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<h2>HL Paper 2</h2><div class="specification">
<p>A wind turbine is designed so that the rotation of the blades generates electricity. The turbine is built on horizontal ground and is made up of a vertical tower and three blades.</p>
<p>The point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> is on the base of the tower directly below point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math> at the top of the tower. The height of the tower, <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>AB</mtext></math>, is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>90</mn><mo> </mo><mtext>m</mtext></math>. The blades of the turbine are centred at <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math> and are each of length <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>40</mn><mo> </mo><mtext>m</mtext></math>. This is shown in the following diagram.</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" 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"></p>
<p>The end of one of the blades of the turbine is represented by point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>C</mtext></math> on the diagram. Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math> be the height of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>C</mtext></math> above the ground, measured in metres, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math> varies as the blade rotates.</p>
</div>
<div class="specification">
<p>Find the</p>
</div>
<div class="specification">
<p>The blades of the turbine complete <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>12</mn></math> rotations per minute under normal conditions, moving at a constant rate.</p>
</div>
<div class="specification">
<p>The height, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math>, of point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>C</mtext></math> can be modelled by the following function. Time, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>, is measured from the instant when the blade <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>[BC]</mtext></math> first passes <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>[AB]</mtext></math> and is measured in seconds.</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mfenced><mi>t</mi></mfenced><mo>=</mo><mn>90</mn><mo>-</mo><mn>40</mn><mo> </mo><mi>cos</mi><mfenced><mrow><mn>72</mn><mi>t</mi><mo>°</mo></mrow></mfenced><mo>,</mo><mo> </mo><mi>t</mi><mo>≥</mo><mn>0</mn></math></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>maximum value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</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>minimum value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the time, in seconds, it takes for the blade <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>[BC]</mtext></math> to make one complete rotation under these conditions.</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>Calculate the angle, in degrees, that the blade <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>[BC]</mtext></math> turns through in one second.</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>Write down the amplitude of the function.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the period of the function.</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>Sketch the function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>(</mo><mi>t</mi><mo>)</mo></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>≤</mo><mi>t</mi><mo>≤</mo><mn>5</mn></math>, clearly labelling the coordinates of the maximum and minimum points.</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 <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>C</mtext></math> above the ground when <math xmlns="http://www.w3.org/1998/Math/MathML"><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>Find the time, in seconds, that point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>C</mtext></math> is above a height of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>100</mn><mo> </mo><mtext>m</mtext></math>, during each complete rotation.</p>
<div class="marks">[3]</div>
<div class="question_part_label">e.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The wind speed increases and the blades rotate faster, but still at a constant rate.</p>
<p>Given that point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>C</mtext></math> is now higher than <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>110</mn><mo> </mo><mtext>m</mtext></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math> second during each complete rotation, find the time for one complete rotation.</p>
<div class="marks">[5]</div>
<div class="question_part_label">f.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>maximum <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>=</mo><mn>130</mn></math> metres <strong><em>A1</em></strong></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>minimum <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>=</mo><mn>50</mn></math> metres <strong><em>A1</em></strong></p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>60</mn><mo>÷</mo><mn>12</mn><mo>=</mo></mrow></mfenced><mo> </mo><mo> </mo><mn>5</mn><mo> </mo><mtext>seconds</mtext></math> <strong><em>A1</em></strong></p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>360</mn><mo>÷</mo><mn>5</mn></math> <em><strong>(M1)</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>360</mn></math> divided by their time for one revolution.<br><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>72</mn><mo>°</mo></math> <strong><em>A1</em></strong></p>
<p> </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>(amplitude =) <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>40</mn></math> <strong><em>A1</em></strong></p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>(period <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>360</mn><mn>72</mn></mfrac><mo>=</mo></math>) <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn></math> <strong><em>A1</em></strong></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><img src="data:image/png;base64,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"></p>
<p>Maximum point labelled with correct coordinates. <strong><em>A1</em></strong></p>
