
<p><br><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo><</mo><mn>0</mn></math> <em><strong>R1</strong></em></p>
<p><strong><br>Note:</strong> Award <em><strong>R1</strong></em> for stating that in <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced></math>, the numerator is negative, and the denominator is positive.</p>
<p><br>so <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> is decreasing for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo><</mo><mn>0</mn></math> <em><strong>AG</strong></em></p>
<p><strong><br>Note:</strong> Do not accept a graphical solution</p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mtext>arcsin</mtext><mfenced><mfrac><mrow><msup><mi>y</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow><mrow><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfrac></mfenced></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>sin</mtext><mo> </mo><mi>x</mi><mo>=</mo><mfrac><mrow><msup><mi>y</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow><mrow><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfrac><mo>⇒</mo><msup><mi>y</mi><mn>2</mn></msup><mo> </mo><mtext>sin</mtext><mo> </mo><mi>x</mi><mo>+</mo><mtext>sin</mtext><mo> </mo><mi>x</mi><mo>=</mo><msup><mi>y</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mfrac><mrow><mn>1</mn><mo>+</mo><mtext>sin</mtext><mo> </mo><mi>x</mi></mrow><mrow><mn>1</mn><mo>-</mo><mtext>sin</mtext><mo> </mo><mi>x</mi></mrow></mfrac></math> <em><strong>A1</strong></em></p>
<p>domain of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi><mo>,</mo><mo> </mo><mi>x</mi><mo>≥</mo><mn>0</mn></math> and so the range of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>g</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math> must be <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi><mo>,</mo><mo> </mo><mi>y</mi><mo>≥</mo><mn>0</mn></math></p>
<p>hence the positive root is taken (or the negative root is rejected) <em><strong>R1</strong></em></p>
<p><br><strong>Note:</strong> The <em><strong>R1</strong></em> is dependent on the above<em><strong> A1</strong></em>.</p>
<p><br>so <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><msup><mi>g</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mi>x</mi></mfenced><mo>=</mo></mrow></mfenced><msqrt><mfrac><mrow><mn>1</mn><mo>+</mo><mtext>sin</mtext><mo> </mo><mi>x</mi></mrow><mrow><mn>1</mn><mo>-</mo><mtext>sin</mtext><mo> </mo><mi>x</mi></mrow></mfrac></msqrt></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> The final <em><strong>A1</strong></em> is not dependent on <em><strong>R1</strong></em> mark.</p>
<p> </p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>domain is <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac><mo>≤</mo><mi>x</mi><mo><</mo><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Accept correct alternative notations, for example, <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⌊</mo><mo>-</mo><mstyle displaystyle="false"><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></mstyle><mo>,</mo><mo> </mo><mstyle displaystyle="false"><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></mstyle><mo>⌊</mo></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⌊</mo><mo>-</mo><mstyle displaystyle="false"><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></mstyle><mo>,</mo><mo> </mo><mstyle displaystyle="false"><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac><mo>)</mo></mstyle></math>.<br>Accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>[</mo><mo>-</mo><mn>1</mn><mo>.</mo><mn>57</mn><mo>,</mo><mo> </mo><mn>1</mn><mo>.</mo><mn>57</mn><mo>[</mo></math> if correct to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math> s.f.</p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="padding-left:60px;"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAEjCAYAAAAc4VcXAAAgAElEQVR4Ae2dh3tUVfrH9x/Y365l1dVVAeuqiAokoW0oUiTS1FUUggqCFBHpHRJASgihBZDeBZJA6FVqSCiB0HsJEDohnTTA9/ec0cwmkyn3nlvmnnO/PM88ZGbue+493/c938/cdu5fCP+gABSAAlAACnAo8BeOGIRAASgABaAAFCAABEUABaAAFIACXAoAIFyyIQgKQAEoAAUAENQAFIACUAAKcCkAgHDJhiAoAAWgABQAQFADUAAKQAEowKUAAMIlG4KgABSAAlAAAEENQAEoAAWgAJcCAAiXbAiCAlAACkABAAQ1AAWgABSAAlwKACBcsiEICkABKAAFABDUABSAAlAACnApwAeQ3x8T4QUNUAOoAdSAtWqACwP8QQAIBoC1BgDygXygBvhrgJ8FXJEACIqVv1ihHbRDDVirBrgwwB8EgGAAWGsAIB/IB2qAvwb4WcAVCYCgWPmLFdpBO9SAtWqACwP8QQAIBoC1BgDygXygBvhrgJ8FXJEACIqVv1gtrt3vjx9RUlIiRUZG0tEjRxz9fPzoIUWMG0d7E/ZI229cIWnjq0S5MMAfBIBY3ARhBvxmwGCxc8cOevutt2jYsKEOYDwsLqIWzZtT587fAyCofflqgJ8FXJEACAaRfIOoVE7ZXkhoaCiFtm1L7G8G5Js3rtOc2bOl7jd+ePD/8BBaOy4M8AcBIKXMRujCQT88AmHggAHUqFFDYnskxUWFNHnyJLqffs/j8qgDm5qvDGOInwVckQCIDEWDPniFQeT48VStWjVih6/2JiTQ9bRrXpcHQAAQYWuACwP8QQAIzFd6M120cAFVrFiR8nJzKGbFCun7K6z5YSxqr01+FnBFAiAoWu1Fa3ENd+3aSc89+yzNmDGdbt++JX1/ARAb70FxYYA/CACxuPnBDLSbwZkzpykoMJBOnzoJeKDe5a4BfhZwRQIgGFBSD6hHD4speupUun3rptT9LP1Dg11tVnLFWenP8bf2HyOW15ALA/xBAAgAIqWx5uZk0/30dNq9exddvnypTB+ZucasWE7Xrl4t87nlzaFUrW7dtpWSEhPdbn92ViZt27bV7Xdm93Hu7JkU0rQJNWncsNwr/0GeJbbRbE0MXR8/C7giAZBSg9LQxGI9ppoFu/ej6Ucf0Z7du8ut98Tx41SpUiXH/SHs0l4R8x4ZOZ6mTpnidtvZJcrLly1z+52Zfd352zYKrl2LunT6jlo2C6HeP/WgmoHVqWZggOPvwoJ8v2+jmXqYsi4uDPAHASAwdikH8bmzZ+jypUvlDuWw+0CaNG7sAEjFChVo/74kIfs/ZMhgWrt2jdttZwCJi41x+50pJvbnmJo1YwYV5D+ga1dSacXyP4DWvWsXGjKwv9+3zUwdTF0XPwu4IgEQAMRWg3nJksXUt08f6tSpE82ePZu+bteO2HkSUwe5DjXH+rB9+3a32832qqz0635t/Epaszresa1fh7al6CmT3W63aDmw5PZyYYA/CADRYTBbspDQr3Imxc4NtGzZwvGruGvXrpSQsIfYTYbToqPL7alYPacd2renI0dSyvXRits9KjyM2OEsdu6p9ef/pZkzpgux3VbU0uc28bOAKxIAgdHaZjCPGzeWZkyf5uhvCUDS0q7R22+/TVevpAqlw1dftqZzZ89afpuLCvKp2vtV6Pixow6ANG3SiGZMi6aM9HuOaWV8GiLGp7occ2GAPwgAQYGqK1BB9Tp29IhjVt70e3cd/S0BCDMwNjdWmzZfCaVDq1at6Mb16263mR2+SnW58sxfRr1k4QJ69h9POaePYYewGtavS+NG/0y4CsuAy4r5WcAVCYAIaoj+MgRR13vjehodPpTsPFRVGiDsRO/mTZvcmrFV+9u4USPKy811u81WuQqLadeuzZfUo/sPjoks2fttWzZTcJ1aNLBfX2curKqxkNvFhQH+IAAEAHFrQkIOHhW5LA0QEftav149j3mzEkBE1FbobeZnAVckAKLCdIQuLPSzjOGKDpCQkJAy/SldmwCIAYeGRBk/XBjgDwJARCkMbKdHwyxtnkr/Fh0gTZo08agHAAKA8CNBXSQAAmP2aERKzVjE5UQHSMMPP/SYN3ZOh91I6S0v69euocBqH9CEiLGO5Xbu2E5vvvYqRf753lssvrMwoNT5v+alARAAxKvRyGoWIgOEXb3UrFkzTXlj92SsjF1Br71SkVbFxlDU+AhaE7+Kkg/sL9PuzRtp1PX7Tl5fu3fuKBMja80I0S/NSFDXAAACgNhy8IsMEHaI6vP//ldz3q6mXqZ/PPUE9e/dS3Nbpc113do1FDF2jKLXvDmzdF136e2w5d/q/F/z0gAIAGLLASwyQNjswt936uQxb2wqk6LCAo/flxjro+IieqXiyxQZMc7jsqwtdkjM24s9KrikTfb/1i2bKXrqFEWvZUuXlIkt3Q7+5jhUphkJ6hoAQAAQWw5gkQFy6eJF6tK5s8e8ZdxPp5VxcR6/LzHmY0dSqGGD+lSvTm2Py148f46qvveu19eKZb96jC9ZF/7ngAGPN6nzf81LAyA8SUKM8IYhMkAOHUqmEeHhHnPg6yqszPR7jskW58+ZTetWx9OTf/8/Onv6FG1cu8ZSkzACOhzQ0YwEdQ0AIICBRyOSeQCLDJBtW7fShMjxHvPmDSAHkxLpqSf+Rq0/+4Ru37xBWRn36c3XX6Wg6lUpKaH8s1NkrgEp+6bO/zUvDYAAIB6NSMoB9me+RQYIO0k9ZbLnKdG9AYTldF/CHsrNynTm/fyZ03Qz7Zrzvcx5l75vmpGgrgEABACxpXGIDBD2tMH58+Z5zFtOdhYl7LHO3gS7ZJhNpZ+TlemcE0t6I/eXr6jzf81LAyD+SjTW69EAzTAXkQHy69KltGjRQr/qpzRH2ZkZ9H3H7+jlF1+gii+/SB3bf0uZ99OF2HalfbTUcpqRoK4BAARGbsvBLDJA5s6ZQ+wwlqWMy804Ypf3hrb5kqIiIxyz8P48Ipyee+Zpql0jECfr3eilSz7V+b/mpQEQoxKJdi1tcCIDJDp6KiUm7rW0vswMDyQl0gYX0A3o14ee+Ntf6byPqVZ0MVM7jkHNSFDXAABixyJDn0lkgEycGEVHUjw/zpadb9j+2zZLAmbp4oWOq8Du3bltye0THlzq/F/z0gAIzNSWA1lkgLB7QC5dvOAxb76uwvKnSQ4ZNMDxQCnXu9f9uU1SrVszEtQ1AIAAIB6NSKqB5ZJnkQEyaOBAunnzhse8WRUgaVevUK2gQDqcfNDjtstcc6b0TZ3/a14aAHExFlOSjHX63UBEBkif3r0oq9R9HK41a0WAFBXkU4O6wbRvb4Lfc++ql1TvNSNBXQMACMzclgNaZID80K2b15yxubDi41d5XcZs01y0YD7Fx8VaapvM1sCU9anzf81LAyAAiC0HtcgA6d1L3+nXjTa2a1ev0IjhQ8vUWV5Odpn3Rm+DbdrXjAR1DQAgAIgtB7LIABk0aJAwOWPw+KhRQ0q9fImyMjMcr1MnjtOM6KnC9EEo+Kjzf81LAyAAiC0HstgAGShEznb8to0CqlWl11+tRG++/prz9WqlCrRPgPtYhAJHiY9pRoK6BgCQEuHxvxCmpNegFhkggwcP9pqrvNwcSjl8yOsyeumIdjimXDfSa9T5v+alARAjk4m2LWtiIgNk2NCy5xNcTdyKV2G5biPeGwQezUhQ1wAAApO3rMkbaTIiA8Tbw6SYZgCIQeYsgleo83/NSwMgIhQFtlF3yIkMkNGjR3vVAwABQDSTQWEDAAjM2asZGbkX4M+2RQbIsGG+D2GtWL7Mlnn1Z01ZYt0KjV+vxQAQAMSWRiMqQNhEiSNHjPCas+KiQrp7947XZSxhdhh7+udILzIobAcAQRHrX8QCaCoqQK6npdHYMWNsmTNAT8GhOYXGr9diAIgAZoeBo2DgqMyjqAA5duwoRU+dAoCozLdtxpBeZFDYDgCCQrSlGYkKkJ07d9CC+Z6fh86M8tHDYsczyG1jmhjD/xvDCo1fr8UAEBTf/4rPRlqIChD2ONs4H5MSsquwcBJd/71WIYCsFxkUtgOA2Mg0hRgAJuVDVIBMjIqitS6PiXXNKy7jtSk82Ngx+R8AYpJhuQ5yvPfvIBcVIKNGjaItmzd73WsEQPxbW34d2wCIjZMPmHk1Rj0HpqgA6dO7N+1LSvSqEwBiYw8BQGycfADEqzECII+p43ff0dmzZ7zq9CAvl1IOH/a6jJ5aoi0LeRYAYqFkwNClNSFR90BaNG9O7F4QmDZ8wm0NACAoDLeFAZjpapoiAuRhcRF92KABpaff01UL1JtEngOASJRMmL5ljU5EgBQW5FOtmjUpP/+BZXUFjPzsXwCInxMA07eFOYkIEPagqDp16vjMD1vu8KFkn8vB7CX0GgBEwqQCSpYzMxEBknE/nUJCQnxqiauwbOwhAIiNkw/Q+DRHvX41iwiQa1ev0BdffOFTIwDExh4CgNg4+QCIT3O0M0COHkmhzp07+9QIALGxhwAgNk4+AOLTHO0MkE2bNtKYMd6fRsj0ycy473O6E710RDsW8ysAxGIJgambZupmmpGIh7BmzvyF2GSKZuqEdQnmRwCIYAkDYIQ0NBEBMiI8nGJiYoTUGyAyydcAEJOEhvHb2ohEBEiPH3+kLVu8T6QIo7a5fwAgNi8AgM0UsIkIkDZt2tCB/ft96sMeKJWTk+1zOcBGQq8BQCRMKqBgOTMTESCNGjak1NTLPrXEVVg29hAAxMbJB2h8mqNev5pFA0j+gzwKDAig9Ht3fWoEgNjYQwAQGycfAPFpjnYFyL17d6l69erE5sPypQEAYmMPAUBsnHwAxKc5+jJPpd+Ltgdy5UqqYyZeJf0DQGzsIQCIjZMPgAAgHmog+eABCg0NVaQP20u5fOmiomWVAAnLCORJAIhAyfIw2DHgrJ9D0fZA1q9bR/369gUUMOa81wAAYn3zASDEz5FoAFm6ZAlFT53i3TxgrtAHABHfnAAY6+dQNIBMmjiR4uNXKTLI3x8/osePHipaFrVq/VpVlSMARLKE4lehJY1MNID07duHkhITFWnJnhuyZnW8omVVmRNq2fqaAiAACAa18TUgGkD++9lndPVKqiIDw1VYxtePZccoAGLj5OMXniKD1GPwigaQunXrUlZmhiJ9ABAbewgAYuPkAyCKDNJuACnIz3fchc7uRlfSdwDExh4CgNg4+QCIIoNUYqK+lhFpD+Tundv0Yf369LC4SJE+bE9l08YNipb1pRO+F8yPABDBEgbTF9KoRALIyRMn6NtvvlGsM67CsrGHACA2Tj5gpNgktf4yFgkgO7Zvp0GDBpqmjVZtEe9HDwNA/Cg+DNw2JiUSQNatXUMToybYJjcAkAYPBEA0iAcAwGQU1oBIAFm4YD5tVHFOIy83h1JSDqMWFNaCVMACQAAQqQraooNYJIBMnjSRTp86pRgIuArLxh4CgNg4+RY1WxmBJhJARoSHK3oOSEmeABAbewgAYuPkAyCKf2WXmCXv/yIBZPiwYap0AUBs7CEAiI2TD4CoMkpeeLA4kQDSv18/VboAIDb2EADExskHQFQZpV0A0q+fuueAsBsOMzLum6alljwgVme/A0B0FhSmDCNxUwOi7IEUFxVR165dkEM3OQR83HglAOJGFBQPDETnGhAFIKmpl2nI4EHIv875lxY+AAgAIm1xW8gERAHIju2/0bToqaoAUlxUSHdu31IVg5qTxHcAEEkSaSGzhDmUrylRALJg/nxij7NVk0OcRC+fbzX6Cb0sAGLj5AM6qoxSy0AXBSBjxoymHTu2q9IFALGxhwAgNk4+AKLKKO0AkLZt21Ja2jVVugAgNvYQAMTGyQdAVBmlHQDSoEEDevzooSpdABAbewgAYuPkAyCqjFJ2gLBJEevVq0fs+R5q+pqfl0snTxxXFaOmfSxrYY8CQCycHBi8NKYkwjmQIykp9GXr1qoBAoO3sYcAIDZOPgBlGqBEAMiihQtJ7TxYgIfN/QMAsXkBACKmQEQEgLApTBYvWqRaD3bIS+nz0wEcyfwGAJEsoQCCagM0w9SsDhB24rx5s4/pBMe5DHYSPSZmhSV1NyO3tl4HAAKA2HoAmARcqwMk/0EeBQUFETuRrrYecBWWjT0EALFx8k0yT7WGJOPyVgdI+r271KRJY9XwYLkCQGzsIQCIjZMPgHAZJg/grA6Q337bRoMGDuTSAwCxsYcAIDZOPgDCZZgyAiRs+HCK5TyPkZOdRbt37TJNSx79EWOQzwEgBgkLc4ahlKoBq++BfPH555R88AByVipngI4CbwRAFIiEooKxaKwBKwOkoCCfAgMC6DamZEedq61zAAQAwS8t42vAygC5dvUqBQcH06OHxTBQtQZq9+UBEOPNAwYNja0MkI0bNlDvXr244cEuAT518gR3PMaHwOMDABE4eXb/9SNQ/60MkFEjR9Lq+HhuAOAqLBt7CABi4+QLZMCi/0q1MkA+++wzSr18CQDBeFBfAwAIACK6OYuw/VYFSGFBPlWvVo3rDvQS3bEHYmMPAUBsnHz84lL/i4tTM6sC5OTJExTatq2mKdwz7qfTypVxpmlZAi78bwHvAkAskAROU8IAEid3VgXIqpUracqUyZrMn03EWFRYoKkN1LI4tVwmVwCIoIkDdIQyLKsCZGJUFJ08gSuoypgixpbysQWAACAYPMbXgFUBMnTIEOVmAWOFVq41AIAYbx4waGhsVYD07dtXsymyB0oVFxVpbgfjRMBxAoAImDTXXwF4b3nzsiJA2NVTPXv+pFk7dhI9ftUqze0AIAJ6EQAiYNIADOHMyooAWb9uHc2YPl2zlriM18YeAoDYOPkAkWbzVPqr2YoACQsbTjt3bNesAQBiYw8BQGycfABEs3mKDJCQkBDKyszQrAEAYmMPAUBsnHwARLN5igqQ8+fOUssWLTTdQFjSd/Yc9ZTDh0zTsmS9+N8C3gWAWCAJMHLpzcdqh7DCw8JoymRtNxDCwOEdJvOD/sK1Qhis9AYruxlZCSDsuR8hTZvSiRPHUVfwFm01wGXo/EEACApWW8EKqp+VAHL3zm2qW7cuHiAlaC1Z6scWPwu4IgEQFC0A4ucaYBMfDh8+XLc85OZk04H9+3Rrz1IG6edcWV4LLgzwBwEgKEhbGo1V9kDY4atWrVrRoeSDuuUBV2HZ+FwIPwu4IgEQAEQ347L8r7NSubYKQNjzz+vVq0dsBl299ANAABAuGnAEASClTEWvAYx2rD+ArQKQbVu3UtSECbrBg9UeAGL9+jPMIzggoCUEAAFAdDUvwwaGznmyCkBGjhhBly5e0DUHAAgAogUKamIBEJ2NSRQDtft2WgEgxUWF9P333+t+9RUeKAWAqIGAlmUBEABE11+/ooDJCgA5knKYIsaNs6X+otSJcNuphQYcsQAIAGJLA7MCQNjDo/bvS7Kl/sIZsyg+wQEBLSEAiCiFge3U1Wj9DZDMjPsUEBBAOdnZuvaLGXNhQT6lXr6ke7swfQEOjWmhAUcsAAJjtqXR+BsgixYupC6dvzdEe5xEF8DojfIdDghoCQFAjEok2jXEHPX6FexPgLDZchs1akjX09IM0QgAAUC0QEFNLAACozfExPQyeqPa8SdA2Inz8DD9pi5x1QgAAUDUQEDLsgAIAAKAmFgDbJ6qWjVr0vW0a4bpDoAAIFqgoCYWADHRPFx/KeK9/wa6v/ZA5s+bR4MGDTIMHqymigoLKO3aVUPXgdr1X+161V6N++uwLAACgNjSaPwBkFu3btKHDRrQzZs3bKm5V+PDONSnJnSAgpomABAUrj6FK5iO/gBIVNQE2rJlsy31BjxM2mNR4/46LAuACGZ8GIj6DESzAVKQ/4Datm1LbPoSo3PIpohnV3oZvR60r08t6qqjDlBQ0wQAAoDY0mjMBkhsTAxNi442RWt2En3F8mWmrEtX88NY1J4zNe6vw7IACIpWe9EKqKGZALlz+xYFBgQQ+98Mw8VVWBbcMzBrjOgABTVNACBmJRbrMcU8lRq0WQApLiygtm3aUFxsrGn9B0AAEDUQ0LIsAAJjN83YlJq7GcuZAZDfHz+iAf3707ChQ03VGAABQLRAQU0sAAKAmGpuZsBByTrMAMi5s2ccj6vNzso0VeOc7Czal5Ro6jqVaI5lTACbGvfXYVkABAAxzWjYg47Yr3KjjIS1rfTZ4kYDhG1L3759KHFvgmH9NUpHtGuC0RvlOzpAQU0TAIhRiUS7TuNkZrp5w3paOHcOPX5Y7Pxcb6N6WFxEEyLG0aED+32CykiAsP6uW7eWRoSH+9wOvTVAewKbvx6eocb9dVgWANEjaWjDKxQWzJ1NY0aN9LqMXsbHINL+63a0a8d2r+szEiAHDx6gmjVrktmHrko0VLMnVhKD/yUBjw5QUNMEAALz92q0Wo3l7u1b9H6VypSRfs/terZs2kAjwobRhXNn3X7vbf0Piwpp7qyZ5Z4pfvb0KXr/3crETiZ7ijcKILdu3qA6tWvT/v37PK7b0zbp9XlWZgZt27rFb+vXqx9ohwNqatxfh2UBEADEUKNhBh82dHC5dbC7pXv16E5fh7ahpIQ9FFitKu3ZtbPccp5M5MjhQ/TxR03o+eeeIbbX4brcV198ThMjx5f7vGQ5IwDCJjFs/cUXFD11qsf1lqzfyP9xFRaH8criAzpAQU0TAIgshWPRfjQLaeoAhKthLl4wn17457OU++cVSosWzKc3Xn2F0u/eUWS+N65dpSmTJnoEyLIli6nCS/+i/Ad5btvTGyDs5H2vnj2pV69efj/vAYAAIGogoGVZAMSixutquCK+ZzB48/VX6e6d22VMPC8nm95643Xq9F175+eply7R888+Q2N/HuX8zFefly5a4BEgF8+fo2eeftLjXo2eAGHnHGbMmE7hYWGO55H72m6jvwdAABAtUFATC4AAIIoNW63xsSuv3nn73+UurT1/9gw98be/0tif/3dine0pvP3vN6hWYIDi7fEGkPv37tJrlSrQjGj3h5P0AgiDx6pVKx17H0ovIVaro9rlM+6n08q4OMU6qm0fy1sYUGrcX4dlARAAxDCjmTp5ElX74P1y7e/ds9sBkJnTy04u+MH7VajCSy+WW96TYXkDCDvHUvX996hPz5/ctqcXQHbu2O54wiA7ee5pO83+nEHN3Xkhs7cD6/MDaHSAgpomABAAxDDjGzSgP9WpGVSufXaJLdsDmTd7ZpnvGEBefOH5Mp95MyFvAGFxtWoEUcdSh8lKt6UHQDZuWE/BwcF08eIFxdtcehvwtx8MVvbxrsb9dVgWAJG9oPzYv7atP6f/1KpZzlyPHE52AGTalMnO7/44L/IaffBuZednvgzWF0DqBdehb9qFum1PK0AWLlhAnTp2NG2GXV9a4HvAyFEDOkBBTRMAiB8NVvZB37H9N1SnVo1yBn7vzm164blnaPDAAc7vrl1JdXw2oH9f52e+9FECkPbftHPbHi9A2KGhyZMmOc55+No+f33PziedOXPabb/9tU1Yr0mAU+P+OiwLgAAghhlNxLgxVL1q+XMg7GRzSJPG1LBeXee6Dx7YT0/9/f9o+583wLFp0Nl9Fd6MxxdAagQFULcund22wQOQ3Jxs6tatK33drh2xv71tmz+/w1VYJpm1Fb1DByioaQIAsWIRSLJN7F6P9959x63RHj96hP71z+eI3RDIzPZHZsxt2xC7u5y9D/3qS/qwbjDd9nJymrX/3LP/cHvCuDD/AVWp/DaFD3M/lbpagFw4f44+/vhjGjFiBOXnP3DbJ39Co/S6ARAARA0EtCwLgEhi1qUNxCp/nzpxnF57paLHX+vxcbHUPKQp9f6pB/3QpTNlZdx3GnONgOpU8aUXac6ssifaS/p27PAh6tT+W3rztVdo7MgRdObkCWcsW+bOrZuO+F+XLC7zeUm8GoAkHzxAtWvVotjYGL/fJFiy/d7+B0AAEC1QUBMLgAAgbg3Wm0Ep/a4g/wF98N67dNnLVUrscBY7r8AuPS3dLrsMN2zoENq6aWOZz0uWYXFsmZKX6z0YbM/m6Sf/TqddwFISrwQgt2/dpKgJE+inHj3o5o3rbrejpD0r/Q+AACBqIKBlWQAEADHUGAf060NzZv6ieh1r41fRpAmRquNKjDwqcjyFfNTYY7w3gLBzL/PnzXM8x3zZr796bKNkXVb7v7iokO7duyvcdltNRyG3RwsNOGIBEADEUKM5nHzQcSmv6x6Cr8F56fy5cnslvmJKvmd7JdXer0KJCXs89s0TQNjJ8R+6daP27b8lK90cWNI3/G/jvQslXsUBAS0hAIiSpGAZj0bsy9DY4akvPvuUNqxdw92Gr3W4fh8Xs4K+a/8NPXIzS2/Jsq4AYYCLiVnhmIqd7X2ULIf/YdhC1YAWGnDEAiCAg+FmmZOV6ZgkcfWqldx7FUoGMYNA7IrlFDHmZ4+XAF9Pu0btQkPplUqVKCgokGbPnkVjRo+m7zp0oOipU+hK6mXD9VDSFy3LsD0w9lx0LW0gVlBwckBASwgAAoCYZjTnTp8q9/AnPY2K7e2cPXXSa3+GDRtKzz37rPP10ksv0dYtWwzdLj37qKQtnEQX1Pz18CItNOCIBUD0SBra8GraSkzPrGU6ftfBCQ8Gkn+98IIw265UIwAEAOFgAVcIAALzl85A3RltZsZ9mjVrJlWuXLkMQOrXqydd/wEQAISLBhxBAAgAIp2BlgbIlSup1KtXT6pRowaFh4fTkSMp9FGTJg6IVKxQgdatXStd/wEQAISDBVwhAAgAIpWBPn5YTAcO7KeJUVGOy3GHDhlCv23bSnm5uc5+spsWO3XqRDt2bHd+Vho6ov/NLkXevy9Jyr6JnhvDt58LA/xBAAgAIoXRsHs2IsePd1yG27VLF9q0caPXx8u6XsZr+MBGnUlRZ5avE34WcEUCIBjYwg5sdl5jzerV1KFDB6oRFCkppvcAABD1SURBVEQ//zxK8eWrAIiND/PIPOa5MMAfBIDIXEwS9q2osJBOnzpFYWHDqXbt2tTjxx8pce/eMoeolPxKBEAAECV1Itwy/CzgigRAJDRZ4YpeQQ5OnzpJgwYOpLrBwdSieXOaM3s23biexr33JDNAHuTl0skTx7m1kbF+bNMnLgzwBwEgCszLNsVnIS2OHT1C06ZFU48fu1Pv3r1p7JgxtGXLZkrXaZJAmQGCq7BsvHfFzwKuSADEQqZpZ1AxMMTHr6LevXpRcHCw43nja9euoazMDEN+SQMgNjZZmcc8Fwb4gwAQmYvJwn1jl9KyK6e2bd1CXbp0pmrVqlHHjh1p+bJllJ2VaQg0SgMaAAFASteDNH/zs4ArEgCxsMlKU9R/aswm+cvMyKBFixZS82bNKDAwgLp06UIJCXsoLzfH0IkWXbUEQAAQ15qQ4j0XBviDABAAxJBf+2xmXDYj7IH9+5w39XXu/D0NGTyYVsbF0YUL573ep2H0YJYZIGxSyYz76Ybk1ei8oH2NYOdnAVckAAKA6Go0DAzhYWHUrFkzx/QhI0aE0/59+0w5LKXGfGQGiBodsKxGwzbIPxIT9/KNSy4M8AcBIAYVgF0GJrviZ/OmjRQeHkYhTZtSyxYtKHrqVEo+eNDjMzmsoA0AYk3jtEJtWGEb2DNrfuzeXf1TMflZwBUJgAAgqn7psMMj165eoY0bNlCXzp0pKDCQQkNDacb06XTzxnVVbflzoMoMEHauyYwLEfyZP9nXXViQ77gisUqVKhQbG0MPi4uVjS0uDPAHASAAiNfCZFdLFRUWOJ7UFxU1gRo2bEi1a9VyFHdSUqLj5LeIg1lmgLC9wtiYFV7zKmLO7LbN7Dzi4UPJ1KB+fcdNtMVFhb5zys8CrkgugMydO4fYKzp6Kl4Sa9C/X1/6pFUrev+996jCyy9Tq5YtHYU8efIk4fPeoEF96t27l/D9cDcGx0dEOO6jcfcdPhPPsyZERlKtmjUdE4WuWR3vHSJcGOAP4gLI8ePHiL1SDh/CCxoIWQNftm5N8+fNE3LbfY27XTt3UMS4sVL2zVffZfyeXcnI9kI+/eQTOnf2jPgAsduuJPor3wlX2Q9hsRsyUbfi1+2ePbupYcMPKWz4cGLnH33mlH9ngiuSaw/EZydwXsF3oqGRXzWSGSDsgVIHD+z3q77wCG3wYhdBDB8+jKpUeZdWrFjuzCU7D7J71y76qUcPx6wNTOcTJ45Tq1atHCfbuSigIQgAgZE7i9NOg15mgNgpj7L2tVu3rjR3zpxyV9Oxi1oYRPbuTaCQkBBKTj5I06dPp3nz5tLehD0aUMAXCoAAIAAIasCWNWBl+LBLsX1tX53atWnZr7+WnQKIjwPcUQAIzMNnofoqZBG/xx6ItkMsIuZctm3u1PE72rRxQ9nxy40CvkAARHCAsF3a1atWUsGDvLKFZOF+JbHJE3Oy/bq9MgOETYG/adNGv+orm1lbrT+PHxZTy5YtadTIkWXzzMcB7igAxMJGq6Rohw0eRNu3bilbRBbv0420a/T5p59QyqFkv223zADBA6Xk3rtiPxrj4uJo6pQpjlmt2furV6/8cSiLGwV8gQCIxc3WE0Tu3b1D7b/5mhJ27/KbCXvaNiWfsz2QwQMH0MTI8cTuuFUSo+cyAIjcJqtnrVilrePHjlL/fv0oNibGcYf6+XNnHTf4Lpg/nxL3JgAgVkmU1beDHa5q0vBD2r5NrD0PV13ZL6d+vXtRfFwsAKLjDxnsgcgJR3Zp7/r168pM1b8vKZGup6X9b/zw7UhwR2EPRMeB62qQRr2fM+sX6tmju19+uevdJ/bcim9C25a9ksSEnMi8B5KZcZ/WrlnzP1MxQU+96wLtcUKQGwV8gQCIYIMrJyuTAqu+T/fu3JbGIAb260uXL14wtT8yAwTmy2m+gnmB2zzzcYA7CgARrGgmRIyjoYMG6mq27IqO40ePUPKB/eVet2/d1HVd7oo+MWEPtf+6neHrKb1uAMTGJivYmC9dtz7/5kYBXyAAIlAxsZPNDeoFU9qVVF3NtnvXzvSPp55wvJ575ml65ukn6ekn/07s71m/zNB1Xe4GAJsu/t133jJ1rwoAAUDc1aLwn/FxgDsKABEIIIX5Dyiw6ge6GvqeXTupd8+fHCeyJ0RGOE7QLV2ymIYOGeT4uzA/X9f1eRqgnTt2oMPJB01ZF9sGmQGSl5tDyQcPmKalp5zicz9AmhsFfIEAiEAAOXr4EP3QtbOuxnDh3Fl6VFxE69espiULFzja/nXpEho/bqyu6/FlJmxPZ86smaatU2aA4CosPxi3VXyEjwPcUQCIVRKvYDvYtAWjR44wxGR/XbKYNm9Y52g7avw4ihw/zpD1eALJru2/0dDBg0xbJwBiY5NVMNY81anlP+dGAV8gACJQMS1dvIhWLFtqiMmGDx9GaVf/OLcybPBAmhQVach6PA3AM6dPUatmIaatEwABQDzVotCf83GAOwoAEQggM6dPo62bN+lusvkP8qhGQHUqLPjjfMeosOEUNnwoscNbq2JjdF+fuwHK7qqtH1zHlHWx9csOkNLPkHCnNz6TFKDcKOALBEAAEJo/dw4FBVRz3pjIoPHiC/+k9yq/Q+fPnOY29fR7dykl+SAV5D/w2QYAop+hsanA2Yl0QEI/TYXRko8D3FEACABCcTEraHLUBKfhPMjLpbatP6dZM6Y7oaJ2AC1bspiaNm7oaOOjRg3p4vlzzvbdtQWA2NDsBBp77mrWkp9xo4AvEAARqIiNOoSl50Bg81stWjCf3ni1Et26cd0BjckTIimg2gd083qpOXtcdAdAABA969C2bfFxgDsKAHExMisXnggAYY/bfOuN16jtV1869zjY4RR2c+KQgQOcn7nqDIDoBxB2LuuKzjebuuYL7/XLl65acqOALxAAAUA8mjpPYV+5dJGe+NtfqW+vns52maExqDRt3MjjITEARD9Dwn0g+mnJMwb8GsPHAe4oAAQAcRq9HoW/e+cOB0CiIiOc7bK9kqBqH9A7/36D2N307tYDgOhnegCIflq6q1VLf8aNAr5AAAQAcWvovINky+aNDoDMnTPb2W4JQF5/paLHR+8CIPqZHgCin5a848BvcXwc4I4CQCQCCPt136vHj9Sxw7ceX9v+fPztujWric0/pfR15fIlJxC8DQ42txY7hDVtymTn8uwQVpW336IP3q1MRX/ea+LaBgCin+kBIPpp6Vqnln/PjQK+QABEIoCoKe6sjPt09fIlxS82Y66S9tkDothMvr1/+tG5PLtR8aUXnqfQUifWXdsCQPQzPXbfzbmzZ5z6u2qN9/ppbTkt+TjAHQWA2BQgRhU+u4mt3n9qU6sWzZ1PGbxw9oxjr2R+qcNarusHQCQ2NYHGmGtdCveeGwV8gQCIQMVthct4k/Ym0MdNGtPqlXEef+HGLF/muJOd7Y2wATgqPIxqBQUQOxfiaUACIACIp9rA5ypqg48D3FEACADi0dTdDdwJ4yOo8ttv0Uv/et55o6C75WJXLKeO7b+lsaN/pkH9+9ENLzcRsngARIVJ+KhZdjNncXGRqry6yyE+0y8npmnJjQK+QADEx2A0LfEKtsMKeyBMD3aM/T+1alJeTrZuJgWA6GdWbM9vdfwq3XJjpTGAbfFRJ3wc4I4CQBQYt1WK1kiA3L55g36ZHk0TIyNoX+Jej+ZT8CCPIsaOobgVyz0uw6MXAOLDGFTUKa7C0k9Lnlr2aww3CvgCARAVA9OvhfH7YzIKINmZGVSrRiBVevlFxxVU7DLchfPnuQXExvXr6NTxY26/06IPAKKf6QEg+mmppab9EsvHAe4oAAQAoajxETRv9izKysygMydPUI3AAKr81pu6Q8LbgAJA9DM9AEQ/Lb3VrCW/40YBXyAAIhhANm1Yr7uxL120oMwcVcdSDjsuu72WquzmQT0GErtvgV3+q0dbStqQ+YFS7IfApo0bTdNSid5YxiSo8XGAOwoAEQggy5cuoV+mTTXcGNhNhi+/+ILh6yltKvuT9lLrzz41bZ0yA6S0rvjbJOO2io9wo4AvEACxSuIVbAebJmT40CGGm+zqVSupY4f2hq+ntLmxB1CxS4RLf2bk3wCIzYxVwfgyst5Ma5uPA9xRAIhAhXXn1k0K/aq1oSb7+NFD+qRFc9q2ZbOh63EdUCPDh1Ps8mWmrRMAAUBca1CK99wo4AsEQAQCCJsmpE7NIMpIv2eI0bL5roYPGUxTJ090TkNixqB6VFxEQdWq0vVrVw3pl7s+yAwQNvfYqVMnTdPSnb74zE+A5uMAdxQAIhBA2KDs17sXTZ4Ypbs5PCwuosH9+xG7TNfswX/y+DFq2qSRqdCSGSC4CstP5m0FL+FGAV8gAGKFpKvYhpTkg1Q/+D9e55XiAUB8XCyx6UfYNBgsnv1/6OAB+v3RQ8OBwu45WThvruHrKa0LAGJjk1Ux3krXjBB/83GAOwoAEbCYPm3VgrZs0u8yTbYHEFy7Jo39eSSNGz3K8erRvRsNGzzIFFPv0qmjE1xmDVIABAAxq9ZMXQ83CvgCARABAZJyKJmCqlelzIz7mg2eTWHCHjf7SoWXyr2OHErW3L6vwbM/KZEG9u9r+HpctwMAAUBca0KK93wc4I4CQAQECCv09WvX0DehbYgd79ZS+OzE/IO8XLcvLe36imWHyNjz0z9r1ZLS797R1Adf63L3vcwAYU+AvJJ62XRN3emMz0wGNTcK+AIBEEEBwgbmmZMnqWf3H4gZhkgDlV0qvHjBfPo6tC09yM3xy7bLDBCRagHbqjNg+DjAHQWACAwQNvhSDidT/z69KTc7yy9GzGMA7IFT7MZBNi08T7weMQCIzsYl+DjSo6Ys0QY3CvgCARAJCr/Qj0bMM2iKLLDHJDNA2OHBxw+L/QZnnppAjE5A5+MAdxQAIgFAMPjUDz6ZAcLOi8Xo/LwW1Jj6GvOLZtwo4AsEQAAQW/5SlR0gy5eZNy2MX4wS49b9uOXjAHcUAIJCdF+IkusCgAjyi1ryOtQdvtwo4AsEQFCgAIhkNYCpTGwMRz4OcEcBIJKZh+6/aCTVR+Y9kOysTNqxfbstfxjYvv65UcAXCIBIapC2H0g+8iozQJB77IHw4UB9FADiw2gwGOUcjACInHm1/XhVzwBNEQAIAGLLQx0ACAAiJWw04UB9MAACgAAgktVATnYWJezZY8u8SgkFNfWpngGaIgAQNcnBstKYksx7ILgKy8Z7V5pwoD4YAAEUpIGCml+fQ4YMpgMH9kvZdwAEAFGPAr4IAAQAkdJEfcEEALGxyco85vk4wB3FBxDu1SEQCkABKAAFZFEAAJElk+gHFIACUMBkBQAQkwXH6qAAFIACsigAgMiSSfQDCkABKGCyAgCIyYJjdVAACkABWRQAQGTJJPoBBaAAFDBZAQDEZMGxOigABaCALAoAILJkEv2AAlAACpisAABisuBYHRSAAlBAFgUAEFkyiX5AASgABUxWAAAxWXCsDgpAASggiwIAiCyZRD+gABSAAiYrAICYLDhWBwWgABSQRQEARJZMoh9QAApAAZMVAEBMFhyrgwJQAArIogAAIksm0Q8oAAWggMkKACAmC47VQQEoAAVkUQAAkSWT6AcUgAJQwGQFABCTBcfqoAAUgAKyKACAyJJJ9AMKQAEoYLICAIjJgmN1UAAKQAFZFABAZMkk+gEFoAAUMFkBAMRkwbE6KAAFoIAsCgAgsmQS/YACUAAKmKwAAGKy4FgdFIACUEAWBQAQWTKJfkABKAAFTFYAADFZcKwOCkABKCCLAv8PyyTVfJnblvsAAAAASUVORK5CYII="> <em><strong>A1</strong></em><em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p><strong>Note:<em> A1</em></strong> for correct domain and correct range and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-intercept at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>1</mn></math><br><em><strong> A1</strong></em> for asymptotic behaviour <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>→</mo><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></math><br><em><strong> A1</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></math><br> Coordinates are not required. <br> Do not accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>1</mn><mo>.</mo><mn>57</mn></math> or other inexact values.</p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">f.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">f.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cot}}\,2\theta = \frac{{1 - {\text{ta}}{{\text{n}}^2}\,\theta }}{{2\,{\text{tan}}\,\theta }}">
<mrow>
<mtext>cot</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mi>θ</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mrow>
<mtext>ta</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
</mrow>
<mrow>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>tan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
</mrow>
</mfrac>
</math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Verify that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = {\text{tan}}\,\theta ">
<mi>x</mi>
<mo>=</mo>
<mrow>
<mtext>tan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = - \,{\text{cot}}\,\theta ">
<mi>x</mi>
<mo>=</mo>
<mo>−</mo>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cot</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
</math></span> satisfy the equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^2} + \left( {2\,{\text{cot}}\,2\theta } \right)x - 1 = 0">
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cot</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mi>θ</mi>
</mrow>
<mo>)</mo>
</mrow>
<mi>x</mi>
<mo>−</mo>
<mn>1</mn>
<mo>=</mo>
<mn>0</mn>
</math></span>.</p>
<div class="marks">[7]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, or otherwise, show that the exact value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{tan}}\frac{\pi }{{12}} = 2 - \sqrt 3 ">
<mrow>
<mtext>tan</mtext>
</mrow>
<mfrac>
<mi>π</mi>
<mrow>
<mn>12</mn>
</mrow>
</mfrac>
<mo>=</mo>
<mn>2</mn>
<mo>−</mo>
<msqrt>
<mn>3</mn>
</msqrt>
</math></span>.</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>Using the results from parts (b) and (c) find the exact value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{tan}}\frac{\pi }{{24}} - {\text{cot}}\frac{\pi }{{24}}">
<mrow>
<mtext>tan</mtext>
</mrow>
<mfrac>
<mi>π</mi>
<mrow>
<mn>24</mn>
</mrow>
</mfrac>
<mo>−</mo>
<mrow>
<mtext>cot</mtext>
</mrow>
<mfrac>
<mi>π</mi>
<mrow>
<mn>24</mn>
</mrow>
</mfrac>
</math></span>.</p>
<p>Give your answer in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a + b\sqrt 3 ">
<mi>a</mi>
<mo>+</mo>
<mi>b</mi>
<msqrt>
<mn>3</mn>
</msqrt>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b \in \mathbb{Z}">
<mi>b</mi>
<mo>∈</mo>
<mrow>
<mi mathvariant="double-struck">Z</mi>
</mrow>
</math></span>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>stating the relationship between <span style="display: inline !important;float: none;background-color: #ffffff;color: #000000;font-family: Verdana,Arial,Helvetica,sans-serif;font-size: 14px;font-style: normal;font-variant: normal;font-weight: 400;letter-spacing: normal;text-align: left;text-decoration: none;text-indent: 0px;white-space: normal;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\cot ">
<mi>cot</mi>
</math></span></span> and <span style="background-color: #ffffff;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\tan ">
<mi>tan</mi>
</math></span></span> and stating the identity for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{tan}}\,2\theta ">
<mrow>
<mtext>tan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mi>θ</mi>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cot}}\,2\theta = \frac{1}{{{\text{tan}}\,2\theta }}">
<mrow>
<mtext>cot</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mi>θ</mi>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mrow>
<mtext>tan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mi>θ</mi>
</mrow>
</mfrac>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{tan}}\,2\theta = \frac{{2\,{\text{tan}}\,\theta }}{{1 - {\text{ta}}{{\text{n}}^2}\,\theta }}">
<mrow>
<mtext>tan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mi>θ</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>tan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
</mrow>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mrow>
<mtext>ta</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
</mrow>
</mfrac>
</math></span></p>
<p>⇒ <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cot}}\,2\theta = \frac{{1 - {\text{ta}}{{\text{n}}^2}\,\theta }}{{2\,{\text{tan}}\,\theta }}">
<mrow>
<mtext>cot</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mi>θ</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mrow>
<mtext>ta</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
</mrow>
<mrow>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>tan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
</mrow>
</mfrac>
</math></span> <em><strong>AG</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><em><strong>METHOD 1</strong></em></p>
<p>attempting to substitute <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{tan}}\,\theta ">
<mrow>
<mtext>tan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
</math></span> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> and using the result from (a) <em><strong>M1</strong></em></p>
<p>LHS = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ta}}{{\text{n}}^2}\,\theta + 2\,{\text{tan}}\,\theta \left( {\frac{{1 - {\text{ta}}{{\text{n}}^2}\,\theta }}{{2\,{\text{tan}}\,\theta }}} \right) - 1">
<mrow>
<mtext>ta</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
<mo>+</mo>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>tan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mrow>
<mtext>ta</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
</mrow>
<mrow>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>tan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mn>1</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ta}}{{\text{n}}^2}\,\theta + 1 - {\text{ta}}{{\text{n}}^2}\,\theta - 1 = 0">
<mrow>
<mtext>ta</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
<mo>+</mo>
<mn>1</mn>
<mo>−</mo>
<mrow>
<mtext>ta</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
<mo>−</mo>
<mn>1</mn>
<mo>=</mo>
<mn>0</mn>
</math></span>(= RHS) <em><strong>A1</strong></em></p>
<p>so <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = {\text{tan}}\,\theta ">
<mi>x</mi>
<mo>=</mo>
<mrow>
<mtext>tan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
</math></span> satisfies the equation <em><strong>AG</strong></em></p>
<p>attempting to substitute <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - \,{\text{cot}}\,\theta ">
<mo>−</mo>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cot</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
</math></span> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> and using the result from (a) <em><strong>M1</strong></em></p>
<p>LHS = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{co}}{{\text{t}}^2}\,\theta - 2\,{\text{cot}}\,\theta \left( {\frac{{1 - {\text{ta}}{{\text{n}}^2}\,\theta }}{{2\,{\text{tan}}\,\theta }}} \right) - 1">
<mrow>
<mtext>co</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>t</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
<mo>−</mo>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cot</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mrow>
<mtext>ta</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
</mrow>
<mrow>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>tan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mn>1</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{{{\text{ta}}{{\text{n}}^2}\,\theta }} - \left( {\frac{{1 - {\text{ta}}{{\text{n}}^2}\,\theta }}{{\,{\text{ta}}{{\text{n}}^2}\,\theta }}} \right) - 1">
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mrow>
<mtext>ta</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
</mrow>
</mfrac>
<mo>−</mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mrow>
<mtext>ta</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
</mrow>
<mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>ta</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mn>1</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \frac{1}{{{\text{ta}}{{\text{n}}^2}\,\theta }} - \frac{1}{{{\text{ta}}{{\text{n}}^2}\,\theta }} + 1 - 1 = 0">
<mfrac>
<mn>1</mn>
<mrow>
<mrow>
<mtext>ta</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
</mrow>
</mfrac>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mrow>
<mtext>ta</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
</mrow>
</mfrac>
<mo>+</mo>
<mn>1</mn>
<mo>−</mo>
<mn>1</mn>
<mo>=</mo>
<mn>0</mn>
</math></span>(= RHS) <em><strong>A1</strong></em></p>
<p>so <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = - \,{\text{cot}}\,\theta ">
<mi>x</mi>
<mo>=</mo>
<mo>−</mo>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cot</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
</math></span> satisfies the equation <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>METHOD 2</strong></em></p>
<p>let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\alpha = {\text{tan}}\,\theta ">
<mi>α</mi>
<mo>=</mo>
<mrow>
<mtext>tan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\beta = - \,{\text{cot}}\,\theta ">
<mi>β</mi>
<mo>=</mo>
<mo>−</mo>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cot</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
</math></span></p>
<p>attempting to find the sum of roots <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\alpha + \beta = {\text{tan}}\,\theta - \frac{1}{{{\text{tan}}\,\theta }}">
<mi>α</mi>
<mo>+</mo>
<mi>β</mi>
<mo>=</mo>
<mrow>
<mtext>tan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mrow>
<mtext>tan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
</mrow>
</mfrac>
</math></span></p>
<p> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{{\text{ta}}{{\text{n}}^2}\,\theta - 1}}{{{\text{tan}}\,\theta }}">
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<mtext>ta</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mrow>
<mrow>
<mtext>tan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
</mrow>
</mfrac>
</math></span><span style="display: inline !important;float: none;background-color: #ffffff;color: #000000;font-family: Verdana,Arial,Helvetica,sans-serif;font-size: 14px;font-style: normal;font-variant: normal;font-weight: 400;letter-spacing: normal;text-align: left;text-decoration: none;text-indent: 0px;white-space: normal;"> </span><em style="color: #000000;font-family: Verdana,Arial,Helvetica,sans-serif;font-size: 14px;font-style: italic;font-variant: normal;font-weight: 400;letter-spacing: normal;text-align: left;text-decoration: none;text-indent: 0px;white-space: normal;"><strong>A1</strong></em></p>
<p> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = - 2\,{\text{cot}}\,2\theta ">
<mo>=</mo>
<mo>−</mo>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cot</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mi>θ</mi>
</math></span> (from part (a)) <em><strong>A1</strong></em></p>
<p>attempting to find the product of roots <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\alpha \beta = {\text{tan}}\,\theta \times \left( { - \,{\text{cot}}\,\theta } \right)">
<mi>α</mi>
<mi>β</mi>
<mo>=</mo>
<mrow>
<mtext>tan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
<mo>×</mo>
<mrow>
<mo>(</mo>
<mrow>
<mo>−</mo>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cot</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p>= −1 <em><strong>A1</strong></em></p>
<p>the coefficient of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> and the constant term in the quadratic are <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{2\,{\text{cot}}\,2\theta }">
<mrow>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cot</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mi>θ</mi>
</mrow>
</math></span> and −1 respectively <em><strong>R1</strong></em></p>
<p>hence the two roots are <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\alpha = {\text{tan}}\,\theta ">
<mi>α</mi>
<mo>=</mo>
<mrow>
<mtext>tan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\beta = - \,{\text{cot}}\,\theta ">
<mi>β</mi>
<mo>=</mo>
<mo>−</mo>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cot</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
</math></span> <em><strong>AG</strong></em></p>
<p><em><strong>[7 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><em><strong>METHOD 1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = {\text{tan}}\frac{\pi }{{12}}">
<mi>x</mi>
<mo>=</mo>
<mrow>
<mtext>tan</mtext>
</mrow>
<mfrac>
<mi>π</mi>
<mrow>
<mn>12</mn>
</mrow>
</mfrac>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = - {\text{cot}}\frac{\pi }{{12}}">
<mi>x</mi>
<mo>=</mo>
<mo>−</mo>
<mrow>
<mtext>cot</mtext>
</mrow>
<mfrac>
<mi>π</mi>
<mrow>
<mn>12</mn>
</mrow>
</mfrac>
</math></span> are roots of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^2} + \left( {2\,{\text{cot}}\frac{\pi }{{6}}} \right)x - 1 = 0">
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cot</mtext>
</mrow>
<mfrac>
<mi>π</mi>
<mrow>
<mn>6</mn>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mi>x</mi>
<mo>−</mo>
<mn>1</mn>
<mo>=</mo>
<mn>0</mn>
</math></span> <em><strong>R1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>R1</strong> </em>if only <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = {\text{tan}}\frac{\pi }{{12}}">
<mi>x</mi>
<mo>=</mo>
<mrow>
<mtext>tan</mtext>
</mrow>
<mfrac>
<mi>π</mi>
<mrow>
<mn>12</mn>
</mrow>
</mfrac>
</math></span> is stated as a root of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^2} + \left( {2\,{\text{cot}}\frac{\pi }{{6}}} \right)x - 1 = 0">
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cot</mtext>
</mrow>
<mfrac>
<mi>π</mi>
<mrow>
<mn>6</mn>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mi>x</mi>
<mo>−</mo>
<mn>1</mn>
<mo>=</mo>
<mn>0</mn>
</math></span>.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^2} + 2\sqrt 3 x - 1 = 0">
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>2</mn>
<msqrt>
<mn>3</mn>
</msqrt>
<mi>x</mi>
<mo>−</mo>
<mn>1</mn>
<mo>=</mo>
<mn>0</mn>
</math></span> <em><strong>A1</strong></em></p>
<p>attempting to solve <strong>their</strong> quadratic equation <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = - \sqrt 3 \pm 2">
<mi>x</mi>
<mo>=</mo>
<mo>−</mo>
<msqrt>
<mn>3</mn>
</msqrt>
<mo>±</mo>
<mn>2</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{tan}}\frac{\pi }{{12}} > 0">
<mrow>
<mtext>tan</mtext>
</mrow>
<mfrac>
<mi>π</mi>
<mrow>
<mn>12</mn>
</mrow>
</mfrac>
<mo>></mo>
<mn>0</mn>
</math></span> (<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - {\text{cot}}\frac{\pi }{{12}} < 0">
<mo>−</mo>
<mrow>
<mtext>cot</mtext>
</mrow>
<mfrac>
<mi>π</mi>
<mrow>
<mn>12</mn>
</mrow>
</mfrac>
<mo><</mo>
<mn>0</mn>
</math></span>) <em><strong>R1</strong></em></p>
<p>so <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{tan}}\frac{\pi }{{12}} = 2 - \sqrt 3 ">
<mrow>
<mtext>tan</mtext>
</mrow>
<mfrac>
<mi>π</mi>
<mrow>
<mn>12</mn>
</mrow>
</mfrac>
<mo>=</mo>
<mn>2</mn>
<mo>−</mo>
<msqrt>
<mn>3</mn>
</msqrt>
</math></span> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>METHOD 2</strong></em></p>
<p>attempting to substitute <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\theta = \frac{\pi }{{12}}">
<mi>θ</mi>
<mo>=</mo>
<mfrac>
<mi>π</mi>
<mrow>
<mn>12</mn>
</mrow>
</mfrac>
</math></span> into the identity for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{tan}}\,2\theta ">
<mrow>
<mtext>tan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mi>θ</mi>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{tan}}\frac{\pi }{6} = \frac{{2\,{\text{tan}}\frac{\pi }{{12}}}}{{1 - {\text{ta}}{{\text{n}}^2}\frac{\pi }{{12}}}}">
<mrow>
<mtext>tan</mtext>
</mrow>
<mfrac>
<mi>π</mi>
<mn>6</mn>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>tan</mtext>
</mrow>
<mfrac>
<mi>π</mi>
<mrow>
<mn>12</mn>
</mrow>
</mfrac>
</mrow>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mrow>
<mtext>ta</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mfrac>
<mi>π</mi>
<mrow>
<mn>12</mn>
</mrow>
</mfrac>
</mrow>
</mfrac>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ta}}{{\text{n}}^2}\frac{\pi }{{12}} + 2\sqrt 3 \,{\text{tan}}\frac{\pi }{{12}} - 1 = 0">
<mrow>
<mtext>ta</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mfrac>
<mi>π</mi>
<mrow>
<mn>12</mn>
</mrow>
</mfrac>
<mo>+</mo>
<mn>2</mn>
<msqrt>
<mn>3</mn>
</msqrt>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>tan</mtext>
</mrow>
<mfrac>
<mi>π</mi>
<mrow>
<mn>12</mn>
</mrow>
</mfrac>
<mo>−</mo>
<mn>1</mn>
<mo>=</mo>
<mn>0</mn>
</math></span> <em><strong>A1</strong></em></p>
<p>attempting to solve <strong>their </strong>quadratic equation <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{tan}}\frac{\pi }{{12}} = - \sqrt 3 \pm 2">
<mrow>
<mtext>tan</mtext>
</mrow>
<mfrac>
<mi>π</mi>
<mrow>
<mn>12</mn>
</mrow>
</mfrac>
<mo>=</mo>
<mo>−</mo>
<msqrt>
<mn>3</mn>
</msqrt>
<mo>±</mo>
<mn>2</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{tan}}\frac{\pi }{{12}} > 0">
<mrow>
<mtext>tan</mtext>
</mrow>
<mfrac>
<mi>π</mi>
<mrow>
<mn>12</mn>
</mrow>
</mfrac>
<mo>></mo>
<mn>0</mn>
</math></span> <em><strong>R1</strong></em></p>
<p>so <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{tan}}\frac{\pi }{{12}} = 2 - \sqrt 3 ">
<mrow>
<mtext>tan</mtext>
</mrow>
<mfrac>
<mi>π</mi>
<mrow>
<mn>12</mn>
</mrow>
</mfrac>
<mo>=</mo>
<mn>2</mn>
<mo>−</mo>
<msqrt>
<mn>3</mn>
</msqrt>
</math></span> <em><strong>AG</strong></em></p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{tan}}\frac{\pi }{{24}} - {\text{cot}}\frac{\pi }{{24}}">
<mrow>
<mtext>tan</mtext>
</mrow>
<mfrac>
<mi>π</mi>
<mrow>
<mn>24</mn>
</mrow>
</mfrac>
<mo>−</mo>
<mrow>
<mtext>cot</mtext>
</mrow>
<mfrac>
<mi>π</mi>
<mrow>
<mn>24</mn>
</mrow>
</mfrac>
</math></span> is the sum of the roots of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^2} + \left( {2\,{\text{cot}}\frac{\pi }{{12}}} \right)x - 1 = 0">
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cot</mtext>
</mrow>
<mfrac>
<mi>π</mi>
<mrow>
<mn>12</mn>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mi>x</mi>
<mo>−</mo>
<mn>1</mn>
<mo>=</mo>
<mn>0</mn>
</math></span> <em><strong>R1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{tan}}\frac{\pi }{{24}} - {\text{cot}}\frac{\pi }{{24}} = - 2\,{\text{cot}}\frac{\pi }{{12}}">
<mrow>
<mtext>tan</mtext>
</mrow>
<mfrac>
<mi>π</mi>
<mrow>
<mn>24</mn>
</mrow>
</mfrac>
<mo>−</mo>
<mrow>
<mtext>cot</mtext>
</mrow>
<mfrac>
<mi>π</mi>
<mrow>
<mn>24</mn>
</mrow>
</mfrac>
<mo>=</mo>
<mo>−</mo>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cot</mtext>
</mrow>
<mfrac>
<mi>π</mi>
<mrow>
<mn>12</mn>
</mrow>
</mfrac>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{ - 2}}{{2 - \sqrt 3 }}">
<mo>=</mo>
<mfrac>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
<mrow>
<mn>2</mn>
<mo>−</mo>
<msqrt>
<mn>3</mn>
</msqrt>
</mrow>
</mfrac>
</math></span> <em><strong>A1</strong></em></p>
<p>attempting to rationalise <strong>their</strong> denominator <em><strong> (M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = - 4 - 2\sqrt 3 ">
<mo>=</mo>
<mo>−</mo>
<mn>4</mn>
<mo>−</mo>
<mn>2</mn>
<msqrt>
<mn>3</mn>
</msqrt>
</math></span> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>A continuous random variable <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> has a probability density function given by</p>
<p style="padding-left: 180px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfenced open="{" close><mtable><mtr><mtd><mtext>arccos</mtext><mo> </mo><mi>x</mi><mo> </mo></mtd><mtd><mn>0</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mn>1</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mtext>otherwise</mtext></mtd></mtr></mtable></mfenced></math></p>
<p>The median of this distribution is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</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>Given that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mfenced><mrow><mfenced open="|" close="|"><mrow><mi>X</mi><mo>-</mo><mi>m</mi></mrow></mfenced><mo>≤</mo><mi>a</mi></mrow></mfenced><mo>=</mo><mn>0</mn><mo>.</mo><mn>3</mn></math>, determine the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>recognises that <math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∫</mo><mn>0</mn><mi>m</mi></munderover><mtext>arccos</mtext><mo> </mo><mi>x</mi><mo> </mo><mo>d</mo><mi>x</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>5</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo> </mo><mtext>arccos</mtext><mo> </mo><mi>m</mi><mo>-</mo><msqrt><mn>1</mn><mo>-</mo><msup><mi>m</mi><mn>2</mn></msup></msqrt><mo>-</mo><mfenced><mrow><mn>0</mn><mo>-</mo><msqrt><mn>1</mn></msqrt></mrow></mfenced><mo>=</mo><mn>0</mn><mo>.</mo><mn>5</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>360034</mn><mo>…</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>360</mn></math> <em><strong>A1</strong></em></p>