<p>At least one minimum point labelled. Coordinates seen for any minimum points must be correct. <strong><em>A1</em></strong></p>
<p>Correct shape with an attempt at symmetry and “concave up" evident as it approaches the minimum points. Graph must be drawn in the given domain. <strong><em>A1</em></strong></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><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>=</mo><mn>90</mn><mo>-</mo><mn>40</mn><mo> </mo><mi>cos</mi><mfenced><mrow><mn>144</mn><mo>°</mo></mrow></mfenced></math> <strong><em>(M1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>h</mi><mo>=</mo></mrow></mfenced><mo> </mo><mn>122</mn><mo> </mo><mfenced><mtext>m</mtext></mfenced><mo> </mo><mo> </mo><mfenced><mrow><mn>122</mn><mo>.</mo><mn>3606</mn><mo>…</mo></mrow></mfenced></math> <strong><em>A1</em></strong></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">e.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>evidence of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>=</mo><mn>100</mn></math> on graph <strong>OR </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>100</mn><mo>=</mo><mn>90</mn><mo>-</mo><mn>40</mn><mo> </mo><mi>cos</mi><mfenced><mrow><mn>72</mn><mi>t</mi></mrow></mfenced></math> <strong><em>(M1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> coordinates <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>.</mo><mn>55</mn><mo> </mo><mo>(</mo><mn>3</mn><mo>.</mo><mn>54892</mn><mo>.</mo><mo>.</mo><mo>.</mo><mo>)</mo></math> <strong>OR </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.</mo><mn>45</mn><mo> </mo><mo>(</mo><mn>1</mn><mo>.</mo><mn>45107</mn><mo>.</mo><mo>.</mo><mo>.</mo><mo>)</mo></math> or equivalent <strong><em>(A1)</em></strong></p>
<p><br><strong>Note:</strong> Award <em><strong>A1</strong></em> for either <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>-coordinate seen.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>2</mn><mo>.</mo><mn>10</mn></math> seconds <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>2</mn><mo>.</mo><mn>09784</mn><mo>…</mo></mrow></mfenced></math> <strong><em>A1</em></strong></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">e.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>90</mn><mo>-</mo><mn>40</mn><mo> </mo><mi>cos</mi><mfenced><mrow><mi>a</mi><mi>t</mi><mo>°</mo></mrow></mfenced><mo>=</mo><mn>110</mn></math> <strong><em>(M1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mfenced><mrow><mi>a</mi><mi>t</mi><mo>°</mo></mrow></mfenced><mo>=</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>5</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mi>t</mi><mo>°</mo><mo>=</mo><mn>120</mn><mo>,</mo><mo> </mo><mn>240</mn></math> <strong><em>(A1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>=</mo><mfrac><mn>240</mn><mi>a</mi></mfrac><mo>-</mo><mfrac><mn>120</mn><mi>a</mi></mfrac></math> <strong><em>(M1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>120</mn></math> <strong><em>(A1)</em></strong></p>
<p>period <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>360</mn><mn>120</mn></mfrac><mo>=</mo><mn>3</mn></math> seconds <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><img src="data:image/png;base64,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"></p>
<p>attempt at diagram <strong><em>(M1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mo> </mo><mi>α</mi><mo>=</mo><mfrac><mn>20</mn><mn>40</mn></mfrac></math> (or recognizing special triangle) <strong><em>(M1)</em></strong></p>
<p>angle made by <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>C</mtext></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>α</mi><mo>=</mo><mn>120</mn><mo>°</mo></math> <strong><em>(A1)</em></strong></p>
<p>one third of a revolution in <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math> second <strong><em>(M1)</em></strong></p>
<p>hence one revolution <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>3</mn></math> seconds <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>METHOD 3</strong></p>
<p>considering <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mfenced><mi>t</mi></mfenced><mo>=</mo><mn>110</mn></math> on original function <strong><em>(M1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mfrac><mn>5</mn><mn>3</mn></mfrac></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>10</mn><mn>3</mn></mfrac></math> <strong><em>(A1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>10</mn><mn>3</mn></mfrac><mo>-</mo><mfrac><mn>5</mn><mn>3</mn></mfrac><mo>=</mo><mfrac><mn>5</mn><mn>3</mn></mfrac></math> <strong><em>(A1)</em></strong></p>
<p><br><strong>Note:</strong> Accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>1</mn><mo>.</mo><mn>67</mn></math> or equivalent.</p>
<p><br>so period is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>3</mn><mn>5</mn></mfrac></math> of original period <strong><em>(R1)</em></strong></p>
<p>so new period is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math> seconds <strong><em>A1</em></strong></p>
<p> </p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">f.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>This was perhaps the question with the best responses on the paper. Many candidates got close to full marks on this problem. The issues associated with the question were sometimes due to a lack of understanding of the definitions of amplitude and period. A good number of candidates solved both parts of part (e) suggesting that they had a good understanding of the concept of a function and how it can be applied to mathematical models. Part (f) was also well done by a surprisingly large number of candidates using a variety of approaches. This is evidence that candidates had good problem-solving skills.</p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>This was perhaps the question with the best responses on the paper. Many candidates got close to full marks on this problem. The issues associated with the question were sometimes due to a lack of understanding of the definitions of amplitude and period. A good number of candidates solved both parts of part (e) suggesting that they had a good understanding of the concept of a function and how it can be applied to mathematical models. Part (f) was also well done by a surprisingly large number of candidates using a variety of approaches. This is evidence that candidates had good problem-solving skills.</p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>This was perhaps the question with the best responses on the paper. Many candidates got close to full marks on this problem. The issues associated with the question were sometimes due to a lack of understanding of the definitions of amplitude and period. A good number of candidates solved both parts of part (e) suggesting that they had a good understanding of the concept of a function and how it can be applied to mathematical models. Part (f) was also well done by a surprisingly large number of candidates using a variety of approaches. This is evidence that candidates had good problem-solving skills.</p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>This was perhaps the question with the best responses on the paper. Many candidates got close to full marks on this problem. The issues associated with the question were sometimes due to a lack of understanding of the definitions of amplitude and period. A good number of candidates solved both parts of part (e) suggesting that they had a good understanding of the concept of a function and how it can be applied to mathematical models. Part (f) was also well done by a surprisingly large number of candidates using a variety of approaches. This is evidence that candidates had good problem-solving skills.</p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>This was perhaps the question with the best responses on the paper. Many candidates got close to full marks on this problem. The issues associated with the question were sometimes due to a lack of understanding of the definitions of amplitude and period. A good number of candidates solved both parts of part (e) suggesting that they had a good understanding of the concept of a function and how it can be applied to mathematical models. Part (f) was also well done by a surprisingly large number of candidates using a variety of approaches. This is evidence that candidates had good problem-solving skills.</p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>This was perhaps the question with the best responses on the paper. Many candidates got close to full marks on this problem. The issues associated with the question were sometimes due to a lack of understanding of the definitions of amplitude and period. A good number of candidates solved both parts of part (e) suggesting that they had a good understanding of the concept of a function and how it can be applied to mathematical models. Part (f) was also well done by a surprisingly large number of candidates using a variety of approaches. This is evidence that candidates had good problem-solving skills.</p>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>This was perhaps the question with the best responses on the paper. Many candidates got close to full marks on this problem. The issues associated with the question were sometimes due to a lack of understanding of the definitions of amplitude and period. A good number of candidates solved both parts of part (e) suggesting that they had a good understanding of the concept of a function and how it can be applied to mathematical models. Part (f) was also well done by a surprisingly large number of candidates using a variety of approaches. This is evidence that candidates had good problem-solving skills.</p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>This was perhaps the question with the best responses on the paper. Many candidates got close to full marks on this problem. The issues associated with the question were sometimes due to a lack of understanding of the definitions of amplitude and period. A good number of candidates solved both parts of part (e) suggesting that they had a good understanding of the concept of a function and how it can be applied to mathematical models. Part (f) was also well done by a surprisingly large number of candidates using a variety of approaches. This is evidence that candidates had good problem-solving skills.</p>
<div class="question_part_label">e.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>This was perhaps the question with the best responses on the paper. Many candidates got close to full marks on this problem. The issues associated with the question were sometimes due to a lack of understanding of the definitions of amplitude and period. A good number of candidates solved both parts of part (e) suggesting that they had a good understanding of the concept of a function and how it can be applied to mathematical models. Part (f) was also well done by a surprisingly large number of candidates using a variety of approaches. This is evidence that candidates had good problem-solving skills.</p>
<div class="question_part_label">e.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>This was perhaps the question with the best responses on the paper. Many candidates got close to full marks on this problem. The issues associated with the question were sometimes due to a lack of understanding of the definitions of amplitude and period. A good number of candidates solved both parts of part (e) suggesting that they had a good understanding of the concept of a function and how it can be applied to mathematical models. Part (f) was also well done by a surprisingly large number of candidates using a variety of approaches. This is evidence that candidates had good problem-solving skills.</p>
<div class="question_part_label">f.</div>
</div>
<br><hr><br><div class="specification">
<p>The cross-sectional view of a tunnel is shown on the axes below. The line <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>[</mo><mtext>AB</mtext><mo>]</mo></math> represents a vertical wall located at the left side of the tunnel. The height, in metres, of the tunnel above the horizontal ground is modelled by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>1</mn><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mo> </mo><mn>0</mn><mo>.</mo><mn>8</mn><msup><mi>x</mi><mn>2</mn></msup><mo>,</mo><mo> </mo><mn>2</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mn>8</mn></math>, relative to an origin <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math>.</p>
<p style="text-align: center;"><img 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"></p>