<p><br><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>attempts to find at least one endpoint (limit) both in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math> (or their <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math>) and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mrow><mi>m</mi><mo>-</mo><mi>a</mi><mo>≤</mo><mi>X</mi><mo>≤</mo><mi>m</mi><mo>+</mo><mi>a</mi></mrow></mfenced><mo>=</mo><mn>0</mn><mo>.</mo><mn>3</mn></math> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∫</mo><mrow><mn>0</mn><mo>.</mo><mn>360034</mn><mo>…</mo><mo>-</mo><mi>a</mi></mrow><mrow><mn>0</mn><mo>.</mo><mn>360034</mn><mo>…</mo><mo>+</mo><mi>a</mi></mrow></munderover><mtext>arccos</mtext><mo> </mo><mi>x</mi><mo> </mo><mo>d</mo><mi>x</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>3</mn></math> <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"><munderover><mo>∫</mo><mrow><mi>m</mi><mo>-</mo><mi>a</mi></mrow><mrow><mi>m</mi><mo>+</mo><mi>a</mi></mrow></munderover><mtext>arccos</mtext><mo> </mo><mi>x</mi><mo> </mo><mo>d</mo><mi>x</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>3</mn></math>.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mfenced open="[" close="]"><mrow><mi>x</mi><mo> </mo><mtext>arccos</mtext><mo> </mo><mi>x</mi><mo>-</mo><msqrt><mn>1</mn><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup></msqrt></mrow></mfenced><mrow><mn>0</mn><mo>.</mo><mn>360034</mn><mo>…</mo><mo>-</mo><mi>a</mi></mrow><mrow><mn>0</mn><mo>.</mo><mn>360034</mn><mo>…</mo><mo>+</mo><mi>a</mi></mrow></msubsup></math></p>
<p>attempts to solve their equation for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> <em><strong>(M1)</strong></em></p>
<p><strong><br>Note:</strong> The above <em><strong>(M1)</strong></em> is dependent on the first <em><strong>(M1)</strong></em>.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>124861</mn><mo>…</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>125</mn></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"><munderover><mo>∫</mo><mrow><mo>-</mo><mi>a</mi></mrow><mi>a</mi></munderover><mtext>arccos </mtext><menclose notation="left"><mi>x</mi><mo>-</mo><mn>0</mn><mo>.</mo><mn>360034</mn><mo>…</mo><menclose notation="left"><mo> </mo><mo>d</mo><mi>x</mi><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mn>0</mn><mo>.</mo><mn>3</mn></mrow></mfenced></menclose></menclose></math> <em><strong>(M1)(A1)</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Only award <em><strong>(M1)</strong></em> if at least one limit has been translated correctly.</p>
<p><strong>Note:</strong> Award <em><strong>(M1)(A1)</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∫</mo><mrow><mo>-</mo><mi>a</mi></mrow><mi>a</mi></munderover><mtext>arccos </mtext><menclose notation="left"><mi>x</mi><mo>-</mo><mi>m</mi><menclose notation="left"><mo> </mo><mo>d</mo><mi>x</mi><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mn>0</mn><mo>.</mo><mn>3</mn></mrow></mfenced></menclose></menclose></math>.</p>
<p><br>attempts to solve their equation for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>124861</mn><mo>…</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>125</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>METHOD 3</strong></p>
<p><strong>EITHER </strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∫</mo><mrow><mo>-</mo><mi>a</mi></mrow><mi>a</mi></munderover><mtext>arccos </mtext><mfenced><mrow><mi>x</mi><mo>+</mo><mn>0</mn><mo>.</mo><mn>360034</mn><mo>…</mo></mrow></mfenced><mo> </mo><mo>d</mo><mi>x</mi><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mn>0</mn><mo>.</mo><mn>3</mn></mrow></mfenced></math> <em><strong>(M1)(A1)</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Only award <em><strong>(M1)</strong></em> if at least one limit has been translated correctly.</p>
<p><strong>Note:</strong> Award <em><strong>(M1)(A1)</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∫</mo><mrow><mo>-</mo><mi>a</mi></mrow><mi>a</mi></munderover><mtext>arccos </mtext><mfenced><mrow><mi>x</mi><mo>+</mo><mi>m</mi></mrow></mfenced><mo> </mo><mo>d</mo><mi>x</mi><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mn>0</mn><mo>.</mo><mn>3</mn></mrow></mfenced></math>.</p>
<p><br><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∫</mo><mrow><mn>2</mn><mfenced><mrow><mn>0</mn><mo>.</mo><mn>360034</mn><mo>…</mo></mrow></mfenced><mo>-</mo><mi>a</mi></mrow><mrow><mn>2</mn><mfenced><mrow><mn>0</mn><mo>.</mo><mn>360034</mn><mo>…</mo></mrow></mfenced><mo>+</mo><mi>a</mi></mrow></munderover><mtext>arccos </mtext><mfenced><mrow><mi>x</mi><mo>-</mo><mn>0</mn><mo>.</mo><mn>360034</mn><mo>…</mo></mrow></mfenced><mo> </mo><mo>d</mo><mi>x</mi><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mn>0</mn><mo>.</mo><mn>3</mn></mrow></mfenced></math> <em><strong>(M1)(A1)</strong></em></p>
<p><strong><br>Note:</strong> Only award <em><strong>(M1)</strong></em> if at least one limit has been translated correctly.</p>
<p><strong>Note:</strong> Award <em><strong>(M1)(A1)</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∫</mo><mrow><mn>2</mn><mi>m</mi><mo>-</mo><mi>a</mi></mrow><mrow><mn>2</mn><mi>m</mi><mo>+</mo><mi>a</mi></mrow></munderover><mtext>arccos </mtext><mfenced><mrow><mi>x</mi><mo>-</mo><mi>m</mi></mrow></mfenced><mo> </mo><mo>d</mo><mi>x</mi><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mn>0</mn><mo>.</mo><mn>3</mn></mrow></mfenced></math>.</p>
<p><br><strong>THEN</strong></p>
<p>attempts to solve their equation for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> <em><strong>(M1)</strong></em></p>
<p><strong><br>Note:</strong> The above <em><strong>(M1)</strong></em> is dependent on the first <em><strong>(M1)</strong></em>.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>124861</mn><mo>…</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>125</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The height of water, in metres, in Dungeness harbour is modelled by the function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>H</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mi>a</mi><mo> </mo><mi>sin</mi><mo>(</mo><mi>b</mi><mo>(</mo><mi>t</mi><mo>-</mo><mi>c</mi><mo>)</mo><mo>)</mo><mo>+</mo><mi>d</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> is the number of hours after midnight, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>,</mo><mo> </mo><mi>b</mi><mo>,</mo><mo> </mo><mi>c</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math> are constants, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>></mo><mn>0</mn><mo>,</mo><mo> </mo><mi>b</mi><mo>></mo><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>></mo><mn>0</mn></math>.</p>
<p>The following graph shows the height of the water for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>13</mn></math> hours, starting at midnight.</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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"></p>
<p>The first high tide occurs at <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>04</mn><mo>:</mo><mn>30</mn></math> and the next high tide occurs <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>12</mn></math> hours later. Throughout the day, the height of the water fluctuates between <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>.</mo><mn>2</mn><mo> </mo><mtext>m</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mo>.</mo><mn>8</mn><mo> </mo><mtext>m</mtext></math>.</p>
<p>All heights are given correct to one decimal place.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></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>Find the smallest possible value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></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>Find the height of the water at <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>12</mn><mo>:</mo><mn>00</mn></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the number of hours, over a 24-hour period, for which the tide is higher than <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn></math> metres.</p>
<div class="marks">[3]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>A fisherman notes that the water height at nearby Folkestone harbour follows the same sinusoidal pattern as that of Dungeness harbour, with the exception that high tides (and low tides) occur <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>50</mn></math> minutes earlier than at Dungeness.</p>
<p>Find a suitable equation that may be used to model the tidal height of water at Folkestone harbour.</p>
<div class="marks">[2]</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"><mn>12</mn><mo>=</mo><mfrac><mrow><mn>2</mn><mi mathvariant="normal">π</mi></mrow><mi>b</mi></mfrac></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>=</mo><mfrac><mrow><mn>2</mn><mi mathvariant="normal">π</mi></mrow><mn>12</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mfrac><mrow><mn>6</mn><mo>.</mo><mn>8</mn><mo>-</mo><mn>2</mn><mo>.</mo><mn>2</mn></mrow><mn>2</mn></mfrac></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mfrac><mrow><mtext>max</mtext><mo>-</mo><mtext>min</mtext></mrow><mn>2</mn></mfrac></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>2</mn><mo>.</mo><mn>3</mn><mo> </mo><mfenced><mtext>m</mtext></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo>=</mo><mfrac><mrow><mn>6</mn><mo>.</mo><mn>8</mn><mo>+</mo><mn>2</mn><mo>.</mo><mn>2</mn></mrow><mn>2</mn></mfrac></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo>=</mo><mfrac><mrow><mtext>max</mtext><mo>+</mo><mtext>min</mtext></mrow><mn>2</mn></mfrac></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>4</mn><mo>.</mo><mn>5</mn><mo> </mo><mfenced><mtext>m</mtext></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>substituting <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>4</mn><mo>.</mo><mn>5</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>H</mi><mo>=</mo><mn>6</mn><mo>.</mo><mn>8</mn></math> for example into their equation for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>H</mi></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mo>.</mo><mn>8</mn><mo>=</mo><mn>2</mn><mo>.</mo><mn>3</mn><mo> </mo><mi>sin</mi><mfenced><mrow><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac><mfenced><mrow><mn>4</mn><mo>.</mo><mn>5</mn><mo>-</mo><mi>c</mi></mrow></mfenced></mrow></mfenced><mo>+</mo><mn>4</mn><mo>.</mo><mn>5</mn></math></p>
<p>attempt to solve their equation <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>=</mo><mn>1</mn><mo>.</mo><mn>5</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>using horizontal translation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>12</mn><mn>4</mn></mfrac></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>.</mo><mn>5</mn><mo>-</mo><mi>c</mi><mo>=</mo><mn>3</mn></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>=</mo><mn>1</mn><mo>.</mo><mn>5</mn></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>H</mi><mo>'</mo><mfenced><mi>t</mi></mfenced><mo>=</mo><mfenced><mrow><mn>2</mn><mo>.</mo><mn>3</mn></mrow></mfenced><mfenced><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></mfenced><mi>cos</mi><mfenced><mrow><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac><mfenced><mrow><mi>t</mi><mo>-</mo><mi>c</mi></mrow></mfenced></mrow></mfenced></math> <em><strong>(A1)</strong></em></p>
<p>attempts to solve their <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>H</mi><mo>'</mo><mfenced><mrow><mn>4</mn><mo>.</mo><mn>5</mn></mrow></mfenced><mo>=</mo><mn>0</mn></math> for <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"><mfenced><mrow><mn>2</mn><mo>.</mo><mn>3</mn></mrow></mfenced><mfenced><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></mfenced><mi>cos</mi><mfenced><mrow><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac><mfenced><mrow><mn>4</mn><mo>.</mo><mn>5</mn><mo>-</mo><mi>c</mi></mrow></mfenced></mrow></mfenced><mo>=</mo><mn>0</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>=</mo><mn>1</mn><mo>.</mo><mn>5</mn></math> <em><strong>A1</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>attempt to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>H</mi></math> when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>12</mn></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn></math>, graphically or algebraically <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>H</mi><mo>=</mo><mn>2</mn><mo>.</mo><mn>87365</mn><mo>…</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>H</mi><mo>=</mo><mn>2</mn><mo>.</mo><mn>87</mn><mo> </mo><mfenced><mtext>m</mtext></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to solve <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mo>=</mo><mn>2</mn><mo>.</mo><mn>3</mn><mo> </mo><mi>sin</mi><mfenced><mrow><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac><mfenced><mrow><mi>t</mi><mo>-</mo><mn>1</mn><mo>.</mo><mn>5</mn></mrow></mfenced></mrow></mfenced><mo>+</mo><mn>4</mn><mo>.</mo><mn>5</mn></math> <em><strong>(M1)</strong></em></p>
<p>times are <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>1</mn><mo>.</mo><mn>91852</mn><mo>…</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>7</mn><mo>.</mo><mn>08147</mn><mo>…</mo><mo> </mo><mo>,</mo><mo> </mo><mfenced><mrow><mi>t</mi><mo>=</mo><mn>13</mn><mo>.</mo><mn>9185</mn><mo>…</mo><mo>,</mo><mo> </mo><mi>t</mi><mo>=</mo><mn>19</mn><mo>.</mo><mn>0814</mn><mo>…</mo></mrow></mfenced></math> <em><strong>(A1)</strong></em></p>
<p>total time is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>×</mo><mfenced><mrow><mn>7</mn><mo>.</mo><mn>081</mn><mo>…</mo><mo>-</mo><mn>1</mn><mo>.</mo><mn>919</mn><mo>…</mo></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn><mo>.</mo><mn>3258</mn><mo>…</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>10</mn><mo>.</mo><mn>3</mn></math> (hours) <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn></math>.</p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>substitutes <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mfrac><mn>11</mn><mn>3</mn></mfrac></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>H</mi><mo>=</mo><mn>6</mn><mo>.</mo><mn>8</mn></math> into their equation for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>H</mi></math> and attempts to solve for <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"><mn>6</mn><mo>.</mo><mn>8</mn><mo>=</mo><mn>2</mn><mo>.</mo><mn>3</mn><mo> </mo><mi>sin</mi><mfenced><mrow><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac><mfenced><mrow><mfrac><mn>11</mn><mn>3</mn></mfrac><mo>-</mo><mi>c</mi></mrow></mfenced></mrow></mfenced><mo>+</mo><mn>4</mn><mo>.</mo><mn>5</mn><mo>⇒</mo><mi>c</mi><mo>=</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>H</mi><mfenced><mi>t</mi></mfenced><mo>=</mo><mn>2</mn><mo>.</mo><mn>3</mn><mo> </mo><mi>sin</mi><mfenced><mrow><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac><mfenced><mrow><mi>t</mi><mo>-</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></mrow></mfenced></mrow></mfenced><mo>+</mo><mn>4</mn><mo>.</mo><mn>5</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong><br>uses their horizontal translation <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mfrac><mn>12</mn><mn>4</mn></mfrac><mo>=</mo><mn>3</mn></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>11</mn><mn>3</mn></mfrac><mo>-</mo><mi>c</mi><mo>=</mo><mn>3</mn><mo>⇒</mo><mi>c</mi><mo>=</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>H</mi><mfenced><mi>t</mi></mfenced><mo>=</mo><mn>2</mn><mo>.</mo><mn>3</mn><mo> </mo><mi>sin</mi><mfenced><mrow><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac><mfenced><mrow><mi>t</mi><mo>-</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></mrow></mfenced></mrow></mfenced><mo>+</mo><mn>4</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">g.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">g.</div>
</div>
<br><hr><br><div class="specification">
<p>The voltage <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v">
<mi>v</mi>
</math></span> in a circuit is given by the equation</p>
<p style="text-align: center;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v\left( t \right) = 3\,{\text{sin}}\left( {100\pi t} \right)">
<mi>v</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>3</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>100</mn>
<mi>π<!-- π --></mi>
<mi>t</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t \geqslant 0">
<mi>t</mi>
<mo>⩾<!-- ⩾ --></mo>
<mn>0</mn>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> is measured in seconds.</p>
</div>
<div class="specification">
<p>The current <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="i">
<mi>i</mi>
</math></span> in this circuit is given by the equation</p>
<p style="text-align: center;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="i\left( t \right) = 2\,{\text{sin}}\left( {100\pi \left( {t + 0.003} \right)} \right)">
<mi>i</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>100</mn>
<mi>π<!-- π --></mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>t</mi>
<mo>+</mo>
<mn>0.003</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
</div>
<div class="specification">
<p>The power <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
<mi>p</mi>
</math></span> in this circuit is given by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p\left( t \right) = v\left( t \right) \times i\left( t \right)">
<mi>p</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mi>v</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>×<!-- × --></mo>
<mi>i</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</math></span>.</p>
</div>
<div class="specification">
<p>The average power <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{p_{av}}">
<mrow>
<msub>
<mi>p</mi>
<mrow>
<mi>a</mi>
<mi>v</mi>
</mrow>
</msub>
</mrow>
</math></span> in this circuit from <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t = 0">
<mi>t</mi>
<mo>=</mo>
<mn>0</mn>
</math></span> to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t = T">
<mi>t</mi>
<mo>=</mo>
<mi>T</mi>
</math></span> is given by the equation</p>
<p style="text-align: center;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{p_{av}}\left( T \right) = \frac{1}{T}\int_0^T {p\left( t \right){\text{d}}t} ">
<mrow>
<msub>
<mi>p</mi>
<mrow>
<mi>a</mi>
<mi>v</mi>
</mrow>
</msub>
</mrow>
<mrow>
<mo>(</mo>
<mi>T</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mi>T</mi>
</mfrac>
<msubsup>
<mo>∫<!-- ∫ --></mo>
<mn>0</mn>
<mi>T</mi>
</msubsup>
<mrow>
<mi>p</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="T > 0">
<mi>T</mi>
<mo>></mo>
<mn>0</mn>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the maximum and minimum value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v">
<mi>v</mi>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down two transformations that will transform the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = v\left( t \right)">
<mi>y</mi>
<mo>=</mo>
<mi>v</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</math></span> onto the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = i\left( t \right)">
<mi>y</mi>
<mo>=</mo>
<mi>i</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = p\left( t \right)">
<mi>y</mi>
<mo>=</mo>
<mi>p</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</math></span> for 0 ≤ <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> ≤ 0.02 , showing clearly the coordinates of the first maximum and the first minimum.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the total time in the interval 0 ≤ <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> ≤ 0.02 for which <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p\left( t \right)">
<mi>p</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</math></span> ≥ 3.</p>
<p> </p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{p_{av}}">
<mrow>
<msub>
<mi>p</mi>
<mrow>
<mi>a</mi>
<mi>v</mi>
</mrow>
</msub>
</mrow>
</math></span>(0.007).</p>
<p> </p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>With reference to your graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = p\left( t \right)">
<mi>y</mi>
<mo>=</mo>
<mi>p</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</math></span> explain why <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{p_{av}}\left( T \right)">
<mrow>
<msub>
<mi>p</mi>
<mrow>
<mi>a</mi>
<mi>v</mi>
</mrow>
</msub>
</mrow>
<mrow>
<mo>(</mo>
<mi>T</mi>
<mo>)</mo>
</mrow>
</math></span> > 0 for all <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="T">
<mi>T</mi>
</math></span> > 0.</p>
<p> </p>
<div class="marks">[2]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p\left( t \right)">
<mi>p</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</math></span> can be written as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p\left( t \right) = a\,{\text{sin}}\left( {b\left( {t - c} \right)} \right) + d">
<mi>p</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mi>a</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>b</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>t</mi>
<mo>−</mo>
<mi>c</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mi>d</mi>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b">
<mi>b</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c">
<mi>c</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d">
<mi>d</mi>
</math></span> > 0, use your graph to find the values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b">
<mi>b</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c">
<mi>c</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d">
<mi>d</mi>
</math></span>.</p>
<p> </p>
<div class="marks">[6]</div>
<div class="question_part_label">g.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p>3, −3 <em><strong>A1</strong></em><em><strong>A1</strong></em> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>stretch parallel to the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span>-axis (with <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>-axis invariant), scale factor <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{2}{3}">
<mfrac>
<mn>2</mn>
<mn>3</mn>
</mfrac>
</math></span> <em><strong>A1</strong></em></p>
<p>translation of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} { - 0.003} \\ 0 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>0.003</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> (shift to the left by 0.003) <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Can be done in either order.</p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img src="data:image/png;base64,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"></p>
<p>correct shape over correct domain with correct endpoints <em><strong>A1</strong></em><br>first maximum at (0.0035, 4.76) <em><strong>A1</strong></em><br>first minimum at (0.0085, −1.24) <em><strong>A1</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
<mi>p</mi>
</math></span> ≥ 3 between <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> = 0.0016762 and 0.0053238 and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> = 0.011676 and 0.015324 <em><strong>(M1)(A1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1A1</strong></em> for either interval.</p>
<p>= 0.00730 <em><strong>A1</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{p_{av}} = \frac{1}{{0.007}}\int_0^{0.007} {6\,{\text{sin}}\left( {100\pi t} \right)} {\text{sin}}\left( {100\pi \left( {t + 0.003} \right)} \right){\text{d}}t">
<mrow>
<msub>
<mi>p</mi>
<mrow>
<mi>a</mi>
<mi>v</mi>
</mrow>
</msub>
</mrow>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mn>0.007</mn>
</mrow>
</mfrac>
<msubsup>
<mo>∫</mo>
<mn>0</mn>
<mrow>
<mn>0.007</mn>
</mrow>
</msubsup>
<mrow>
<mn>6</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>100</mn>
<mi>π</mi>
<mi>t</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mtext>sin</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>100</mn>
<mi>π</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>t</mi>
<mo>+</mo>
<mn>0.003</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</math></span> <em><strong>(M1)</strong></em></p>
<p>= 2.87 <em><strong>A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>in each cycle the area under the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> axis is smaller than area above the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> axis <em><strong>R1</strong></em></p>
<p>the curve begins with the positive part of the cycle <em><strong>R1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a = \frac{{4.76 - \left( { - 1.24} \right)}}{2}">
<mi>a</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mn>4.76</mn>
<mo>−</mo>
<mrow>
<mo>(</mo>
<mrow>
<mo>−</mo>
<mn>1.24</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</mfrac>
</math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a = 3.00">
<mi>a</mi>
<mo>=</mo>
<mn>3.00</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d = \frac{{4.76 + \left( { - 1.24} \right)}}{2}">
<mi>d</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mn>4.76</mn>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mo>−</mo>
<mn>1.24</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</mfrac>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d = 1.76">
<mi>d</mi>
<mo>=</mo>
<mn>1.76</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b = \frac{{2\pi }}{{0.01}}">
<mi>b</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mn>2</mn>
<mi>π</mi>
</mrow>
<mrow>
<mn>0.01</mn>
</mrow>
</mfrac>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b = 628\left( { = 200\pi } \right)">
<mi>b</mi>
<mo>=</mo>
<mn>628</mn>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mn>200</mn>
<mi>π</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c = 0.0035 - \frac{{0.01}}{4}">
<mi>c</mi>
<mo>=</mo>
<mn>0.0035</mn>
<mo>−</mo>
<mfrac>
<mrow>
<mn>0.01</mn>
</mrow>
<mn>4</mn>
</mfrac>
</math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c = 0.00100">
<mi>c</mi>
<mo>=</mo>
<mn>0.00100</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">g.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">g.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the rectangle OABC such that AB = OC = 10 and BC = OA = 1 , with the points P , Q and R placed on the line OC such that OP = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
<mi>p</mi>
</math></span>, OQ = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q">
<mi>q</mi>
</math></span> and OR = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r">
<mi>r</mi>
</math></span>, such that 0 < <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
<mi>p</mi>
</math></span> < <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q">
<mi>q</mi>
</math></span> < <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r">
<mi>r</mi>
</math></span> < 10.</p>
<p style="text-align: center;"><img 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"></p>
<p style="text-align: left;">Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\theta _p}">
<mrow>
<msub>
<mi>θ<!-- θ --></mi>
<mi>p</mi>
</msub>
</mrow>
</math></span> be the angle APO, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\theta _q}">
<mrow>
<msub>
<mi>θ<!-- θ --></mi>
<mi>q</mi>
</msub>
</mrow>
</math></span> be the angle AQO and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\theta _r}">
<mrow>
<msub>
<mi>θ<!-- θ --></mi>
<mi>r</mi>
</msub>
</mrow>
</math></span> be the angle ARO.</p>
</div>
<div class="specification">
<p>Consider the case when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\theta _p} = {\theta _q} + {\theta _r}">
<mrow>
<msub>
<mi>θ<!-- θ --></mi>
<mi>p</mi>
</msub>
</mrow>
<mo>=</mo>
<mrow>
<msub>
<mi>θ<!-- θ --></mi>
<mi>q</mi>
</msub>
</mrow>
<mo>+</mo>
<mrow>
<msub>
<mi>θ<!-- θ --></mi>
<mi>r</mi>
</msub>
</mrow>
</math></span> and QR = 1.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an expression for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\theta _p}"> <mrow> <msub> <mi>θ</mi> <mi>p</mi> </msub> </mrow> </math></span> in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p"> <mi>p</mi> </math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p = \frac{{{q^2} + q - 1}}{{2q + 1}}"> <mi>p</mi> <mo>=</mo> <mfrac> <mrow> <mrow> <msup> <mi>q</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mi>q</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mn>2</mn> <mi>q</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </math></span>.</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>By sketching the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p"> <mi>p</mi> </math></span> as a function of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q"> <mi>q</mi> </math></span>, determine the range of values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p"> <mi>p</mi> </math></span> for which there are possible values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q"> <mi>q</mi> </math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color:#999;font-size:90%;font-style:italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p><strong>METHOD 1</strong></p>
<p>use of tan <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{tan}}\,{\theta _p} = \frac{1}{p}"> <mrow> <mtext>tan</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <msub> <mi>θ</mi> <mi>p</mi> </msub> </mrow> <mo>=</mo> <mfrac> <mn>1</mn> <mi>p</mi> </mfrac> </math></span> <em><strong>(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\theta _p} = {\text{arctan}}\left( {\frac{1}{p}} \right)"> <mrow> <msub> <mi>θ</mi> <mi>p</mi> </msub> </mrow> <mo>=</mo> <mrow> <mtext>arctan</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>1</mn> <mi>p</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>AP <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \sqrt {{p^2} + 1} "> <mo>=</mo> <msqrt> <mrow> <msup> <mi>p</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>1</mn> </msqrt> </math></span> <em><strong>(A1)</strong></em></p>
<p>use of sin, cos, sine rule or cosine rule using the correct length of AP <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\theta _p} = {\text{arcsin}}\left( {\frac{1}{{\sqrt {{p^2} + 1} }}} \right)"> <mrow> <msub> <mi>θ</mi> <mi>p</mi> </msub> </mrow> <mo>=</mo> <mrow> <mtext>arcsin</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>1</mn> <mrow> <msqrt> <mrow> <msup> <mi>p</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>1</mn> </msqrt> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\theta _p} = {\text{arccos}}\left( {\frac{p}{{\sqrt {{p^2} + 1} }}} \right)"> <mrow> <msub> <mi>θ</mi> <mi>p</mi> </msub> </mrow> <mo>=</mo> <mrow> <mtext>arccos</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mi>p</mi> <mrow> <msqrt> <mrow> <msup> <mi>p</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>1</mn> </msqrt> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>QR = 1 ⇒ <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r = q + 1"> <mi>r</mi> <mo>=</mo> <mi>q</mi> <mo>+</mo> <mn>1</mn> </math></span> <em><strong>(A1)</strong></em></p>
<p><strong>Note:</strong> This may be seen anywhere.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{tan}}\,{\theta _p} = {\text{tan}}\left( {{\theta _q} + {\theta _r}} \right)"> <mrow> <mtext>tan</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <msub> <mi>θ</mi> <mi>p</mi> </msub> </mrow> <mo>=</mo> <mrow> <mtext>tan</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <msub> <mi>θ</mi> <mi>q</mi> </msub> </mrow> <mo>+</mo> <mrow> <msub> <mi>θ</mi> <mi>r</mi> </msub> </mrow> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p>attempt to use compound angle formula for tan <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{tan}}\,{\theta _p} = \frac{{{\text{tan}}\,{\theta _q} + {\text{tan}}\,{\theta _r}}}{{1 - {\text{tan}}\,{\theta _q}\,{\text{tan}}\,{\theta _r}}}"> <mrow> <mtext>tan</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <msub> <mi>θ</mi> <mi>p</mi> </msub> </mrow> <mo>=</mo> <mfrac> <mrow> <mrow> <mtext>tan</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <msub> <mi>θ</mi> <mi>q</mi> </msub> </mrow> <mo>+</mo> <mrow> <mtext>tan</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <msub> <mi>θ</mi> <mi>r</mi> </msub> </mrow> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mrow> <mtext>tan</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <msub> <mi>θ</mi> <mi>q</mi> </msub> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>tan</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <msub> <mi>θ</mi> <mi>r</mi> </msub> </mrow> </mrow> </mfrac> </math></span> <em><strong>(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{p} = \frac{{\frac{1}{q} + \frac{1}{r}}}{{1 - \left( {\frac{1}{q}} \right)\left( {\frac{1}{r}} \right)}}"> <mfrac> <mn>1</mn> <mi>p</mi> </mfrac> <mo>=</mo> <mfrac> <mrow> <mfrac> <mn>1</mn> <mi>q</mi> </mfrac> <mo>+</mo> <mfrac> <mn>1</mn> <mi>r</mi> </mfrac> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>1</mn> <mi>q</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>1</mn> <mi>r</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{p} = \frac{{\frac{1}{q} + \frac{1}{{q + 1}}}}{{1 - \left( {\frac{1}{q}} \right)\left( {\frac{1}{{q + 1}}} \right)}}"> <mfrac> <mn>1</mn> <mi>p</mi> </mfrac> <mo>=</mo> <mfrac> <mrow> <mfrac> <mn>1</mn> <mi>q</mi> </mfrac> <mo>+</mo> <mfrac> <mn>1</mn> <mrow> <mi>q</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>1</mn> <mi>q</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>1</mn> <mrow> <mi>q</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p = \frac{{1 - \left( {\frac{1}{q}} \right)\left( {\frac{1}{{q + 1}}} \right)}}{{\left( {\frac{1}{q}} \right) + \left( {\frac{1}{{q + 1}}} \right)}}"> <mi>p</mi> <mo>=</mo> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>1</mn> <mi>q</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>1</mn> <mrow> <mi>q</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>1</mn> <mi>q</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>1</mn> <mrow> <mi>q</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{p} = \frac{{q + q + 1}}{{q\left( {q + 1} \right) - 1}}"> <mfrac> <mn>1</mn> <mi>p</mi> </mfrac> <mo>=</mo> <mfrac> <mrow> <mi>q</mi> <mo>+</mo> <mi>q</mi> <mo>+</mo> <mn>1</mn> </mrow> <mrow> <mi>q</mi> <mrow> <mo>(</mo> <mrow> <mi>q</mi> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </math></span> <em><strong>M1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1</strong></em> for multiplying top and bottom by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{q\left( {q + 1} \right)}"> <mrow> <mi>q</mi> <mrow> <mo>(</mo> <mrow> <mi>q</mi> <mo>+</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> </math></span>.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p = \frac{{{q^2} + q - 1}}{{2q + 1}}"> <mi>p</mi> <mo>=</mo> <mfrac> <mrow> <mrow> <msup> <mi>q</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mi>q</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mn>2</mn> <mi>q</mi> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </math></span> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img src="data:image/png;base64,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"></p>
<p>increasing function with positive <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q"> <mi>q</mi> </math></span>-intercept <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Accept curves which extend beyond the domain shown above.</p>
<p> </p>
<p>(0.618 <) <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q"> <mi>q</mi> </math></span> < 9 <em><strong>(A1)</strong></em></p>
<p>⇒ range is (0 <) <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p"> <mi>p</mi> </math></span> < 4.68 <strong>(A1)</strong></p>
<p>0 < <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p"> <mi>p</mi> </math></span> < 4.68 <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<p> </p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>The plane <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Π</mi><mn>1</mn></msub></math> has equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mi>x</mi><mo>−</mo><mi>y</mi><mo>+</mo><mi>z</mi><mo>=</mo><mo>−</mo><mn>13</mn></math> and the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi></math> has vector equation</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">r</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mi>λ</mi><mfenced><mtable><mtr><mtd><mo>-</mo><mn>3</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mn>4</mn></mtd></mtr></mtable></mfenced><mo> </mo><mo>,</mo><mo> </mo><mi>λ</mi><mo> </mo><mo>∈</mo><mo> </mo><mi mathvariant="normal">ℝ</mi></math>.</p>
</div>
<div class="specification">
<p>The plane <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Π</mi><mn>2</mn></msub></math> contains the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math> and the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi></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"><mi>L</mi></math> meets <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Π</mi><mn>1</mn></msub></math> at the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math>, find the coordinates of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the shortest distance from the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext><mo>(</mo><mn>0</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Π</mi><mn>1</mn></msub></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the equation of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Π</mi><mn>2</mn></msub></math>, giving your answer in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">r</mi><mo mathvariant="bold">.</mo><mi mathvariant="bold-italic">n</mi><mo>=</mo><mi>d</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the acute angle between <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Π</mi><mn>1</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Π</mi><mn>2</mn></msub></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mfenced><mrow><mn>1</mn><mo>-</mo><mn>3</mn><mi>λ</mi></mrow></mfenced><mo>-</mo><mfenced><mrow><mn>2</mn><mo>-</mo><mi>λ</mi></mrow></mfenced><mo>+</mo><mfenced><mrow><mo>-</mo><mn>2</mn><mo>+</mo><mn>4</mn><mi>λ</mi></mrow></mfenced><mo>=</mo><mo>-</mo><mn>13</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi><mo>=</mo><mn>3</mn></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">r</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mn>3</mn><mfenced><mtable><mtr><mtd><mo>-</mo><mn>3</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mn>4</mn></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mtable><mtr><mtd><mn>8</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mn>10</mn></mtd></mtr></mtable></mfenced></math> <em><strong>(M1)</strong></em></p>
<p>so <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mrow><mo>-</mo><mn>8</mn><mo>,</mo><mo> </mo><mo>-</mo><mn>1</mn><mo>,</mo><mo> </mo><mn>10</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><strong><br>Note:</strong> Do not award the final <em><strong>A1</strong></em> if a vector given instead of coordinates</p>
<p><em><strong><br>[4 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">r</mi><mo>=</mo><mi>μ</mi><mfenced><mtable><mtr><mtd><mn>3</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr></mtable></mfenced></math></p>
<p>substituting into equation of the plane <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9</mn><mi>μ</mi><mo>+</mo><mi>μ</mi><mo>+</mo><mi>μ</mi><mo>=</mo><mo>-</mo><mn>13</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>μ</mi><mo>=</mo><mo>-</mo><mfrac><mn>13</mn><mn>11</mn></mfrac><mo> </mo><mfenced><mrow><mo>=</mo><mo>-</mo><mn>1</mn><mo>.</mo><mn>18</mn><mo>…</mo></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p>distance <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mn>13</mn><msqrt><msup><mn>3</mn><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mo>-</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mn>1</mn><mn>2</mn></msup></msqrt></mrow><mn>11</mn></mfrac></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>13</mn><msqrt><mn>11</mn></msqrt></mfrac><mfenced><mrow><mo>=</mo><mfrac><mrow><mn>13</mn><msqrt><mn>11</mn></msqrt></mrow><mn>11</mn></mfrac><mo>=</mo><mn>3</mn><mo>.</mo><mn>92</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><br><strong><br>METHOD 2</strong></p>
<p>choice of any point on the plane, eg <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>-</mo><mn>8</mn><mo>,</mo><mo> </mo><mo>-</mo><mn>1</mn><mo>,</mo><mo> </mo><mn>10</mn></mrow></mfenced></math> to use in distance formula <em><strong>(M1)</strong></em></p>
<p>so distance <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mfenced><mtable><mtr><mtd><mo>-</mo><mn>8</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mn>10</mn></mtd></mtr></mtable></mfenced><mo>·</mo><mfenced><mtable><mtr><mtd><mo>-</mo><mn>3</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mrow><msqrt><msup><mfenced><mrow><mo>-</mo><mn>3</mn></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mn>1</mn><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mo>-</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup></msqrt></mfrac></math> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>A1</strong></em> for numerator, <em><strong>A1</strong></em> for denominator.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mn>24</mn><mo>-</mo><mn>1</mn><mo>-</mo><mn>10</mn></mrow><msqrt><mn>11</mn></msqrt></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>13</mn><msqrt><mn>11</mn></msqrt></mfrac><mfenced><mrow><mo>=</mo><mfrac><mrow><mn>13</mn><msqrt><mn>11</mn></msqrt></mrow><mn>11</mn></mfrac><mo>=</mo><mn>3</mn><mo>.</mo><mn>92</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><em><strong><br>[4 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p>identify two vectors <em><strong>(A1)</strong></em></p>