<p style="text-align: left;">Point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> has coordinates <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>2</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math>, point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math> has coordinates <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>2</mn><mo>,</mo><mo> </mo><mn>2</mn><mo>.</mo><mn>4</mn><mo>)</mo></math>, and point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>C</mtext></math> has coordinates <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>8</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math>.</p>
</div>
<div class="specification">
<p>Find the height of the tunnel when</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac></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 maximum height of the tunnel.</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><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>4</mn></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>6</mn></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use the trapezoidal rule, with three intervals, to estimate the cross-sectional area of the tunnel.</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>Write down the integral which can be used to find the cross-sectional area of the tunnel.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find the cross-sectional area of the tunnel.</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>evidence of power rule (at least one correct term seen) <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn><mo>.</mo><mn>6</mn><mi>x</mi></math> <em><strong>A1</strong></em></p>
<p><strong><br></strong><em><strong>[2 marks]</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"><mo>-</mo><mn>0</mn><mo>.</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn><mo>.</mo><mn>6</mn><mi>x</mi><mo>=</mo><mn>0</mn></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>5</mn><mo>.</mo><mn>33</mn><mo> </mo><mfenced><mrow><mn>5</mn><mo>.</mo><mn>33333</mn><mo>…</mo><mo>,</mo><mo> </mo><mfrac><mn>16</mn><mn>3</mn></mfrac></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>1</mn><mo>×</mo><mn>5</mn><mo>.</mo><mn>33333</mn><msup><mo>…</mo><mn>3</mn></msup><mo>+</mo><mn>0</mn><mo>.</mo><mn>8</mn><mo>×</mo><mn>5</mn><mo>.</mo><mn>33333</mn><msup><mo>…</mo><mn>2</mn></msup></math> <em><strong>(M1)</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>M1</strong></em> for substituting their zero for <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo> </mo><mfenced><mrow><mn>5</mn><mo>.</mo><mn>333</mn><mo>…</mo></mrow></mfenced></math> into <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn><mo>.</mo><mn>59</mn><mo> </mo><mo> </mo><mi mathvariant="normal">m</mi><mo> </mo><mo> </mo><mfenced><mrow><mn>7</mn><mo>.</mo><mn>58519</mn><mo>…</mo></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>M0A0M0A0</strong></em> for an unsupported <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn><mo>.</mo><mn>59</mn></math>. <br>Award at most <em><strong>M0A0M1A0</strong></em> if only the last two lines in the solution are seen. <br>Award at most <em><strong>M1A0M1A1</strong></em> if their <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>5</mn><mo>.</mo><mn>33</mn></math> is not seen.</p>
<p><strong><br></strong><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>One correct substitution seen <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mo>.</mo><mn>4</mn><mo> </mo><mtext>m</mtext></math> <em><strong>A1</strong></em></p>
<p><strong><br></strong><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn><mo>.</mo><mn>2</mn><mo> </mo><mtext>m</mtext></math> <em><strong>A1</strong></em></p>
<p><strong><br></strong><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>×</mo><mn>2</mn><mfenced><mrow><mfenced><mrow><mn>2</mn><mo>.</mo><mn>4</mn><mo>+</mo><mn>0</mn></mrow></mfenced><mo>+</mo><mn>2</mn><mfenced><mrow><mn>6</mn><mo>.</mo><mn>4</mn><mo>+</mo><mn>7</mn><mo>.</mo><mn>2</mn></mrow></mfenced></mrow></mfenced></math> <em><strong>(A1)(M1)</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>=</mo><mn>2</mn></math> seen. Award <em><strong>M1</strong></em> for correct substitution into the trapezoidal rule (the zero can be omitted in working).</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>29</mn><mo>.</mo><mn>6</mn><mo> </mo><msup><mtext>m</mtext><mn>2</mn></msup></math> <em><strong>A1</strong></em></p>
<p><strong><br></strong><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><msubsup><mo>∫</mo><mn>2</mn><mn>8</mn></msubsup><mo>-</mo><mn>0</mn><mo>.</mo><mn>1</mn><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>0</mn><mo>.</mo><mn>8</mn><msup><mi>x</mi><mn>2</mn></msup><mo> </mo><mo>d</mo><mi>x</mi></math> <strong>OR </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><msubsup><mo>∫</mo><mn>2</mn><mn>8</mn></msubsup><mi>y</mi><mo> </mo><mo>d</mo><mi>x</mi></math> <em><strong>A1A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for a correct integral, <em><strong>A</strong><strong>1</strong></em> for correct limits in the correct location. Award at most <em><strong>A0A1</strong></em> if <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>d</mtext><mi>x</mi></math> is omitted.</p>
<p><strong><br></strong><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mn>32</mn><mo>.</mo><mn>4</mn><mo> </mo><msup><mtext>m</mtext><mn>2</mn></msup></math> <em><strong>A2</strong></em></p>
<p><strong><br>Note:</strong> As per the marking instructions, <em><strong>FT</strong></em> from their integral in part (d)(i). Award at most <em><strong>A1FTA0</strong></em> if their area is <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>></mo><mn>48</mn></math>, this is outside the constraints of the question (a <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mo>×</mo><mn>8</mn></math> rectangle).</p>
<p><strong><br></strong><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.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.</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 student investigating the relationship between chemical reactions and temperature finds the Arrhenius equation on the internet.</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mi>A</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mfrac><mi>c</mi><mi>T</mi></mfrac></mrow></msup></math></p>
<p>This equation links a variable <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> with the temperature <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math> are positive constants and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mo>></mo><mn>0</mn></math>.</p>
</div>
<div class="specification">
<p>The Arrhenius equation predicts that the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mi>k</mi></math> against <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mi>T</mi></mfrac></math> is a straight line.</p>
</div>
<div class="specification">
<p>Write down</p>
</div>
<div class="specification">
<p>The following data are found for a particular reaction, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math> is measured in Kelvin and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> is measured in <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>cm</mtext><mn>3</mn></msup><mo> </mo><msup><mtext>mol</mtext><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo> </mo><msup><mtext>s</mtext><mrow><mo>−</mo><mn>1</mn></mrow></msup></math>:</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<div class="specification">
<p>Find an estimate of</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>k</mi></mrow><mrow><mo>d</mo><mi>T</mi></mrow></mfrac></math> is always positive.</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>Given that <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi>lim</mi><mrow><mi>T</mi><mo>→</mo><mo>∞</mo></mrow></munder><mi>k</mi><mo>=</mo><mi>A</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi>lim</mi><mrow><mi>T</mi><mo>→</mo><mn>0</mn></mrow></munder><mi>k</mi><mo>=</mo><mn>0</mn></math>, sketch the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> against <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></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>(i) the gradient of this line in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math>;</p>
<p>(ii) the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-intercept of this line in terms 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">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the equation of the regression line for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mi>k</mi></math> on <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mi>T</mi></mfrac></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math>.</p>
<p>It is not required to state units for this value.</p>
<div class="marks">[1]</div>
<div class="question_part_label">e.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math>.</p>
<p>It is not required to state units for this value.</p>
<div class="marks">[2]</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>attempt to use chain rule, including the differentiation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mi>T</mi></mfrac></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>k</mi></mrow><mrow><mo>d</mo><mi>T</mi></mrow></mfrac><mo>=</mo><mi>A</mi><mo>×</mo><mfrac><mi>c</mi><msup><mi>T</mi><mn>2</mn></msup></mfrac><mo>×</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mfrac><mi>c</mi><mi>T</mi></mfrac></mrow></msup></math> <em><strong>A1</strong></em></p>
<p>this is the product of positive quantities so must be positive <em><strong>R1</strong></em></p>
<p><br><strong>Note:</strong> The <em><strong>R1</strong> </em>may be awarded for correct argument from <strong>their</strong> derivative. <em><strong>R1</strong> </em>is not possible if their derivative is not always positive.</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><img src="data:image/png;base64,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"> <em><strong>A1A1A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for an increasing graph, entirely in first quadrant, becoming concave down for larger values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math>, <em><strong>A1</strong></em> for tending towards the origin and <em><strong>A1</strong> </em>for asymptote labelled at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mi>A</mi></math>.</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>taking <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi></math> of both sides <strong>OR</strong> substituting <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>ln</mi><mo> </mo><mi>x</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mn>1</mn><mi>T</mi></mfrac></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mi>k</mi><mo>=</mo><mi>ln</mi><mo> </mo><mi>A</mi><mo>-</mo><mfrac><mi>c</mi><mi>T</mi></mfrac></math> OR <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>-</mo><mi>c</mi><mi>x</mi><mo>+</mo><mi>ln</mi><mo> </mo><mi>A</mi></math> <em><strong>(A1)</strong></em></p>
<p><br>(i) so gradient is <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mi>c</mi></math> <em><strong>A1</strong></em></p>
<p><br>(ii) <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-intercept is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mi>A</mi></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> The implied <em><strong>(M1)</strong></em> and <em><strong>(A1)</strong></em> can only be awarded if <strong>both</strong> correct answers are seen. Award zero if only one value is correct <strong>and</strong> no working is seen.</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>an attempt to convert data to <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mi>T</mi></mfrac></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mi>k</mi></math> <em><strong>(M1)</strong></em></p>
<p>e.g. at least one correct row in the following table</p>
<p><img src="data:image/png;base64,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"></p>
<p>line is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mi>k</mi><mo>=</mo><mo>-</mo><mn>13400</mn><mo>×</mo><mfrac><mn>1</mn><mi>T</mi></mfrac><mo>+</mo><mn>15</mn><mo>.</mo><mn>0</mn><mo> </mo><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mo>-</mo><mn>13383</mn><mo>.</mo><mn>1</mn><mo>…</mo><mo>×</mo><mfrac><mn>1</mn><mi>T</mi></mfrac><mo>+</mo><mn>15</mn><mo>.</mo><mn>0107</mn><mo>…</mo></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>=</mo><mn>13400</mn><mo> </mo><mo> </mo><mo> </mo><mfenced><mrow><mn>13383</mn><mo>.</mo><mn>1</mn><mo>…</mo></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">e.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to rearrange or solve graphically <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mi>A</mi><mo>=</mo><mn>15</mn><mo>.</mo><mn>0107</mn><mo>…</mo></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mn>3</mn><mo> </mo><mn>300</mn><mo> </mo><mn>000</mn><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mfenced><mrow><mn>3</mn><mo> </mo><mn>304</mn><mo> </mo><mn>258</mn><mo>…</mo></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> <strong>Note</strong>: Accept an <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3269017</mn></math>… from use of <math xmlns="http://www.w3.org/1998/Math/MathML" class="wrs_chemistry"><mn>3</mn><mi>sf</mi></math> value.</p>