<p><em>eg</em>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr></mtable></mfenced></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mo>-</mo><mn>3</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mn>4</mn></mtd></mtr></mtable></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">n</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr></mtable></mfenced><mo>×</mo><mfenced><mtable><mtr><mtd><mo>-</mo><mn>3</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mn>4</mn></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mtable><mtr><mtd><mn>6</mn></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr><mtr><mtd><mn>5</mn></mtd></mtr></mtable></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><br><strong>OR</strong></p>
<p><br>identify three points in the plane <em><strong>(A1)</strong></em></p>
<p><em>eg</em> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn></math> gives <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr></mtable></mfenced></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable></mfenced></math></p>
<p>solving system of equations <em><strong>(M1)</strong></em></p>
<p><br><strong>THEN</strong></p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Π</mi><mn>2</mn></msub><mo> </mo><mo>:</mo><mo> </mo><mi mathvariant="bold-italic">r</mi><mo>.</mo><mfenced><mtable><mtr><mtd><mn>6</mn></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr><mtr><mtd><mn>5</mn></mtd></mtr></mtable></mfenced><mo>=</mo><mn>0</mn></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mi>x</mi><mo>+</mo><mn>2</mn><mi>y</mi><mo>+</mo><mn>5</mn><mi>z</mi><mo>=</mo><mn>0</mn></math>.</p>
<p><em><strong><br>[3 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>vector normal to <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Π</mi><mn>1</mn></msub></math> is eg <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="bold-italic">n</mi><mn mathvariant="bold-italic">1</mn></msub><mo>=</mo><mfenced><mtable><mtr><mtd><mn>3</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr></mtable></mfenced></math></p>
<p>vector normal to <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Π</mi><mn>2</mn></msub></math> is eg <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="bold-italic">n</mi><mn mathvariant="bold-italic">2</mn></msub><mo>=</mo><mfenced><mtable><mtr><mtd><mn>6</mn></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr><mtr><mtd><mn>5</mn></mtd></mtr></mtable></mfenced></math> <em><strong>(A1)</strong></em></p>
<p>required angle is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mo> </mo><mi>θ</mi><mfrac><mrow><mfenced><mtable><mtr><mtd><mn>3</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr></mtable></mfenced><mo>·</mo><mfenced><mtable><mtr><mtd><mn>6</mn></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr><mtr><mtd><mn>5</mn></mtd></mtr></mtable></mfenced></mrow><mrow><msqrt><mn>11</mn></msqrt><msqrt><mn>65</mn></msqrt></mrow></mfrac></math> <em><strong>M1A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mo> </mo><mi>θ</mi><mo>=</mo><mfrac><mn>21</mn><mrow><msqrt><mn>11</mn></msqrt><msqrt><mn>65</mn></msqrt></mrow></mfrac><mo>=</mo><mn>0</mn><mo>.</mo><mn>785</mn><mo>…</mo></math> <em><strong>(A1)</strong></em></p>
<p style="padding-left:30px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>667526</mn><mo>…</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>668</mn><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mn>38</mn><mo>.</mo><mn>2</mn><mo>°</mo></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Award the penultimate <em><strong>(A1)</strong></em> but not the final <em><strong>A1</strong></em> for the obtuse angle <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>.</mo><mn>47406</mn><mo>…</mo></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>142</mn><mo>°</mo></math>.</p>
<p><em><strong><br>[5 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>A particle <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> moves in a straight line such that after time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> seconds, its velocity, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi></math> in <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>m s</mtext><mrow><mo>-</mo><mn>1</mn></mrow></msup></math>, is given by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mo>=</mo><msup><mtext>e</mtext><mrow><mo>−</mo><mn>3</mn><mi>t</mi></mrow></msup><mo> </mo><mi>sin</mi><mo> </mo><mn>6</mn><mo> </mo><mi>t</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo><</mo><mi>t</mi><mo><</mo><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></math>.</p>
</div>
<div class="specification">
<p>At time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> has displacement <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mo>(</mo><mi>t</mi><mo>)</mo></math>; at time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mo>(</mo><mn>0</mn><mo>)</mo><mo>=</mo><mn>0</mn></math>.</p>
</div>
<div class="specification">
<p>At successive times when the acceleration of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> is<math xmlns="http://www.w3.org/1998/Math/MathML"><mo> </mo><mn>0</mn><mo> </mo><msup><mtext>m s</mtext><mrow><mo>−</mo><mn>2</mn></mrow></msup><mo> </mo></math>, the velocities of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> form a geometric sequence. The acceleration of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> is zero at times <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>t</mi><mn>1</mn></msub><mo>,</mo><mo> </mo><msub><mi>t</mi><mn>2</mn></msub><mo>,</mo><mo> </mo><msub><mi>t</mi><mn>3</mn></msub></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>t</mi><mn>1</mn></msub><mo><</mo><msub><mi>t</mi><mn>2</mn></msub><mo><</mo><msub><mi>t</mi><mn>3</mn></msub></math> and the respective velocities are <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>v</mi><mn>1</mn></msub><mo>,</mo><mo> </mo><msub><mi>v</mi><mn>2</mn></msub><mo>,</mo><mo> </mo><msub><mi>v</mi><mn>3</mn></msub></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the times when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> comes to instantaneous rest.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>.</p>
<div class="marks">[7]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the maximum displacement of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math>, in metres, from its initial position.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the total distance travelled by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> in the first <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.</mo><mn>5</mn></math> seconds of its motion.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that, at these times, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tan</mi><mo> </mo><mn>6</mn><mi>t</mi><mo>=</mo><mn>2</mn></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">e.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msub><mi>v</mi><mn>2</mn></msub><msub><mi>v</mi><mn>1</mn></msub></mfrac><mo>=</mo><mfrac><msub><mi>v</mi><mn>3</mn></msub><msub><mi>v</mi><mn>2</mn></msub></mfrac><mo>=</mo><mo>-</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></mrow></msup></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">e.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac><mfenced><mrow><mo>=</mo><mn>0</mn><mo>.</mo><mn>524</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi mathvariant="normal">π</mi><mn>3</mn></mfrac><mfenced><mrow><mo>=</mo><mn>1</mn><mo>.</mo><mn>05</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><em><strong><br>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to use integration by parts <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mo>=</mo><mo>∫</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mi>t</mi></mrow></msup><mtext> </mtext><mi>sin</mi><mo> </mo><mn>6</mn><mi>t</mi><mo> </mo><mtext>d</mtext><mi>t</mi></math></p>
<p><strong><br>EITHER</strong></p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><mfrac><mrow><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mi>t</mi></mrow></msup><mtext> </mtext><mi>sin</mi><mo> </mo><mn>6</mn><mi>t</mi></mrow><mn>3</mn></mfrac><mo>-</mo><mo>∫</mo><mo>-</mo><mn>2</mn><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mi>t</mi></mrow></msup><mtext> cos</mtext><mo> </mo><mn>6</mn><mi>t</mi><mo> </mo><mtext>d</mtext><mi>t</mi></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><mfrac><mrow><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mi>t</mi></mrow></msup><mtext> </mtext><mi>sin</mi><mo> </mo><mn>6</mn><mi>t</mi></mrow><mn>3</mn></mfrac><mo>-</mo><mfenced><mrow><mfrac><mrow><mn>2</mn><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mi>t</mi></mrow></msup><mtext> cos</mtext><mo> </mo><mn>6</mn><mi>t</mi></mrow><mn>3</mn></mfrac><mo>-</mo><mo>∫</mo><mo>-</mo><mn>4</mn><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mi>t</mi></mrow></msup><mtext> sin</mtext><mo> </mo><mn>6</mn><mi>t</mi><mo> </mo><mtext>d</mtext><mi>t</mi></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><mfrac><mrow><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mi>t</mi></mrow></msup><mtext> </mtext><mi>sin</mi><mo> </mo><mn>6</mn><mi>t</mi></mrow><mn>3</mn></mfrac><mo>-</mo><mfenced><mrow><mfrac><mrow><mn>2</mn><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mi>t</mi></mrow></msup><mtext> cos</mtext><mo> </mo><mn>6</mn><mi>t</mi></mrow><mn>3</mn></mfrac><mo>+</mo><mn>4</mn><mi>s</mi></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mi>s</mi><mo>=</mo><mfrac><mrow><mo>-3</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mi>t</mi></mrow></msup><mtext> </mtext><mi>sin</mi><mo> </mo><mn>6</mn><mi>t</mi><mo>-</mo><mn>6</mn><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mi>t</mi></mrow></msup><mtext> cos</mtext><mo> </mo><mn>6</mn><mi>t</mi></mrow><mn>9</mn></mfrac></math> <em><strong>M1</strong></em></p>
<p><br><strong>OR</strong></p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><mfrac><mrow><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mi>t</mi></mrow></msup><mtext> cos</mtext><mo> </mo><mn>6</mn><mi>t</mi></mrow><mn>6</mn></mfrac><mo>-</mo><mo>∫</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mi>t</mi></mrow></msup><mtext> cos</mtext><mo> </mo><mn>6</mn><mi>t</mi><mo> </mo><mtext>d</mtext><mi>t</mi></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><mfrac><mrow><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mi>t</mi></mrow></msup><mtext> cos</mtext><mo> </mo><mn>6</mn><mi>t</mi></mrow><mn>6</mn></mfrac><mo>-</mo><mfenced><mrow><mfrac><mrow><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mi>t</mi></mrow></msup><mtext> sin</mtext><mo> </mo><mn>6</mn><mi>t</mi></mrow><mn>12</mn></mfrac><mo>+</mo><mo>∫</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mi>t</mi></mrow></msup><mtext> sin</mtext><mo> </mo><mn>6</mn><mi>t</mi><mo> </mo><mtext>d</mtext><mi>t</mi></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><mfrac><mrow><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mi>t</mi></mrow></msup><mtext> cos</mtext><mo> </mo><mn>6</mn><mi>t</mi></mrow><mn>6</mn></mfrac><mo>-</mo><mfenced><mrow><mfrac><mrow><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mi>t</mi></mrow></msup><mtext> sin</mtext><mo> </mo><mn>6</mn><mi>t</mi></mrow><mn>12</mn></mfrac><mo>+</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mi>s</mi></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>5</mn><mn>4</mn></mfrac><mi>s</mi><mo>=</mo><mfrac><mrow><mo>-</mo><mn>2</mn><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mi>t</mi></mrow></msup><mtext> cos</mtext><mo> </mo><mn>6</mn><mi>t</mi><mo>-</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mi>t</mi></mrow></msup><mtext> sin</mtext><mo> </mo><mn>6</mn><mi>t</mi></mrow><mn>12</mn></mfrac></math> <em><strong>M1</strong></em></p>
<p><br><strong>THEN</strong></p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mo>=</mo><mo>-</mo><mfrac><mrow><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mi>t</mi></mrow></msup><mtext> </mtext><mfenced><mrow><mtext> sin</mtext><mo> </mo><mn>6</mn><mi>t</mi><mo>+</mo><mn>2</mn><mo> </mo><mtext>cos</mtext><mo> </mo><mn>6</mn><mi>t</mi></mrow></mfenced></mrow><mn>15</mn></mfrac><mfenced><mrow><mo>+</mo><mi>c</mi></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p>at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn><mo>,</mo><mo> </mo><mi>s</mi><mo>=</mo><mn>0</mn><mo>⇒</mo><mn>0</mn><mo>=</mo><mo>-</mo><mfrac><mn>2</mn><mn>15</mn></mfrac><mo>+</mo><mi>c</mi></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>=</mo><mfrac><mn>2</mn><mn>15</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mo>=</mo><mfrac><mn>2</mn><mn>15</mn></mfrac><mo>-</mo><mfrac><mrow><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mi>t</mi></mrow></msup><mtext> </mtext><mfenced><mrow><mtext> sin</mtext><mo> </mo><mn>6</mn><mi>t</mi><mo>+</mo><mn>2</mn><mo> </mo><mtext>cos</mtext><mo> </mo><mn>6</mn><mi>t</mi></mrow></mfenced></mrow><mn>15</mn></mfrac></math></p>
<p><em><strong><br>[7 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p>substituting <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></math> into their equation for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>s</mi><mo>=</mo><mfrac><mn>2</mn><mn>15</mn></mfrac><mo>-</mo><mfrac><mrow><msup><mtext>e</mtext><mrow><mo>-</mo><mstyle displaystyle="true"><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></mstyle></mrow></msup><mtext> </mtext><mfenced><mrow><mtext> sin</mtext><mo> </mo><mi mathvariant="normal">π</mi><mo>+</mo><mn>2</mn><mo> </mo><mtext>cos</mtext><mo> </mo><mi mathvariant="normal">π</mi></mrow></mfenced></mrow><mn>15</mn></mfrac></mrow></mfenced></math></p>
<p><br><strong>OR</strong><br><br></p>
<p>using GDC to find maximum value <em><strong>(M1)</strong></em><br><br></p>
<p><strong>OR</strong></p>
<p>evaluating <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mn>0</mn><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></msubsup><mi>v</mi><mtext>d</mtext><mi>t</mi></math> <em><strong>(M1)</strong></em></p>
<p><br><strong>THEN</strong></p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>0</mn><mo>.</mo><mn>161</mn><mfenced><mrow><mo>=</mo><mfrac><mn>2</mn><mn>15</mn></mfrac><mfenced><mrow><mn>1</mn><mo>+</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mstyle displaystyle="false"><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></mstyle></mrow></msup></mrow></mfenced></mrow></mfenced></math> <em><strong>A1</strong></em> </p>
<p><em><strong><br>[2 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1 </strong></p>
<p><strong><br>EITHER</strong></p>
<p>distance required <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><munderover><mo>∫</mo><mn>0</mn><mrow><mn>1</mn><mo>.</mo><mn>5</mn></mrow></munderover><mfenced open="|" close="|"><mrow><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mi>t</mi></mrow></msup><mo> </mo><mi>sin</mi><mo> </mo><mn>6</mn><mi>t</mi></mrow></mfenced><mo> </mo><mtext>d</mtext><mi>t</mi></math> <em><strong>(M1)</strong></em></p>
<p><br><strong>OR</strong></p>
<p>distance required <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><munderover><mo>∫</mo><mn>0</mn><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></munderover><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mi>t</mi></mrow></msup><mo> </mo><mi>sin</mi><mo> </mo><mn>6</mn><mi>t</mi><mo> </mo><mtext>d</mtext><mi>t</mi><mo>+</mo><mfenced open="|" close="|"><mrow><munderover><mo>∫</mo><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac><mfrac><mi mathvariant="normal">π</mi><mn>3</mn></mfrac></munderover><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mi>t</mi></mrow></msup><mo> </mo><mi>sin</mi><mo> </mo><mn>6</mn><mi>t</mi><mo> </mo><mtext>d</mtext><mi>t</mi></mrow></mfenced><mo>+</mo><munderover><mo>∫</mo><mfrac><mi mathvariant="normal">π</mi><mn>3</mn></mfrac><mrow><mn>1</mn><mo>.</mo><mn>5</mn></mrow></munderover><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mi>t</mi></mrow></msup><mo> </mo><mi>sin</mi><mo> </mo><mn>6</mn><mi>t</mi><mo> </mo><mtext>d</mtext><mi>t</mi></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>=</mo><mn>0</mn><mo>.</mo><mn>16105</mn><mo>…</mo><mo>+</mo><mn>0</mn><mo>.</mo><mn>033479</mn><mo>…</mo><mo>+</mo><mn>0</mn><mo>.</mo><mn>006806</mn><mo>…</mo></mrow></mfenced></math></p>
<p><br><strong>THEN</strong></p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>0</mn><mo>.</mo><mn>201</mn><mo> </mo><mfenced><mtext>m</mtext></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><br>using successive minimum and maximum values on the displacement graph <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>16105</mn><mo>…</mo><mo>+</mo><mfenced><mrow><mn>0</mn><mo>.</mo><mn>16105</mn><mo>…</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>12757</mn><mo>…</mo></mrow></mfenced><mo>+</mo><mfenced><mrow><mn>0</mn><mo>.</mo><mn>13453</mn><mo>…</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>12757</mn><mo>…</mo></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>0</mn><mo>.</mo><mn>201</mn><mo> </mo><mfenced><mtext>m</mtext></mfenced></math> <em><strong>A1</strong></em></p>
<p><em><strong><br>[2 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>valid attempt to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mtext>d</mtext><mi>v</mi></mrow><mrow><mtext>d</mtext><mi>t</mi></mrow></mfrac></math> using product rule and set <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mtext>d</mtext><mi>v</mi></mrow><mrow><mtext>d</mtext><mi>t</mi></mrow></mfrac><mo>=</mo><mn>0</mn></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mtext>d</mtext><mi>v</mi></mrow><mrow><mtext>d</mtext><mi>t</mi></mrow></mfrac><mo>=</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mi>t</mi></mrow></msup><mn>6</mn><mo> </mo><mi>cos</mi><mo> </mo><mn>6</mn><mi>t</mi><mo>-</mo><mn>3</mn><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mi>t</mi></mrow></msup><mo> </mo><mi>sin</mi><mo> </mo><mn>6</mn><mi>t</mi></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mtext>d</mtext><mi>v</mi></mrow><mrow><mtext>d</mtext><mi>t</mi></mrow></mfrac><mo>=</mo><mn>0</mn><mo>⇒</mo><mi>tan</mi><mo> </mo><mn>6</mn><mi>t</mi><mo>=</mo><mn>2</mn></math> <em><strong>AG</strong></em></p>
<p><em><strong><br>[2 marks]</strong></em></p>
<div class="question_part_label">e.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to evaluate <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>t</mi><mn>1</mn></msub><mo>,</mo><mo> </mo><msub><mi>t</mi><mn>2</mn></msub><mo>,</mo><mo> </mo><msub><mi>t</mi><mn>3</mn></msub></math> in exact form <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><msub><mi>t</mi><mn>1</mn></msub><mo>=</mo><mtext>arctan</mtext><mo> </mo><mn>2</mn><mfenced><mrow><mo>⇒</mo><msub><mi>t</mi><mn>1</mn></msub><mo>=</mo><mfrac><mn>1</mn><mn>6</mn></mfrac><mo> </mo><mtext>arctan</mtext><mo> </mo><mn>2</mn></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><msub><mi>t</mi><mn>2</mn></msub><mo>=</mo><mi mathvariant="normal">π</mi><mo>+</mo><mtext>arctan</mtext><mo> </mo><mn>2</mn><mfenced><mrow><mo>⇒</mo><msub><mi>t</mi><mn>2</mn></msub><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac><mo>+</mo><mfrac><mn>1</mn><mn>6</mn></mfrac><mo> </mo><mtext>arctan</mtext><mo> </mo><mn>2</mn></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><msub><mi>t</mi><mn>3</mn></msub><mo>=</mo><mn>2</mn><mi mathvariant="normal">π</mi><mo>+</mo><mtext>arctan</mtext><mo> </mo><mn>2</mn><mfenced><mrow><mo>⇒</mo><msub><mi>t</mi><mn>3</mn></msub><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>3</mn></mfrac><mo>+</mo><mfrac><mn>1</mn><mn>6</mn></mfrac><mo> </mo><mtext>arctan</mtext><mo> </mo><mn>2</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> The <em><strong>A1</strong></em> is for any two consecutive correct, or showing that <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><msub><mi>t</mi><mn>2</mn></msub><mo>=</mo><mi mathvariant="normal">π</mi><mo>+</mo><mn>6</mn><msub><mi>t</mi><mn>1</mn></msub></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><msub><mi>t</mi><mn>3</mn></msub><mo>=</mo><mi mathvariant="normal">π</mi><mo>+</mo><mn>6</mn><msub><mi>t</mi><mn>2</mn></msub></math>.</p>
<p><br>showing that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mo> </mo><mn>6</mn><msub><mi>t</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><mo>-</mo><mi>sin</mi><mo> </mo><mn>6</mn><msub><mi>t</mi><mi>n</mi></msub></math></p>
<p>eg <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tan</mi><mo> </mo><mn>6</mn><mi>t</mi><mo>=</mo><mn>2</mn><mo>⇒</mo><mi>sin</mi><mo> </mo><mn>6</mn><mi>t</mi><mo>=</mo><mo>±</mo><mfrac><mn>2</mn><msqrt><mn>5</mn></msqrt></mfrac></math> <em><strong>M1A1</strong></em></p>
<p>showing that <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><msub><mi>t</mi><mrow><mi>n</mi><mo>+1</mo></mrow></msub></mrow></msup><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><msub><mi>t</mi><mi>n</mi></msub></mrow></msup></mfrac><mo>=</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></mrow></msup></math> <em><strong>M1</strong></em></p>
<p>eg <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mfenced><mrow><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac><mo>+</mo><mi>k</mi></mrow></mfenced></mrow></msup><mo>÷</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mn>3</mn><mi>k</mi></mrow></msup><mo>=</mo><msup><mtext>e</mtext><mrow><mi>-</mi><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></mrow></msup></math></p>
<p><br><strong>Note:</strong> Award the <em><strong>A1</strong></em> for any two consecutive terms.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msub><mi>v</mi><mn>3</mn></msub><msub><mi>v</mi><mn>2</mn></msub></mfrac><mo>=</mo><mfrac><msub><mi>v</mi><mn>2</mn></msub><msub><mi>v</mi><mn>1</mn></msub></mfrac><mo>=</mo><mo>-</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></mrow></msup></math> <em><strong>AG</strong></em></p>
<p><em><strong><br>[5 marks]</strong></em></p>
<div class="question_part_label">e.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.ii.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the set of values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k"> <mi>k</mi> </math></span> that satisfy the inequality <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{k^2} - k - 12 < 0"> <mrow> <msup> <mi>k</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mi>k</mi> <mo>−</mo> <mn>12</mn> <mo><</mo> <mn>0</mn> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The triangle ABC is shown in the following diagram. Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\cos B < \frac{1}{4}"> <mi>cos</mi> <mo></mo> <mi>B</mi> <mo><</mo> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </math></span>, find the range of possible values for AB.</p>
<p><img src="images/Schermafbeelding_2017-08-09_om_18.13.24.png" alt="M17/5/MATHL/HP2/ENG/TZ2/04.b"></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{k^2} - k - 12 < 0"> <mrow> <msup> <mi>k</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mi>k</mi> <mo>−</mo> <mn>12</mn> <mo><</mo> <mn>0</mn> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(k - 4)(k + 3) < 0"> <mo stretchy="false">(</mo> <mi>k</mi> <mo>−</mo> <mn>4</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo><</mo> <mn>0</mn> </math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 3 < k < 4"> <mo>−</mo> <mn>3</mn> <mo><</mo> <mi>k</mi> <mo><</mo> <mn>4</mn> </math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\cos B = \frac{{{2^2} + {c^2} - {4^2}}}{{4c}}{\text{ }}({\text{or }}16 = {2^2} + {c^2} - 4c\cos B)"> <mi>cos</mi> <mo></mo> <mi>B</mi> <mo>=</mo> <mfrac> <mrow> <mrow> <msup> <mn>2</mn> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mi>c</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mrow> <msup> <mn>4</mn> <mn>2</mn> </msup> </mrow> </mrow> <mrow> <mn>4</mn> <mi>c</mi> </mrow> </mfrac> <mrow> <mtext> </mtext> </mrow> <mo stretchy="false">(</mo> <mrow> <mtext>or </mtext> </mrow> <mn>16</mn> <mo>=</mo> <mrow> <msup> <mn>2</mn> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mi>c</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>4</mn> <mi>c</mi> <mi>cos</mi> <mo></mo> <mi>B</mi> <mo stretchy="false">)</mo> </math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow \frac{{{c^2} - 12}}{{4c}} < \frac{1}{4}"> <mo stretchy="false">⇒</mo> <mfrac> <mrow> <mrow> <msup> <mi>c</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>12</mn> </mrow> <mrow> <mn>4</mn> <mi>c</mi> </mrow> </mfrac> <mo><</mo> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow {c^2} - c - 12 < 0"> <mo stretchy="false">⇒</mo> <mrow> <msup> <mi>c</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mi>c</mi> <mo>−</mo> <mn>12</mn> <mo><</mo> <mn>0</mn> </math></span></p>
<p>from result in (a)</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0 < {\text{AB}} < 4"> <mn>0</mn> <mo><</mo> <mrow> <mtext>AB</mtext> </mrow> <mo><</mo> <mn>4</mn> </math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 3 < {\text{AB}} < 4"> <mo>−</mo> <mn>3</mn> <mo><</mo> <mrow> <mtext>AB</mtext> </mrow> <mo><</mo> <mn>4</mn> </math></span> <strong><em>(A1)</em></strong></p>
<p>but AB must be at least 2</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow 2 < {\text{AB}} < 4"> <mo stretchy="false">⇒</mo> <mn>2</mn> <mo><</mo> <mrow> <mtext>AB</mtext> </mrow> <mo><</mo> <mn>4</mn> </math></span> <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Allow <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \leqslant {\text{AB}}"> <mo>⩽</mo> <mrow> <mtext>AB</mtext> </mrow> </math></span> for either of the final two <strong><em>A </em></strong>marks.</p>
<p> </p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Two airplanes, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math>, have position vectors with respect to an origin <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math> given respectively by</p>
<p style="padding-left: 180px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="bold-italic">r</mi><mtext mathvariant="bold-italic">A</mtext></msub><mo>=</mo><mfenced><mtable><mtr><mtd><mn>19</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mi>t</mi><mfenced><mtable><mtr><mtd><mo>-</mo><mn>6</mn></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr><mtr><mtd><mn>4</mn></mtd></mtr></mtable></mfenced></math></p>
<p style="padding-left: 180px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="bold-italic">r</mi><mi mathvariant="bold-italic">B</mi></msub><mo>=</mo><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>12</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mi>t</mi><mfenced><mtable><mtr><mtd><mn>4</mn></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr></mtable></mfenced></math></p>
<p>where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> represents the time in minutes and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>≤</mo><mi>t</mi><mo>≤</mo><mn>2</mn><mo>.</mo><mn>5</mn></math>.</p>
<p>Entries in each column vector give the displacement east of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math>, the displacement north of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math> and the distance above sea level, all measured in kilometres.</p>
</div>
<div class="specification">
<p>The two airplanes’ lines of flight cross at point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the three-figure bearing on which airplane <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> is travelling.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that airplane <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> travels at a greater speed than airplane <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the acute angle between the two airplanes’ lines of flight. Give your answer in degrees.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the coordinates of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the length of time between the first airplane arriving at <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math> and the second airplane arriving at <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mo>(</mo><mi>t</mi><mo>)</mo></math> represent the distance between airplane <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> and airplane <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>≤</mo><mi>t</mi><mo>≤</mo><mn>2</mn><mo>.</mo><mn>5</mn></math>.</p>
<p>Find the minimum value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mo>(</mo><mi>t</mi><mo>)</mo></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ϕ</mi></math> be the required angle (bearing)</p>
<p><strong><br>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ϕ</mi><mo>=</mo><mn>90</mn><mo>°</mo><mo>-</mo><mtext>arctan</mtext><mfrac><mn>1</mn><mn>2</mn></mfrac><mo> </mo><mfenced><mrow><mo>=</mo><mtext>arctan</mtext><mo> </mo><mn>2</mn></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>M1</strong> </em>for a labelled sketch.</p>
<p><br><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mo> </mo><mi>ϕ</mi><mo>=</mo><mfrac><mrow><mfenced><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr></mtable></mfenced><mo>·</mo><mfenced><mtable><mtr><mtd><mn>4</mn></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable></mfenced></mrow><mrow><msqrt><mn>1</mn></msqrt><mo>×</mo><msqrt><mn>20</mn></msqrt></mrow></mfrac><mo> </mo><mfenced><mrow><mo>=</mo><mn>0</mn><mo>.</mo><mn>4472</mn><mo>…</mo><mo>,</mo><mo>=</mo><mfrac><mn>1</mn><msqrt><mn>5</mn></msqrt></mfrac></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ϕ</mi><mo>=</mo><mtext>arccos</mtext><mfenced><mrow><mn>0</mn><mo>.</mo><mn>4472</mn><mo>…</mo></mrow></mfenced></math></p>
<p><br><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>063</mn><mo>°</mo></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Do not accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>063</mn><mo>.</mo><mn>6</mn><mo>°</mo></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>63</mn><mo>.</mo><mn>4</mn><mo>°</mo></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.</mo><msup><mn>10</mn><mi>c</mi></msup></math>.</p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>let <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><msub><mi mathvariant="bold-italic">b</mi><mi>A</mi></msub></mfenced></math> be the speed of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> and let <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><msub><mi mathvariant="bold-italic">b</mi><mi>B</mi></msub></mfenced></math> be the speed of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math></p>
<p>attempts to find the speed of one of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><msub><mi mathvariant="bold-italic">b</mi><mi>A</mi></msub></mfenced><mo>=</mo><msqrt><msup><mfenced><mrow><mo>-</mo><mn>6</mn></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mn>2</mn><mn>2</mn></msup><mo>+</mo><msup><mn>4</mn><mn>2</mn></msup></msqrt></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><msub><mi mathvariant="bold-italic">b</mi><mi>B</mi></msub></mfenced><mo>=</mo><msqrt><msup><mn>4</mn><mn>2</mn></msup><mo>+</mo><msup><mn>2</mn><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mo>-</mo><mn>2</mn></mrow></mfenced><mn>2</mn></msup></msqrt></math></p>
<p><br><strong>Note:</strong> Award <em><strong>M0</strong> </em>for <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><msub><mi mathvariant="bold-italic">b</mi><mi>A</mi></msub></mfenced><mo>=</mo><msqrt><msup><mn>19</mn><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mo>-</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mn>1</mn><mn>2</mn></msup></msqrt></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><msub><mi mathvariant="bold-italic">b</mi><mi>B</mi></msub></mfenced><mo>=</mo><msqrt><msup><mn>1</mn><mn>2</mn></msup><mo>+</mo><msup><mn>0</mn><mn>2</mn></msup><mo>+</mo><msup><mn>12</mn><mn>2</mn></msup></msqrt></math>.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><msub><mi mathvariant="bold-italic">b</mi><mi>A</mi></msub></mfenced><mo>=</mo><mn>7</mn><mo>.</mo><mn>48</mn><mo>…</mo><mo> </mo><mfenced><mrow><mo>=</mo><msqrt><mn>56</mn></msqrt></mrow></mfenced></math> (km min<sup>-1</sup>) and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><msub><mi mathvariant="bold-italic">b</mi><mi>B</mi></msub></mfenced><mo>=</mo><mn>4</mn><mo>.</mo><mn>89</mn><mo>…</mo><mo> </mo><mfenced><mrow><mo>=</mo><msqrt><mn>24</mn></msqrt></mrow></mfenced></math> (km min<sup>-1</sup>) <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><msub><mi mathvariant="bold-italic">b</mi><mi>A</mi></msub></mfenced><mo>></mo><mfenced open="|" close="|"><msub><mi mathvariant="bold-italic">b</mi><mi>B</mi></msub></mfenced></math> so <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> travels at a greater speed than <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>attempts to use <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>speed</mtext><mo>=</mo><mfrac><mtext>distance</mtext><mtext>time</mtext></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>speed</mtext><mi>A</mi></msub><mo>=</mo><mfrac><mfenced open="|" close="|"><mrow><msub><mi>r</mi><mi>A</mi></msub><mfenced><msub><mi>t</mi><mn>2</mn></msub></mfenced><mo>-</mo><msub><mi>r</mi><mi>A</mi></msub><mfenced><msub><mi>t</mi><mn>1</mn></msub></mfenced></mrow></mfenced><mrow><msub><mi>t</mi><mn>2</mn></msub><mo>-</mo><msub><mi>t</mi><mn>1</mn></msub></mrow></mfrac></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>speed</mtext><mi>B</mi></msub><mo>=</mo><mfrac><mfenced open="|" close="|"><mrow><msub><mi>r</mi><mi>B</mi></msub><mfenced><msub><mi>t</mi><mn>2</mn></msub></mfenced><mo>-</mo><msub><mi>r</mi><mi>B</mi></msub><mfenced><msub><mi>t</mi><mn>1</mn></msub></mfenced></mrow></mfenced><mrow><msub><mi>t</mi><mn>2</mn></msub><mo>-</mo><msub><mi>t</mi><mn>1</mn></msub></mrow></mfrac></math> <em><strong>(M1)</strong></em></p>
<p>for example:</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>speed</mtext><mi>A</mi></msub><mo>=</mo><mfrac><mfenced open="|" close="|"><mrow><msub><mi>r</mi><mi>A</mi></msub><mfenced><mn>1</mn></mfenced><mo>-</mo><msub><mi>r</mi><mi>A</mi></msub><mfenced><mn>0</mn></mfenced></mrow></mfenced><mn>1</mn></mfrac></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>speed</mtext><mi>B</mi></msub><mo>=</mo><mfrac><mfenced open="|" close="|"><mrow><msub><mi>r</mi><mi>B</mi></msub><mfenced><mn>1</mn></mfenced><mo>-</mo><msub><mi>r</mi><mi>B</mi></msub><mfenced><mn>0</mn></mfenced></mrow></mfenced><mn>1</mn></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>speed</mtext><mi>A</mi></msub><mo>=</mo><mfrac><msqrt><msup><mfenced><mrow><mo>-</mo><mn>6</mn></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mn>2</mn><mn>2</mn></msup><mo>+</mo><msup><mn>4</mn><mn>2</mn></msup></msqrt><mn>1</mn></mfrac></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>speed</mtext><mi>B</mi></msub><mo>=</mo><mfrac><msqrt><msup><mn>4</mn><mn>2</mn></msup><mo>+</mo><msup><mn>2</mn><mn>2</mn></msup><mo>+</mo><msup><mn>2</mn><mn>2</mn></msup></msqrt><mn>1</mn></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>speed</mtext><mi>A</mi></msub><mo>=</mo><mn>7</mn><mo>.</mo><mn>48</mn><mo>…</mo><mfenced><mrow><mn>2</mn><msqrt><mn>14</mn></msqrt></mrow></mfenced></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>speed</mtext><mi>B</mi></msub><mo>=</mo><mn>4</mn><mo>.</mo><mn>89</mn><mo>…</mo><mfenced><msqrt><mn>24</mn></msqrt></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>speed</mtext><mi>A</mi></msub><mo>></mo><msub><mtext>speed</mtext><mi>B</mi></msub></math> so <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> travels at a greater speed than <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempts to use the angle between two direction vectors formula <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mo> </mo><mi>θ</mi><mo>=</mo><mfrac><mrow><mfenced><mrow><mo>-</mo><mn>6</mn></mrow></mfenced><mfenced><mn>4</mn></mfenced><mo>+</mo><mfenced><mn>2</mn></mfenced><mfenced><mn>2</mn></mfenced><mo>+</mo><mfenced><mn>4</mn></mfenced><mfenced><mrow><mo>-</mo><mn>2</mn></mrow></mfenced></mrow><mrow><msqrt><msup><mfenced><mrow><mo>-</mo><mn>6</mn></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mn>2</mn><mn>2</mn></msup><mo>+</mo><msup><mn>4</mn><mn>2</mn></msup></msqrt><msqrt><msup><mn>4</mn><mn>2</mn></msup><mo>+</mo><msup><mn>2</mn><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mo>-</mo><mn>2</mn></mrow></mfenced><mn>2</mn></msup></msqrt></mrow></mfrac></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mo> </mo><mi>θ</mi><mo>=</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>7637</mn><mo>…</mo><mo> </mo><mfenced><mrow><mo>=</mo><mo>-</mo><mfrac><mn>7</mn><msqrt><mn>84</mn></msqrt></mfrac></mrow></mfenced></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi><mo>=</mo><mtext>arccos</mtext><mfenced><mrow><mo>-</mo><mn>0</mn><mo>.</mo><mn>7637</mn><mo>…</mo></mrow></mfenced><mo> </mo><mfenced><mrow><mo>=</mo><mn>2</mn><mo>.</mo><mn>4399</mn><mo>…</mo></mrow></mfenced></math></p>
<p>attempts to find the acute angle <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>180</mn><mo>°</mo><mo>-</mo><mi>θ</mi></math> using their value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>40</mn><mo>.</mo><mn>2</mn><mo>°</mo></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>for example, sets <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="bold-italic">r</mi><mi mathvariant="bold-italic">A</mi></msub><mfenced><msub><mi>t</mi><mn>1</mn></msub></mfenced><mo>=</mo><msub><mi mathvariant="bold-italic">r</mi><mi mathvariant="bold-italic">B</mi></msub><mfenced><msub><mi>t</mi><mn>2</mn></msub></mfenced></math> and forms at least two equations <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>19</mn><mo>-</mo><mn>6</mn><msub><mi>t</mi><mn>1</mn></msub><mo>=</mo><mn>1</mn><mo>+</mo><mn>4</mn><msub><mi>t</mi><mn>2</mn></msub></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>1</mn><mo>+</mo><mn>2</mn><msub><mi>t</mi><mn>1</mn></msub><mo>=</mo><mn>2</mn><msub><mi>t</mi><mn>2</mn></msub></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>+</mo><mn>4</mn><msub><mi>t</mi><mn>1</mn></msub><mo>=</mo><mn>12</mn><mo>-</mo><mn>2</mn><msub><mi>t</mi><mn>2</mn></msub></math></p>
<p><br><strong>Note:</strong> Award <em><strong>M0</strong> </em>for equations involving <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> only.</p>
<p><br><strong>EITHER</strong></p>
<p>attempts to solve the system of equations for one of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>t</mi><mn>1</mn></msub></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>t</mi><mn>2</mn></msub></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>t</mi><mn>1</mn></msub><mo>=</mo><mn>2</mn></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>t</mi><mn>2</mn></msub><mo>=</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p><br><strong>OR</strong></p>
<p>attempts to solve the system of equations for <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>t</mi><mn>1</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>t</mi><mn>2</mn></msub></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>t</mi><mn>1</mn></msub><mo>=</mo><mn>2</mn></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>t</mi><mn>2</mn></msub><mo>=</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p><br><strong>THEN</strong></p>
<p>substitutes their <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>t</mi><mn>1</mn></msub></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>t</mi><mn>2</mn></msub></math> value into the corresponding <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="bold-italic">r</mi><mi mathvariant="bold-italic">A</mi></msub></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="bold-italic">r</mi><mi mathvariant="bold-italic">B</mi></msub></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mrow><mn>7</mn><mo>,</mo><mn>3</mn><mo>,</mo><mn>9</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mtext>OP</mtext><mo>→</mo></mover><mo>=</mo><mfenced><mtable><mtr><mtd><mn>7</mn></mtd></mtr><mtr><mtd><mn>3</mn></mtd></mtr><mtr><mtd><mn>9</mn></mtd></mtr></mtable></mfenced></math>. Accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn></math> km east of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math> km north of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9</mn></math> km above sea level.</p>