<p><em><strong>[2 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;">
<p>This question caused significant difficulties for many candidates and many did not even attempt the question. Very few candidates were able to differentiate the expression in part (a) resulting in difficulties for part (b). Responses to parts (c) to (e) illustrated a lack of understanding of linearizing a set of data. Those candidates that were able to do part (d) frequently lost a mark as their answer was given in <em>x</em> and <em>y</em>.</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.</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="specification">
<p>Consider the expression <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = {\text{tan}}\left( {x + \frac{\pi }{4}} \right){\text{cot}}\left( {\frac{\pi }{4} - x} \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>
<mfrac>
<mi>π<!-- π --></mi>
<mn>4</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mtext>cot</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mi>π<!-- π --></mi>
<mn>4</mn>
</mfrac>
<mo>−<!-- − --></mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
</div>
<div class="specification">
<p>The expression <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> can be written as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g\left( t \right)">
<mi>g</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t = {\text{tan}}\,x">
<mi>t</mi>
<mo>=</mo>
<mrow>
<mtext>tan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</math></span>.</p>
</div>
<div class="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\alpha ">
<mi>α<!-- α --></mi>
</math></span>, <em>β</em> be the roots of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g\left( t \right) = k">
<mi>g</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mi>k</mi>
</math></span>, where 0 < <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k">
<mi>k</mi>
</math></span> < 1.</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="y = f\left( x \right)">
<mi>y</mi>
<mo>=</mo>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - \frac{{5\pi }}{8} \leqslant x \leqslant \frac{\pi }{8}">
<mo>−</mo>
<mfrac>
<mrow>
<mn>5</mn>
<mi>π</mi>
</mrow>
<mn>8</mn>
</mfrac>
<mo>⩽</mo>
<mi>x</mi>
<mo>⩽</mo>
<mfrac>
<mi>π</mi>
<mn>8</mn>
</mfrac>
</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>With reference to your graph, explain why <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> is a function on the given domain.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Explain why <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> has no inverse on the given domain.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Explain why <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> is not a function for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - \frac{{3\pi }}{4} \leqslant x \leqslant \frac{\pi }{4}">
<mo>−</mo>
<mfrac>
<mrow>
<mn>3</mn>
<mi>π</mi>
</mrow>
<mn>4</mn>
</mfrac>
<mo>⩽</mo>
<mi>x</mi>
<mo>⩽</mo>
<mfrac>
<mi>π</mi>
<mn>4</mn>
</mfrac>
</math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.iv.</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="g\left( t \right) = {\left( {\frac{{1 + t}}{{1 - t}}} \right)^2}">
<mi>g</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>t</mi>
</mrow>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>t</mi>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</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>Sketch the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = g\left( t \right)">
<mi>y</mi>
<mo>=</mo>
<mi>g</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</math></span> for <em>t</em> ≤ 0. Give the coordinates of any intercepts and the equations of any asymptotes.</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 <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\alpha ">
<mi>α</mi>
</math></span> and <em>β</em> in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k">
<mi>k</mi>
</math></span>.</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>Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\alpha ">
<mi>α</mi>
</math></span> + <em>β</em> < −2.</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><img src="data:image/png;base64,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"> <em><strong>A1A1</strong></em></p>
<p><em><strong>A1</strong> </em>for correct concavity, many to one graph, symmetrical about the midpoint of the domain and with two axes intercepts.</p>
<p><strong>Note:</strong> Axes intercepts and scales not required.</p>
<p><strong>A1</strong> for correct domain</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>for each value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> there is a unique value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right)">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Accept “passes the vertical line test” or equivalent.</p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>no inverse because the function fails the horizontal line test or equivalent <em><strong>R1</strong></em></p>
<p><strong>Note:</strong> No <strong>FT</strong> if the graph is in degrees (one-to-one).</p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>the expression is not valid at either of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \frac{\pi }{4}\,\,\left( {{\text{or}} - \frac{{3\pi }}{4}} \right)">
<mi>x</mi>
<mo>=</mo>
<mfrac>
<mi>π</mi>
<mn>4</mn>
</mfrac>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>or</mtext>
</mrow>
<mo>−</mo>
<mfrac>
<mrow>
<mn>3</mn>
<mi>π</mi>
</mrow>
<mn>4</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>R1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.iv.</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="f\left( x \right) = \frac{{{\text{tan}}\left( {x + \frac{\pi }{4}} \right)}}{{{\text{tan}}\left( {\frac{\pi }{4} - x} \right)}}">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<mtext>tan</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mfrac>
<mi>π</mi>
<mn>4</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mrow>
<mtext>tan</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mi>π</mi>
<mn>4</mn>
</mfrac>
<mo>−</mo>
<mi>x</mi>
</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{{\frac{{{\text{tan}}\,x + {\text{tan}}\,\frac{\pi }{4}}}{{1 - {\text{tan}}\,x\,{\text{tan}}\,\frac{\pi }{4}}}}}{{\frac{{{\text{tan}}\,\frac{\pi }{4} - {\text{tan}}\,x}}{{1 + {\text{tan}}\,\frac{\pi }{4}{\text{tan}}\,x}}}}">
<mo>=</mo>
<mfrac>
<mrow>
<mfrac>
<mrow>
<mrow>
<mtext>tan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>+</mo>
<mrow>
<mtext>tan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mfrac>
<mi>π</mi>
<mn>4</mn>
</mfrac>
</mrow>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mrow>
<mtext>tan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>tan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mfrac>
<mi>π</mi>
<mn>4</mn>
</mfrac>
</mrow>
</mfrac>
</mrow>
<mrow>
<mfrac>
<mrow>
<mrow>
<mtext>tan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mfrac>
<mi>π</mi>
<mn>4</mn>
</mfrac>
<mo>−</mo>
<mrow>
<mtext>tan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mrow>
<mtext>tan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mfrac>
<mi>π</mi>
<mn>4</mn>
</mfrac>
<mrow>
<mtext>tan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
</mfrac>
</mrow>
</mfrac>
</math></span> <em><strong>M1A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {\left( {\frac{{1 + t}}{{1 - t}}} \right)^2}">
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>t</mi>
</mrow>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>t</mi>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</math></span> <em><strong>AG</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = {\text{tan}}\left( {x + \frac{\pi }{4}} \right){\text{tan}}\left( {\frac{\pi }{2} - \frac{\pi }{4} + x} \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>
<mfrac>
<mi>π</mi>
<mn>4</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mtext>tan</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
<mo>−</mo>
<mfrac>
<mi>π</mi>
<mn>4</mn>
</mfrac>
<mo>+</mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong> (M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {\text{ta}}{{\text{n}}^2}\left( {x + \frac{\pi }{4}} \right)">
<mo>=</mo>
<mrow>
<mtext>ta</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mfrac>
<mi>π</mi>
<mn>4</mn>
</mfrac>
</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="g\left( t \right) = {\left( {\frac{{{\text{tan}}\,x + {\text{tan}}\,\frac{\pi }{4}}}{{1 - {\text{tan}}\,x\,{\text{tan}}\,\frac{\pi }{4}}}} \right)^2}">
<mi>g</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mrow>
<mtext>tan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>+</mo>
<mrow>
<mtext>tan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mfrac>
<mi>π</mi>
<mn>4</mn>
</mfrac>
</mrow>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mrow>
<mtext>tan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>tan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mfrac>
<mi>π</mi>
<mn>4</mn>
</mfrac>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {\left( {\frac{{1 + t}}{{1 - t}}} \right)^2}">
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>t</mi>
</mrow>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>t</mi>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</math></span> <em><strong>AG</strong></em></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> </p>
<p><img src="data:image/png;base64,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"></p>
<p>for <em>t</em> ≤ 0, correct concavity with two axes intercepts and with asymptote <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span> = 1 <em><strong>A1</strong></em></p>
<p><em>t</em> intercept at (−1, 0) <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span> intercept at (0, 1) <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><strong>METHOD 1</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\alpha ">
<mi>α</mi>
</math></span>, <em>β</em> satisfy <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{{\left( {1 + t} \right)}^2}}}{{{{\left( {1 - t} \right)}^2}}} = k">
<mfrac>
<mrow>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>t</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>t</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
<mo>=</mo>
<mi>k</mi>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1 + {t^2} + 2t = k\left( {1 + {t^2} - 2t} \right)">
<mn>1</mn>
<mo>+</mo>
<mrow>
<msup>
<mi>t</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>2</mn>
<mi>t</mi>
<mo>=</mo>
<mi>k</mi>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mrow>
<msup>
<mi>t</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>2</mn>
<mi>t</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="\left( {k - 1} \right){t^2} - 2\left( {k + 1} \right)t + \left( {k - 1} \right) = 0">
<mrow>
<mo>(</mo>
<mrow>
<mi>k</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<msup>
<mi>t</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>2</mn>
<mrow>
<mo>(</mo>
<mrow>
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mi>t</mi>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>k</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span> <em><strong>A1</strong></em></p>
<p>attempt at using quadratic formula <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\alpha ">
<mi>α</mi>
</math></span>, <em>β </em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{k + 1 \pm 2\sqrt k }}{{k - 1}}">
<mo>=</mo>
<mfrac>
<mrow>
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
<mo>±</mo>
<mn>2</mn>
<msqrt>
<mi>k</mi>
</msqrt>
</mrow>
<mrow>
<mi>k</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mfrac>
</math></span> or equivalent <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\alpha ">
<mi>α</mi>
</math></span>, <em>β</em> satisfy <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{1 + t}}{{1 - t}} = \left( \pm \right)\sqrt k ">
<mfrac>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>t</mi>
</mrow>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>t</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mo>±</mo>
<mo>)</mo>
</mrow>
<msqrt>
<mi>k</mi>
</msqrt>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t + \sqrt k t = \sqrt k - 1">
<mi>t</mi>
<mo>+</mo>
<msqrt>
<mi>k</mi>
</msqrt>
<mi>t</mi>
<mo>=</mo>
<msqrt>
<mi>k</mi>
</msqrt>
<mo>−</mo>