<p> </p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempts to find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>t</mi><mn>1</mn></msub><mo>-</mo><msub><mi>t</mi><mn>2</mn></msub></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>t</mi><mn>1</mn></msub><mo>-</mo><msub><mi>t</mi><mn>2</mn></msub><mo>=</mo><mn>2</mn><mo>-</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>5</mn></math> minutes (<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>30</mn></math> seconds) <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">d.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p>attempts to find <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="bold-italic">r</mi><mi mathvariant="bold-italic">B</mi></msub><mo>-</mo><msub><mi mathvariant="bold-italic">r</mi><mi mathvariant="bold-italic">A</mi></msub></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="bold-italic">r</mi><mi mathvariant="bold-italic">B</mi></msub><mo>-</mo><msub><mi mathvariant="bold-italic">r</mi><mi mathvariant="bold-italic">A</mi></msub><mo>=</mo><mfenced><mtable><mtr><mtd><mo>-</mo><mn>18</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>11</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mi>t</mi><mfenced><mtable><mtr><mtd><mn>10</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>6</mn></mtd></mtr></mtable></mfenced></math></p>
<p>attempts to find their <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mo>(</mo><mi>t</mi><mo>)</mo></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><msqrt><msup><mfenced><mrow><mn>10</mn><mi>t</mi><mo>-</mo><mn>18</mn></mrow></mfenced><mn>2</mn></msup><mo>+</mo><mn>1</mn><mo>+</mo><msup><mfenced><mrow><mn>11</mn><mo>-</mo><mn>6</mn><mi>t</mi></mrow></mfenced><mn>2</mn></msup></msqrt></math> <em><strong>A1</strong></em></p>
<p><strong><br>OR</strong></p>
<p>attempts to find <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="bold-italic">r</mi><mi mathvariant="bold-italic">A</mi></msub><mo>-</mo><msub><mi mathvariant="bold-italic">r</mi><mi mathvariant="bold-italic">B</mi></msub></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="bold-italic">r</mi><mi mathvariant="bold-italic">A</mi></msub><mo>-</mo><msub><mi mathvariant="bold-italic">r</mi><mi mathvariant="bold-italic">B</mi></msub><mo>=</mo><mfenced><mtable><mtr><mtd><mn>18</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>11</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mi>t</mi><mfenced><mtable><mtr><mtd><mo>-</mo><mn>10</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>6</mn></mtd></mtr></mtable></mfenced></math></p>
<p>attempts to find their <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mo>(</mo><mi>t</mi><mo>)</mo></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><msqrt><msup><mfenced><mrow><mn>18</mn><mo>-</mo><mn>10</mn><mi>t</mi></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mo>-</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mo>-</mo><mn>11</mn><mo>+</mo><mn>6</mn><mi>t</mi></mrow></mfenced><mn>2</mn></msup></msqrt></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>M0M0A0</strong></em> for expressions using two different time parameters.</p>
<p><br><strong>THEN</strong></p>
<p>either attempts to find the local minimum point of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mo>(</mo><mi>t</mi><mo>)</mo></math> or attempts to find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mo>'</mo><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mn>0</mn></math> (or equivalent) <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>1</mn><mo>.</mo><mn>8088</mn><mo>…</mo><mo> </mo><mfenced><mrow><mo>=</mo><mfrac><mn>123</mn><mn>68</mn></mfrac></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mn>1</mn><mo>.</mo><mn>01459</mn><mo>…</mo></math></p>
<p>minimum value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mo>(</mo><mi>t</mi><mo>)</mo></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.</mo><mn>01</mn><mo> </mo><mfenced><mrow><mo>=</mo><mfrac><msqrt><mn>1190</mn></msqrt><mn>34</mn></mfrac></mrow></mfenced></math> (km) <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>M0</strong> </em>for attempts at the shortest distance between two lines.</p>
<p> </p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>General comment about this question: many candidates were not exposed to this setting of vectors question and were rather lost.</p>
<p>Part (a) Probably the least answered question on the whole paper. Many candidates left it blank, others tried using 3D vectors. Out of those who calculated the angle correctly, only a small percentage were able to provide the correct true bearing as a 3-digit figure.</p>
<p>Part (b) Well done by many candidates who used the direction vectors to calculate and compare the speeds. A number of candidates tried to use the average rate of change but mostly unsuccessfully.</p>
<p>Part (c) Most candidates used the correct vectors and the formula to obtain the obtuse angle. Then only some read the question properly to give the acute angle in degrees, as requested.</p>
<p>Part (d) Well done by many candidates who used two different parameters. They were able to solve and obtain two values for time, the difference in minutes and the correct point of intersection. A number of candidates only had one parameter, thus scoring no marks in part (d) (i). The frequent error in part (d)(ii) was providing incorrect units.</p>
<p>Part (e) Many correct answers were seen with an efficient way of setting the question and using their GDC to obtain the answer, graphically or numerically. Some gave time only instead of actually giving the minimal distance. A number of candidates tried to find the distance between two skew lines ignoring the fact that the lines intersect.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p>A water trough which is 10 metres long has a uniform cross-section in the shape of a semicircle with radius 0.5 metres. It is partly filled with water as shown in the following diagram of the cross-section. The centre of the circle is O and the angle KOL is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\theta ">
<mi>θ<!-- θ --></mi>
</math></span> radians.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-09_om_11.09.30.png" alt="M17/5/MATHL/HP2/ENG/TZ1/08"></p>
</div>
<div class="specification">
<p>The volume of water is increasing at a constant rate of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0.0008{\text{ }}{{\text{m}}^3}{{\text{s}}^{ - 1}}">
<mn>0.0008</mn>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>m</mtext>
</mrow>
<mn>3</mn>
</msup>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>s</mtext>
</mrow>
<mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an expression for the volume of water <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="V{\text{ }}({{\text{m}}^3})">
<mi>V</mi>
<mrow>
<mtext> </mtext>
</mrow>
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mrow>
<mtext>m</mtext>
</mrow>
<mn>3</mn>
</msup>
</mrow>
<mo stretchy="false">)</mo>
</math></span> in the trough in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\theta ">
<mi>θ</mi>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}\theta }}{{{\text{d}}t}}">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>θ</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
</math></span> when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\theta = \frac{\pi }{3}">
<mi>θ</mi>
<mo>=</mo>
<mfrac>
<mi>π</mi>
<mn>3</mn>
</mfrac>
</math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p>area of segment <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{2} \times {0.5^2} \times (\theta - \sin \theta )">
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mo>×</mo>
<mrow>
<msup>
<mn>0.5</mn>
<mn>2</mn>
</msup>
</mrow>
<mo>×</mo>
<mo stretchy="false">(</mo>
<mi>θ</mi>
<mo>−</mo>
<mi>sin</mi>
<mo></mo>
<mi>θ</mi>
<mo stretchy="false">)</mo>
</math></span> <strong><em>M1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="V = {\text{area of segment}} \times 10">
<mi>V</mi>
<mo>=</mo>
<mrow>
<mtext>area of segment</mtext>
</mrow>
<mo>×</mo>
<mn>10</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="V = \frac{5}{4}(\theta - \sin \theta )">
<mi>V</mi>
<mo>=</mo>
<mfrac>
<mn>5</mn>
<mn>4</mn>
</mfrac>
<mo stretchy="false">(</mo>
<mi>θ</mi>
<mo>−</mo>
<mi>sin</mi>
<mo></mo>
<mi>θ</mi>
<mo stretchy="false">)</mo>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}V}}{{{\text{d}}t}} = \frac{5}{4}(1 - \cos \theta )\frac{{{\text{d}}\theta }}{{{\text{d}}t}}">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>V</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mn>5</mn>
<mn>4</mn>
</mfrac>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>−</mo>
<mi>cos</mi>
<mo></mo>
<mi>θ</mi>
<mo stretchy="false">)</mo>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>θ</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
</math></span> <strong><em>M1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0.0008 = \frac{5}{4}\left( {1 - \cos \frac{\pi }{3}} \right)\frac{{{\text{d}}\theta }}{{{\text{d}}t}}">
<mn>0.0008</mn>
<mo>=</mo>
<mfrac>
<mn>5</mn>
<mn>4</mn>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>cos</mi>
<mo></mo>
<mfrac>
<mi>π</mi>
<mn>3</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>θ</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
</math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}\theta }}{{{\text{d}}t}} = 0.00128{\text{ }}({\text{rad}}\,{s^{ - 1}})">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>θ</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mn>0.00128</mn>
<mrow>
<mtext> </mtext>
</mrow>
<mo stretchy="false">(</mo>
<mrow>
<mtext>rad</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<msup>
<mi>s</mi>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mo stretchy="false">)</mo>
</math></span> <strong><em>A1</em></strong></p>
<p><strong>METHOD 2</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}\theta }}{{{\text{d}}t}} = \frac{{{\text{d}}\theta }}{{{\text{d}}V}} \times \frac{{{\text{d}}V}}{{{\text{d}}t}}">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>θ</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>θ</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>V</mi>
</mrow>
</mfrac>
<mo>×</mo>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>V</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
</math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}V}}{{{\text{d}}\theta }} = \frac{5}{4}(1 - \cos \theta )">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>V</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>θ</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mn>5</mn>
<mn>4</mn>
</mfrac>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>−</mo>
<mi>cos</mi>
<mo></mo>
<mi>θ</mi>
<mo stretchy="false">)</mo>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}\theta }}{{{\text{d}}t}} = \frac{{4 \times 0.0008}}{{5\left( {1 - \cos \frac{\pi }{3}} \right)}}">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>θ</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mn>4</mn>
<mo>×</mo>
<mn>0.0008</mn>
</mrow>
<mrow>
<mn>5</mn>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>cos</mi>
<mo></mo>
<mfrac>
<mi>π</mi>
<mn>3</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
</mfrac>
</math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}\theta }}{{{\text{d}}t}} = 0.00128\left( {\frac{4}{{3125}}} \right)({\text{rad }}{s^{ - 1}})">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>θ</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mn>0.00128</mn>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>4</mn>
<mrow>
<mn>3125</mn>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo stretchy="false">(</mo>
<mrow>
<mtext>rad </mtext>
</mrow>
<mrow>
<msup>
<mi>s</mi>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mo stretchy="false">)</mo>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> defined by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = 3x\arccos (x)">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>3</mn>
<mi>x</mi>
<mi>arccos</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 1 \leqslant x \leqslant 1">
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo>⩽<!-- ⩽ --></mo>
<mi>x</mi>
<mo>⩽<!-- ⩽ --></mo>
<mn>1</mn>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> indicating clearly any intercepts with the axes 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>State the range of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Solve the inequality <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left| {3x\arccos (x)} \right| > 1">
<mrow>
<mo>|</mo>
<mrow>
<mn>3</mn>
<mi>x</mi>
<mi>arccos</mi>
<mo></mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mrow>
<mo>|</mo>
</mrow>
<mo>></mo>
<mn>1</mn>
</math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><img src="images/Schermafbeelding_2017-03-01_om_06.12.12.png" alt="N16/5/MATHL/HP2/ENG/TZ0/05.a/M"></p>
<p>correct shape passing through the origin and correct domain <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Endpoint coordinates are not required. The domain can be indicated by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 1">
<mo>−</mo>
<mn>1</mn>
</math></span> and 1 marked on the axis.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(0.652,{\text{ }}1.68)">
<mo stretchy="false">(</mo>
<mn>0.652</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>1.68</mn>
<mo stretchy="false">)</mo>
</math></span> <strong><em>A1</em></strong></p>
<p>two correct intercepts (coordinates not required) <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>A graph passing through the origin is sufficient for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(0,{\text{ }}0)">
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>0</mn>
<mo stretchy="false">)</mo>
</math></span>.</p>
<p> </p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="[-9.42,{\text{ }}1.68]{\text{ }}({\text{or }} - 3\pi ,{\text{ }}1.68])">
<mo stretchy="false">[</mo>
<mo>−</mo>
<mn>9.42</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>1.68</mn>
<mo stretchy="false">]</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mo stretchy="false">(</mo>
<mrow>
<mtext>or </mtext>
</mrow>
<mo>−</mo>
<mn>3</mn>
<mi>π</mi>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>1.68</mn>
<mo stretchy="false">]</mo>
<mo stretchy="false">)</mo>
</math></span> <strong><em>A1A1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>A1A0 </em></strong>for open or semi-open intervals with correct endpoints. Award <strong><em>A1A0 </em></strong>for closed intervals with one correct endpoint.</p>
<p> </p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempting to solve either <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left| {3x\arccos (x)} \right| > 1">
<mrow>
<mo>|</mo>
<mrow>
<mn>3</mn>
<mi>x</mi>
<mi>arccos</mi>
<mo></mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mrow>
<mo>|</mo>
</mrow>
<mo>></mo>
<mn>1</mn>
</math></span> (or equivalent) or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left| {3x\arccos (x)} \right| = 1">
<mrow>
<mo>|</mo>
<mrow>
<mn>3</mn>
<mi>x</mi>
<mi>arccos</mi>
<mo></mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mrow>
<mo>|</mo>
</mrow>
<mo>=</mo>
<mn>1</mn>
</math></span> (or equivalent) (<em>eg</em>. graphically) <strong><em>(M1)</em></strong></p>
<p><img src="images/Schermafbeelding_2017-03-01_om_06.22.47.png" alt="N16/5/MATHL/HP2/ENG/TZ0/05.c/M"></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = - 0.189,{\text{ }}0.254,{\text{ }}0.937">
<mi>x</mi>
<mo>=</mo>
<mo>−</mo>
<mn>0.189</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>0.254</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>0.937</mn>
</math></span> <strong><em>(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 1 \leqslant x < - 0.189{\text{ or }}0.254 < x < 0.937">
<mo>−</mo>
<mn>1</mn>
<mo>⩽</mo>
<mi>x</mi>
<mo><</mo>
<mo>−</mo>
<mn>0.189</mn>
<mrow>
<mtext> or </mtext>
</mrow>
<mn>0.254</mn>
<mo><</mo>
<mi>x</mi>
<mo><</mo>
<mn>0.937</mn>
</math></span> <strong><em>A1A1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>A0 </em></strong>for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x < - 0.189">
<mi>x</mi>
<mo><</mo>
<mo>−</mo>
<mn>0.189</mn>
</math></span>.</p>
<p> </p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = 2{\sin ^2}x + 7\sin 2x + \tan x - 9,{\text{ }}0 \leqslant x < \frac{\pi }{2}">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>2</mn>
<mrow>
<msup>
<mi>sin</mi>
<mn>2</mn>
</msup>
</mrow>
<mi>x</mi>
<mo>+</mo>
<mn>7</mn>
<mi>sin</mi>
<mo><!-- --></mo>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mi>tan</mi>
<mo><!-- --></mo>
<mi>x</mi>
<mo>−<!-- − --></mo>
<mn>9</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>0</mn>
<mo>⩽<!-- ⩽ --></mo>
<mi>x</mi>
<mo><</mo>
<mfrac>
<mi>π<!-- π --></mi>
<mn>2</mn>
</mfrac>
</math></span>.</p>
</div>
<div class="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u = \tan x">
<mi>u</mi>
<mo>=</mo>
<mi>tan</mi>
<mo><!-- --></mo>
<mi>x</mi>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine an expression for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f’(x)"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></span> in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch a graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f’(x)"> <mi>y</mi> <mo>=</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></span> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0 \leqslant x < \frac{\pi }{2}"> <mn>0</mn> <mo>⩽</mo> <mi>x</mi> <mo><</mo> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>-coordinate(s) of the point(s) of inflexion of the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f(x)"> <mi>y</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></span>, labelling these clearly on the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f’(x)"> <mi>y</mi> <mo>=</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Express <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sin x"> <mi>sin</mi> <mo></mo> <mi>x</mi> </math></span> in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mu "><mi>u</mi></math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Express <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sin 2x"> <mi>sin</mi> <mo></mo> <mn>2</mn> <mi>x</mi> </math></span> in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u"> <mi>u</mi> </math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = 0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </math></span> can be expressed as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u^3} - 7{u^2} + 15u - 9 = 0"> <mrow> <msup> <mi>u</mi> <mn>3</mn> </msup> </mrow> <mo>−</mo> <mn>7</mn> <mrow> <msup> <mi>u</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>15</mn> <mi>u</mi> <mo>−</mo> <mn>9</mn> <mo>=</mo> <mn>0</mn> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Solve the equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = 0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </math></span>, giving your answers in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\arctan k"> <mi>arctan</mi> <mo></mo> <mi>k</mi> </math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k \in \mathbb{Z}"> <mi>k</mi> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> </math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f’(x) = 4\sin x\cos x + 14\cos 2x + {\sec ^2}x"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>4</mn> <mi>sin</mi> <mo></mo> <mi>x</mi> <mi>cos</mi> <mo></mo> <mi>x</mi> <mo>+</mo> <mn>14</mn> <mi>cos</mi> <mo></mo> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mrow> <msup> <mi>sec</mi> <mn>2</mn> </msup> </mrow> <mi>x</mi> </math></span> (or equivalent) <strong><em>(M1)A1</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img src="images/Schermafbeelding_2018-02-08_om_16.47.49.png" alt="N17/5/MATHL/HP2/ENG/TZ0/11.a.ii/M"> <strong><em>A1A1A1A1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>A1 </em></strong>for correct behaviour at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 0"> <mi>x</mi> <mo>=</mo> <mn>0</mn> </math></span>, <strong><em>A1 </em></strong>for correct domain and correct behaviour for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x \to \frac{\pi }{2}"> <mi>x</mi> <mo stretchy="false">→</mo> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </math></span>, <strong><em>A1 </em></strong>for two clear intersections with <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>-axis and minimum point, <strong><em>A1 </em></strong>for clear maximum point.</p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 0.0736"> <mi>x</mi> <mo>=</mo> <mn>0.0736</mn> </math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 1.13"> <mi>x</mi> <mo>=</mo> <mn>1.13</mn> </math></span> <strong><em>A1</em></strong></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to write <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sin x"> <mi>sin</mi> <mo></mo> <mi>x</mi> </math></span> in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u"> <mi>u</mi> </math></span> only <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sin x = \frac{u}{{\sqrt {1 + {u^2}} }}"> <mi>sin</mi> <mo></mo> <mi>x</mi> <mo>=</mo> <mfrac> <mi>u</mi> <mrow> <msqrt> <mn>1</mn> <mo>+</mo> <mrow> <msup> <mi>u</mi> <mn>2</mn> </msup> </mrow> </msqrt> </mrow> </mfrac> </math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\cos x = \frac{1}{{\sqrt {1 + {u^2}} }}"> <mi>cos</mi> <mo></mo> <mi>x</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <msqrt> <mn>1</mn> <mo>+</mo> <mrow> <msup> <mi>u</mi> <mn>2</mn> </msup> </mrow> </msqrt> </mrow> </mfrac> </math></span> <strong><em>(A1)</em></strong></p>
<p>attempt to use <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sin 2x = 2\sin x\cos x{\text{ }}\left( { = 2\frac{u}{{\sqrt {1 + {u^2}} }}\frac{1}{{\sqrt {1 + {u^2}} }}} \right)"> <mi>sin</mi> <mo></mo> <mn>2</mn> <mi>x</mi> <mo>=</mo> <mn>2</mn> <mi>sin</mi> <mo></mo> <mi>x</mi> <mi>cos</mi> <mo></mo> <mi>x</mi> <mrow> <mtext> </mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mn>2</mn> <mfrac> <mi>u</mi> <mrow> <msqrt> <mn>1</mn> <mo>+</mo> <mrow> <msup> <mi>u</mi> <mn>2</mn> </msup> </mrow> </msqrt> </mrow> </mfrac> <mfrac> <mn>1</mn> <mrow> <msqrt> <mn>1</mn> <mo>+</mo> <mrow> <msup> <mi>u</mi> <mn>2</mn> </msup> </mrow> </msqrt> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sin 2x = \frac{{2u}}{{1 + {u^2}}}"> <mi>sin</mi> <mo></mo> <mn>2</mn> <mi>x</mi> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> <mi>u</mi> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mrow> <msup> <mi>u</mi> <mn>2</mn> </msup> </mrow> </mrow> </mfrac> </math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2{\sin ^2}x + 7\sin 2x + \tan x - 9 = 0"> <mn>2</mn> <mrow> <msup> <mi>sin</mi> <mn>2</mn> </msup> </mrow> <mi>x</mi> <mo>+</mo> <mn>7</mn> <mi>sin</mi> <mo></mo> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mi>tan</mi> <mo></mo> <mi>x</mi> <mo>−</mo> <mn>9</mn> <mo>=</mo> <mn>0</mn> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{2{u^2}}}{{1 + {u^2}}} + \frac{{14u}}{{1 + {u^2}}} + u - 9{\text{ }}( = 0)"> <mfrac> <mrow> <mn>2</mn> <mrow> <msup> <mi>u</mi> <mn>2</mn> </msup> </mrow> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mrow> <msup> <mi>u</mi> <mn>2</mn> </msup> </mrow> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mn>14</mn> <mi>u</mi> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mrow> <msup> <mi>u</mi> <mn>2</mn> </msup> </mrow> </mrow> </mfrac> <mo>+</mo> <mi>u</mi> <mo>−</mo> <mn>9</mn> <mrow> <mtext> </mtext> </mrow> <mo stretchy="false">(</mo> <mo>=</mo> <mn>0</mn> <mo stretchy="false">)</mo> </math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{2{u^2} + 14u + u(1 + {u^2}) - 9(1 + {u^2})}}{{1 + {u^2}}} = 0"> <mfrac> <mrow> <mn>2</mn> <mrow> <msup> <mi>u</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>14</mn> <mi>u</mi> <mo>+</mo> <mi>u</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mrow> <msup> <mi>u</mi> <mn>2</mn> </msup> </mrow> <mo stretchy="false">)</mo> <mo>−</mo> <mn>9</mn> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mrow> <msup> <mi>u</mi> <mn>2</mn> </msup> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>1</mn> <mo>+</mo> <mrow> <msup> <mi>u</mi> <mn>2</mn> </msup> </mrow> </mrow> </mfrac> <mo>=</mo> <mn>0</mn> </math></span> (or equivalent) <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u^3} - 7{u^2} + 15u - 9 = 0"> <mrow> <msup> <mi>u</mi> <mn>3</mn> </msup> </mrow> <mo>−</mo> <mn>7</mn> <mrow> <msup> <mi>u</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>15</mn> <mi>u</mi> <mo>−</mo> <mn>9</mn> <mo>=</mo> <mn>0</mn> </math></span> <strong><em>AG</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u = 1"> <mi>u</mi> <mo>=</mo> <mn>1</mn> </math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u = 3"> <mi>u</mi> <mo>=</mo> <mn>3</mn> </math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \arctan (1)"> <mi>x</mi> <mo>=</mo> <mi>arctan</mi> <mo></mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \arctan (3)"> <mi>x</mi> <mo>=</mo> <mi>arctan</mi> <mo></mo> <mo stretchy="false">(</mo> <mn>3</mn> <mo stretchy="false">)</mo> </math></span> <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Only accept answers given the required form.</p>
<p> </p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Two submarines A and B have their routes planned so that their positions at time <em>t</em> hours, 0 ≤ <em>t</em> < 20 , would be defined by the position vectors <em><strong>r</strong><sub>A</sub></em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left( \begin{gathered} \,2 \hfill \\ \,4 \hfill \\ - 1 \hfill \\ \end{gathered} \right) + t\left( \begin{gathered} - 1 \hfill \\ \,1 \hfill \\ - 0.15 \hfill \\ \end{gathered} \right)">
<mo>=</mo>
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mn>4</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mi>t</mi>
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>−<!-- − --></mo>
<mn>0.15</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
</math></span> and <em><strong>r</strong><sub>B</sub></em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left( \begin{gathered} \,0 \hfill \\ \,3.2 \hfill \\ - 2 \hfill \\ \end{gathered} \right) + t\left( \begin{gathered} - 0.5 \hfill \\ \,1.2 \hfill \\ \,0.1 \hfill \\ \end{gathered} \right)">
<mo>=</mo>
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mn>3.2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>−<!-- − --></mo>
<mn>2</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mi>t</mi>
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mo>−<!-- − --></mo>
<mn>0.5</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mn>1.2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mn>0.1</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
</math></span> relative to a fixed point on the surface of the ocean (all lengths are in kilometres).</p>
</div>
<div class="specification">
<p>To avoid the collision submarine B adjusts its velocity so that its position vector is now given by</p>
<p style="padding-left: 120px;"><em><strong>r</strong><sub>B</sub></em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left( \begin{gathered} \,0 \hfill \\ \,3.2 \hfill \\ - 2 \hfill \\ \end{gathered} \right) + t\left( \begin{gathered} - 0.45 \hfill \\ \,1.08 \hfill \\ \,0.09 \hfill \\ \end{gathered} \right)">
<mo>=</mo>
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mn>3.2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>−<!-- − --></mo>
<mn>2</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mi>t</mi>
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mo>−<!-- − --></mo>
<mn>0.45</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mn>1.08</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mn>0.09</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the two submarines would collide at a point P and write down the coordinates of P.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that submarine B travels in the same direction as originally planned.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <em>t</em> when submarine B passes through P.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an expression for the distance between the two submarines in terms of <em>t</em>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <em>t</em> when the two submarines are closest together.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the distance between the two submarines at this time.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.iii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p 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><em><strong>r</strong><sub>A </sub>= <strong>r</strong><sub>B <strong>(M1)</strong></sub></em></p>
<p>2 − <em>t</em> = − 0.5t ⇒ <em>t</em> = 4 <strong>A1</strong></p>
<p>checking <em>t</em> = 4 satisfies 4 + <em>t</em> = 3.2 + 1.2<em>t</em> and − 1 − 0.15<em>t</em> = − 2 + 0.1<em>t <strong>R1</strong></em></p>
<p>P(−2, 8, −1.6) <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Do not award final <em><strong>A1</strong></em> if answer given as column vector.</p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0.9 \times \left( \begin{gathered} - 0.5 \hfill \\ \,1.2 \hfill \\ \,0.1 \hfill \\ \end{gathered} \right) = \left( \begin{gathered} - 0.45 \hfill \\ \,1.08 \hfill \\ \,0.09 \hfill \\ \end{gathered} \right)">
<mn>0.9</mn>
<mo>×</mo>
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mo>−</mo>
<mn>0.5</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mn>1.2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mn>0.1</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mo>−</mo>
<mn>0.45</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mn>1.08</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mn>0.09</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Accept use of cross product equalling zero.</p>
<p>hence in the same direction <em><strong>AG</strong></em></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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( \begin{gathered} \, - 0.45t \hfill \\ 3.2 + 1.08t \hfill \\ - 2 + 0.09t \hfill \\ \end{gathered} \right) = \left( \begin{gathered} - 2 \hfill \\ \,8 \hfill \\ - 1.6 \hfill \\ \end{gathered} \right)">
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mo>−</mo>
<mn>0.45</mn>
<mi>t</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>3.2</mn>
<mo>+</mo>
<mn>1.08</mn>
<mi>t</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>−</mo>
<mn>2</mn>
<mo>+</mo>
<mn>0.09</mn>
<mi>t</mi>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mo>−</mo>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mn>8</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>−</mo>
<mn>1.6</mn>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
</math></span> <em><strong>M1</strong></em></p>
<p><strong>Note:</strong> The <strong><em>M1</em></strong> can be awarded for any one of the resultant equations.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow t = \frac{{40}}{9} = 4.44 \ldots ">
<mo stretchy="false">⇒</mo>
<mi>t</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mn>40</mn>
</mrow>
<mn>9</mn>
</mfrac>
<mo>=</mo>
<mn>4.44</mn>
<mo>…</mo>
</math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><em><strong>r</strong><sub>A</sub></em> − <em><strong>r</strong><sub>B</sub></em> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( \begin{gathered} \,2 - t \hfill \\ \,4 + t \hfill \\ - 1 - 0.15t \hfill \\ \end{gathered} \right) - \left( \begin{gathered} \, - 0.45t \hfill \\ 3.2 + 1.08t \hfill \\ - 2 + 0.09t \hfill \\ \end{gathered} \right)">
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mo>−</mo>
<mi>t</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mn>4</mn>
<mo>+</mo>
<mi>t</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>−</mo>
<mn>1</mn>
<mo>−</mo>
<mn>0.15</mn>
<mi>t</mi>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mo>−</mo>
<mn>0.45</mn>
<mi>t</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>3.2</mn>
<mo>+</mo>
<mn>1.08</mn>
<mi>t</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mo>−</mo>
<mn>2</mn>
<mo>+</mo>
<mn>0.09</mn>
<mi>t</mi>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
</math></span> <em><strong> (M1)(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left( \begin{gathered} \,2 - 0.55t \hfill \\ \,0.8 - 0.08t \hfill \\ 1 - 0.24t \hfill \\ \end{gathered} \right)">
<mo>=</mo>
<mrow>
<mo>(</mo>
<mtable rowspacing="3pt" columnspacing="1em" displaystyle="true">
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mo>−</mo>
<mn>0.55</mn>
<mi>t</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mspace width="thinmathspace"></mspace>
<mn>0.8</mn>
<mo>−</mo>
<mn>0.08</mn>
<mi>t</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
<mo>−</mo>
<mn>0.24</mn>
<mi>t</mi>
</mtd>
</mtr>
</mtable>
<mo>)</mo>
</mrow>
</math></span> <em><strong>(A1)</strong></em></p>
<p><strong>Note</strong>: Accept <em><strong>r</strong><sub>A</sub></em> − <em><strong>r</strong><sub>B</sub></em>.</p>
<p>distance <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="D = \sqrt {{{\left( {2 - 0.55t} \right)}^2} + {{\left( {0.8 - 0.08t} \right)}^2} + {{\left( {1 - 0.24t} \right)}^2}} ">
<mi>D</mi>
<mo>=</mo>
<msqrt>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mo>−</mo>
<mn>0.55</mn>
<mi>t</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>0.8</mn>
<mo>−</mo>
<mn>0.08</mn>
<mi>t</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mn>0.24</mn>
<mi>t</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
</math></span> <em><strong> M1A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( { = \sqrt {8.64 - 2.688t + 0.317{t^2}} } \right)">
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<msqrt>
<mn>8.64</mn>
<mo>−</mo>
<mn>2.688</mn>
<mi>t</mi>
<mo>+</mo>
<mn>0.317</mn>
<mrow>
<msup>
<mi>t</mi>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>minimum when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}D}}{{{\text{d}}t}} = 0">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>D</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mn>0</mn>
</math></span> <em><strong>(M1)</strong></em></p>
<p><em>t</em> = 3.83 <em><strong>A1</strong></em></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>0.511 (km) <em><strong>A1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">c.iii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.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>The points A, B and C have the following position vectors with respect to an origin O.</p>
<p style="text-align: center;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\rm{OA}}} = 2">
<mover>
<mrow>
<mrow>
<mi mathvariant="normal">O</mi>
<mi mathvariant="normal">A</mi>
</mrow>
</mrow>
<mo>→<!-- → --></mo>
</mover>
<mo>=</mo>
<mn>2</mn>
</math></span><strong><em>i</em></strong> + <strong><em>j</em></strong> – 2<strong><em>k</em></strong></p>
<p style="text-align: center;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\rm{OB}}} = 2">
<mover>
<mrow>
<mrow>
<mi mathvariant="normal">O</mi>
<mi mathvariant="normal">B</mi>
</mrow>
</mrow>
<mo>→<!-- → --></mo>
</mover>
<mo>=</mo>
<mn>2</mn>
</math></span><strong><em>i</em></strong> – <strong><em>j</em></strong> + 2<strong><em>k</em></strong></p>
<p style="text-align: center;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\rm{OC}}} = ">
<mover>
<mrow>
<mrow>
<mi mathvariant="normal">O</mi>
<mi mathvariant="normal">C</mi>
</mrow>
</mrow>
<mo>→<!-- → --></mo>
</mover>
<mo>=</mo>
</math></span> <strong><em>i</em></strong> + 3<strong><em>j</em></strong> + 3<strong><em>k</em></strong></p>
</div>
<div class="specification">
<p>The plane <em>Π</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="_2">
<msub>
<mi></mi>
<mn>2</mn>
</msub>
</math></span> contains the points O, A and B and the plane <em>Π</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="_3">
<msub>
<mi></mi>
<mn>3</mn>
</msub>
</math></span> contains the points O, A and C.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the vector equation of the line (BC).</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine whether or not the lines (OA) and (BC) intersect.</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>Find the Cartesian equation of the plane <em>Π</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="_1">
<msub>
<mi></mi>
<mn>1</mn>
</msub>
</math></span>, which passes through C and is perpendicular to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\rm{OA}}} ">
<mover>
<mrow>
<mrow>
<mi mathvariant="normal">O</mi>
<mi mathvariant="normal">A</mi>
</mrow>
</mrow>
<mo>→</mo>
</mover>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the line (BC) lies in the plane <em>Π</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="_1">
<msub>
<mi></mi>
<mn>1</mn>
</msub>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Verify that 2<strong><em>j </em></strong>+ <strong><em>k </em></strong>is perpendicular to the plane <em>Π</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="_2">
<msub>
<mi></mi>
<mn>2</mn>
</msub>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find a vector perpendicular to the plane <em>Π</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="_3">
<msub>
<mi></mi>
<mn>3</mn>
</msub>
</math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the acute angle between the planes <em>Π</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="_2">
<msub>
<mi></mi>
<mn>2</mn>
</msub>
</math></span> and <em>Π</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="_3">
<msub>
<mi></mi>
<mn>3</mn>
</msub>
</math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">g.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\rm{BC}}} ">
<mover>
<mrow>
<mrow>
<mi mathvariant="normal">B</mi>
<mi mathvariant="normal">C</mi>
</mrow>
</mrow>
<mo>→</mo>
</mover>
</math></span> = (<strong><em>i</em></strong> + 3<strong><em>j</em></strong> + 3<strong><em>k</em></strong>) <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - ">
<mo>−</mo>
</math></span> (2<strong><em>i</em></strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - ">
<mo>−</mo>
</math></span> <strong><em>j</em></strong> + 2<strong><em>k</em></strong>) = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - ">
<mo>−</mo>
</math></span><strong><em>i</em></strong> + 4<strong><em>j</em></strong> + <strong><em>k</em></strong> <strong><em>(A1)</em></strong></p>
<p><strong><em>r</em></strong> = (2<strong><em>i</em></strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - ">
<mo>−</mo>
</math></span> <strong><em>j</em></strong> + 2<strong><em>k</em></strong>) + <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\lambda ">
<mi>λ</mi>
</math></span>(<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - ">
<mo>−</mo>
</math></span><strong><em>i</em></strong> + 4<strong><em>j</em></strong> + <strong><em>k</em></strong>)</p>
<p>(or <strong><em>r</em></strong> = (<strong><em>i</em></strong> + 3<strong><em>j</em></strong> + 3<strong><em>k</em></strong>) + <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\lambda ">
<mi>λ</mi>
</math></span>(<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - ">
<mo>−</mo>
</math></span><strong><em>i</em></strong> + 4<strong><em>j </em></strong>+ <strong><em>k</em></strong>) <strong><em>(M1)A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Do not award <strong><em>A1 </em></strong>unless <strong><em>r </em></strong>= or equivalent correct notation seen.</p>
<p> </p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to write in parametric form using two different parameters <strong>AND </strong>equate <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2\mu = 2 - \lambda ">
<mn>2</mn>
<mi>μ</mi>
<mo>=</mo>
<mn>2</mn>
<mo>−</mo>
<mi>λ</mi>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mu = - 1 + 4\lambda ">
<mi>μ</mi>
<mo>=</mo>
<mo>−</mo>
<mn>1</mn>
<mo>+</mo>
<mn>4</mn>
<mi>λ</mi>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 2\mu = 2 + \lambda ">
<mo>−</mo>
<mn>2</mn>
<mi>μ</mi>
<mo>=</mo>
<mn>2</mn>
<mo>+</mo>
<mi>λ</mi>
</math></span> <strong><em>A1</em></strong></p>
<p>attempt to solve first pair of simultaneous equations for two parameters <strong><em>M1</em></strong></p>
<p>solving first two equations gives <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\lambda = \frac{4}{9},{\text{ }}\mu = \frac{7}{9}">
<mi>λ</mi>
<mo>=</mo>
<mfrac>
<mn>4</mn>
<mn>9</mn>
</mfrac>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>μ</mi>
<mo>=</mo>
<mfrac>
<mn>7</mn>
<mn>9</mn>
</mfrac>
</math></span> <strong><em>(A1)</em></strong></p>
<p>substitution of these two values in third equation <strong><em>(M1)</em></strong></p>
<p>since the values do not fit, the lines do not intersect <strong><em>R1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Candidates may note that adding the first and third equations immediately leads to a contradiction and hence they can immediately deduce that the lines do not intersect.</p>
<p> </p>
<p><strong><em>[6 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>plane is of the form <strong><em>r</em></strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \bullet ">
<mo>∙</mo>
</math></span> (2<strong><em>i</em></strong> + <strong><em>j</em></strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - ">
<mo>−</mo>
</math></span> 2<strong><em>k</em></strong>) = <em>d</em> <strong><em>(A1)</em></strong></p>
<p><em>d </em>= (<strong><em>i</em></strong> + 3<strong><em>j</em></strong> + 3<strong><em>k</em></strong>) <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \bullet ">
<mo>∙</mo>
</math></span> (2<strong><em>i</em></strong> + <strong><em>j</em></strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - ">
<mo>−</mo>
</math></span> 2<strong><em>k</em></strong>) = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - ">
<mo>−</mo>
</math></span>1 <strong><em>(M1)</em></strong></p>
<p>hence Cartesian form of plane is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2x + y - 2z = - 1">
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mi>y</mi>
<mo>−</mo>
<mn>2</mn>
<mi>z</mi>
<mo>=</mo>
<mo>−</mo>
<mn>1</mn>
</math></span> <strong><em>A1</em></strong></p>
<p><strong>METHOD 2</strong></p>
<p>plane is of the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2x + y - 2z = d">
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mi>y</mi>
<mo>−</mo>
<mn>2</mn>
<mi>z</mi>
<mo>=</mo>
<mi>d</mi>
</math></span> <strong><em>(A1)</em></strong></p>
<p>substituting <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(1,{\text{ }}3,{\text{ }}3)">
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>3</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>3</mn>
<mo stretchy="false">)</mo>
</math></span> (to find gives <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2 + 3 - 6 = - 1">
<mn>2</mn>
<mo>+</mo>
<mn>3</mn>
<mo>−</mo>
<mn>6</mn>
<mo>=</mo>
<mo>−</mo>
<mn>1</mn>
</math></span>) <strong><em>(M1)</em></strong></p>
<p>hence Cartesian form of plane is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2x + y - 2z = - 1">
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mi>y</mi>
<mo>−</mo>
<mn>2</mn>
<mi>z</mi>
<mo>=</mo>
<mo>−</mo>
<mn>1</mn>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>attempt scalar product of direction vector BC with normal to plane <strong><em>M1</em></strong></p>
<p>(<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - ">
<mo>−</mo>
</math></span><strong><em>i</em></strong> + 4<strong><em>j</em></strong> + <strong><em>k</em></strong>) <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \bullet ">
<mo>∙</mo>
</math></span> (2<strong><em>i</em></strong> + <strong><em>j</em></strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - ">
<mo>−</mo>
</math></span> 2<strong><em>k</em></strong>) <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = - 2 + 4 - 2">
<mo>=</mo>
<mo>−</mo>
<mn>2</mn>
<mo>+</mo>
<mn>4</mn>
<mo>−</mo>
<mn>2</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 0">
<mo>=</mo>
<mn>0</mn>
</math></span> <strong><em>A1</em></strong></p>
<p>hence BC lies in <em>Π</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="_1">
<msub>
<mi></mi>
<mn>1</mn>
</msub>
</math></span> <strong><em>AG</em></strong></p>
<p><strong>METHOD 2</strong></p>
<p>substitute eqn of line into plane <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{line }}r = \left( {\begin{array}{*{20}{c}} 2 \\ { - 1} \\ 2 \end{array}} \right) + \lambda \left( {\begin{array}{*{20}{c}} { - 1} \\ 4 \\ 1 \end{array}} \right).{\text{ Plane }}{\pi _1}:2x + y - 2z = - 1">
<mrow>
<mtext>line </mtext>
</mrow>
<mi>r</mi>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mi>λ</mi>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>.</mo>
<mrow>
<mtext> Plane </mtext>
</mrow>
<mrow>
<msub>
<mi>π</mi>
<mn>1</mn>
</msub>
</mrow>
<mo>:</mo>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mi>y</mi>
<mo>−</mo>
<mn>2</mn>
<mi>z</mi>
<mo>=</mo>
<mo>−</mo>
<mn>1</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2(2 - \lambda ) + ( - 1 + 4\lambda ) - 2(2 + \lambda )">
<mn>2</mn>
<mo stretchy="false">(</mo>
<mn>2</mn>
<mo>−</mo>
<mi>λ</mi>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mo stretchy="false">(</mo>
<mo>−</mo>
<mn>1</mn>
<mo>+</mo>
<mn>4</mn>
<mi>λ</mi>
<mo stretchy="false">)</mo>
<mo>−</mo>
<mn>2</mn>
<mo stretchy="false">(</mo>
<mn>2</mn>
<mo>+</mo>
<mi>λ</mi>
<mo stretchy="false">)</mo>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = - 1">
<mo>=</mo>
<mo>−</mo>
<mn>1</mn>
</math></span> <strong><em>A1</em></strong></p>
<p>hence BC lies in <em>Π</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="_1">
<msub>
<mi></mi>
<mn>1</mn>
</msub>
</math></span> <strong><em>AG</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Candidates may also just substitute <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2i - j + 2k">
<mn>2</mn>
<mi>i</mi>
<mo>−</mo>
<mi>j</mi>
<mo>+</mo>
<mn>2</mn>
<mi>k</mi>
</math></span> into the plane since they are told C lies on <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\pi _1}">
<mrow>
<msub>
<mi>π</mi>
<mn>1</mn>
</msub>
</mrow>
</math></span>.</p>
<p> </p>
<p><strong>Note:</strong> Do not award <strong><em>A1FT</em></strong>.</p>
<p> </p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>applying scalar product to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\rm{OA}}} ">
<mover>
<mrow>
<mrow>
<mi mathvariant="normal">O</mi>
<mi mathvariant="normal">A</mi>
</mrow>
</mrow>
<mo>→</mo>
</mover>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\rm{OB}}} ">
<mover>
<mrow>
<mrow>
<mi mathvariant="normal">O</mi>
<mi mathvariant="normal">B</mi>
</mrow>
</mrow>
<mo>→</mo>
</mover>
</math></span> <strong><em>M1</em></strong></p>
<p>(2<strong><em>j</em></strong> + <strong><em>k</em></strong>) <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \bullet ">
<mo>∙</mo>
</math></span> (2<strong><em>i</em></strong> + <strong><em>j </em></strong><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - ">
<mo>−</mo>
</math></span> 2<strong><em>k</em></strong>) = 0 <strong><em>A1</em></strong></p>
<p>(2<strong><em>j</em></strong> + <strong><em>k</em></strong>) <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \bullet ">
<mo>∙</mo>
</math></span> (2<strong><em>i</em></strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - ">
<mo>−</mo>
</math></span> <strong><em>j</em></strong> + 2<strong><em>k</em></strong>) =0 <strong><em>A1</em></strong></p>
<p><strong>METHOD 2</strong></p>
<p>attempt to find cross product of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\rm{OA}}} ">
<mover>
<mrow>
<mrow>
<mi mathvariant="normal">O</mi>
<mi mathvariant="normal">A</mi>
</mrow>
</mrow>
<mo>→</mo>
</mover>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\rm{OB}}} ">
<mover>
<mrow>
<mrow>
<mi mathvariant="normal">O</mi>
<mi mathvariant="normal">B</mi>
</mrow>
</mrow>
<mo>→</mo>
</mover>
</math></span> <strong><em>M1</em></strong></p>
<p>plane <em>Π</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="_2">
<msub>
<mi></mi>
<mn>2</mn>
</msub>
</math></span> has normal <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{OA}}} \times \overrightarrow {{\text{OB}}} ">
<mover>
<mrow>
<mtext>OA</mtext>
</mrow>
<mo>→</mo>
</mover>
<mo>×</mo>
<mover>
<mrow>
<mtext>OB</mtext>
</mrow>
<mo>→</mo>
</mover>
</math></span> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - ">
<mo>−</mo>
</math></span> 8<strong><em>j </em></strong><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - ">
<mo>−</mo>
</math></span> 4<strong><em>k</em></strong> <strong><em>A1</em></strong></p>
<p>since <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - ">
<mo>−</mo>
</math></span>8<strong><em>j </em></strong><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - ">
<mo>−</mo>
</math></span> 4<strong><em>k </em></strong>= <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - ">
<mo>−</mo>
</math></span>4(2<strong><em>j</em></strong> + <strong><em>k</em></strong>), 2<strong><em>j </em></strong>+ <strong><em>k </em></strong>is perpendicular to the plane <em>Π</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="_2">
<msub>
<mi></mi>
<mn>2</mn>
</msub>
</math></span> <strong><em>R1</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>plane <em>Π</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="_3">
<msub>
<mi></mi>
<mn>3</mn>
</msub>
</math></span> has normal <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{OA}}} \times \overrightarrow {{\text{OC}}} ">
<mover>
<mrow>
<mtext>OA</mtext>
</mrow>
<mo>→</mo>
</mover>
<mo>×</mo>
<mover>
<mrow>
<mtext>OC</mtext>
</mrow>
<mo>→</mo>
</mover>
</math></span> = 9<strong><em>i</em></strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - ">
<mo>−</mo>
</math></span> 8<strong><em>j</em></strong> + 5<strong><em>k</em></strong> <strong><em>A1</em></strong></p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to use dot product of normal vectors <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\cos \theta = \frac{{(2j + k) \bullet (9i - 8j + 5k)}}{{\left| {2j + k} \right|\left| {9i - 8j + 5k} \right|}}">
<mi>cos</mi>
<mo></mo>
<mi>θ</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mo stretchy="false">(</mo>
<mn>2</mn>
<mi>j</mi>
<mo>+</mo>
<mi>k</mi>
<mo stretchy="false">)</mo>
<mo>∙</mo>
<mo stretchy="false">(</mo>
<mn>9</mn>
<mi>i</mi>
<mo>−</mo>
<mn>8</mn>
<mi>j</mi>
<mo>+</mo>
<mn>5</mn>
<mi>k</mi>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<mrow>
<mo>|</mo>
<mrow>
<mn>2</mn>
<mi>j</mi>
<mo>+</mo>
<mi>k</mi>
</mrow>
<mo>|</mo>
</mrow>
<mrow>
<mo>|</mo>
<mrow>
<mn>9</mn>
<mi>i</mi>
<mo>−</mo>
<mn>8</mn>
<mi>j</mi>
<mo>+</mo>
<mn>5</mn>
<mi>k</mi>
</mrow>
<mo>|</mo>
</mrow>
</mrow>
</mfrac>
</math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{ - 11}}{{\sqrt 5 \sqrt {170} }}\,\,\,( = - 0.377 \ldots )">
<mo>=</mo>
<mfrac>
<mrow>
<mo>−</mo>
<mn>11</mn>
</mrow>
<mrow>
<msqrt>
<mn>5</mn>
</msqrt>
<msqrt>
<mn>170</mn>
</msqrt>
</mrow>
</mfrac>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mo stretchy="false">(</mo>
<mo>=</mo>
<mo>−</mo>
<mn>0.377</mn>
<mo>…</mo>
<mo stretchy="false">)</mo>
</math></span> <strong><em>(A1)</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Accept <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{11}}{{\sqrt 5 \sqrt {170} }}">
<mfrac>
<mrow>
<mn>11</mn>
</mrow>
<mrow>
<msqrt>
<mn>5</mn>
</msqrt>
<msqrt>
<mn>170</mn>
</msqrt>
</mrow>
</mfrac>
</math></span>. acute angle between planes <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 67.8^\circ \,\,\,{\text{(}} = 1.18^\circ )">
<mo>=</mo>
<msup>
<mn>67.8</mn>
<mo>∘</mo>
</msup>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>(</mtext>
</mrow>
<mo>=</mo>
<msup>
<mn>1.18</mn>
<mo>∘</mo>
</msup>
<mo stretchy="false">)</mo>
</math></span> <strong><em>A1</em></strong></p>
<p> </p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">g.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">g.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the planes <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>Π</mtext><mn>1</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>Π</mtext><mn>2</mn></msub></math> with the following equations.</p>
<p style="padding-left: 60px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>Π</mtext><mn>1</mn></msub><mtext>: </mtext><mn>3</mn><mi>x</mi><mo>+</mo><mn>2</mn><mi>y</mi><mo>+</mo><mi>z</mi><mo>=</mo><mn>6</mn></math></p>
<p style="padding-left: 60px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>Π</mtext><mn>2</mn></msub><mtext>: </mtext><mi>x</mi><mo>-</mo><mn>2</mn><mi>y</mi><mo>+</mo><mi>z</mi><mo>=</mo><mn>4</mn></math></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find a Cartesian equation of the plane <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>Π</mtext><mn>3</mn></msub></math> which is perpendicular to <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>Π</mtext><mn>1</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>Π</mtext><mn>2</mn></msub></math> and passes through the origin <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>0</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></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>Find the coordinates of the point where <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>Π</mtext><mn>1</mn></msub></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>Π</mtext><mn>2</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>Π</mtext><mn>3</mn></msub></math> intersect.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>attempt to find a vector perpendicular to <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>Π</mtext><mn>1</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>Π</mtext><mn>2</mn></msub></math> using a cross product <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>3</mn></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr></mtable></mfenced><mo>×</mo><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mrow><mn>2</mn><mo>-</mo><mfenced><mrow><mo>-</mo><mn>2</mn></mrow></mfenced></mrow></mfenced><mi mathvariant="bold-italic">i</mi><mo>+</mo><mfenced><mrow><mn>1</mn><mo>-</mo><mn>3</mn></mrow></mfenced><mi mathvariant="bold-italic">j</mi><mo>+</mo><mfenced><mrow><mo>-</mo><mn>6</mn><mo>-</mo><mn>2</mn></mrow></mfenced><mi mathvariant="bold-italic">k</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced><mtable><mtr><mtd><mn>4</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>8</mn></mtd></mtr></mtable></mfenced><mfenced><mrow><mo>=</mo><mn>2</mn><mfenced><mtable><mtr><mtd><mn>2</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>4</mn></mtd></mtr></mtable></mfenced></mrow></mfenced></math> <em><strong>(A1)</strong></em></p>
<p>equation is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mi>x</mi><mo>-</mo><mn>2</mn><mi>y</mi><mo>-</mo><mn>8</mn><mi>z</mi><mo>=</mo><mn>0</mn><mfenced><mrow><mo>⇒</mo><mn>2</mn><mi>x</mi><mo>-</mo><mi>y</mi><mo>-</mo><mn>4</mn><mi>z</mi><mo>=</mo><mn>0</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to solve <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math> simultaneous equations in <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math> variables <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mfrac><mn>41</mn><mn>21</mn></mfrac><mo>,</mo><mo> </mo><mo>-</mo><mfrac><mn>10</mn><mn>21</mn></mfrac><mo>,</mo><mo> </mo><mfrac><mn>23</mn><mn>21</mn></mfrac></mrow></mfenced><mfenced><mrow><mo>=</mo><mfenced><mrow><mn>1</mn><mo>.</mo><mn>95</mn><mo>,</mo><mo> </mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>476</mn><mo>,</mo><mo> </mo><mn>1</mn><mo>.</mo><mn>10</mn></mrow></mfenced></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</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>Iqbal attempts three practice papers in mathematics. The probability that he passes the first paper is 0.6. Whenever he gains a pass in a paper, his confidence increases so that the probability of him passing the next paper increases by 0.1. Whenever he fails a paper the probability of him passing the next paper is 0.6.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Complete the given probability tree diagram for Iqbal’s three attempts, labelling each branch with the correct probability.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate the probability that Iqbal passes at least two of the papers he attempts.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the probability that Iqbal passes his third paper, given that he passed only one previous paper.</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 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><em><img 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"> <strong>A1A1A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for each correct column of probabilities.</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>probability (at least twice) =</p>
<p><strong>EITHER</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {0.6 \times 0.7 \times 0.8} \right) + \left( {0.6 \times 0.7 \times 0.2} \right) + \left( {0.6 \times 0.3 \times 0.6} \right) + \left( {0.4 \times 0.6 \times 0.7} \right)"> <mrow> <mo>(</mo> <mrow> <mn>0.6</mn> <mo>×</mo> <mn>0.7</mn> <mo>×</mo> <mn>0.8</mn> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mn>0.6</mn> <mo>×</mo> <mn>0.7</mn> <mo>×</mo> <mn>0.2</mn> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mn>0.6</mn> <mo>×</mo> <mn>0.3</mn> <mo>×</mo> <mn>0.6</mn> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mn>0.4</mn> <mo>×</mo> <mn>0.6</mn> <mo>×</mo> <mn>0.7</mn> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>(M1)</strong></em></p>
<p><strong>OR</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {0.6 \times 0.7} \right) + \left( {0.6 \times 0.3 \times 0.6} \right) + \left( {0.4 \times 0.6 \times 0.7} \right)"> <mrow> <mo>(</mo> <mrow> <mn>0.6</mn> <mo>×</mo> <mn>0.7</mn> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mn>0.6</mn> <mo>×</mo> <mn>0.3</mn> <mo>×</mo> <mn>0.6</mn> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mn>0.4</mn> <mo>×</mo> <mn>0.6</mn> <mo>×</mo> <mn>0.7</mn> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>(M1)</strong></em></p>
<p><strong>Note:</strong> Award<em><strong> M1</strong></em> for summing all required probabilities.</p>
<p><strong>THEN</strong></p>
<p>= 0.696<em> <strong>A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>P(passes third paper given only one paper passed before)</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{{\text{P}}\,\left( {{\text{passes third AND only one paper passed before}}} \right)}}{{{\text{P}}\,\left( {{\text{passes once in first two papers}}} \right)}}"> <mo>=</mo> <mfrac> <mrow> <mrow> <mtext>P</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>passes third AND only one paper passed before</mtext> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mrow> <mtext>P</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>passes once in first two papers</mtext> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{\left( {0.6 \times 0.3 \times 0.6} \right) + \left( {0.4 \times 0.6 \times 0.7} \right)}}{{\left( {0.6 \times 0.3} \right) + \left( {0.4 \times 0.6} \right)}}"> <mo>=</mo> <mfrac> <mrow> <mrow> <mo>(</mo> <mrow> <mn>0.6</mn> <mo>×</mo> <mn>0.3</mn> <mo>×</mo> <mn>0.6</mn> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mn>0.4</mn> <mo>×</mo> <mn>0.6</mn> <mo>×</mo> <mn>0.7</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mrow> <mo>(</mo> <mrow> <mn>0.6</mn> <mo>×</mo> <mn>0.3</mn> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mn>0.4</mn> <mo>×</mo> <mn>0.6</mn> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </math></span> <em><strong>A1</strong></em></p>
<p>= 0.657<em> <strong>A1</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>The diagram shows two circles with centres at the points A and B and radii <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2r">
<mn>2</mn>
<mi>r</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r">
<mi>r</mi>
</math></span>, respectively. The point B lies on the circle with centre A. The circles intersect at the points C and D.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-02-28_om_17.29.37.png" alt="N16/5/MATHL/HP2/ENG/TZ0/09"></p>
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\alpha ">
<mi>α<!-- α --></mi>
</math></span> be the measure of the angle CAD and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\theta ">
<mi>θ<!-- θ --></mi>
</math></span> be the measure of the angle CBD in radians.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an expression for the shaded area in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\alpha ">
<mi>α</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\theta ">
<mi>θ</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r">
<mi>r</mi>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\alpha = 4\arcsin \frac{1}{4}">
<mi>α</mi>
<mo>=</mo>
<mn>4</mn>
<mi>arcsin</mi>
<mo></mo>
<mfrac>
<mn>1</mn>
<mn>4</mn>
</mfrac>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r">
<mi>r</mi>
</math></span> given that the shaded area is equal to 4.</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 style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A = 2(\alpha - \sin \alpha ){r^2} + \frac{1}{2}(\theta - \sin \theta ){r^2}">
<mi>A</mi>
<mo>=</mo>
<mn>2</mn>
<mo stretchy="false">(</mo>
<mi>α</mi>
<mo>−</mo>
<mi>sin</mi>
<mo></mo>
<mi>α</mi>
<mo stretchy="false">)</mo>
<mrow>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mo stretchy="false">(</mo>
<mi>θ</mi>
<mo>−</mo>
<mi>sin</mi>
<mo></mo>
<mi>θ</mi>
<mo stretchy="false">)</mo>
<mrow>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
</math></span> <strong><em>M1A1A1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>M1A1A1 </em></strong>for alternative correct expressions <em>eg</em>. <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A = 4\left( {\frac{\alpha }{2} - \sin \frac{\alpha }{2}} \right){r^2} + \frac{1}{2}\theta {r^2}">
<mi>A</mi>
<mo>=</mo>
<mn>4</mn>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mi>α</mi>
<mn>2</mn>
</mfrac>
<mo>−</mo>
<mi>sin</mi>
<mo></mo>
<mfrac>
<mi>α</mi>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mi>θ</mi>
<mrow>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
</math></span>.</p>
<p> </p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>consider for example triangle ADM where M is the midpoint of BD <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sin \frac{\alpha }{4} = \frac{1}{4}">
<mi>sin</mi>
<mo></mo>
<mfrac>
<mi>α</mi>
<mn>4</mn>
</mfrac>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>4</mn>
</mfrac>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{\alpha }{4} = \arcsin \frac{1}{4}">
<mfrac>
<mi>α</mi>
<mn>4</mn>
</mfrac>
<mo>=</mo>
<mi>arcsin</mi>
<mo></mo>
<mfrac>
<mn>1</mn>
<mn>4</mn>
</mfrac>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\alpha = 4\arcsin \frac{1}{4}">
<mi>α</mi>
<mo>=</mo>
<mn>4</mn>
<mi>arcsin</mi>
<mo></mo>
<mfrac>
<mn>1</mn>
<mn>4</mn>
</mfrac>
</math></span> <strong><em>AG</em></strong></p>
<p><strong>METHOD 2</strong></p>
<p>attempting to use the cosine rule (to obtain <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1 - \cos \frac{\alpha }{2} = \frac{1}{8}">
<mn>1</mn>
<mo>−</mo>
<mi>cos</mi>
<mo></mo>
<mfrac>
<mi>α</mi>
<mn>2</mn>
</mfrac>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>8</mn>
</mfrac>
</math></span>) <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sin \frac{\alpha }{4} = \frac{1}{4}">
<mi>sin</mi>
<mo></mo>
<mfrac>
<mi>α</mi>
<mn>4</mn>
</mfrac>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>4</mn>
</mfrac>
</math></span> (obtained from <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sin \frac{\alpha }{4} = \sqrt {\frac{{1 - \cos \frac{\alpha }{2}}}{2}} ">
<mi>sin</mi>
<mo></mo>
<mfrac>
<mi>α</mi>
<mn>4</mn>
</mfrac>
<mo>=</mo>
<msqrt>
<mfrac>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>cos</mi>
<mo></mo>
<mfrac>
<mi>α</mi>
<mn>2</mn>
</mfrac>
</mrow>
<mn>2</mn>
</mfrac>
</msqrt>
</math></span>) <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{\alpha }{4} = \arcsin \frac{1}{4}">
<mfrac>
<mi>α</mi>
<mn>4</mn>
</mfrac>
<mo>=</mo>
<mi>arcsin</mi>
<mo></mo>
<mfrac>
<mn>1</mn>
<mn>4</mn>
</mfrac>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\alpha = 4\arcsin \frac{1}{4}">
<mi>α</mi>
<mo>=</mo>
<mn>4</mn>
<mi>arcsin</mi>
<mo></mo>
<mfrac>
<mn>1</mn>
<mn>4</mn>
</mfrac>
</math></span> <strong><em>AG</em></strong></p>
<p><strong>METHOD 3</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sin \left( {\frac{\pi }{2} - \frac{\alpha }{4}} \right) = 2\sin \frac{\alpha }{2}">
<mi>sin</mi>
<mo></mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
<mo>−</mo>
<mfrac>
<mi>α</mi>
<mn>4</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>2</mn>
<mi>sin</mi>
<mo></mo>
<mfrac>
<mi>α</mi>
<mn>2</mn>
</mfrac>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{\theta }{2} = \frac{\pi }{2} - \frac{\alpha }{4}">
<mfrac>
<mi>θ</mi>
<mn>2</mn>
</mfrac>
<mo>=</mo>
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
<mo>−</mo>
<mfrac>
<mi>α</mi>
<mn>4</mn>
</mfrac>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\cos \frac{\alpha }{4} = 4\sin \frac{\alpha }{4}\cos \frac{\alpha }{4}">
<mi>cos</mi>
<mo></mo>
<mfrac>
<mi>α</mi>
<mn>4</mn>
</mfrac>
<mo>=</mo>
<mn>4</mn>
<mi>sin</mi>
<mo></mo>
<mfrac>
<mi>α</mi>
<mn>4</mn>
</mfrac>
<mi>cos</mi>
<mo></mo>
<mfrac>
<mi>α</mi>
<mn>4</mn>
</mfrac>
</math></span> <strong><em>M1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>M1 </em></strong>either for use of the double angle formula or the conversion from sine to cosine.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{4} = \sin \frac{\alpha }{4}">
<mfrac>
<mn>1</mn>
<mn>4</mn>
</mfrac>
<mo>=</mo>
<mi>sin</mi>
<mo></mo>
<mfrac>
<mi>α</mi>
<mn>4</mn>
</mfrac>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{\alpha }{4} = \arcsin \frac{1}{4}">
<mfrac>
<mi>α</mi>
<mn>4</mn>
</mfrac>
<mo>=</mo>
<mi>arcsin</mi>
<mo></mo>
<mfrac>
<mn>1</mn>
<mn>4</mn>
</mfrac>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\alpha = 4\arcsin \frac{1}{4}">
<mi>α</mi>
<mo>=</mo>
<mn>4</mn>
<mi>arcsin</mi>
<mo></mo>
<mfrac>
<mn>1</mn>
<mn>4</mn>
</mfrac>
</math></span> <strong><em>AG</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>(from triangle ADM), <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\theta = \pi - \frac{\alpha }{2}{\text{ }}\left( { = \pi - 2\arcsin \frac{1}{4} = 2\arcsin \frac{1}{4} = 2.6362 \ldots } \right)">
<mi>θ</mi>
<mo>=</mo>
<mi>π</mi>
<mo>−</mo>
<mfrac>
<mi>α</mi>
<mn>2</mn>
</mfrac>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mi>π</mi>
<mo>−</mo>
<mn>2</mn>
<mi>arcsin</mi>
<mo></mo>
<mfrac>
<mn>1</mn>
<mn>4</mn>
</mfrac>
<mo>=</mo>
<mn>2</mn>
<mi>arcsin</mi>
<mo></mo>
<mfrac>
<mn>1</mn>
<mn>4</mn>
</mfrac>
<mo>=</mo>
<mn>2.6362</mn>
<mo>…</mo>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>A1</em></strong></p>
<p>attempting to solve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2(\alpha - \sin \alpha ){r^2} + \frac{1}{2}(\theta - \sin \theta ){r^2} = 4">
<mn>2</mn>
<mo stretchy="false">(</mo>
<mi>α</mi>
<mo>−</mo>
<mi>sin</mi>
<mo></mo>
<mi>α</mi>
<mo stretchy="false">)</mo>
<mrow>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mo stretchy="false">(</mo>
<mi>θ</mi>
<mo>−</mo>
<mi>sin</mi>
<mo></mo>
<mi>θ</mi>
<mo stretchy="false">)</mo>
<mrow>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>4</mn>
</math></span></p>
<p>with <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\alpha = 4\arcsin \frac{1}{4}">
<mi>α</mi>
<mo>=</mo>
<mn>4</mn>
<mi>arcsin</mi>
<mo></mo>
<mfrac>
<mn>1</mn>
<mn>4</mn>
</mfrac>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\theta = \pi - \frac{\alpha }{2}{\text{ }}\left( { = 2\arccos \frac{1}{4}} \right)">
<mi>θ</mi>
<mo>=</mo>
<mi>π</mi>
<mo>−</mo>
<mfrac>
<mi>α</mi>
<mn>2</mn>
</mfrac>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mn>2</mn>
<mi>arccos</mi>
<mo></mo>
<mfrac>
<mn>1</mn>
<mn>4</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r">
<mi>r</mi>
</math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r = 1.69">
<mi>r</mi>
<mo>=</mo>
<mn>1.69</mn>
</math></span> <strong><em>A1</em></strong></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.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>The plane <em>П</em><sub>1</sub> contains the points P(1, 6, −7) , Q(0, 1, 1) and R(2, 0, −4).</p>
</div>
<div class="specification">
<p>The Cartesian equation of the plane <em>П</em><sub>2</sub> is given by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x - 3y - z = 3">
<mi>x</mi>
<mo>−<!-- − --></mo>
<mn>3</mn>
<mi>y</mi>
<mo>−<!-- − --></mo>
<mi>z</mi>
<mo>=</mo>
<mn>3</mn>
</math></span>.</p>
</div>
<div class="specification">
<p>The Cartesian equation of the plane <em>П</em><sub>3</sub> is given by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="ax + by + cz = 1">
<mi>a</mi>
<mi>x</mi>
<mo>+</mo>
<mi>b</mi>
<mi>y</mi>
<mo>+</mo>
<mi>c</mi>
<mi>z</mi>
<mo>=</mo>
<mn>1</mn>
</math></span>.</p>
</div>
<div class="specification">
<p>Consider the case that <em>П</em><sub>3</sub> contains <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="L">
<mi>L</mi>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the Cartesian equation of the plane containing P, Q and R.</p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that <em>П</em><sub>1</sub> and <em>П</em><sub>2</sub> meet in a line <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="L">
<mi>L</mi>
</math></span>, verify that the vector equation of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="L">
<mi>L</mi>
</math></span> can be given by <em><strong>r</strong></em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left( {\begin{array}{*{20}{c}} {\frac{5}{4}} \\ 0 \\ { - \frac{7}{4}} \end{array}} \right) + \lambda \left( {\begin{array}{*{20}{c}} {\frac{1}{2}} \\ 1 \\ { - \frac{5}{2}} \end{array}} \right)">
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mfrac>
<mn>5</mn>
<mn>4</mn>
</mfrac>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mfrac>
<mn>7</mn>
<mn>4</mn>
</mfrac>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mi>λ</mi>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mfrac>
<mn>5</mn>
<mn>2</mn>
</mfrac>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that <em>П</em><sub>3</sub> is parallel to the line <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="L">
<mi>L</mi>
</math></span>, show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a + 2b - 5c = 0">
<mi>a</mi>
<mo>+</mo>
<mn>2</mn>
<mi>b</mi>
<mo>−</mo>
<mn>5</mn>
<mi>c</mi>
<mo>=</mo>
<mn>0</mn>
</math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="5a - 7c = 4">
<mn>5</mn>
<mi>a</mi>
<mo>−</mo>
<mn>7</mn>
<mi>c</mi>
<mo>=</mo>
<mn>4</mn>
</math></span>.</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>Given that <em>П</em><sub>3</sub> is equally inclined to both <em>П</em><sub>1</sub> and <em>П</em><sub>2</sub>, determine two distinct possible Cartesian equations for <em>П</em><sub>3</sub>.</p>
<div class="marks">[7]</div>
<div class="question_part_label">d.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><strong>METHOD 1</strong></p>
<p>for example</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{PQ}}} = \left( {\begin{array}{*{20}{c}} { - 1} \\ { - 5} \\ 8 \end{array}} \right)">
<mover>
<mrow>
<mtext>PQ</mtext>
</mrow>
<mo>→</mo>
</mover>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>5</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>8</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{PR}}} = \left( {\begin{array}{*{20}{c}} 1 \\ { - 6} \\ 3 \end{array}} \right)">
<mover>
<mrow>
<mtext>PR</mtext>
</mrow>
<mo>→</mo>
</mover>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>6</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{PQ}}} \times \overrightarrow {{\text{PR}}} ">
<mover>
<mrow>
<mtext>PQ</mtext>
</mrow>
<mo>→</mo>
</mover>
<mo>×</mo>
<mover>
<mrow>
<mtext>PR</mtext>
</mrow>
<mo>→</mo>
</mover>
</math></span> = 33<em><strong>i</strong></em> + 11<em><strong>j</strong></em> + 11<em><strong>k (M1)A1</strong></em></p>
<p><em><strong>r.n</strong></em> = <em><strong>a.n</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="33x + 11y + 11z = \left( {\begin{array}{*{20}{c}} 0 \\ 1 \\ 1 \end{array}} \right) \cdot \left( {\begin{array}{*{20}{c}} {33} \\ {11} \\ {11} \end{array}} \right) = 22">
<mn>33</mn>
<mi>x</mi>
<mo>+</mo>
<mn>11</mn>
<mi>y</mi>
<mo>+</mo>
<mn>11</mn>
<mi>z</mi>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>⋅</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>33</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>11</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>11</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>22</mn>
</math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow 3x + y + z = 2">
<mo stretchy="false">⇒</mo>
<mn>3</mn>
<mi>x</mi>
<mo>+</mo>
<mi>y</mi>
<mo>+</mo>
<mi>z</mi>
<mo>=</mo>
<mn>2</mn>
</math></span> or equivalent <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>assume plane can be written as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="ax + by + cz = 1">
<mi>a</mi>
<mi>x</mi>
<mo>+</mo>
<mi>b</mi>
<mi>y</mi>
<mo>+</mo>
<mi>c</mi>
<mi>z</mi>
<mo>=</mo>
<mn>1</mn>
</math></span> <em><strong>M1</strong></em></p>
<p>substituting each set of coordinates gives the system of equations:</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a + 6b - 7c = 1">
<mi>a</mi>
<mo>+</mo>
<mn>6</mn>
<mi>b</mi>
<mo>−</mo>
<mn>7</mn>
<mi>c</mi>
<mo>=</mo>
<mn>1</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0a + b + c = 1">
<mn>0</mn>
<mi>a</mi>
<mo>+</mo>
<mi>b</mi>
<mo>+</mo>
<mi>c</mi>
<mo>=</mo>
<mn>1</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2a + 0b - 4c = 1">
<mn>2</mn>
<mi>a</mi>
<mo>+</mo>
<mn>0</mn>
<mi>b</mi>
<mo>−</mo>
<mn>4</mn>
<mi>c</mi>
<mo>=</mo>
<mn>1</mn>
</math></span> <em><strong>A1</strong></em></p>
<p>solving by GDC <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a = \frac{3}{2}">
<mi>a</mi>
<mo>=</mo>
<mfrac>
<mn>3</mn>
<mn>2</mn>
</mfrac>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b = \frac{1}{2}">
<mi>b</mi>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="C = \frac{1}{2}">
<mi>C</mi>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</math></span> <em><strong>A1</strong></em><em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow \frac{3}{2}x + \frac{1}{2}y + \frac{1}{2}z = 1">
<mo stretchy="false">⇒</mo>
<mfrac>
<mn>3</mn>
<mn>2</mn>
</mfrac>
<mi>x</mi>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mi>y</mi>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mi>z</mi>
<mo>=</mo>
<mn>1</mn>
</math></span> or equivalent</p>
<p> </p>
<p><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>substitution of equation of line into both equations of planes <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3\left( {\frac{5}{4} + \frac{\lambda }{2}} \right) + \left( { - \frac{7}{4} - \frac{{5\lambda }}{2}} \right) = 2">
<mn>3</mn>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>5</mn>
<mn>4</mn>
</mfrac>
<mo>+</mo>
<mfrac>
<mi>λ</mi>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mo>−</mo>
<mfrac>
<mn>7</mn>
<mn>4</mn>
</mfrac>
<mo>−</mo>
<mfrac>
<mrow>
<mn>5</mn>
<mi>λ</mi>
</mrow>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>2</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\frac{5}{4} + \frac{\lambda }{2}} \right) - 3\lambda - \left( { - \frac{7}{4} - \frac{{5\lambda }}{2}} \right) = 3">
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>5</mn>
<mn>4</mn>
</mfrac>
<mo>+</mo>
<mfrac>
<mi>λ</mi>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mn>3</mn>
<mi>λ</mi>
<mo>−</mo>
<mrow>
<mo>(</mo>
<mrow>
<mo>−</mo>
<mfrac>
<mn>7</mn>
<mn>4</mn>
</mfrac>
<mo>−</mo>
<mfrac>
<mrow>
<mn>5</mn>
<mi>λ</mi>
</mrow>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>3</mn>
</math></span> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>adding <em>Π</em><sub>1</sub> and <em>Π</em><sub>2</sub> gives <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="4x - 2y = 5">
<mn>4</mn>
<mi>x</mi>
<mo>−</mo>
<mn>2</mn>
<mi>y</mi>
<mo>=</mo>
<mn>5</mn>
</math></span> <em><strong>M1</strong></em></p>
<p>given <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = \lambda \Rightarrow x = \frac{5}{4} + \frac{\lambda }{2}">
<mi>y</mi>
<mo>=</mo>
<mi>λ</mi>
<mo stretchy="false">⇒</mo>
<mi>x</mi>
<mo>=</mo>
<mfrac>
<mn>5</mn>
<mn>4</mn>
</mfrac>
<mo>+</mo>
<mfrac>
<mi>λ</mi>
<mn>2</mn>
</mfrac>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z = 2 - y - 3x = - \frac{7}{4} - \frac{{5\lambda }}{2}">
<mi>z</mi>
<mo>=</mo>
<mn>2</mn>
<mo>−</mo>
<mi>y</mi>
<mo>−</mo>
<mn>3</mn>
<mi>x</mi>
<mo>=</mo>
<mo>−</mo>
<mfrac>
<mn>7</mn>
<mn>4</mn>
</mfrac>
<mo>−</mo>
<mfrac>
<mrow>
<mn>5</mn>
<mi>λ</mi>
</mrow>
<mn>2</mn>
</mfrac>
</math></span> <em><strong>A1</strong></em></p>
<p>⇒<em><strong>r</strong></em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left( {\begin{array}{*{20}{c}} {\frac{5}{4}} \\ 0 \\ { - \frac{7}{4}} \end{array}} \right) + \lambda \left( {\begin{array}{*{20}{c}} {\frac{1}{2}} \\ 1 \\ { - \frac{5}{2}} \end{array}} \right)">
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mfrac>
<mn>5</mn>
<mn>4</mn>
</mfrac>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mfrac>
<mn>7</mn>
<mn>4</mn>
</mfrac>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mi>λ</mi>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mfrac>
<mn>5</mn>
<mn>2</mn>
</mfrac>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>METHOD 3</strong></p>
<p><em><strong>n</strong></em><sub>1</sub> × <em><strong>n</strong></em><sub>2</sub> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 2 \\ 4 \\ { - 10} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>10</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 4\left( {\begin{array}{*{20}{c}} {\frac{1}{2}} \\ 1 \\ { - \frac{5}{2}} \end{array}} \right)">
<mo>=</mo>
<mn>4</mn>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mfrac>
<mn>5</mn>
<mn>2</mn>
</mfrac>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>R1</strong></em></p>
<p>common point <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{5}{4} - 3\left( 0 \right) - \left( { - \frac{7}{4}} \right) = 3">
<mfrac>
<mn>5</mn>
<mn>4</mn>
</mfrac>
<mo>−</mo>
<mn>3</mn>
<mrow>
<mo>(</mo>
<mn>0</mn>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mrow>
<mo>(</mo>
<mrow>
<mo>−</mo>
<mfrac>
<mn>7</mn>
<mn>4</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>3</mn>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 3\left( {\frac{5}{4}} \right) - 0 - \left( { - \frac{7}{4}} \right) = - 2">
<mo>−</mo>
<mn>3</mn>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>5</mn>
<mn>4</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mn>0</mn>
<mo>−</mo>
<mrow>
<mo>(</mo>
<mrow>
<mo>−</mo>
<mfrac>
<mn>7</mn>
<mn>4</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mo>−</mo>
<mn>2</mn>
</math></span> <em><strong>A1</strong></em> </p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>normal to <em>П</em><sub>3</sub> is perpendicular to direction of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="L">
<mi>L</mi>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow \left( {\begin{array}{*{20}{c}} a \\ b \\ c \end{array}} \right) \cdot \left( {\begin{array}{*{20}{c}} 1 \\ 2 \\ { - 5} \end{array}} \right) = 0">
<mo stretchy="false">⇒</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>a</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>b</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>c</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>⋅</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>5</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><strong>⇒</strong><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a + 2b - 5c = 0">
<mi>a</mi>
<mo>+</mo>
<mn>2</mn>
<mi>b</mi>
<mo>−</mo>
<mn>5</mn>
<mi>c</mi>
<mo>=</mo>
<mn>0</mn>
</math></span> <em><strong>AG</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>substituting <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} {\frac{5}{4}} \\ 0 \\ { - \frac{7}{4}} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mfrac>
<mn>5</mn>
<mn>4</mn>
</mfrac>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mfrac>
<mn>7</mn>
<mn>4</mn>
</mfrac>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> into <em>П</em><sub>3</sub>: <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{5a}}{4} - \frac{{7c}}{4} = 1">
<mfrac>
<mrow>
<mn>5</mn>
<mi>a</mi>
</mrow>
<mn>4</mn>
</mfrac>
<mo>−</mo>
<mfrac>
<mrow>
<mn>7</mn>
<mi>c</mi>
</mrow>
<mn>4</mn>
</mfrac>
<mo>=</mo>
<mn>1</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="5a - 7c = 4">
<mn>5</mn>
<mi>a</mi>
<mo>−</mo>
<mn>7</mn>
<mi>c</mi>
<mo>=</mo>
<mn>4</mn>
</math></span> <em><strong>AG</strong></em></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>attempt to find scalar products for <em>П</em><sub>1</sub> and <em>П</em><sub>3</sub>, <em>П</em><sub>2</sub> and <em>П</em><sub>3</sub>.</p>
<p>and equating <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{3a + b + C}}{{\sqrt {11} \sqrt {{a^2} + {b^2} + {c^2}} }} = \frac{{a - 3b - c}}{{\sqrt {11} \sqrt {{a^2} + {b^2} + {c^2}} }}">
<mfrac>
<mrow>
<mn>3</mn>
<mi>a</mi>
<mo>+</mo>
<mi>b</mi>
<mo>+</mo>
<mi>C</mi>
</mrow>
<mrow>
<msqrt>
<mn>11</mn>
</msqrt>
<msqrt>
<mrow>
<msup>
<mi>a</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mi>b</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mi>c</mi>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mi>a</mi>
<mo>−</mo>
<mn>3</mn>
<mi>b</mi>
<mo>−</mo>
<mi>c</mi>
</mrow>
<mrow>
<msqrt>
<mn>11</mn>
</msqrt>
<msqrt>
<mrow>
<msup>
<mi>a</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mi>b</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mi>c</mi>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
</mrow>
</mfrac>
</math></span> <em><strong>M1</strong></em></p>
<p><strong>Note:</strong> Accept <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3a + b + c = a - 3b - c">
<mn>3</mn>
<mi>a</mi>
<mo>+</mo>
<mi>b</mi>
<mo>+</mo>
<mi>c</mi>
<mo>=</mo>
<mi>a</mi>
<mo>−</mo>
<mn>3</mn>
<mi>b</mi>
<mo>−</mo>
<mi>c</mi>
</math></span>.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow a + 2b + c = 0">
<mo stretchy="false">⇒</mo>
<mi>a</mi>
<mo>+</mo>
<mn>2</mn>
<mi>b</mi>
<mo>+</mo>
<mi>c</mi>
<mo>=</mo>
<mn>0</mn>
</math></span> <em><strong>A1</strong></em></p>
<p>attempt to solve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a + 2b + c = 0">
<mi>a</mi>
<mo>+</mo>
<mn>2</mn>
<mi>b</mi>
<mo>+</mo>
<mi>c</mi>
<mo>=</mo>
<mn>0</mn>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a + 2b - 5c = 0">
<mi>a</mi>
<mo>+</mo>
<mn>2</mn>
<mi>b</mi>
<mo>−</mo>
<mn>5</mn>
<mi>c</mi>
<mo>=</mo>
<mn>0</mn>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="5a - 7c = 4">
<mn>5</mn>
<mi>a</mi>
<mo>−</mo>
<mn>7</mn>
<mi>c</mi>
<mo>=</mo>
<mn>4</mn>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow a = \frac{4}{5}{\text{,}}\,\,b = - \frac{2}{5}{\text{,}}\,\,c = 0">
<mo stretchy="false">⇒</mo>
<mi>a</mi>
<mo>=</mo>
<mfrac>
<mn>4</mn>
<mn>5</mn>
</mfrac>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>b</mi>
<mo>=</mo>
<mo>−</mo>
<mfrac>
<mn>2</mn>
<mn>5</mn>
</mfrac>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>c</mi>
<mo>=</mo>
<mn>0</mn>
</math></span> <em><strong>A1</strong></em></p>
<p>hence equation is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{4x}}{5} - \frac{{2y}}{5} = 1">
<mfrac>
<mrow>
<mn>4</mn>
<mi>x</mi>
</mrow>
<mn>5</mn>
</mfrac>
<mo>−</mo>
<mfrac>
<mrow>
<mn>2</mn>
<mi>y</mi>
</mrow>
<mn>5</mn>
</mfrac>
<mo>=</mo>
<mn>1</mn>
</math></span></p>
<p>for second equation:</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{3a + b + C}}{{\sqrt {11} \sqrt {{a^2} + {b^2} + {c^2}} }} = - \frac{{a - 3b - c}}{{\sqrt {11} \sqrt {{a^2} + {b^2} + {c^2}} }}">
<mfrac>
<mrow>
<mn>3</mn>
<mi>a</mi>
<mo>+</mo>
<mi>b</mi>
<mo>+</mo>
<mi>C</mi>
</mrow>
<mrow>
<msqrt>
<mn>11</mn>
</msqrt>
<msqrt>
<mrow>
<msup>
<mi>a</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mi>b</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mi>c</mi>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
</mrow>
</mfrac>
<mo>=</mo>
<mo>−</mo>
<mfrac>
<mrow>
<mi>a</mi>
<mo>−</mo>
<mn>3</mn>
<mi>b</mi>
<mo>−</mo>
<mi>c</mi>
</mrow>
<mrow>
<msqrt>
<mn>11</mn>
</msqrt>
<msqrt>
<mrow>
<msup>
<mi>a</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mi>b</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mi>c</mi>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
</mrow>
</mfrac>
</math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow 2a - b = 0">
<mo stretchy="false">⇒</mo>
<mn>2</mn>
<mi>a</mi>
<mo>−</mo>
<mi>b</mi>
<mo>=</mo>
<mn>0</mn>
</math></span></p>
<p>attempt to solve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2a - b = 0">
<mn>2</mn>
<mi>a</mi>
<mo>−</mo>
<mi>b</mi>
<mo>=</mo>
<mn>0</mn>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a + 2b - 5c = 0">
<mi>a</mi>
<mo>+</mo>
<mn>2</mn>
<mi>b</mi>
<mo>−</mo>
<mn>5</mn>
<mi>c</mi>
<mo>=</mo>
<mn>0</mn>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="5a - 7c = 4">
<mn>5</mn>
<mi>a</mi>
<mo>−</mo>
<mn>7</mn>
<mi>c</mi>
<mo>=</mo>
<mn>4</mn>
</math></span> <em><strong>M1</strong></em></p>
<p>⇒<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a = - 2">
<mi>a</mi>
<mo>=</mo>
<mo>−</mo>
<mn>2</mn>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b = - 4">
<mi>b</mi>
<mo>=</mo>
<mo>−</mo>
<mn>4</mn>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c = - 2">
<mi>c</mi>
<mo>=</mo>
<mo>−</mo>
<mn>2</mn>
</math></span> <em><strong>A1</strong></em></p>
<p>hence equation is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="- 2x - 4y - 2z = 1">
<mo>−</mo>
<mn>2</mn>
<mi>x</mi>
<mo>−</mo>
<mn>4</mn>
<mi>y</mi>
<mo>−</mo>
<mn>2</mn>
<mi>z</mi>
<mo>=</mo>
<mn>1</mn>
</math></span></p>
<p><em><strong>[7 marks]</strong></em></p>
<div class="question_part_label">d.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>A scientist conducted a nine-week experiment on two plants, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math>, of the same species. He wanted to determine the effect of using a new plant fertilizer. Plant <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> was given fertilizer regularly, while Plant <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> was not.</p>
<p>The scientist found that the height of Plant <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>,</mo><mo> </mo><msub><mi>h</mi><mi>A</mi></msub><mo> </mo><mtext>cm</mtext></math>, at time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> weeks can be modelled by the function <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>h</mi><mi>A</mi></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mi>sin</mi><mo>(</mo><mn>2</mn><mi>t</mi><mo>+</mo><mn>6</mn><mo>)</mo><mo>+</mo><mn>9</mn><mi>t</mi><mo>+</mo><mn>27</mn></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>≤</mo><mi>t</mi><mo>≤</mo><mn>9</mn></math>.</p>
<p>The scientist found that the height of Plant <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mo>,</mo><mo> </mo><msub><mi>h</mi><mi>B</mi></msub><mo> </mo><mtext>cm</mtext></math>, at time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> weeks can be modelled by the function <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>h</mi><mi>B</mi></msub><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mn>8</mn><mi>t</mi><mo>+</mo><mn>32</mn></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>≤</mo><mi>t</mi><mo>≤</mo><mn>9</mn></math>.</p>
</div>
<div class="specification">
<p>Use the scientist’s models to find the initial height of</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Plant <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Plant <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> correct to three significant figures.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> when <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>h</mi><mi>A</mi></msub><mfenced><mi>t</mi></mfenced><mo>=</mo><msub><mi>h</mi><mi>B</mi></msub><mfenced><mi>t</mi></mfenced></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>For <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>></mo><mn>6</mn></math>, prove that Plant <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> was always taller than Plant <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>For <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>≤</mo><mi>t</mi><mo>≤</mo><mn>9</mn></math>, find the total amount of time when the rate of growth of Plant <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> was greater than the rate of growth of Plant <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>32</mn></math> (cm) <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>h</mi><mi>A</mi></msub><mfenced><mn>0</mn></mfenced><mo>=</mo><mi>sin</mi><mfenced><mn>6</mn></mfenced><mo>+</mo><mn>27</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>26</mn><mo>.</mo><mn>7205</mn><mo>…</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>26</mn><mo>.</mo><mn>7</mn></math> (cm) <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempts to solve <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>h</mi><mi>A</mi></msub><mfenced><mi>t</mi></mfenced><mo>=</mo><msub><mi>h</mi><mi>B</mi></msub><mfenced><mi>t</mi></mfenced></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>4</mn><mo>.</mo><mn>0074</mn><mo>…</mo><mo>,</mo><mn>4</mn><mo>.</mo><mn>7034</mn><mo>…</mo><mo>,</mo><mn>5</mn><mo>.</mo><mn>88332</mn><mo>…</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>4</mn><mo>.</mo><mn>01</mn><mo>,</mo><mn>4</mn><mo>.</mo><mn>70</mn><mo>,</mo><mn>5</mn><mo>.</mo><mn>88</mn></math> (weeks) <em><strong>A2</strong></em></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>h</mi><mi>A</mi></msub><mfenced><mi>t</mi></mfenced><mo>-</mo><msub><mi>h</mi><mi>B</mi></msub><mfenced><mi>t</mi></mfenced><mo>=</mo><mi>sin</mi><mfenced><mrow><mn>2</mn><mi>t</mi><mo>+</mo><mn>6</mn></mrow></mfenced><mo>+</mo><mi>t</mi><mo>-</mo><mn>5</mn></math> <em><strong>A1</strong></em></p>
<p><br><strong>EITHER</strong></p>
<p>for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>></mo><mn>6</mn><mo>,</mo><mo> </mo><mi>t</mi><mo>-</mo><mn>5</mn><mo>></mo><mn>1</mn></math> <em><strong>A1</strong></em></p>
<p>and as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mfenced><mrow><mn>2</mn><mi>t</mi><mo>+</mo><mn>6</mn></mrow></mfenced><mo>≥</mo><mo>-</mo><mn>1</mn><mo>⇒</mo><msub><mi>h</mi><mi>A</mi></msub><mfenced><mi>t</mi></mfenced><mo>-</mo><msub><mi>h</mi><mi>B</mi></msub><mfenced><mi>t</mi></mfenced><mo>></mo><mn>0</mn></math> <em><strong>R1</strong></em></p>
<p><br><strong>OR</strong></p>
<p>the minimum value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mfenced><mrow><mn>2</mn><mi>t</mi><mo>+</mo><mn>6</mn></mrow></mfenced><mo>=</mo><mo>-</mo><mn>1</mn></math> <em><strong>R1</strong></em></p>
<p>so for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>></mo><mn>6</mn><mo>,</mo><mo> </mo><msub><mi>h</mi><mi>A</mi></msub><mfenced><mi>t</mi></mfenced><mo>-</mo><msub><mi>h</mi><mi>B</mi></msub><mfenced><mi>t</mi></mfenced><mo>=</mo><mi>t</mi><mo>-</mo><mn>6</mn><mo>></mo><mn>0</mn></math> <em><strong>A1</strong></em></p>
<p><br><strong>THEN</strong></p>
<p>hence for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>></mo><mn>6</mn></math>, Plant <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> was always taller than Plant <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>recognises that <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>h</mi><mi>A</mi></msub><mo>'</mo><mfenced><mi>t</mi></mfenced></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>h</mi><mi>B</mi></msub><mo>'</mo><mfenced><mi>t</mi></mfenced></math> are required <em><strong>(M1)</strong></em></p>
<p>attempts to solve <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>h</mi><mi>A</mi></msub><mo>'</mo><mfenced><mi>t</mi></mfenced><mo>=</mo><msub><mi>h</mi><mi>B</mi></msub><mo>'</mo><mfenced><mi>t</mi></mfenced></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>1</mn><mo>.</mo><mn>18879</mn><mo>…</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>.</mo><mn>23598</mn><mo>…</mo></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>.</mo><mn>33038</mn><mo>…</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mo>.</mo><mn>37758</mn><mo>…</mo></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn><mo>.</mo><mn>47197</mn><mo>…</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn><mo>.</mo><mn>51917</mn><mo>…</mo></math> <em><strong>(A1)</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award full marks for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mfrac><mrow><mn>4</mn><mi mathvariant="normal">π</mi></mrow><mn>3</mn></mfrac><mo>-</mo><mn>3</mn><mo>,</mo><mo> </mo><mfrac><mrow><mn>5</mn><mi mathvariant="normal">π</mi></mrow><mn>3</mn></mfrac><mo>-</mo><mn>3</mn><mo>,</mo><mo> </mo><mfenced><mrow><mfrac><mrow><mn>7</mn><mi mathvariant="normal">π</mi></mrow><mn>3</mn></mfrac><mo>-</mo><mn>3</mn><mo>,</mo><mo> </mo><mfrac><mrow><mn>8</mn><mi mathvariant="normal">π</mi></mrow><mn>3</mn></mfrac><mo>-</mo><mn>3</mn><mo> </mo><mfrac><mrow><mn>10</mn><mi mathvariant="normal">π</mi></mrow><mn>3</mn></mfrac><mo>-</mo><mn>3</mn><mo>,</mo><mo> </mo><mfrac><mrow><mn>11</mn><mi mathvariant="normal">π</mi></mrow><mn>3</mn></mfrac><mo>-</mo><mn>3</mn></mrow></mfenced></math>.</p>
<p><em>Award</em> subsequent marks for correct use of these exact values.</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.</mo><mn>18879</mn><mo>…</mo><mo><</mo><mi>t</mi><mo><</mo><mn>2</mn><mo>.</mo><mn>23598</mn><mo>…</mo></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>.</mo><mn>33038</mn><mo>…</mo><mo><</mo><mi>t</mi><mo><</mo><mn>5</mn><mo>.</mo><mn>37758</mn><mo>…</mo></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn><mo>.</mo><mn>47197</mn><mo>…</mo><mo><</mo><mi>t</mi><mo><</mo><mn>8</mn><mo>.</mo><mn>51917</mn><mo>…</mo></math> <em><strong>(A1)</strong></em></p>
<p>attempts to calculate the total amount of time <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mfenced><mrow><mn>2</mn><mo>.</mo><mn>2359</mn><mo>…</mo><mo>-</mo><mn>1</mn><mo>.</mo><mn>1887</mn><mo>…</mo></mrow></mfenced><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mn>3</mn><mfenced><mrow><mfenced><mrow><mfrac><mrow><mn>5</mn><mi mathvariant="normal">π</mi></mrow><mn>3</mn></mfrac><mo>-</mo><mn>3</mn></mrow></mfenced><mo>-</mo><mfenced><mrow><mfrac><mrow><mn>4</mn><mi mathvariant="normal">π</mi></mrow><mn>3</mn></mfrac><mo>-</mo><mn>3</mn></mrow></mfenced></mrow></mfenced></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>3</mn><mo>.</mo><mn>14</mn><mo> </mo><mfenced><mrow><mo>=</mo><mi mathvariant="normal">π</mi></mrow></mfenced></math> (weeks) <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>Part (a) In general, very well done, most students scored full marks. Some though had an incorrect answer for part(a)(ii) because they had their GDC in degrees.</p>
<p>Part (b) Well attempted. Some accuracy errors and not all candidates listed all three values.</p>
<p>Part (c) Most students tried a graphical approach (but this would only get them one out of three marks) and only some provided a convincing algebraic justification. Many candidates tried to explain in words without a convincing mathematical justification or used numerical calculations with specific time values. Some arrived at the correct simplified equation for the difference in heights but could not do much with it. Then only a few provided a correct mathematical proof.</p>
<p>Part (d) In general, well attempted by many candidates. The common error was giving the answer as 3.15 due to the pre-mature rounding. Some candidates only provided the values of time when the rates are equal, some intervals rather than the total time.</p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the following diagram.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<p>The sides of the equilateral triangle ABC have lengths 1 m. The midpoint of [AB] is denoted by P. The circular arc AB has centre, M, the midpoint of [CP].</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find AM.</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="{\text{A}}\mathop {\text{M}}\limits^ \wedge {\text{P}}">
<mrow>
<mtext>A</mtext>
</mrow>
<mover>
<mrow>
<mtext>M</mtext>
</mrow>
<mo>∧</mo>
</mover>
<mo></mo>
<mrow>
<mtext>P</mtext>
</mrow>
</math></span> in radians.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the area of the shaded region.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>PC <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{\sqrt 3 }}{2}">
<mo>=</mo>
<mfrac>
<mrow>
<msqrt>
<mn>3</mn>
</msqrt>
</mrow>
<mn>2</mn>
</mfrac>
</math></span> or 0.8660 <em><strong>(M1)</strong></em></p>
<p>PM <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{2}">
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</math></span>PC <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{\sqrt 3 }}{4}">
<mo>=</mo>
<mfrac>
<mrow>
<msqrt>
<mn>3</mn>
</msqrt>
</mrow>
<mn>4</mn>
</mfrac>
</math></span> or 0.4330 <strong>(A1)</strong></p>
<p>AM <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \sqrt {\frac{1}{4} + \frac{3}{{16}}} ">
<mo>=</mo>
<msqrt>
<mfrac>
<mn>1</mn>
<mn>4</mn>
</mfrac>
<mo>+</mo>
<mfrac>
<mn>3</mn>
<mrow>
<mn>16</mn>
</mrow>
</mfrac>
</msqrt>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{\sqrt 7 }}{4}">
<mo>=</mo>
<mfrac>
<mrow>
<msqrt>
<mn>7</mn>
</msqrt>
</mrow>
<mn>4</mn>
</mfrac>
</math></span> or 0.661 (m) <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>using the cosine rule</p>
<p>AM<sup>2</sup> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {1^2} + {\left( {\frac{{\sqrt 3 }}{4}} \right)^2} - 2 \times \frac{{\sqrt 3 }}{4} \times {\text{cos}}\left( {30^\circ } \right)">
<mo>=</mo>
<mrow>
<msup>
<mn>1</mn>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<msqrt>
<mn>3</mn>
</msqrt>
</mrow>
<mn>4</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>2</mn>
<mo>×</mo>
<mfrac>
<mrow>
<msqrt>
<mn>3</mn>
</msqrt>
</mrow>
<mn>4</mn>
</mfrac>
<mo>×</mo>
<mrow>
<mtext>cos</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<msup>
<mn>30</mn>
<mo>∘</mo>
</msup>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>M1A1</strong></em></p>
<p>AM <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{\sqrt 7 }}{4}">
<mo>=</mo>
<mfrac>
<mrow>
<msqrt>
<mn>7</mn>
</msqrt>
</mrow>
<mn>4</mn>
</mfrac>
</math></span> or 0.661 (m) <em><strong>A1</strong></em></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>tan (<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{A}}\mathop {\text{M}}\limits^ \wedge {\text{P}}">
<mrow>
<mtext>A</mtext>
</mrow>
<mover>
<mrow>
<mtext>M</mtext>
</mrow>
<mo>∧</mo>
</mover>
<mo></mo>
<mrow>
<mtext>P</mtext>
</mrow>
</math></span>) <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{2}{{\sqrt 3 }}">
<mo>=</mo>
<mfrac>
<mn>2</mn>
<mrow>
<msqrt>
<mn>3</mn>
</msqrt>
</mrow>
</mfrac>
</math></span> or equivalent <em><strong>(M1)</strong></em></p>
<p>= 0.857 <em><strong>A1</strong></em></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><strong>EITHER</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{2}{\text{A}}{{\text{M}}^2}\left( {2\,{\text{A}}\mathop {\text{M}}\limits^ \wedge {\text{P}} - {\text{sin}}\left( {2\,{\text{A}}\mathop {\text{M}}\limits^ \wedge {\text{P}}} \right)} \right)">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mrow>
<mtext>A</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>M</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>A</mtext>
</mrow>
<mover>
<mrow>
<mtext>M</mtext>
</mrow>
<mo>∧</mo>
</mover>
<mo></mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo>−</mo>
<mrow>
<mtext>sin</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>A</mtext>
</mrow>
<mover>
<mrow>
<mtext>M</mtext>
</mrow>
<mo>∧</mo>
</mover>
<mo></mo>
<mrow>
<mtext>P</mtext>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>(M1)A1</strong></em></p>
<p><strong>OR </strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{2}{\text{A}}{{\text{M}}^2} \times 2\,{\text{A}}\mathop {\text{M}}\limits^ \wedge {\text{P}} - = \frac{{\sqrt 3 }}{8}">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mrow>
<mtext>A</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>M</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>×</mo>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>A</mtext>
</mrow>
<mover>
<mrow>
<mtext>M</mtext>
</mrow>
<mo>∧</mo>
</mover>
<mo></mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo>−</mo>
<mo>=</mo>
<mfrac>
<mrow>
<msqrt>
<mn>3</mn>
</msqrt>
</mrow>
<mn>8</mn>
</mfrac>
</math></span> <em><strong>(M1)A1</strong></em></p>
<p>= 0.158(m<sup>2</sup>) <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1</strong> </em>for attempting to calculate area of a sector minus area of a triangle.</p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.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>
<br><hr><br><div class="question">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = {\text{tan}}\left( {x + \pi } \right){\text{cos}}\left( {x - \frac{\pi }{2}} \right)">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mtext>tan</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mi>π</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mtext>cos</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0 < x < \frac{\pi }{2}">
<mn>0</mn>
<mo><</mo>
<mi>x</mi>
<mo><</mo>
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
</math></span>.</p>
<p>Express <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right)">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span> in terms of sin <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> and cos <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{tan}}\left( {x + \pi } \right) = \tan x\left( { = \frac{{{\text{sin}}\,x}}{{{\text{cos}}\,x}}} \right)">
<mrow>
<mtext>tan</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mi>π</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mi>tan</mi>
<mo></mo>
<mi>x</mi>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mrow>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span><em><strong> (M1)A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\left( {x - \frac{\pi }{2}} \right) = {\text{sin}}\,x">
<mrow>
<mtext>cos</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</math></span><em><strong> (M1)A1</strong></em></p>
<p><strong>Note:</strong> The two <em><strong>M1</strong></em>s can be awarded for observation or for expanding.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{tan}}\left( {x + \pi } \right) = {\text{cos}}\left( {x - \frac{\pi }{2}} \right) = \frac{{{\text{si}}{{\text{n}}^2}\,x}}{{{\text{cos}}\,x}}">
<mrow>
<mtext>tan</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mi>π</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mtext>cos</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<mtext>si</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mrow>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
</mfrac>
</math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[5 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>Consider the vectors <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">a</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">b</mi></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">a</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mn>12</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>5</mn></mtd></mtr></mtable></mfenced></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mi mathvariant="bold-italic">b</mi></mfenced><mo>=</mo><mn>15</mn></math>.</p>
</div>
<div class="specification">
<p>Consider the vector <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">p</mi></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">p</mi><mo>=</mo><mi mathvariant="bold-italic">a</mi><mo>+</mo><mi mathvariant="bold-italic">b</mi></math>.</p>
</div>
<div class="specification">
<p>Consider the vector <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">q</mi></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">q</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mi>x</mi></mtd></mtr><mtr><mtd><mi>y</mi></mtd></mtr></mtable></mfenced></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>,</mo><mo> </mo><mi>y</mi><mo>∈</mo><msup><mi mathvariant="normal">ℝ</mi><mo>+</mo></msup></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the possible range of values for <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mrow><mi mathvariant="bold-italic">a</mi><mo>+</mo><mi mathvariant="bold-italic">b</mi></mrow></mfenced></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>Given that <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mrow><mi mathvariant="bold-italic">a</mi><mo>+</mo><mi mathvariant="bold-italic">b</mi></mrow></mfenced></math> is a minimum, find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">p</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">q</mi></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>|</mo><mi mathvariant="bold-italic">q</mi><mo>|</mo><mo>=</mo><mo>|</mo><mi mathvariant="bold-italic">b</mi><mo>|</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">q</mi></math> is perpendicular to <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">a</mi></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mi mathvariant="bold-italic">a</mi></mfenced><mo>=</mo><msqrt><msup><mn>12</mn><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mo>-</mo><mn>5</mn></mrow></mfenced><mn>2</mn></msup></msqrt><mfenced><mrow><mo>=</mo><mn>13</mn></mrow></mfenced></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>≤</mo><mfenced open="|" close="|"><mrow><mi mathvariant="bold-italic">a</mi><mo>+</mo><mi mathvariant="bold-italic">b</mi></mrow></mfenced><mo>≤</mo><mn>28</mn></math> (accept min <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math> and max <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>28</mn></math>) <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>(A1)A0</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>28</mn></math> seen with no indication that they are the endpoints of an interval.</p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>recognition that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">p</mi></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">b</mi></math> is a negative multiple of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">a</mi></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">p</mi><mo>=</mo><mo>-</mo><mn>2</mn><mover><mi mathvariant="bold-italic">a</mi><mo mathvariant="bold">^</mo></mover></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">b</mi><mo>=</mo><mo>-</mo><mfrac><mn>15</mn><mn>13</mn></mfrac><mi mathvariant="bold-italic">a</mi><mfenced><mrow><mo>=</mo><mo>-</mo><mfrac><mn>15</mn><mn>13</mn></mfrac><mfenced><mtable><mtr><mtd><mn>12</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>5</mn></mtd></mtr></mtable></mfenced></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">p</mi><mo>=</mo><mo>-</mo><mfrac><mn>2</mn><mn>13</mn></mfrac><mfenced><mtable><mtr><mtd><mn>12</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>5</mn></mtd></mtr></mtable></mfenced><mfenced><mrow><mo>=</mo><mfenced><mtable><mtr><mtd><mo>-</mo><mn>1</mn><mo>.</mo><mn>85</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo>.</mo><mn>769</mn></mtd></mtr></mtable></mfenced></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">q</mi></math> is perpendicular to <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>12</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>5</mn></mtd></mtr></mtable></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi mathvariant="bold-italic">q</mi></math> is in the direction <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>5</mn></mtd></mtr><mtr><mtd><mn>12</mn></mtd></mtr></mtable></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">q</mi><mo>=</mo><mi>k</mi><mfenced><mtable><mtr><mtd><mn>5</mn></mtd></mtr><mtr><mtd><mn>12</mn></mtd></mtr></mtable></mfenced></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mfenced open="|" close="|"><mi mathvariant="bold-italic">q</mi></mfenced><mo>=</mo></mrow></mfenced><msqrt><msup><mfenced><mrow><mn>5</mn><mi>k</mi></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mn>12</mn><mi>k</mi></mrow></mfenced><mn>2</mn></msup></msqrt><mo>=</mo><mn>15</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mfrac><mn>15</mn><mn>13</mn></mfrac></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">q</mi><mo>=</mo><mfrac><mn>15</mn><mn>13</mn></mfrac><mfenced><mtable><mtr><mtd><mn>5</mn></mtd></mtr><mtr><mtd><mn>12</mn></mtd></mtr></mtable></mfenced><mfenced><mrow><mo>=</mo><mfenced><mtable><mtr><mtd><mfrac><mn>75</mn><mn>13</mn></mfrac></mtd></mtr><mtr><mtd><mfrac><mn>180</mn><mn>13</mn></mfrac></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mtable><mtr><mtd><mn>5</mn><mo>.</mo><mn>77</mn></mtd></mtr><mtr><mtd><mn>13</mn><mo>.</mo><mn>8</mn></mtd></mtr></mtable></mfenced></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 mathvariant="bold-italic">q</mi></math> is perpendicular to <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>12</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>5</mn></mtd></mtr></mtable></mfenced></math></p>
<p>attempt to set scalar product <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">q</mi><mo mathvariant="bold">.</mo><mi mathvariant="bold-italic">a</mi><mo>=</mo><mn>0</mn></math> OR product of gradients <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><mn>1</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>12</mn><mi>x</mi><mo>-</mo><mn>5</mn><mi>y</mi><mo>=</mo><mn>0</mn></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mfenced open="|" close="|"><mi mathvariant="bold-italic">q</mi></mfenced><mo>=</mo></mrow></mfenced><msqrt><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup></msqrt><mo>=</mo><mn>15</mn></math></p>
<p>attempt to solve simultaneously to find a quadratic in <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> or <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"><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mfenced><mfrac><mrow><mn>12</mn><mi>x</mi></mrow><mn>5</mn></mfrac></mfenced><mn>2</mn></msup><mo>=</mo><msup><mn>15</mn><mn>2</mn></msup></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mfrac><mrow><mn>5</mn><mi>y</mi></mrow><mn>12</mn></mfrac></mfenced><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><msup><mn>15</mn><mn>2</mn></msup></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">q</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mfrac><mn>75</mn><mn>13</mn></mfrac></mtd></mtr><mtr><mtd><mfrac><mn>180</mn><mn>13</mn></mfrac></mtd></mtr></mtable></mfenced><mfenced><mrow><mo>=</mo><mfenced><mtable><mtr><mtd><mn>5</mn><mo>.</mo><mn>77</mn></mtd></mtr><mtr><mtd><mn>13</mn><mo>.</mo><mn>8</mn></mtd></mtr></mtable></mfenced></mrow></mfenced></math> <em><strong>A1A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> independently for each value. Accept values given as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mn>75</mn><mn>13</mn></mfrac></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfrac><mn>180</mn><mn>13</mn></mfrac></math> or equivalent.</p>
<p> </p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Three points <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext><mfenced><mrow><mn>3</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>,</mo><mo> </mo><mn>0</mn></mrow></mfenced><mo>,</mo><mo> </mo><mtext>B</mtext><mfenced><mrow><mn>0</mn><mo>,</mo><mo> </mo><mo>-</mo><mn>2</mn><mo>,</mo><mo> </mo><mn>0</mn></mrow></mfenced></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>C</mtext><mfenced><mrow><mn>1</mn><mo>,</mo><mo> </mo><mn>1</mn><mo>,</mo><mo> </mo><mo>-</mo><mn>7</mn></mrow></mfenced></math> lie on the plane <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Π</mi><mn>1</mn></msub></math>.</p>
</div>
<div class="specification">
<p>Plane <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Π</mi><mn>2</mn></msub></math> has equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mi>x</mi><mo>-</mo><mi>y</mi><mo>+</mo><mn>2</mn><mi>z</mi><mo>=</mo><mn>2</mn></math>.</p>
</div>
<div class="specification">
<p>The plane <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Π</mi><mn>3</mn></msub></math> is given by <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>x</mi><mo>-</mo><mn>2</mn><mi>z</mi><mo>=</mo><mn>3</mn></math>. The line <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi></math> and the plane <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Π</mi><mn>3</mn></msub></math> intersect at the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math>.</p>
</div>
<div class="specification">
<p>The point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext><mo>(</mo><mn>0</mn><mo>,</mo><mo> </mo><mo>-</mo><mn>2</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math> lies on <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the vector <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mtext>AB</mtext><mo>→</mo></mover></math> and the vector <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mtext>AC</mtext><mo>→</mo></mover></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>Hence find the equation of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Π</mi><mn>1</mn></msub></math>, expressing your answer in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi><mi>y</mi><mo>+</mo><mi>c</mi><mi>z</mi><mo>=</mo><mi>d</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>,</mo><mo> </mo><mi>b</mi><mo>,</mo><mo> </mo><mi>c</mi><mo>,</mo><mo> </mo><mi>d</mi><mo>∈</mo><mi mathvariant="normal">ℤ</mi></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The line <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi></math> is the intersection of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Π</mi><mn>1</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Π</mi><mn>2</mn></msub></math>. Verify that the vector equation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi></math> can be written as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">r</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mi>λ</mi><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr></mtable></mfenced></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that at the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mo>,</mo><mo> </mo><mi>λ</mi><mo>=</mo><mfrac><mn>3</mn><mn>4</mn></mfrac></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find the coordinates of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the reflection of the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math> in the plane <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Π</mi><mn>3</mn></msub></math>.</p>
<div class="marks">[7]</div>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find the vector equation of the line formed when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi></math> is reflected in the plane <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Π</mi><mn>3</mn></msub></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>attempts to find either <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mtext>AB</mtext><mo>→</mo></mover></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mtext>AC</mtext><mo>→</mo></mover></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mtext>AB</mtext><mo>→</mo></mover><mo>=</mo><mfenced><mtable><mtr><mtd><mo>-</mo><mn>3</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr></mtable></mfenced></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mtext>AC</mtext><mo>→</mo></mover><mo>=</mo><mfenced><mtable><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>7</mn></mtd></mtr></mtable></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>attempts to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mtext>AB</mtext><mo>→</mo></mover><mo>×</mo><mover><mtext>AC</mtext><mo>→</mo></mover></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mtext>AB</mtext><mo>→</mo></mover><mo>×</mo><mover><mtext>AC</mtext><mo>→</mo></mover><mo>=</mo><mfenced><mtable><mtr><mtd><mn>14</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>21</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>7</mn></mtd></mtr></mtable></mfenced></math> <em><strong>A1</strong></em></p>
<p><br><strong>EITHER</strong></p>
<p>equation of plane is of the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>14</mn><mi>x</mi><mo>-</mo><mn>21</mn><mi>y</mi><mo>-</mo><mn>7</mn><mi>z</mi><mo>=</mo><mi>d</mi><mo> </mo><mo> </mo><mo> </mo><mfenced><mrow><mn>2</mn><mi>x</mi><mo>-</mo><mn>3</mn><mi>y</mi><mo>-</mo><mi>z</mi><mo>=</mo><mi>d</mi></mrow></mfenced></math> <em><strong>(A1)</strong></em></p>
<p>substitutes a valid point e.g <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>3</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>,</mo><mo> </mo><mn>0</mn></mrow></mfenced></math> to obtain a value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo>=</mo><mn>42</mn><mo> </mo><mo> </mo><mfenced><mrow><mi>d</mi><mo>=</mo><mn>6</mn></mrow></mfenced></math></p>
<p><br><strong>OR</strong></p>
<p>attempts to use <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">r</mi><mo>·</mo><mi mathvariant="bold-italic">n</mi><mo>=</mo><mi mathvariant="bold-italic">a</mi><mo>·</mo><mi mathvariant="bold-italic">n</mi></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">r</mi><mo>·</mo><mfenced><mtable><mtr><mtd><mn>14</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>21</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>7</mn></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mtable><mtr><mtd><mn>3</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr></mtable></mfenced><mo>·</mo><mfenced><mtable><mtr><mtd><mn>14</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>21</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>7</mn></mtd></mtr></mtable></mfenced><mo> </mo><mo> </mo><mfenced><mrow><mi mathvariant="bold-italic">r</mi><mo>·</mo><mfenced><mtable><mtr><mtd><mn>14</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>21</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>7</mn></mtd></mtr></mtable></mfenced><mo>=</mo><mn>42</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">r</mi><mo>·</mo><mfenced><mtable><mtr><mtd><mn>2</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>3</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mtable><mtr><mtd><mn>3</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr></mtable></mfenced><mo>·</mo><mfenced><mtable><mtr><mtd><mn>2</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>3</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr></mtable></mfenced><mo> </mo><mo> </mo><mfenced><mrow><mi mathvariant="bold-italic">r</mi><mo>·</mo><mfenced><mtable><mtr><mtd><mn>2</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>3</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr></mtable></mfenced><mo>=</mo><mn>6</mn></mrow></mfenced></math></p>
<p><br><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>14</mn><mi>x</mi><mo>-</mo><mn>21</mn><mi>y</mi><mo>-</mo><mn>7</mn><mi>z</mi><mo>=</mo><mn>42</mn><mo> </mo><mo> </mo><mfenced><mrow><mn>2</mn><mi>x</mi><mo>-</mo><mn>3</mn><mi>y</mi><mo>-</mo><mi>z</mi><mo>=</mo><mn>6</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>equation of plane is of the form <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><mtr><mtd><mi>z</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mtable><mtr><mtd><mn>3</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mi>s</mi><mfenced><mtable><mtr><mtd><mo>-</mo><mn>3</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mi>t</mi><mfenced><mtable><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>7</mn></mtd></mtr></mtable></mfenced></math> <em><strong>A1</strong></em></p>
<p>attempts to form equations for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>,</mo><mo> </mo><mi>y</mi><mo>,</mo><mo> </mo><mi>z</mi><mo> </mo></math>in terms of their parameters <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>3</mn><mo>-</mo><mn>3</mn><mi>s</mi><mo>-</mo><mn>2</mn><mi>t</mi><mo> </mo><mo>,</mo><mo> </mo><mi>y</mi><mo>=</mo><mo>-</mo><mn>2</mn><mi>s</mi><mo>+</mo><mi>t</mi><mo> </mo><mo>,</mo><mo> </mo><mi>z</mi><mo>=</mo><mo>-</mo><mn>7</mn><mi>t</mi></math> <em><strong>A1</strong></em></p>
<p>eliminates at least one of their parameters <em><strong>(M1)</strong></em></p>
<p>for example, <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>x</mi><mo>-</mo><mn>3</mn><mi>y</mi><mo>=</mo><mn>6</mn><mo>-</mo><mn>7</mn><mi>t</mi><mfenced><mrow><mo>⇒</mo><mn>2</mn><mi>x</mi><mo>-</mo><mn>3</mn><mi>y</mi><mo>=</mo><mn>6</mn><mo>+</mo><mi>z</mi></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>x</mi><mo>-</mo><mn>3</mn><mi>y</mi><mo>-</mo><mi>z</mi><mo>=</mo><mn>6</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[5 marks]</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>substitutes <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">r</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mi>λ</mi><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr></mtable></mfenced></math> into their <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Π</mi><mn>1</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Π</mi><mn>2</mn></msub></math> (given) <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Π</mi><mn>1</mn></msub><mo>:</mo><mo> </mo><mn>2</mn><mi>λ</mi><mo>-</mo><mn>3</mn><mfenced><mrow><mo>-</mo><mn>2</mn><mo>+</mo><mi>λ</mi></mrow></mfenced><mo>-</mo><mfenced><mrow><mo>-</mo><mi>λ</mi></mrow></mfenced><mo>=</mo><mn>6</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Π</mi><mn>2</mn></msub><mo>:</mo><mo> </mo><mn>3</mn><mi>λ</mi><mo>-</mo><mn>3</mn><mfenced><mrow><mo>-</mo><mn>2</mn><mo>+</mo><mi>λ</mi></mrow></mfenced><mo>+</mo><mn>2</mn><mfenced><mrow><mo>-</mo><mi>λ</mi></mrow></mfenced><mo>=</mo><mn>2</mn></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>(M1)A0</strong></em> for correct verification using a specific value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi></math>.</p>
<p>so the vector equation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi></math> can be written as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">r</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mi>λ</mi><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr></mtable></mfenced></math> <em><strong>AG</strong></em></p>
<p><br><strong>METHOD 2</strong></p>
<p><strong>EITHER</strong></p>
<p>attempts to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>2</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>3</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr></mtable></mfenced><mo>×</mo><mfenced><mtable><mtr><mtd><mn>3</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable></mfenced></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced><mtable><mtr><mtd><mo>-</mo><mn>7</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>7</mn></mtd></mtr><mtr><mtd><mn>7</mn></mtd></mtr></mtable></mfenced></math></p>
<p><br><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>2</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>3</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr></mtable></mfenced><mo>·</mo><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mrow><mn>2</mn><mo>-</mo><mn>3</mn><mo>+</mo><mn>1</mn></mrow></mfenced><mo>=</mo><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>3</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable></mfenced><mo>·</mo><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mrow><mn>3</mn><mo>-</mo><mn>1</mn><mo>-</mo><mn>2</mn></mrow></mfenced><mo>=</mo><mn>0</mn></math> <em><strong>M1</strong></em></p>
<p><br><strong>THEN</strong></p>
<p>substitutes <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>0</mn><mo>,</mo><mo>-</mo><mn>2</mn><mo>,</mo><mn>0</mn></mrow></mfenced></math> into <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Π</mi><mn>1</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Π</mi><mn>2</mn></msub></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Π</mi><mn>1</mn></msub><mo>:</mo><mo> </mo><mn>2</mn><mfenced><mn>0</mn></mfenced><mo>-</mo><mn>3</mn><mfenced><mrow><mo>-</mo><mn>2</mn></mrow></mfenced><mo>-</mo><mfenced><mn>0</mn></mfenced><mo>=</mo><mn>6</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Π</mi><mn>2</mn></msub><mo>:</mo><mo> </mo><mn>3</mn><mfenced><mn>0</mn></mfenced><mo>-</mo><mfenced><mrow><mo>-</mo><mn>2</mn></mrow></mfenced><mo>+</mo><mn>2</mn><mfenced><mn>0</mn></mfenced><mo>=</mo><mn>2</mn></math> <em><strong>A1</strong></em></p>
<p>so the vector equation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi></math> can be written as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">r</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mi>λ</mi><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr></mtable></mfenced></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>METHOD 3</strong></p>
<p>attempts to solve <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>x</mi><mo>-</mo><mn>3</mn><mi>y</mi><mo>-</mo><mi>z</mi><mo>=</mo><mn>6</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mi>x</mi><mo>-</mo><mi>y</mi><mo>+</mo><mn>2</mn><mi>z</mi><mo>=</mo><mn>2</mn></math> <em><strong>(M1)</strong></em></p>
<p>for example, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mo>-</mo><mi>λ</mi><mo>,</mo><mo> </mo><mi>y</mi><mo>=</mo><mo>-</mo><mn>2</mn><mo>-</mo><mi>λ</mi><mo>,</mo><mo> </mo><mi>z</mi><mo>=</mo><mi>λ</mi></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Award <strong>A1</strong> for substituting <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn></math> (or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>-</mo><mn>2</mn></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>=</mo><mn>0</mn></math>) into <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Π</mi><mn>1</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Π</mi><mn>2</mn></msub></math> and solving simultaneously. For example, solving <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>3</mn><mi>y</mi><mo>-</mo><mi>z</mi><mo>=</mo><mn>6</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mi>y</mi><mo>+</mo><mn>2</mn><mi>z</mi><mo>=</mo><mn>2</mn></math> to obtain <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>-</mo><mn>2</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>=</mo><mn>0</mn></math>.</p>
<p>so the vector equation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi></math> can be written as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">r</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mi>λ</mi><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr></mtable></mfenced></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>substitutes the equation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi></math> into the equation of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Π</mi><mn>3</mn></msub></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>λ</mi><mo>+</mo><mn>2</mn><mi>λ</mi><mo>=</mo><mn>3</mn><mo>⇒</mo><mn>4</mn><mi>λ</mi><mo>=</mo><mn>3</mn></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi><mo>=</mo><mfrac><mn>3</mn><mn>4</mn></mfrac></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math> has coordinates <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mfrac><mn>3</mn><mn>4</mn></mfrac><mo>,</mo><mo>-</mo><mfrac><mn>5</mn><mn>4</mn></mfrac><mo>,</mo><mo>-</mo><mfrac><mn>3</mn><mn>4</mn></mfrac></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>normal to <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Π</mi><mn>3</mn></msub></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">n</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mn>2</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr></mtable></mfenced></math> <em><strong>(A1)</strong></em></p>
<p><br><strong>Note:</strong> May be seen or used anywhere.</p>
<p><br>considers the line normal to <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Π</mi><mn>3</mn></msub></math> passing through <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext><mfenced><mrow><mn>0</mn><mo>,</mo><mo>-</mo><mn>2</mn><mo>,</mo><mn>0</mn></mrow></mfenced></math> <em><strong>(M1)</strong></em><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">r</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mi>μ</mi><mfenced><mtable><mtr><mtd><mn>2</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr></mtable></mfenced></math> <em><strong>A1</strong></em></p>
<p><strong><br>EITHER</strong></p>
<p>finding the point on the normal line that intersects <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Π</mi><mn>3</mn></msub></math><br>attempts to solve simultaneously with plane <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>x</mi><mo>-</mo><mn>2</mn><mi>z</mi><mo>=</mo><mn>3</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mi>μ</mi><mo>+</mo><mn>4</mn><mi>μ</mi><mo>=</mo><mn>3</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>μ</mi><mo>=</mo><mfrac><mn>3</mn><mn>8</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p>point is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mfrac><mn>3</mn><mn>4</mn></mfrac><mo>,</mo><mo> </mo><mo>-</mo><mn>2</mn><mo>,</mo><mo>-</mo><mfrac><mn>3</mn><mn>4</mn></mfrac></mrow></mfenced></math></p>
<p><br><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mfenced><mtable><mtr><mtd><mn>2</mn><mi>μ</mi></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn><mi>μ</mi></mtd></mtr></mtable></mfenced><mo>-</mo><mfenced><mtable><mtr><mtd><mstyle displaystyle="false"><mfrac><mn>3</mn><mn>4</mn></mfrac></mstyle></mtd></mtr><mtr><mtd><mo>-</mo><mstyle displaystyle="false"><mfrac><mn>5</mn><mn>4</mn></mfrac></mstyle></mtd></mtr><mtr><mtd><mo>-</mo><mstyle displaystyle="false"><mfrac><mn>3</mn><mn>4</mn></mfrac></mstyle></mtd></mtr></mtable></mfenced></mrow></mfenced><mo>·</mo><mfenced><mtable><mtr><mtd><mn>2</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr></mtable></mfenced><mo>=</mo><mn>0</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mi>μ</mi><mo>-</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mo>+</mo><mn>4</mn><mi>μ</mi><mo>-</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mo>=</mo><mn>0</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>μ</mi><mo>=</mo><mfrac><mn>3</mn><mn>8</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p><br><strong>OR</strong></p>
<p>attempts to find the equation of the plane parallel to <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Π</mi><mn>3</mn></msub></math> containing <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext><mo>'</mo><mo> </mo><mfenced><mrow><mi>x</mi><mo>-</mo><mi>z</mi><mo>=</mo><mn>3</mn></mrow></mfenced></math> and solve simultaneously with <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>μ</mi><mo>'</mo><mo>+</mo><mn>2</mn><mi>μ</mi><mo>'</mo><mo>=</mo><mn>3</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>μ</mi><mo>'</mo><mo>=</mo><mfrac><mn>3</mn><mn>4</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p><br><strong>THEN</strong></p>
<p>so, another point on the reflected line is given by</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">r</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mfrac><mn>3</mn><mn>4</mn></mfrac><mfenced><mtable><mtr><mtd><mn>2</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr></mtable></mfenced></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mtext>B</mtext><mo>'</mo><mfenced><mrow><mfrac><mn>3</mn><mn>2</mn></mfrac><mo>,</mo><mo>-</mo><mn>2</mn><mo>,</mo><mo>-</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[7 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>attempts to find the direction vector of the reflected line using their <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B'</mtext></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mrow><mtext>PB</mtext><mo>'</mo></mrow><mo>→</mo></mover><mo>=</mo><mfenced><mtable><mtr><mtd><mstyle displaystyle="false"><mfrac><mn>3</mn><mn>4</mn></mfrac></mstyle></mtd></mtr><mtr><mtd><mo>-</mo><mstyle displaystyle="false"><mfrac><mn>3</mn><mn>4</mn></mfrac></mstyle></mtd></mtr><mtr><mtd><mo>-</mo><mstyle displaystyle="false"><mfrac><mn>3</mn><mn>4</mn></mfrac></mstyle></mtd></mtr></mtable></mfenced></math></p>
<p><br><strong>OR</strong></p>
<p>attempts to find their direction vector of the reflected line using a vector approach <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mrow><mtext>PB</mtext><mo>'</mo></mrow><mo>→</mo></mover><mo>=</mo><mover><mtext>PB</mtext><mo>→</mo></mover><mo>+</mo><mover><mrow><mtext>BB</mtext><mo>'</mo></mrow><mo>→</mo></mover><mo>=</mo><mo>-</mo><mfrac><mn>3</mn><mn>4</mn></mfrac><mfenced><mtable><mtr><mtd><mstyle displaystyle="false"><mn>1</mn></mstyle></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mfrac><mn>3</mn><mn>2</mn></mfrac><mfenced><mtable><mtr><mtd><mstyle displaystyle="false"><mn>1</mn></mstyle></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr></mtable></mfenced></math></p>
<p><br><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">r</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mstyle displaystyle="false"><mfrac><mn>3</mn><mn>2</mn></mfrac></mstyle></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr><mtr><mtd><mo>-</mo><mstyle displaystyle="false"><mfrac><mn>3</mn><mn>2</mn></mfrac></mstyle></mtd></mtr></mtable></mfenced><mo>+</mo><mi>λ</mi><mfenced><mtable><mtr><mtd><mstyle displaystyle="false"><mfrac><mn>3</mn><mn>4</mn></mfrac></mstyle></mtd></mtr><mtr><mtd><mo>-</mo><mstyle displaystyle="false"><mfrac><mn>3</mn><mn>4</mn></mfrac></mstyle></mtd></mtr><mtr><mtd><mo>-</mo><mstyle displaystyle="false"><mfrac><mn>3</mn><mn>4</mn></mfrac></mstyle></mtd></mtr></mtable></mfenced></math> (or equivalent) <em><strong>A1</strong></em></p>
<p><strong><br>Note:</strong> Award <em><strong>A0</strong></em> for either '<math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">r</mi><mo>=</mo></math>' or '<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><mtr><mtd><mi>z</mi></mtd></mtr></mtable></mfenced><mo>=</mo></math>' not stated. Award <em><strong>A0 </strong></em>for '<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi><mo>'</mo><mo>=</mo></math>'</p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">d.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.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">d.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>The points <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext><mfenced><mrow><mn>5</mn><mo>,</mo><mo> </mo><mo>-</mo><mn>2</mn><mo>,</mo><mo> </mo><mn>5</mn></mrow></mfenced></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext><mfenced><mrow><mn>5</mn><mo>,</mo><mo> </mo><mn>4</mn><mo>,</mo><mo> </mo><mo>-</mo><mn>1</mn></mrow></mfenced></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>C</mtext><mfenced><mrow><mo>-</mo><mn>1</mn><mo>,</mo><mo> </mo><mo>-</mo><mn>2</mn><mo>,</mo><mo> </mo><mo>-</mo><mn>1</mn></mrow></mfenced></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>D</mtext><mfenced><mrow><mn>7</mn><mo>,</mo><mo> </mo><mo>-</mo><mn>4</mn><mo>,</mo><mo> </mo><mo>-</mo><mn>3</mn></mrow></mfenced></math> are the vertices of a right-pyramid.</p>
</div>
<div class="specification">
<p>The line <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi></math> passes through the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>D</mtext></math> and is perpendicular to <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Π</mi></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the vectors <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mtext>AB</mtext><mo>→</mo></mover></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mtext>AC</mtext><mo>→</mo></mover></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>Use a vector method to show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext><mover><mtext>A</mtext><mo>^</mo></mover><mtext>C</mtext><mo>=</mo><mn>60</mn><mo>°</mo></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the Cartesian equation of the plane <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Π</mi></math> that contains the triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>ABC</mtext></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo>+</mo><mi>z</mi><mo>=</mo><mo>-</mo><mn>2</mn></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>Find a vector equation of the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>L</mi></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence determine the minimum distance, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>d</mi><mtext>min</mtext></msub></math>, from <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>D</mtext></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Π</mi></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the volume of right-pyramid <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>ABCD</mtext></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color:#999;font-size:90%;font-style:italic;">* This sample question was produced by experienced DP mathematics senior examiners to aid teachers in preparing for external assessment in the new MAA course. There may be minor differences in formatting compared to formal exam papers.</p>
<p style="color:#999;font-size:90%;font-style:italic;"> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mtext>AB</mtext><mo>→</mo></mover><mo>=</mo><mfenced><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>6</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>6</mn></mtd></mtr></mtable></mfenced><mo> </mo><mfenced><mrow><mo>=</mo><mn>6</mn><mfenced><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mrow></mfenced></math> <strong>A1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mtext>AC</mtext><mo>→</mo></mover><mo>=</mo><mfenced><mtable><mtr><mtd><mo>-</mo><mn>6</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>6</mn></mtd></mtr></mtable></mfenced><mo> </mo><mfenced><mrow><mo>=</mo><mn>6</mn><mfenced><mtable><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mrow></mfenced></math> <strong>A1</strong></p>
<p> </p>
<p><strong>[2 marks]</strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempts to use <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mo> </mo><mtext>B</mtext><mover><mtext>A</mtext><mo>^</mo></mover><mtext>C</mtext><mo>=</mo><mfrac><mrow><mover><mtext>AB</mtext><mo>→</mo></mover><mo>·</mo><mover><mtext>AC</mtext><mo>→</mo></mover></mrow><mrow><mfenced open="|" close="|"><mover><mtext>AB</mtext><mo>→</mo></mover></mfenced><mfenced open="|" close="|"><mover><mtext>AC</mtext><mo>→</mo></mover></mfenced></mrow></mfrac></math> <strong>(M1)</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mfenced><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>6</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>6</mn></mtd></mtr></mtable></mfenced><mo>·</mo><mfenced><mtable><mtr><mtd><mo>-</mo><mn>6</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>6</mn></mtd></mtr></mtable></mfenced></mrow><mrow><msqrt><mn>72</mn></msqrt><mo>×</mo><msqrt><mn>72</mn></msqrt></mrow></mfrac></math> <strong>A1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math> <strong>A1</strong></p>
<p>so <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext><mover><mtext>A</mtext><mo>^</mo></mover><mtext>C</mtext><mo>=</mo><mn>60</mn><mo>°</mo></math> <strong>AG</strong></p>
<p> </p>
<p><strong>[3 marks]</strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempts to find a vector normal to <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Π</mi></math> <strong>M1</strong></p>
<p>for example, <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mtext>AB</mtext><mo>→</mo></mover><mo>×</mo><mover><mtext>AC</mtext><mo>→</mo></mover><mo>=</mo><mfenced><mtable><mtr><mtd><mo>-</mo><mn>36</mn></mtd></mtr><mtr><mtd><mn>36</mn></mtd></mtr><mtr><mtd><mn>36</mn></mtd></mtr></mtable></mfenced><mo> </mo><mfenced><mrow><mo>=</mo><mn>36</mn><mfenced><mtable><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr></mtable></mfenced></mrow></mfenced></math> leading to <strong>A1</strong></p>
<p>a vector normal to <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Π</mi></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">n</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr></mtable></mfenced></math></p>
<p> </p>
<p><strong>EITHER</strong></p>
<p>substitutes <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>5</mn><mo>,</mo><mo> </mo><mo>-</mo><mn>2</mn><mo>,</mo><mo> </mo><mo>-</mo><mn>5</mn></mrow></mfenced></math> (or <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>5</mn><mo>,</mo><mo> </mo><mn>4</mn><mo>,</mo><mo> </mo><mo>-</mo><mn>1</mn></mrow></mfenced></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>-</mo><mn>1</mn><mo>,</mo><mo> </mo><mo>-</mo><mn>2</mn><mo>,</mo><mo> </mo><mo>-</mo><mn>1</mn></mrow></mfenced></math>) into <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo>+</mo><mi>z</mi><mo>=</mo><mi>d</mi></math> and attempts to find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math></p>
<p>for example, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo>=</mo><mo>-</mo><mn>5</mn><mo>-</mo><mn>2</mn><mo>+</mo><mn>5</mn><mo> </mo><mfenced><mrow><mo>=</mo><mo>-</mo><mn>2</mn></mrow></mfenced></math> <strong>M1</strong></p>
<p> </p>
<p><strong>OR</strong></p>
<p>attempts to use <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">r</mi><mo mathvariant="bold">·</mo><mi mathvariant="bold-italic">n</mi><mo>=</mo><mi mathvariant="bold-italic">a</mi><mo mathvariant="bold">·</mo><mi mathvariant="bold-italic">n</mi></math> <strong>M1</strong></p>
<p>for example, <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><mtr><mtd><mi>z</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><mtr><mtd><mn>1</mn></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mtable><mtr><mtd><mn>5</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr><mtr><mtd><mn>5</mn></mtd></mtr></mtable></mfenced><mo>·</mo><mfenced><mtable><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr></mtable></mfenced></math></p>
<p> </p>
<p><strong>THEN</strong></p>
<p>leading to the Cartesian equation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Π</mi></math> as <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo>+</mo><mi>z</mi><mo>=</mo><mo>-</mo><mn>2</mn></math> <strong>AG</strong></p>
<p> </p>
<p><strong>[3 marks]</strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">r</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mn>7</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>4</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>3</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mi>λ</mi><mfenced><mtable><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr></mtable></mfenced><mo> </mo><mfenced><mrow><mi>λ</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></mrow></mfenced></math> <strong>A1</strong></p>
<p> </p>
<p><strong>[1 mark]</strong></p>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>substitutes <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>7</mn><mo>-</mo><mi>λ</mi><mo>,</mo><mo> </mo><mi>y</mi><mo>=</mo><mo>-</mo><mn>4</mn><mo>+</mo><mi>λ</mi><mo>,</mo><mo> </mo><mi>z</mi><mo>=</mo><mo>-</mo><mn>3</mn><mo>+</mo><mi>λ</mi></math> into <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo>+</mo><mi>z</mi><mo>=</mo><mo>-</mo><mn>2</mn></math> <strong>(M1)</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfenced><mrow><mn>7</mn><mo>-</mo><mi>λ</mi></mrow></mfenced><mo>+</mo><mfenced><mrow><mo>-</mo><mn>4</mn><mo>+</mo><mi>λ</mi></mrow></mfenced><mo>+</mo><mfenced><mrow><mo>-</mo><mn>3</mn><mo>+</mo><mi>λ</mi></mrow></mfenced><mo>=</mo><mo>-</mo><mn>2</mn><mo> </mo><mfenced><mrow><mn>3</mn><mi>λ</mi><mo>=</mo><mn>12</mn></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi><mo>=</mo><mn>4</mn></math> <strong>A1</strong></p>
<p>shows a correct calculation for finding <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>d</mi><mtext>min</mtext></msub></math>, for example, attempts to find</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mrow><mn>4</mn><mfenced><mtable><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr></mtable></mfenced></mrow></mfenced></math> <strong>M1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>d</mi><mtext>min</mtext></msub><mo>=</mo><mn>4</mn><msqrt><mn>3</mn></msqrt><mo> </mo><mfenced><mrow><mo>=</mo><mn>6</mn><mo>.</mo><mn>93</mn></mrow></mfenced></math> <strong>A1</strong></p>
<p> </p>
<p><strong>[4 marks]</strong></p>
<div class="question_part_label">d.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>let the area of triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>ABC</mtext></math> be <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math></p>
<p> </p>
<p><strong>EITHER</strong></p>
<p>attempts to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mfenced open="|" close="|"><mrow><mover><mtext>AB</mtext><mo>→</mo></mover><mo>×</mo><mover><mtext>AC</mtext><mo>→</mo></mover></mrow></mfenced></math>, for example <strong>M1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mfenced open="|" close="|"><mfenced><mtable><mtr><mtd><mo>-</mo><mn>36</mn></mtd></mtr><mtr><mtd><mn>36</mn></mtd></mtr><mtr><mtd><mn>36</mn></mtd></mtr></mtable></mfenced></mfenced></math></p>
<p> </p>
<p><strong>OR</strong></p>
<p>attempts to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac><mfenced open="|" close="|"><mover><mtext>AB</mtext><mo>→</mo></mover></mfenced><mfenced open="|" close="|"><mover><mtext>AC</mtext><mo>→</mo></mover></mfenced><mi>sin</mi><mo> </mo><mi>θ</mi></math>, for example <strong>M1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>×</mo><mn>6</mn><msqrt><mn>2</mn></msqrt><mo>×</mo><mn>6</mn><msqrt><mn>2</mn></msqrt><mo>×</mo><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac></math> (where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mfrac><mi mathvariant="normal">π</mi><mn>3</mn></mfrac><mo>=</mo><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac></math>)</p>
<p> </p>
<p><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mn>18</mn><msqrt><mn>3</mn></msqrt><mo> </mo><mfenced><mrow><mo>=</mo><mn>31</mn><mo>.</mo><mn>2</mn></mrow></mfenced></math> <strong>A1</strong></p>
<p>uses <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>A</mi><mi>h</mi></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> is the area of triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>ABC</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>=</mo><msub><mi>d</mi><mtext>min</mtext></msub></math> <strong>M1</strong></p>
<p> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mo>×</mo><mn>18</mn><msqrt><mn>3</mn></msqrt><mo>×</mo><mn>4</mn><msqrt><mn>3</mn></msqrt></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>72</mn></math> <strong>A1</strong></p>
<p> </p>
<p><strong>[4 marks]</strong></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="question">
<p>In triangle ABC, AB = 5, BC = 14 and AC = 11.</p>
<p>Find all the interior angles of the triangle. Give your answers in degrees to one decimal place.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p>attempt to apply cosine rule <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{cos}}\,{\text{A}} = \frac{{{5^2} + {{11}^2} - {{14}^2}}}{{2 \times 5 \times 11}} = - 0.4545 \ldots "> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>A</mtext> </mrow> <mo>=</mo> <mfrac> <mrow> <mrow> <msup> <mn>5</mn> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mrow> <mn>11</mn> </mrow> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mrow> <msup> <mrow> <mn>14</mn> </mrow> <mn>2</mn> </msup> </mrow> </mrow> <mrow> <mn>2</mn> <mo>×</mo> <mn>5</mn> <mo>×</mo> <mn>11</mn> </mrow> </mfrac> <mo>=</mo> <mo>−</mo> <mn>0.4545</mn> <mo>…</mo> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow {\text{A}} = 117.03569 \ldots ^\circ "> <mo stretchy="false">⇒</mo> <mrow> <mtext>A</mtext> </mrow> <mo>=</mo> <mn>117.03569</mn> <msup> <mo>…</mo> <mo>∘</mo> </msup> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow {\text{A}} = 117.0^\circ "> <mo stretchy="false">⇒</mo> <mrow> <mtext>A</mtext> </mrow> <mo>=</mo> <msup> <mn>117.0</mn> <mo>∘</mo> </msup> </math></span> <em><strong> A1</strong></em></p>
<p>attempt to apply sine rule or cosine rule: <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{sin}}\,117.03569 \ldots ^\circ }}{{14}} = \frac{{{\text{sin}}\,{\text{B}}}}{{11}}"> <mfrac> <mrow> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mn>117.03569</mn> <msup> <mo>…</mo> <mo>∘</mo> </msup> </mrow> <mrow> <mn>14</mn> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>B</mtext> </mrow> </mrow> <mrow> <mn>11</mn> </mrow> </mfrac> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow {\text{B}} = 44.4153 \ldots ^\circ "> <mo stretchy="false">⇒</mo> <mrow> <mtext>B</mtext> </mrow> <mo>=</mo> <mn>44.4153</mn> <msup> <mo>…</mo> <mo>∘</mo> </msup> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow {\text{B}} = 44.4^\circ "> <mo stretchy="false">⇒</mo> <mrow> <mtext>B</mtext> </mrow> <mo>=</mo> <msup> <mn>44.4</mn> <mo>∘</mo> </msup> </math></span> <em><strong> A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{C}} = 180^\circ - {\text{A}} - {\text{B}}"> <mrow> <mtext>C</mtext> </mrow> <mo>=</mo> <msup> <mn>180</mn> <mo>∘</mo> </msup> <mo>−</mo> <mrow> <mtext>A</mtext> </mrow> <mo>−</mo> <mrow> <mtext>B</mtext> </mrow> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{C}} = 18.5^\circ "> <mrow> <mtext>C</mtext> </mrow> <mo>=</mo> <msup> <mn>18.5</mn> <mo>∘</mo> </msup> </math></span> <em><strong> A1</strong></em></p>
<p><strong>Note:</strong> Candidates may attempt to find angles in any order of their choosing.</p>
<p><em><strong>[5 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>The following shape consists of three arcs of a circle, each with centre at the opposite vertex of an equilateral triangle as shown in the diagram.</p>
<p style="text-align: center;"><img 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"></p>
<p style="text-align: left;">For this shape, calculate</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>the perimeter.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>the area.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>each arc has length <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r\theta = 6 \times \frac{\pi }{3} = 2\pi \,\left( { = 6.283 \ldots } \right)"> <mi>r</mi> <mi>θ</mi> <mo>=</mo> <mn>6</mn> <mo>×</mo> <mfrac> <mi>π</mi> <mn>3</mn> </mfrac> <mo>=</mo> <mn>2</mn> <mi>π</mi> <mspace width="thinmathspace"></mspace> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mn>6.283</mn> <mo>…</mo> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>(M1)</strong></em></p>
<p>perimeter is therefore <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="6\pi \,\left( { = 18.8} \right)"> <mn>6</mn> <mi>π</mi> <mspace width="thinmathspace"></mspace> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mn>18.8</mn> </mrow> <mo>)</mo> </mrow> </math></span> (cm) <em><strong>A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>area of sector, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="s"> <mi>s</mi> </math></span>, is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{2}{r^2}\theta = 18 \times \frac{\pi }{3} = 6\pi \,\left( { = 18.84 \ldots } \right)"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mrow> <msup> <mi>r</mi> <mn>2</mn> </msup> </mrow> <mi>θ</mi> <mo>=</mo> <mn>18</mn> <mo>×</mo> <mfrac> <mi>π</mi> <mn>3</mn> </mfrac> <mo>=</mo> <mn>6</mn> <mi>π</mi> <mspace width="thinmathspace"></mspace> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mn>18.84</mn> <mo>…</mo> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>(A1)</strong></em></p>
<p>area of triangle, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t"> <mi>t</mi> </math></span>, is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{2} \times 6 \times 3\sqrt 3 = 9\sqrt 3 \,\left( { = 15.58 \ldots } \right)"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>×</mo> <mn>6</mn> <mo>×</mo> <mn>3</mn> <msqrt> <mn>3</mn> </msqrt> <mo>=</mo> <mn>9</mn> <msqrt> <mn>3</mn> </msqrt> <mspace width="thinmathspace"></mspace> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mn>15.58</mn> <mo>…</mo> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>(M1)</strong></em><em><strong>(A1)</strong></em></p>
<p><strong>Note:</strong> area of segment, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k"> <mi>k</mi> </math></span>, is 3.261… implies area of triangle</p>
<p>finding <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3s - 2t"> <mn>3</mn> <mi>s</mi> <mo>−</mo> <mn>2</mn> <mi>t</mi> </math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3k + t"> <mn>3</mn> <mi>k</mi> <mo>+</mo> <mi>t</mi> </math></span> or similar</p>
<p>area <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 3s - 2t = 18\pi - 18\sqrt 3 \,\left( { = 25.4} \right)"> <mo>=</mo> <mn>3</mn> <mi>s</mi> <mo>−</mo> <mn>2</mn> <mi>t</mi> <mo>=</mo> <mn>18</mn> <mi>π</mi> <mo>−</mo> <mn>18</mn> <msqrt> <mn>3</mn> </msqrt> <mspace width="thinmathspace"></mspace> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mn>25.4</mn> </mrow> <mo>)</mo> </mrow> </math></span> (cm<sup>2</sup>) <em><strong>(M1)</strong></em><em><strong>A1</strong></em></p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>Barry is at the top of a cliff, standing 80 m above sea level, and observes two yachts in the sea.<br>“<em>Seaview</em>” <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(S)"> <mo stretchy="false">(</mo> <mi>S</mi> <mo stretchy="false">)</mo> </math></span> is at an angle of depression of 25°.<br>“<em>Nauti Buoy</em>” <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(N)"> <mo stretchy="false">(</mo> <mi>N</mi> <mo stretchy="false">)</mo> </math></span> is at an angle of depression of 35°.<br>The following three dimensional diagram shows Barry and the two yachts at S and N.<br>X lies at the foot of the cliff and angle <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{SXN}} = "> <mrow> <mtext>SXN</mtext> </mrow> <mo>=</mo> </math></span> 70°.</p>
<p><img src="images/Schermafbeelding_2018-02-08_om_11.45.43.png" alt="N17/5/MATHL/HP2/ENG/TZ0/05"></p>
<p>Find, to 3 significant figures, the distance between the two yachts.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p>attempt to use tan, or sine rule, in triangle BXN or BXS <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{NX}} = 80\tan 55{\rm{^\circ }}\left( { = \frac{{80}}{{\tan 35{\rm{^\circ }}}} = 114.25} \right)"> <mrow> <mtext>NX</mtext> </mrow> <mo>=</mo> <mn>80</mn> <mi>tan</mi> <mo></mo> <mn>55</mn> <mrow> <mrow> <msup> <mi></mi> <mo>∘</mo> </msup> </mrow> </mrow> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mfrac> <mrow> <mn>80</mn> </mrow> <mrow> <mi>tan</mi> <mo></mo> <mn>35</mn> <mrow> <mrow> <msup> <mi></mi> <mo>∘</mo> </msup> </mrow> </mrow> </mrow> </mfrac> <mo>=</mo> <mn>114.25</mn> </mrow> <mo>)</mo> </mrow> </math></span> <strong><em>(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{SX}} = 80\tan 65{\rm{^\circ }}\left( { = \frac{{80}}{{\tan 25{\rm{^\circ }}}} = 171.56} \right)"> <mrow> <mtext>SX</mtext> </mrow> <mo>=</mo> <mn>80</mn> <mi>tan</mi> <mo></mo> <mn>65</mn> <mrow> <mrow> <msup> <mi></mi> <mo>∘</mo> </msup> </mrow> </mrow> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mfrac> <mrow> <mn>80</mn> </mrow> <mrow> <mi>tan</mi> <mo></mo> <mn>25</mn> <mrow> <mrow> <msup> <mi></mi> <mo>∘</mo> </msup> </mrow> </mrow> </mrow> </mfrac> <mo>=</mo> <mn>171.56</mn> </mrow> <mo>)</mo> </mrow> </math></span> <strong><em>(A1)</em></strong></p>
<p>Attempt to use cosine rule <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{S}}{{\text{N}}^2} = {171.56^2} + {114.25^2} - 2 \times 171.56 \times 114.25\cos 70"> <mrow> <mtext>S</mtext> </mrow> <mrow> <msup> <mrow> <mtext>N</mtext> </mrow> <mn>2</mn> </msup> </mrow> <mo>=</mo> <mrow> <msup> <mn>171.56</mn> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mn>114.25</mn> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>2</mn> <mo>×</mo> <mn>171.56</mn> <mo>×</mo> <mn>114.25</mn> <mi>cos</mi> <mo></mo> <mn>70</mn> </math></span>° <strong><em>(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{SN}} = 171{\text{ }}({\text{m}})"> <mrow> <mtext>SN</mtext> </mrow> <mo>=</mo> <mn>171</mn> <mrow> <mtext> </mtext> </mrow> <mo stretchy="false">(</mo> <mrow> <mtext>m</mtext> </mrow> <mo stretchy="false">)</mo> </math></span> <strong><em>A1</em></strong></p>
<p> </p>
<p>Note: Award final <strong><em>A1 </em></strong>only if the correct answer has been given to 3 significant figures.</p>
<p> </p>
<p><strong><em>[6 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>Consider <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mfrac><mrow><mtext>arctan</mtext><mfenced><mrow><mi>cos</mi><mo> </mo><mi>x</mi></mrow></mfenced><mo>-</mo><mi>k</mi></mrow><msup><mi>x</mi><mn>2</mn></msup></mfrac></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that a finite limit only exists for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Using l’Hôpital’s rule, show algebraically that the value of the limit is <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></math>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>(as <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><msup><mi>x</mi><mn>2</mn></msup><mo>=</mo><mn>0</mn></math>, the indeterminate form <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>0</mn><mn>0</mn></mfrac></math> is required for the limit to exist)</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mfenced><mrow><mtext>arctan</mtext><mfenced><mrow><mi>cos</mi><mo> </mo><mi>x</mi></mrow></mfenced><mo>-</mo><mi>k</mi></mrow></mfenced><mo>=</mo><mn>0</mn></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>arctan</mtext><mo> </mo><mn>1</mn><mo>-</mo><mi>k</mi><mo>=</mo><mn>0</mn><mo> </mo><mo> </mo><mfenced><mrow><mi>k</mi><mo>=</mo><mtext>arctan</mtext><mo> </mo><mn>1</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p>so <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac></math> <em><strong>AG</strong></em></p>
<p><strong><br>Note:</strong> Award <em><strong>M1A0</strong></em> for using <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac></math> to show the limit is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>0</mn><mn>0</mn></mfrac></math>.</p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mfrac><mrow><mtext>arctan</mtext><mfenced><mrow><mi>cos</mi><mo> </mo><mi>x</mi></mrow></mfenced><mo>-</mo><mstyle displaystyle="true"><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac></mstyle></mrow><msup><mi>x</mi><mn>2</mn></msup></mfrac><mfenced><mrow><mo>=</mo><mfrac><mn>0</mn><mn>0</mn></mfrac></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mfrac><mstyle displaystyle="true"><mfrac><mrow><mo>-</mo><mi>sin</mi><mo> </mo><mi>x</mi></mrow><mrow><mn>1</mn><mo>+</mo><msup><mi>cos</mi><mn>2</mn></msup><mo> </mo><mi>x</mi></mrow></mfrac></mstyle><mrow><mn>2</mn><mi>x</mi></mrow></mfrac></math> <em><strong>A1A1</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>A1</strong> </em>for a correct numerator and <em><strong>A1</strong> </em>for a correct denominator.</p>
<p><br>recognises to apply l’Hôpital’s rule again <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mfrac><mstyle displaystyle="true"><mfrac><mrow><mo>-</mo><mi>sin</mi><mo> </mo><mi>x</mi></mrow><mrow><mn>1</mn><mo>+</mo><msup><mi>cos</mi><mn>2</mn></msup><mo> </mo><mi>x</mi></mrow></mfrac></mstyle><mrow><mn>2</mn><mi>x</mi></mrow></mfrac><mo> </mo><mfenced><mrow><mo>=</mo><mfrac><mn>0</mn><mn>0</mn></mfrac></mrow></mfenced></math></p>
<p><strong><br>Note:</strong> Award <em><strong>M0</strong></em> if their limit is not the indeterminate form <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>0</mn><mn>0</mn></mfrac></math>.</p>
<p><strong><br>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mfrac><mstyle displaystyle="true"><mfrac><mrow><mo>-</mo><mi>cos</mi><mo> </mo><mi>x</mi><mfenced><mrow><mn>1</mn><mo>+</mo><msup><mi>cos</mi><mn>2</mn></msup><mo> </mo><mi>x</mi></mrow></mfenced><mo>-</mo><mn>2</mn><mo> </mo><msup><mi>sin</mi><mn>2</mn></msup><mo> </mo><mi>x</mi><mo> </mo><mi>cos</mi><mo> </mo><mi>x</mi></mrow><msup><mfenced><mrow><mn>1</mn><mo>+</mo><msup><mi>cos</mi><mn>2</mn></msup><mo> </mo><mi>x</mi></mrow></mfenced><mn>2</mn></msup></mfrac></mstyle><mn>2</mn></mfrac><mo> </mo></math> <em><strong>A1A1</strong></em></p>
<p><strong><br>Note:</strong> Award <em><strong>A1</strong> </em>for a correct first term in the numerator and <em><strong>A1</strong> </em>for a correct second term in the numerator.</p>
<p><br><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>0</mn></mrow></munder><mfrac><mrow><mo>-</mo><mi>cos</mi><mo> </mo><mi>x</mi></mrow><mrow><mn>2</mn><mfenced><mrow><mn>1</mn><mo>+</mo><msup><mi>cos</mi><mn>2</mn></msup><mo> </mo><mi>x</mi></mrow></mfenced><mo>-</mo><mn>4</mn><mi>x</mi><mo> </mo><mi>sin</mi><mo> </mo><mi>x</mi><mo> </mo><mi>cos</mi><mo> </mo><mi>x</mi></mrow></mfrac><mo> </mo></math> <em><strong>A1A1</strong></em></p>
<p><strong><br>Note:</strong> Award <em><strong>A1</strong> </em>for a correct numerator and <em><strong>A1</strong> </em>for a correct denominator.</p>
<p><strong><br>THEN</strong></p>
<p>substitutes <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn></math> into the correct expression to evaluate the limit <em><strong>A1</strong></em></p>
<p><strong><br>Note:</strong> The final <em><strong>A1</strong> </em>is dependent on all previous marks.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>Part (a) Many candidates recognised the indeterminate form and provided a nice algebraic proof. Some verified by substituting the given value. Therefore, there is a need to teach the candidates the difference between proof and verification. Only a few candidates were able to give a complete 'show that' proof.</p>
<p>Part (b) Many candidates realised that they needed to apply the L'Hôpital's rule twice. There were many mistakes in differentiation using the chain rule. Not all candidates clearly showed the final substitution.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>Given that <strong><em>a</em></strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \times ">
<mo>×</mo>
</math></span> <strong><em>b</em></strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = ">
<mo>=</mo>
</math></span> <strong><em>b</em></strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \times ">
<mo>×</mo>
</math></span> <strong><em>c</em></strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \ne ">
<mo>≠</mo>
</math></span> <strong>0 </strong>prove that <strong><em>a</em></strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" + ">
<mo>+</mo>
</math></span> <strong><em>c</em></strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = ">
<mo>=</mo>
</math></span> <em>s<strong>b </strong></em>where <em>s </em>is a scalar.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><strong>METHOD 1</strong></p>
<p><strong><em>a</em></strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \times ">
<mo>×</mo>
</math></span> <strong><em>b</em></strong> = <strong><em>b</em></strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \times ">
<mo>×</mo>
</math></span> <strong><em>c</em></strong></p>
<p>(<strong><em>a</em></strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \times ">
<mo>×</mo>
</math></span> <strong><em>b</em></strong>) <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - ">
<mo>−</mo>
</math></span> (<strong><em>b</em></strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \times ">
<mo>×</mo>
</math></span> <strong><em>c</em></strong>) = 0</p>
<p>(<strong><em>a</em></strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \times ">
<mo>×</mo>
</math></span> <strong><em>b</em></strong>) + (<strong><em>c</em></strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \times ">
<mo>×</mo>
</math></span> <strong><em>b</em></strong>) = 0 <strong><em>M1A1</em></strong></p>
<p>(<strong><em>a</em></strong> + <strong><em>c</em></strong>) <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \times ">
<mo>×</mo>
</math></span> <strong><em>b</em></strong> = 0 <strong><em>A1</em></strong></p>
<p>(<strong><em>a</em></strong> + <strong><em>c</em></strong>) is parallel to <strong><em>b</em></strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow ">
<mo stretchy="false">⇒</mo>
</math></span> <strong><em>a</em></strong> + <strong><em>c</em></strong> = <em>s<strong>b</strong></em> <strong><em>R1AG</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Condone absence of arrows, underlining, or other otherwise “correct” vector notation throughout this question.</p>
<p> </p>
<p><strong>Note:</strong> Allow “is in the same direction to”, for the final <strong><em>R </em></strong>mark.</p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><strong><em>a</em></strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \times ">
<mo>×</mo>
</math></span> <strong><em>b</em></strong> = <strong><em>b</em></strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \times ">
<mo>×</mo>
</math></span> <strong><em>c</em></strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow \left( {\begin{array}{*{20}{c}} {{a_2}{b_3} - {a_3}{b_2}} \\ {{a_3}{b_1} - {a_1}{b_3}} \\ {{a_1}{b_2} - {a_2}{b_1}} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {{b_2}{c_3} - {b_3}{c_2}} \\ {{b_3}{c_1} - {b_1}{c_3}} \\ {{b_1}{c_2} - {b_2}{c_1}} \end{array}} \right)">
<mo stretchy="false">⇒</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mrow>
<msub>
<mi>a</mi>
<mn>2</mn>
</msub>
</mrow>
<mrow>
<msub>
<mi>b</mi>
<mn>3</mn>
</msub>
</mrow>
<mo>−</mo>
<mrow>
<msub>
<mi>a</mi>
<mn>3</mn>
</msub>
</mrow>
<mrow>
<msub>
<mi>b</mi>
<mn>2</mn>
</msub>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mrow>
<msub>
<mi>a</mi>
<mn>3</mn>
</msub>
</mrow>
<mrow>
<msub>
<mi>b</mi>
<mn>1</mn>
</msub>
</mrow>
<mo>−</mo>
<mrow>
<msub>
<mi>a</mi>
<mn>1</mn>
</msub>
</mrow>
<mrow>
<msub>
<mi>b</mi>
<mn>3</mn>
</msub>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mrow>
<msub>
<mi>a</mi>
<mn>1</mn>
</msub>
</mrow>
<mrow>
<msub>
<mi>b</mi>
<mn>2</mn>
</msub>
</mrow>
<mo>−</mo>
<mrow>
<msub>
<mi>a</mi>
<mn>2</mn>
</msub>
</mrow>
<mrow>
<msub>
<mi>b</mi>
<mn>1</mn>
</msub>
</mrow>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mrow>
<msub>
<mi>b</mi>
<mn>2</mn>
</msub>
</mrow>
<mrow>
<msub>
<mi>c</mi>
<mn>3</mn>
</msub>
</mrow>
<mo>−</mo>
<mrow>
<msub>
<mi>b</mi>
<mn>3</mn>
</msub>
</mrow>
<mrow>
<msub>
<mi>c</mi>
<mn>2</mn>
</msub>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mrow>
<msub>
<mi>b</mi>
<mn>3</mn>
</msub>
</mrow>
<mrow>
<msub>
<mi>c</mi>
<mn>1</mn>
</msub>
</mrow>
<mo>−</mo>
<mrow>
<msub>
<mi>b</mi>
<mn>1</mn>
</msub>
</mrow>
<mrow>
<msub>
<mi>c</mi>
<mn>3</mn>
</msub>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mrow>
<msub>
<mi>b</mi>
<mn>1</mn>
</msub>
</mrow>
<mrow>
<msub>
<mi>c</mi>
<mn>2</mn>
</msub>
</mrow>
<mo>−</mo>
<mrow>
<msub>
<mi>b</mi>
<mn>2</mn>
</msub>
</mrow>
<mrow>
<msub>
<mi>c</mi>
<mn>1</mn>
</msub>
</mrow>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>M1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{a_2}{b_3} - {a_3}{b_2} = {b_2}{c_3} - {b_3}{c_2} \Rightarrow {b_3}({a_2} + {c_2}) = {b_2}({a_3} + {c_3})">
<mrow>
<msub>
<mi>a</mi>
<mn>2</mn>
</msub>
</mrow>
<mrow>
<msub>
<mi>b</mi>
<mn>3</mn>
</msub>
</mrow>
<mo>−</mo>
<mrow>
<msub>
<mi>a</mi>
<mn>3</mn>
</msub>
</mrow>
<mrow>
<msub>
<mi>b</mi>
<mn>2</mn>
</msub>
</mrow>
<mo>=</mo>
<mrow>
<msub>
<mi>b</mi>
<mn>2</mn>
</msub>
</mrow>
<mrow>
<msub>
<mi>c</mi>
<mn>3</mn>
</msub>
</mrow>
<mo>−</mo>
<mrow>
<msub>
<mi>b</mi>
<mn>3</mn>
</msub>
</mrow>
<mrow>
<msub>
<mi>c</mi>
<mn>2</mn>
</msub>
</mrow>
<mo stretchy="false">⇒</mo>
<mrow>
<msub>
<mi>b</mi>
<mn>3</mn>
</msub>
</mrow>
<mo stretchy="false">(</mo>
<mrow>
<msub>
<mi>a</mi>
<mn>2</mn>
</msub>
</mrow>
<mo>+</mo>
<mrow>
<msub>
<mi>c</mi>
<mn>2</mn>
</msub>
</mrow>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow>
<msub>
<mi>b</mi>
<mn>2</mn>
</msub>
</mrow>
<mo stretchy="false">(</mo>
<mrow>
<msub>
<mi>a</mi>
<mn>3</mn>
</msub>
</mrow>
<mo>+</mo>
<mrow>
<msub>
<mi>c</mi>
<mn>3</mn>
</msub>
</mrow>
<mo stretchy="false">)</mo>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{a_3}{b_1} - {a_1}{b_3} = {b_3}{c_1} - {b_1}{c_3} \Rightarrow {b_1}({a_3} + {c_3}) = {b_3}({a_1} + {c_1})">
<mrow>
<msub>
<mi>a</mi>
<mn>3</mn>
</msub>
</mrow>
<mrow>
<msub>
<mi>b</mi>
<mn>1</mn>
</msub>
</mrow>
<mo>−</mo>
<mrow>
<msub>
<mi>a</mi>
<mn>1</mn>
</msub>
</mrow>
<mrow>
<msub>
<mi>b</mi>
<mn>3</mn>
</msub>
</mrow>
<mo>=</mo>
<mrow>
<msub>
<mi>b</mi>
<mn>3</mn>
</msub>
</mrow>
<mrow>
<msub>
<mi>c</mi>
<mn>1</mn>
</msub>
</mrow>
<mo>−</mo>
<mrow>
<msub>
<mi>b</mi>
<mn>1</mn>
</msub>
</mrow>
<mrow>
<msub>
<mi>c</mi>
<mn>3</mn>
</msub>
</mrow>
<mo stretchy="false">⇒</mo>
<mrow>
<msub>
<mi>b</mi>
<mn>1</mn>
</msub>
</mrow>
<mo stretchy="false">(</mo>
<mrow>
<msub>
<mi>a</mi>
<mn>3</mn>
</msub>
</mrow>
<mo>+</mo>
<mrow>
<msub>
<mi>c</mi>
<mn>3</mn>
</msub>
</mrow>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow>
<msub>
<mi>b</mi>
<mn>3</mn>
</msub>
</mrow>
<mo stretchy="false">(</mo>
<mrow>
<msub>
<mi>a</mi>
<mn>1</mn>
</msub>
</mrow>
<mo>+</mo>
<mrow>
<msub>
<mi>c</mi>
<mn>1</mn>
</msub>
</mrow>
<mo stretchy="false">)</mo>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{a_1}{b_2} - {a_2}{b_1} = {b_1}{c_2} - {b_2}{c_1} \Rightarrow {b_2}({a_1} + {c_1}) = {b_1}({a_2} + {c_2})">
<mrow>
<msub>
<mi>a</mi>
<mn>1</mn>
</msub>
</mrow>
<mrow>
<msub>
<mi>b</mi>
<mn>2</mn>
</msub>
</mrow>
<mo>−</mo>
<mrow>
<msub>
<mi>a</mi>
<mn>2</mn>
</msub>
</mrow>
<mrow>
<msub>
<mi>b</mi>
<mn>1</mn>
</msub>
</mrow>
<mo>=</mo>
<mrow>
<msub>
<mi>b</mi>
<mn>1</mn>
</msub>
</mrow>
<mrow>
<msub>
<mi>c</mi>
<mn>2</mn>
</msub>
</mrow>
<mo>−</mo>
<mrow>
<msub>
<mi>b</mi>
<mn>2</mn>
</msub>
</mrow>
<mrow>
<msub>
<mi>c</mi>
<mn>1</mn>
</msub>
</mrow>
<mo stretchy="false">⇒</mo>
<mrow>
<msub>
<mi>b</mi>
<mn>2</mn>
</msub>
</mrow>
<mo stretchy="false">(</mo>
<mrow>
<msub>
<mi>a</mi>
<mn>1</mn>
</msub>
</mrow>
<mo>+</mo>
<mrow>
<msub>
<mi>c</mi>
<mn>1</mn>
</msub>
</mrow>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow>
<msub>
<mi>b</mi>
<mn>1</mn>
</msub>
</mrow>
<mo stretchy="false">(</mo>
<mrow>
<msub>
<mi>a</mi>
<mn>2</mn>
</msub>
</mrow>
<mo>+</mo>
<mrow>
<msub>
<mi>c</mi>
<mn>2</mn>
</msub>
</mrow>
<mo stretchy="false">)</mo>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{({a_1} + {c_1})}}{{{b_1}}} = \frac{{({a_2} + {c_2})}}{{{b_2}}} = \frac{{({a_3} + {c_3})}}{{{b_3}}} = s">
<mfrac>
<mrow>
<mo stretchy="false">(</mo>
<mrow>
<msub>
<mi>a</mi>
<mn>1</mn>
</msub>
</mrow>
<mo>+</mo>
<mrow>
<msub>
<mi>c</mi>
<mn>1</mn>
</msub>
</mrow>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<mrow>
<msub>
<mi>b</mi>
<mn>1</mn>
</msub>
</mrow>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mo stretchy="false">(</mo>
<mrow>
<msub>
<mi>a</mi>
<mn>2</mn>
</msub>
</mrow>
<mo>+</mo>
<mrow>
<msub>
<mi>c</mi>
<mn>2</mn>
</msub>
</mrow>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<mrow>
<msub>
<mi>b</mi>
<mn>2</mn>
</msub>
</mrow>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mo stretchy="false">(</mo>
<mrow>
<msub>
<mi>a</mi>
<mn>3</mn>
</msub>
</mrow>
<mo>+</mo>
<mrow>
<msub>
<mi>c</mi>
<mn>3</mn>
</msub>
</mrow>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<mrow>
<msub>
<mi>b</mi>
<mn>3</mn>
</msub>
</mrow>
</mrow>
</mfrac>
<mo>=</mo>
<mi>s</mi>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow {a_1} + {c_1} = s{b_1}">
<mo stretchy="false">⇒</mo>
<mrow>
<msub>
<mi>a</mi>
<mn>1</mn>
</msub>
</mrow>
<mo>+</mo>
<mrow>
<msub>
<mi>c</mi>
<mn>1</mn>
</msub>
</mrow>
<mo>=</mo>
<mi>s</mi>
<mrow>
<msub>
<mi>b</mi>
<mn>1</mn>
</msub>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow {a_2} + {c_2} = s{b_2}">
<mo stretchy="false">⇒</mo>
<mrow>
<msub>
<mi>a</mi>
<mn>2</mn>
</msub>
</mrow>
<mo>+</mo>
<mrow>
<msub>
<mi>c</mi>
<mn>2</mn>
</msub>
</mrow>
<mo>=</mo>
<mi>s</mi>
<mrow>
<msub>
<mi>b</mi>
<mn>2</mn>
</msub>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow {a_3} + {c_3} = s{b_3}">
<mo stretchy="false">⇒</mo>
<mrow>
<msub>
<mi>a</mi>
<mn>3</mn>
</msub>
</mrow>
<mo>+</mo>
<mrow>
<msub>
<mi>c</mi>
<mn>3</mn>
</msub>
</mrow>
<mo>=</mo>
<mi>s</mi>
<mrow>
<msub>
<mi>b</mi>
<mn>3</mn>
</msub>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow \left( {\begin{array}{*{20}{c}} {{a_1}} \\ {{a_2}} \\ {{a_3}} \end{array}} \right) + \left( {\begin{array}{*{20}{c}} {{c_1}} \\ {{c_2}} \\ {{c_3}} \end{array}} \right) = s\left( {\begin{array}{*{20}{c}} {{b_1}} \\ {{b_2}} \\ {{b_3}} \end{array}} \right)">
<mo stretchy="false">⇒</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mrow>
<msub>
<mi>a</mi>
<mn>1</mn>
</msub>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mrow>
<msub>
<mi>a</mi>
<mn>2</mn>
</msub>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mrow>
<msub>
<mi>a</mi>
<mn>3</mn>
</msub>
</mrow>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mrow>
<msub>
<mi>c</mi>
<mn>1</mn>
</msub>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mrow>
<msub>
<mi>c</mi>
<mn>2</mn>
</msub>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mrow>
<msub>
<mi>c</mi>
<mn>3</mn>
</msub>
</mrow>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mi>s</mi>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mrow>
<msub>
<mi>b</mi>
<mn>1</mn>
</msub>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mrow>
<msub>
<mi>b</mi>
<mn>2</mn>
</msub>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mrow>
<msub>
<mi>b</mi>
<mn>3</mn>
</msub>
</mrow>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow ">
<mo stretchy="false">⇒</mo>
</math></span> <strong><em>a</em></strong> + <strong><em>c</em></strong> = <em>s<strong>b</strong></em> <strong><em>AG</em></strong></p>
<p><strong><em>[4 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="question">
<p>Two ships, A and B , are observed from an origin O. Relative to O, their position vectors at time <em>t</em> hours after midday are given by</p>
<p style="padding-left:180px;"><em><strong>r</strong></em><sub>A</sub> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 4 \\ 3 \end{array}} \right) + t\left( {\begin{array}{*{20}{c}} 5 \\ 8 \end{array}} \right)"> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mn>4</mn> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>t</mi> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mn>5</mn> </mtd> </mtr> <mtr> <mtd> <mn>8</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p style="padding-left:180px;"><em><strong>r</strong></em><sub>B</sub> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 7 \\ { - 3} \end{array}} \right) + t\left( {\begin{array}{*{20}{c}} 0 \\ {12} \end{array}} \right)"> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mn>7</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>t</mi> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mn>12</mn> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p>where distances are measured in kilometres.</p>
<p>Find the minimum distance between the two ships.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p>attempting to find <em><strong>r</strong></em><sub>B</sub> − <em><strong>r</strong></em><sub>A</sub> for example <em><strong>(M1)</strong></em></p>
<p><em><strong>r</strong></em><sub>B</sub> − <em><strong>r</strong></em><sub>A</sub> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 3 \\ { - 6} \end{array}} \right) + t\left( {\begin{array}{*{20}{c}} { - 5} \\ 4 \end{array}} \right)"> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>−</mo> <mn>6</mn> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mi>t</mi> <mrow> <mo>(</mo> <mrow> <mtable columnspacing="1em" rowspacing="4pt"> <mtr> <mtd> <mrow> <mo>−</mo> <mn>5</mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>4</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> </math></span> </p>
<p>attempting to find |<em><strong>r</strong></em><sub>B</sub> − <em><strong>r</strong></em><sub>A</sub>| <em><strong>M1</strong></em></p>
<p>distance <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d\left( t \right) = \sqrt {{{\left( {3 - 5t} \right)}^2} + {{\left( {4t - 6} \right)}^2}} \left( { = \sqrt {41{t^2} - 78t + 45} } \right)"> <mi>d</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <mo>=</mo> <msqrt> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mn>3</mn> <mo>−</mo> <mn>5</mn> <mi>t</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mn>4</mn> <mi>t</mi> <mo>−</mo> <mn>6</mn> </mrow> <mo>)</mo> </mrow> </mrow> <mn>2</mn> </msup> </mrow> </msqrt> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <msqrt> <mn>41</mn> <mrow> <msup> <mi>t</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>78</mn> <mi>t</mi> <mo>+</mo> <mn>45</mn> </msqrt> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p>using a graph to find the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d"> <mi>d</mi> </math></span> − coordinate of the local minimum <em><strong>M1</strong></em></p>
<p>the minimum distance between the ships is 2.81 (km) <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( { = \frac{{11\sqrt {41} }}{{41}}\,\left( {{\text{km}}} \right)} \right)"> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mfrac> <mrow> <mn>11</mn> <msqrt> <mn>41</mn> </msqrt> </mrow> <mrow> <mn>41</mn> </mrow> </mfrac> <mspace width="thinmathspace"></mspace> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>km</mtext> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[5 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>In a triangle <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ABC, AB}} = 4{\text{ cm, BC}} = 3{\text{ cm}}">
<mrow>
<mtext>ABC, AB</mtext>
</mrow>
<mo>=</mo>
<mn>4</mn>
<mrow>
<mtext> cm, BC</mtext>
</mrow>
<mo>=</mo>
<mn>3</mn>
<mrow>
<mtext> cm</mtext>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\rm{B\hat AC}} = \frac{\pi }{9}">
<mrow>
<mrow>
<mi mathvariant="normal">B</mi>
<mrow>
<mover>
<mi mathvariant="normal">A</mi>
<mo stretchy="false">^<!-- ^ --></mo>
</mover>
</mrow>
<mi mathvariant="normal">C</mi>
</mrow>
</mrow>
<mo>=</mo>
<mfrac>
<mi>π<!-- π --></mi>
<mn>9</mn>
</mfrac>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use the cosine rule to find the two possible values for AC.</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the difference between the areas of the two possible triangles ABC.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p><strong>METHOD 1</strong></p>
<p>let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{AC}} = x"> <mrow> <mtext>AC</mtext> </mrow> <mo>=</mo> <mi>x</mi> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{3^2} = {x^2} + {4^2} - 8x\cos \frac{\pi }{9}"> <mrow> <msup> <mn>3</mn> <mn>2</mn> </msup> </mrow> <mo>=</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mn>4</mn> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>8</mn> <mi>x</mi> <mi>cos</mi> <mo></mo> <mfrac> <mi>π</mi> <mn>9</mn> </mfrac> </math></span> <strong><em>M1A1</em></strong></p>
<p>attempting to solve for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 1.09,{\text{ }}6.43"> <mi>x</mi> <mo>=</mo> <mn>1.09</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mn>6.43</mn> </math></span> <strong><em>A1A1</em></strong></p>
<p><strong>METHOD 2</strong></p>
<p>let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{AC}} = x"> <mrow> <mtext>AC</mtext> </mrow> <mo>=</mo> <mi>x</mi> </math></span></p>
<p>using the sine rule to find a value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="C"> <mi>C</mi> </math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{4^2} = {x^2} + {3^2} - 6x\cos (152.869 \ldots ^\circ ) \Rightarrow x = 1.09"> <mrow> <msup> <mn>4</mn> <mn>2</mn> </msup> </mrow> <mo>=</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mn>3</mn> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>6</mn> <mi>x</mi> <mi>cos</mi> <mo></mo> <mo stretchy="false">(</mo> <mn>152.869</mn> <msup> <mo>…</mo> <mo>∘</mo> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">⇒</mo> <mi>x</mi> <mo>=</mo> <mn>1.09</mn> </math></span> <strong><em>(M1)A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{4^2} = {x^2} + {3^2} - 6x\cos (27.131 \ldots ^\circ ) \Rightarrow x = 6.43"> <mrow> <msup> <mn>4</mn> <mn>2</mn> </msup> </mrow> <mo>=</mo> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mn>3</mn> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>6</mn> <mi>x</mi> <mi>cos</mi> <mo></mo> <mo stretchy="false">(</mo> <mn>27.131</mn> <msup> <mo>…</mo> <mo>∘</mo> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">⇒</mo> <mi>x</mi> <mo>=</mo> <mn>6.43</mn> </math></span> <strong><em>(M1)A1</em></strong></p>
<p><strong>METHOD 3</strong></p>
<p>let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{AC}} = x"> <mrow> <mtext>AC</mtext> </mrow> <mo>=</mo> <mi>x</mi> </math></span></p>
<p>using the sine rule to find a value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="B"> <mi>B</mi> </math></span> and a value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="C"> <mi>C</mi> </math></span> <strong><em>M1</em></strong></p>
<p>obtaining <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="B = 132.869 \ldots ^\circ ,{\text{ }}7.131 \ldots ^\circ "> <mi>B</mi> <mo>=</mo> <mn>132.869</mn> <msup> <mo>…</mo> <mo>∘</mo> </msup> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mn>7.131</mn> <msup> <mo>…</mo> <mo>∘</mo> </msup> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="C = 27.131 \ldots ^\circ ,{\text{ }}152.869 \ldots ^\circ "> <mi>C</mi> <mo>=</mo> <mn>27.131</mn> <msup> <mo>…</mo> <mo>∘</mo> </msup> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mn>152.869</mn> <msup> <mo>…</mo> <mo>∘</mo> </msup> </math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(B = 2.319 \ldots ,{\text{ }}0.124 \ldots "> <mo stretchy="false">(</mo> <mi>B</mi> <mo>=</mo> <mn>2.319</mn> <mo>…</mo> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mn>0.124</mn> <mo>…</mo> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="C = 0.473 \ldots ,{\text{ }}2.668 \ldots )"> <mi>C</mi> <mo>=</mo> <mn>0.473</mn> <mo>…</mo> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mn>2.668</mn> <mo>…</mo> <mo stretchy="false">)</mo> </math></span></p>
<p>attempting to find a value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> using the cosine rule <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 1.09,{\text{ }}6.43"> <mi>x</mi> <mo>=</mo> <mn>1.09</mn> <mo>,</mo> <mrow> <mtext> </mtext> </mrow> <mn>6.43</mn> </math></span> <strong><em>A1A1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>M1A0(M1)A1A0 </em></strong>for one correct value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span></p>
<p> </p>
<p><strong><em>[5 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{2} \times 4 \times 6.428 \ldots \times \sin \frac{\pi }{9}"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>×</mo> <mn>4</mn> <mo>×</mo> <mn>6.428</mn> <mo>…</mo> <mo>×</mo> <mi>sin</mi> <mo></mo> <mfrac> <mi>π</mi> <mn>9</mn> </mfrac> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{2} \times 4 \times 1.088 \ldots \times \sin \frac{\pi }{9}"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>×</mo> <mn>4</mn> <mo>×</mo> <mn>1.088</mn> <mo>…</mo> <mo>×</mo> <mi>sin</mi> <mo></mo> <mfrac> <mi>π</mi> <mn>9</mn> </mfrac> </math></span> <strong><em>(A1)</em></strong></p>
<p>(<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="4.39747 \ldots "> <mn>4.39747</mn> <mo>…</mo> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0.744833 \ldots "> <mn>0.744833</mn> <mo>…</mo> </math></span>)</p>
<p>let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="D"> <mi>D</mi> </math></span> be the difference between the two areas</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="D = \frac{1}{2} \times 4 \times 6.428 \ldots \times \sin \frac{\pi }{9} - \frac{1}{2} \times 4 \times 1.088 \ldots \times \sin \frac{\pi }{9}"> <mi>D</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>×</mo> <mn>4</mn> <mo>×</mo> <mn>6.428</mn> <mo>…</mo> <mo>×</mo> <mi>sin</mi> <mo></mo> <mfrac> <mi>π</mi> <mn>9</mn> </mfrac> <mo>−</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>×</mo> <mn>4</mn> <mo>×</mo> <mn>1.088</mn> <mo>…</mo> <mo>×</mo> <mi>sin</mi> <mo></mo> <mfrac> <mi>π</mi> <mn>9</mn> </mfrac> </math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(D = 4.39747 \ldots - 0.744833 \ldots )"> <mo stretchy="false">(</mo> <mi>D</mi> <mo>=</mo> <mn>4.39747</mn> <mo>…</mo> <mo>−</mo> <mn>0.744833</mn> <mo>…</mo> <mo stretchy="false">)</mo> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 3.65{\text{ (c}}{{\text{m}}^2})"> <mo>=</mo> <mn>3.65</mn> <mrow> <mtext> (c</mtext> </mrow> <mrow> <msup> <mrow> <mtext>m</mtext> </mrow> <mn>2</mn> </msup> </mrow> <mo stretchy="false">)</mo> </math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">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>Two boats <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math> travel due north.</p>
<p>Initially, boat <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math> is positioned <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>50</mn></math> metres due east of boat <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math>.</p>
<p>The distances travelled by boat <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> and boat <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math>, after <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> seconds, are <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> metres and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> metres respectively. The angle <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi></math> is the radian measure of the bearing of boat <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math> from boat <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math>. This information is shown on the following diagram.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>x</mi><mo>+</mo><mn>50</mn><mo> </mo><mtext>cot</mtext><mo> </mo><mi>θ</mi></math> .</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>At time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math>, the following conditions are true.</p>
<p style="padding-left:60px;">Boat <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math> has travelled <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn></math> metres further than boat <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math>.<br>Boat <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math> is travelling at double the speed of boat <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math>.<br>The rate of change of the angle <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>0</mn><mo>.</mo><mn>1</mn></math> radians per second.</p>
<p>Find the speed of boat <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> at time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tan</mi><mo> </mo><mi>θ</mi><mo>=</mo><mfrac><mn>50</mn><mrow><mi>y</mi><mo>-</mo><mi>x</mi></mrow></mfrac></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>cot</mtext><mo> </mo><mi>θ</mi><mo>=</mo><mfrac><mrow><mi>y</mi><mo>-</mo><mi>x</mi></mrow><mn>50</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>x</mi><mo>+</mo><mn>50</mn><mo> </mo><mtext>cot</mtext><mo> </mo><mi>θ</mi></math> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>Note:</strong> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>-</mo><mi>x</mi></math> may be identified as a length on a diagram, and not written explicitly.</p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to differentiate with respect to <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>-</mo><mn>50</mn><msup><mfenced><mrow><mtext>cosec</mtext><mo> </mo><mi>θ</mi></mrow></mfenced><mn>2</mn></msup><mfrac><mrow><mo>d</mo><mi>θ</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></math> <em><strong>A1</strong></em></p>
<p>attempt to set speed of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math> equal to double the speed of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>-</mo><mn>50</mn><msup><mfenced><mrow><mtext>cosec</mtext><mo> </mo><mi>θ</mi></mrow></mfenced><mn>2</mn></msup><mfrac><mrow><mo>d</mo><mi>θ</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mo>-</mo><mn>50</mn><msup><mfenced><mrow><mtext>cosec</mtext><mo> </mo><mi>θ</mi></mrow></mfenced><mn>2</mn></msup><mfrac><mrow><mo>d</mo><mi>θ</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi><mo>=</mo><mtext>arctan</mtext><mo> </mo><mn>5</mn><mfenced><mrow><mo>=</mo><mn>1</mn><mo>.</mo><mn>373</mn><mo>…</mo><mo>=</mo><mn>78</mn><mo>.</mo><mn>69</mn><mo>…</mo><mo>°</mo></mrow></mfenced></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>cosec</mtext><mn>2</mn></msup><mo> </mo><mi>θ</mi><mo>=</mo><mn>1</mn><mo>+</mo><msup><mtext>cot</mtext><mn>2</mn></msup><mo> </mo><mi>θ</mi><mo>=</mo><mn>1</mn><mo>+</mo><msup><mfenced><mfrac><mn>1</mn><mn>5</mn></mfrac></mfenced><mn>2</mn></msup><mo>=</mo><mfrac><mn>26</mn><mn>25</mn></mfrac></math> <em><strong>(A1)</strong></em></p>
<p> </p>
<p><strong>Note:</strong> This <em><strong>A1</strong></em> can be awarded independently of previous marks.</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mo>-</mo><mn>50</mn><mfenced><mfrac><mn>26</mn><mn>25</mn></mfrac></mfenced><mo>×</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>1</mn></math></p>
<p>So the speed of boat <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mo>.</mo><mn>2</mn><mo> </mo><mfenced><msup><mtext>ms</mtext><mrow><mo>-</mo><mn>1</mn></mrow></msup></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mo>.</mo><mn>20</mn></math> from the use of inexact values.</p>
<p> </p>
<p><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>This diagram shows a metallic pendant made out of four equal sectors of a larger circle of radius <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{OB}} = 9{\text{ cm}}">
<mrow>
<mtext>OB</mtext>
</mrow>
<mo>=</mo>
<mn>9</mn>
<mrow>
<mtext> cm</mtext>
</mrow>
</math></span> and four equal sectors of a smaller circle of radius <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{OA}} = 3{\text{ cm}}">
<mrow>
<mtext>OA</mtext>
</mrow>
<mo>=</mo>
<mn>3</mn>
<mrow>
<mtext> cm</mtext>
</mrow>
</math></span>.<br>The angle <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{BOC}} = ">
<mrow>
<mtext>BOC</mtext>
</mrow>
<mo>=</mo>
</math></span> 20°.</p>
<p><img src="images/Schermafbeelding_2018-02-08_om_11.16.43.png" alt="N17/5/MATHL/HP2/ENG/TZ0/03"></p>
<p>Find the area of the pendant.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><strong>METHOD 1</strong></p>
<p>area = (four sector areas radius 9) + (four sector areas radius 3) <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 4\left( {\frac{1}{2}{9^2}\frac{\pi }{9}} \right) + 4\left( {\frac{1}{2}{3^2}\frac{{7\pi }}{{18}}} \right)">
<mo>=</mo>
<mn>4</mn>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mrow>
<msup>
<mn>9</mn>
<mn>2</mn>
</msup>
</mrow>
<mfrac>
<mi>π</mi>
<mn>9</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mn>4</mn>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mrow>
<msup>
<mn>3</mn>
<mn>2</mn>
</msup>
</mrow>
<mfrac>
<mrow>
<mn>7</mn>
<mi>π</mi>
</mrow>
<mrow>
<mn>18</mn>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>(A1)(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 18\pi + 7\pi ">
<mo>=</mo>
<mn>18</mn>
<mi>π</mi>
<mo>+</mo>
<mn>7</mn>
<mi>π</mi>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 25\pi {\text{ }}( = 78.5{\text{ c}}{{\text{m}}^2})">
<mo>=</mo>
<mn>25</mn>
<mi>π</mi>
<mrow>
<mtext> </mtext>
</mrow>
<mo stretchy="false">(</mo>
<mo>=</mo>
<mn>78.5</mn>
<mrow>
<mtext> c</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>m</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo stretchy="false">)</mo>
</math></span> <strong><em>A1</em></strong></p>
<p><strong>METHOD 2</strong></p>
<p>area =</p>
<p>(area of circle radius 3) + (four sector areas radius 9) – (four sector areas radius 3) <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\pi {3^2} + 4\left( {\frac{1}{2}{9^2}\frac{\pi }{9}} \right) - 4\left( {\frac{1}{2}{3^2}\frac{\pi }{9}} \right)">
<mi>π</mi>
<mrow>
<msup>
<mn>3</mn>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>4</mn>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mrow>
<msup>
<mn>9</mn>
<mn>2</mn>
</msup>
</mrow>
<mfrac>
<mi>π</mi>
<mn>9</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mn>4</mn>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mrow>
<msup>
<mn>3</mn>
<mn>2</mn>
</msup>
</mrow>
<mfrac>
<mi>π</mi>
<mn>9</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>(A1)(A1)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>A1 </em></strong>for the second term and <strong><em>A1 </em></strong>for the third term.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 9\pi + 18\pi - 2\pi ">
<mo>=</mo>
<mn>9</mn>
<mi>π</mi>
<mo>+</mo>
<mn>18</mn>
<mi>π</mi>
<mo>−</mo>
<mn>2</mn>
<mi>π</mi>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 25\pi {\text{ }}( = {\text{ }}78.5{\text{ c}}{{\text{m}}^2})">
<mo>=</mo>
<mn>25</mn>
<mi>π</mi>
<mrow>
<mtext> </mtext>
</mrow>
<mo stretchy="false">(</mo>
<mo>=</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>78.5</mn>
<mrow>
<mtext> c</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>m</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo stretchy="false">)</mo>
</math></span> <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Accept working in degrees.</p>
<p> </p>
<p><strong><em>[4 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="question">
<p>Find the Cartesian equation of plane <em>Π</em> containing the points <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{A}}\left( {6,{\text{ }}2,{\text{ }}1} \right)">
<mrow>
<mtext>A</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>6</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>2</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{B}}\left( {3,{\text{ }} - 1,{\text{ }}1} \right)">
<mrow>
<mtext>B</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>3</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mo>−</mo>
<mn>1</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span> and perpendicular to the plane <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x + 2y - z - 6 = 0">
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
<mi>y</mi>
<mo>−</mo>
<mi>z</mi>
<mo>−</mo>
<mn>6</mn>
<mo>=</mo>
<mn>0</mn>
</math></span>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><strong>METHOD 1</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\overrightarrow {{\text{AB}}} = \left( {\begin{array}{*{20}{c}} { - 3} \\ { - 3} \\ 0 \end{array}} \right)">
<mover>
<mrow>
<mtext>AB</mtext>
</mrow>
<mo>→</mo>
</mover>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>3</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>3</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} { - 3} \\ { - 3} \\ 0 \end{array}} \right) \times \left( {\begin{array}{*{20}{c}} 1 \\ 2 \\ { - 1} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>3</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>3</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>×</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>M1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left( {\begin{array}{*{20}{c}} 3 \\ { - 3} \\ { - 3} \end{array}} \right)">
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>3</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>3</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x - y - z = k">
<mi>x</mi>
<mo>−</mo>
<mi>y</mi>
<mo>−</mo>
<mi>z</mi>
<mo>=</mo>
<mi>k</mi>
</math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k = 3">
<mi>k</mi>
<mo>=</mo>
<mn>3</mn>
</math></span> equation of plane <em>Π</em> is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x - y - z = 3">
<mi>x</mi>
<mo>−</mo>
<mi>y</mi>
<mo>−</mo>
<mi>z</mi>
<mo>=</mo>
<mn>3</mn>
</math></span> or equivalent <strong><em>A1</em></strong></p>
<p><strong>METHOD 2</strong></p>
<p>let plane <em>Π</em> be <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="ax + by + cz = d">
<mi>a</mi>
<mi>x</mi>
<mo>+</mo>
<mi>b</mi>
<mi>y</mi>
<mo>+</mo>
<mi>c</mi>
<mi>z</mi>
<mo>=</mo>
<mi>d</mi>
</math></span></p>
<p>attempt to form one or more simultaneous equations: <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a + 2b - c = 0">
<mi>a</mi>
<mo>+</mo>
<mn>2</mn>
<mi>b</mi>
<mo>−</mo>
<mi>c</mi>
<mo>=</mo>
<mn>0</mn>
</math></span> (1) <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="6a + 2b + c = d">
<mn>6</mn>
<mi>a</mi>
<mo>+</mo>
<mn>2</mn>
<mi>b</mi>
<mo>+</mo>
<mi>c</mi>
<mo>=</mo>
<mi>d</mi>
</math></span> (2)</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3a - b + c = d">
<mn>3</mn>
<mi>a</mi>
<mo>−</mo>
<mi>b</mi>
<mo>+</mo>
<mi>c</mi>
<mo>=</mo>
<mi>d</mi>
</math></span> (3) <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award second <strong><em>A1 </em></strong>for equations (2) and (3).</p>
<p> </p>
<p>attempt to solve <strong><em>M1</em></strong></p>
<p><strong>EITHER</strong></p>
<p>using GDC gives <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a = \frac{d}{3},{\text{ }}b = - \frac{d}{3},{\text{ }}c = - \frac{d}{3}">
<mi>a</mi>
<mo>=</mo>
<mfrac>
<mi>d</mi>
<mn>3</mn>
</mfrac>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>b</mi>
<mo>=</mo>
<mo>−</mo>
<mfrac>
<mi>d</mi>
<mn>3</mn>
</mfrac>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>c</mi>
<mo>=</mo>
<mo>−</mo>
<mfrac>
<mi>d</mi>
<mn>3</mn>
</mfrac>
</math></span> <strong><em>(A1)</em></strong></p>
<p>equation of plane <em>Π</em> is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x - y - z = 3">
<mi>x</mi>
<mo>−</mo>
<mi>y</mi>
<mo>−</mo>
<mi>z</mi>
<mo>=</mo>
<mn>3</mn>
</math></span> or equivalent <strong><em>A1</em></strong></p>
<p><strong>OR</strong></p>
<p>row reduction <strong><em>M1</em></strong></p>
<p>equation of plane <em>Π</em> is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x - y - z = 3">
<mi>x</mi>
<mo>−</mo>
<mi>y</mi>
<mo>−</mo>
<mi>z</mi>
<mo>=</mo>
<mn>3</mn>
</math></span> or equivalent <strong><em>A1</em></strong></p>
<p><strong><em>[6 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="question">
<p>Find the acute angle between the planes with equations <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x + y + z = 3"> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo>+</mo> <mi>z</mi> <mo>=</mo> <mn>3</mn> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2x - z = 2"> <mn>2</mn> <mi>x</mi> <mo>−</mo> <mi>z</mi> <mo>=</mo> <mn>2</mn> </math></span>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p><strong><em>n</em></strong><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="_1 = \left( {\begin{array}{*{20}{c}} 1 \\ 1 \\ 1 \end{array}} \right)"> <msub> <mi></mi> <mn>1</mn> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> </math></span> and <strong><em>n</em></strong><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="_2 = \left( {\begin{array}{*{20}{c}} 2 \\ 0 \\ { - 1} \end{array}} \right)"> <msub> <mi></mi> <mn>2</mn> </msub> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </mtd> </mtr> </mtable> </mrow> <mo>)</mo> </mrow> </math></span> <strong><em>(A1)(A1)</em></strong></p>
<p><strong>EITHER </strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\theta = \arccos \left( {\frac{{{n_1} \bullet {n_2}}}{{\left| {{n_1}} \right|\left| {{n_2}} \right|}}} \right)\left( {\cos \theta = \frac{{{n_1} \bullet {n_2}}}{{\left| {{n_1}} \right|\left| {{n_2}} \right|}}} \right)"> <mi>θ</mi> <mo>=</mo> <mi>arccos</mi> <mo></mo> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mrow> <msub> <mi>n</mi> <mn>1</mn> </msub> </mrow> <mo>∙</mo> <mrow> <msub> <mi>n</mi> <mn>2</mn> </msub> </mrow> </mrow> <mrow> <mrow> <mo>|</mo> <mrow> <mrow> <msub> <mi>n</mi> <mn>1</mn> </msub> </mrow> </mrow> <mo>|</mo> </mrow> <mrow> <mo>|</mo> <mrow> <mrow> <msub> <mi>n</mi> <mn>2</mn> </msub> </mrow> </mrow> <mo>|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mi>cos</mi> <mo></mo> <mi>θ</mi> <mo>=</mo> <mfrac> <mrow> <mrow> <msub> <mi>n</mi> <mn>1</mn> </msub> </mrow> <mo>∙</mo> <mrow> <msub> <mi>n</mi> <mn>2</mn> </msub> </mrow> </mrow> <mrow> <mrow> <mo>|</mo> <mrow> <mrow> <msub> <mi>n</mi> <mn>1</mn> </msub> </mrow> </mrow> <mo>|</mo> </mrow> <mrow> <mo>|</mo> <mrow> <mrow> <msub> <mi>n</mi> <mn>2</mn> </msub> </mrow> </mrow> <mo>|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \arccos \left( {\frac{{2 + 0 - 1}}{{\sqrt 3 \sqrt 5 }}} \right)\left( {\cos \theta = \frac{{2 + 0 - 1}}{{\sqrt 3 \sqrt 5 }}} \right)"> <mo>=</mo> <mi>arccos</mi> <mo></mo> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mn>2</mn> <mo>+</mo> <mn>0</mn> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <msqrt> <mn>3</mn> </msqrt> <msqrt> <mn>5</mn> </msqrt> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mi>cos</mi> <mo></mo> <mi>θ</mi> <mo>=</mo> <mfrac> <mrow> <mn>2</mn> <mo>+</mo> <mn>0</mn> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <msqrt> <mn>3</mn> </msqrt> <msqrt> <mn>5</mn> </msqrt> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> <strong><em>(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \arccos \left( {\frac{1}{{\sqrt {15} }}} \right)\left( {\cos \theta = \frac{1}{{\sqrt {15} }}} \right)"> <mo>=</mo> <mi>arccos</mi> <mo></mo> <mrow> <mo>(</mo> <mrow> <mfrac> <mn>1</mn> <mrow> <msqrt> <mn>15</mn> </msqrt> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mi>cos</mi> <mo></mo> <mi>θ</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mrow> <msqrt> <mn>15</mn> </msqrt> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p><strong>OR</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\theta = \arcsin \left( {\frac{{\left| {{n_1} \times {n_2}} \right|}}{{\left| {{n_1}} \right|\left| {{n_2}} \right|}}} \right)\left( {\sin \theta = \frac{{\left| {{n_1} \times {n_2}} \right|}}{{\left| {{n_1}} \right|\left| {{n_2}} \right|}}} \right)"> <mi>θ</mi> <mo>=</mo> <mi>arcsin</mi> <mo></mo> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mrow> <mo>|</mo> <mrow> <mrow> <msub> <mi>n</mi> <mn>1</mn> </msub> </mrow> <mo>×</mo> <mrow> <msub> <mi>n</mi> <mn>2</mn> </msub> </mrow> </mrow> <mo>|</mo> </mrow> </mrow> <mrow> <mrow> <mo>|</mo> <mrow> <mrow> <msub> <mi>n</mi> <mn>1</mn> </msub> </mrow> </mrow> <mo>|</mo> </mrow> <mrow> <mo>|</mo> <mrow> <mrow> <msub> <mi>n</mi> <mn>2</mn> </msub> </mrow> </mrow> <mo>|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mi>sin</mi> <mo></mo> <mi>θ</mi> <mo>=</mo> <mfrac> <mrow> <mrow> <mo>|</mo> <mrow> <mrow> <msub> <mi>n</mi> <mn>1</mn> </msub> </mrow> <mo>×</mo> <mrow> <msub> <mi>n</mi> <mn>2</mn> </msub> </mrow> </mrow> <mo>|</mo> </mrow> </mrow> <mrow> <mrow> <mo>|</mo> <mrow> <mrow> <msub> <mi>n</mi> <mn>1</mn> </msub> </mrow> </mrow> <mo>|</mo> </mrow> <mrow> <mo>|</mo> <mrow> <mrow> <msub> <mi>n</mi> <mn>2</mn> </msub> </mrow> </mrow> <mo>|</mo> </mrow> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \arcsin \left( {\frac{{\sqrt {14} }}{{\sqrt 3 \sqrt 5 }}} \right)\left( {\sin \theta = \frac{{\sqrt {14} }}{{\sqrt 3 \sqrt 5 }}} \right)"> <mo>=</mo> <mi>arcsin</mi> <mo></mo> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <msqrt> <mn>14</mn> </msqrt> </mrow> <mrow> <msqrt> <mn>3</mn> </msqrt> <msqrt> <mn>5</mn> </msqrt> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mi>sin</mi> <mo></mo> <mi>θ</mi> <mo>=</mo> <mfrac> <mrow> <msqrt> <mn>14</mn> </msqrt> </mrow> <mrow> <msqrt> <mn>3</mn> </msqrt> <msqrt> <mn>5</mn> </msqrt> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span> <strong><em>(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \arcsin \left( {\frac{{\sqrt {14} }}{{\sqrt {15} }}} \right)\left( {\sin \theta = \frac{{\sqrt {14} }}{{\sqrt {15} }}} \right)"> <mo>=</mo> <mi>arcsin</mi> <mo></mo> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <msqrt> <mn>14</mn> </msqrt> </mrow> <mrow> <msqrt> <mn>15</mn> </msqrt> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <mi>sin</mi> <mo></mo> <mi>θ</mi> <mo>=</mo> <mfrac> <mrow> <msqrt> <mn>14</mn> </msqrt> </mrow> <mrow> <msqrt> <mn>15</mn> </msqrt> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p> </p>
<p><strong>THEN</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 75.0^\circ {\text{ (or 1.31)}}"> <mo>=</mo> <msup> <mn>75.0</mn> <mo>∘</mo> </msup> <mrow> <mtext> (or 1.31)</mtext> </mrow> </math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[5 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="question">
<p>Boat A is situated 10km away from boat B, and each boat has a marine radio transmitter on board. The range of the transmitter on boat A is 7km, and the range of the transmitter on boat B is 5km. The region in which both transmitters can be detected is represented by the shaded region in the following diagram. Find the area of this region.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p style="text-align: center;"><img 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"></p>
<p style="text-align: left;">use of cosine rule <em><strong>(M1)</strong></em></p>
<p style="text-align: left;">CÂB = arccos <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\frac{{49 + 100 - 25}}{{2 \times 7 \times 10}}} \right) = 0.48276 \ldots \left( { = 27.660 \ldots ^\circ } \right)"> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mn>49</mn> <mo>+</mo> <mn>100</mn> <mo>−</mo> <mn>25</mn> </mrow> <mrow> <mn>2</mn> <mo>×</mo> <mn>7</mn> <mo>×</mo> <mn>10</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0.48276</mn> <mo>…</mo> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mn>27.660</mn> <msup> <mo>…</mo> <mo>∘</mo> </msup> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>(A1)</strong></em></p>
<p style="text-align: left;">C<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mathop {\text{B}}\limits^ \wedge "> <mover> <mrow> <mtext>B</mtext> </mrow> <mo>∧</mo> </mover> </math></span>A = arccos <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\frac{{25 + 100 - 49}}{{2 \times 5 \times 10}}} \right) = 0.70748 \ldots \left( { = 40.535 \ldots ^\circ } \right)"> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mn>25</mn> <mo>+</mo> <mn>100</mn> <mo>−</mo> <mn>49</mn> </mrow> <mrow> <mn>2</mn> <mo>×</mo> <mn>5</mn> <mo>×</mo> <mn>10</mn> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0.70748</mn> <mo>…</mo> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mn>40.535</mn> <msup> <mo>…</mo> <mo>∘</mo> </msup> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>(A1)</strong></em></p>
<p style="text-align: left;">attempt to subtract triangle area from sector area <em><strong>(M1)</strong></em></p>
<p style="text-align: left;">area <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{2} \times 49\left( {2{\text{C}}\mathop {\text{A}}\limits^ \wedge {\text{B}} - {\text{sin}}\,{\text{2C}}\mathop {\text{A}}\limits^ \wedge {\text{B}}} \right)\, + \frac{1}{2} \times 25\left( {2{\text{C}}\mathop {\text{B}}\limits^ \wedge {\text{A}} - {\text{sin}}\,{\text{2C}}\mathop {\text{B}}\limits^ \wedge {\text{A}}} \right)"> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>×</mo> <mn>49</mn> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mrow> <mtext>C</mtext> </mrow> <mover> <mrow> <mtext>A</mtext> </mrow> <mo>∧</mo> </mover> <mo></mo> <mrow> <mtext>B</mtext> </mrow> <mo>−</mo> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>2C</mtext> </mrow> <mover> <mrow> <mtext>A</mtext> </mrow> <mo>∧</mo> </mover> <mo></mo> <mrow> <mtext>B</mtext> </mrow> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace"></mspace> <mo>+</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>×</mo> <mn>25</mn> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mrow> <mtext>C</mtext> </mrow> <mover> <mrow> <mtext>B</mtext> </mrow> <mo>∧</mo> </mover> <mo></mo> <mrow> <mtext>A</mtext> </mrow> <mo>−</mo> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>2C</mtext> </mrow> <mover> <mrow> <mtext>B</mtext> </mrow> <mo>∧</mo> </mover> <mo></mo> <mrow> <mtext>A</mtext> </mrow> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p style="text-align: left;">= 3.5079… + 5.3385… <em><strong>(A1)</strong></em></p>
<p style="text-align: left;"><strong>Note:</strong> Award this <em><strong>A1</strong></em> for either of these two values.</p>
<p style="text-align: left;">= 8.85 (km<sup>2</sup>) <em><strong>A1</strong></em></p>
<p style="text-align: left;"><strong>Note:</strong> Accept all answers that round to 8.8 or 8.9.</p>
<p style="text-align: left;"> </p>
<p style="text-align: left;"><em><strong>[6 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br>