<mn>1</mn>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t = \frac{{\sqrt k - 1}}{{\sqrt k + 1}}">
<mi>t</mi>
<mo>=</mo>
<mfrac>
<mrow>
<msqrt>
<mi>k</mi>
</msqrt>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mrow>
<msqrt>
<mi>k</mi>
</msqrt>
<mo>+</mo>
<mn>1</mn>
</mrow>
</mfrac>
</math></span> (or equivalent) <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t - \sqrt k t = - \left( {\sqrt k + 1} \right)">
<mi>t</mi>
<mo>−</mo>
<msqrt>
<mi>k</mi>
</msqrt>
<mi>t</mi>
<mo>=</mo>
<mo>−</mo>
<mrow>
<mo>(</mo>
<mrow>
<msqrt>
<mi>k</mi>
</msqrt>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t = \frac{{\sqrt k + 1}}{{\sqrt k - 1}}">
<mi>t</mi>
<mo>=</mo>
<mfrac>
<mrow>
<msqrt>
<mi>k</mi>
</msqrt>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mrow>
<msqrt>
<mi>k</mi>
</msqrt>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mfrac>
</math></span> (or equivalent) <em><strong>A1</strong></em></p>
<p>so for <em>eg</em>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\alpha = \frac{{\sqrt k - 1}}{{\sqrt k + 1}}">
<mi>α</mi>
<mo>=</mo>
<mfrac>
<mrow>
<msqrt>
<mi>k</mi>
</msqrt>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mrow>
<msqrt>
<mi>k</mi>
</msqrt>
<mo>+</mo>
<mn>1</mn>
</mrow>
</mfrac>
</math></span>, <em>β</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{\sqrt k + 1}}{{\sqrt k - 1}}">
<mo>=</mo>
<mfrac>
<mrow>
<msqrt>
<mi>k</mi>
</msqrt>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mrow>
<msqrt>
<mi>k</mi>
</msqrt>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mfrac>
</math></span></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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\alpha ">
<mi>α</mi>
</math></span> + <em>β </em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 2\frac{{\left( {k + 1} \right)}}{{\left( {k - 1} \right)}}\,\left( { = - 2\frac{{\left( {1 + k} \right)}}{{\left( {1 - k} \right)}}} \right)">
<mo>=</mo>
<mn>2</mn>
<mfrac>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>k</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
</mfrac>
<mspace width="thinmathspace"></mspace>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mo>−</mo>
<mn>2</mn>
<mfrac>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>k</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>k</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p>since <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1 + k > 1 - k">
<mn>1</mn>
<mo>+</mo>
<mi>k</mi>
<mo>></mo>
<mn>1</mn>
<mo>−</mo>
<mi>k</mi>
</math></span> <em><strong>R1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\alpha ">
<mi>α</mi>
</math></span> + <em>β</em> < −2 <em><strong>AG</strong></em></p>
<p><strong>Note:</strong> Accept a valid graphical reasoning.</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">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.iv.</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>Charlotte decides to model the shape of a cupcake to calculate its volume.</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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"></p>
<p>From rotating a photograph of her cupcake she estimates that its cross-section passes through the points <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>0</mn><mo>,</mo><mo> </mo><mn>3</mn><mo>.</mo><mn>5</mn><mo>)</mo><mo>,</mo><mo> </mo><mo>(</mo><mn>4</mn><mo>,</mo><mo> </mo><mn>6</mn><mo>)</mo><mo>,</mo><mo> </mo><mo>(</mo><mn>6</mn><mo>.</mo><mn>5</mn><mo>,</mo><mo> </mo><mn>4</mn><mo>)</mo><mo>,</mo><mo> </mo><mo>(</mo><mn>7</mn><mo>,</mo><mo> </mo><mn>3</mn><mo>)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>7</mn><mo>.</mo><mn>5</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math>, where all units are in centimetres. The cross-section is symmetrical in the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis, as shown below:</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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"></p>
<p>She models the section from <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>0</mn><mo>,</mo><mo> </mo><mn>3</mn><mo>.</mo><mn>5</mn><mo>)</mo></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>4</mn><mo>,</mo><mo> </mo><mn>6</mn><mo>)</mo></math> as a straight line.</p>
</div>
<div class="specification">
<p>Charlotte models the section of the cupcake that passes through the points <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>4</mn><mo>,</mo><mo> </mo><mn>6</mn><mo>)</mo><mo>,</mo><mo> </mo><mo>(</mo><mn>6</mn><mo>.</mo><mn>5</mn><mo>,</mo><mo> </mo><mn>4</mn><mo>)</mo><mo>,</mo><mo> </mo><mo>(</mo><mn>7</mn><mo>,</mo><mo> </mo><mn>3</mn><mo>)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>7</mn><mo>.</mo><mn>5</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math> with a quadratic curve.</p>
</div>
<div class="specification">
<p>Charlotte thinks that a quadratic with a maximum point at <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>4</mn><mo>,</mo><mo> </mo><mn>6</mn><mo>)</mo></math> and that passes through the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>7</mn><mo>.</mo><mn>5</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math> would be a better fit.</p>
</div>
<div class="specification">
<p>Believing this to be a better model for her cupcake, Charlotte finds the volume of revolution about the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis to estimate the volume of the cupcake.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the equation of the line passing through these two points.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the equation of the least squares regression quadratic curve for these four points.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By considering the gradient of this curve when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>4</mn></math>, explain why it may not be a good model.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the equation of the new model.</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>Write down an expression for her estimate of the volume as a sum of two integrals.</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of Charlotte’s estimate.</p>
<div class="marks">[1]</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><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfrac><mn>5</mn><mn>8</mn></mfrac><mi>x</mi><mo>+</mo><mfrac><mn>7</mn><mn>2</mn></mfrac><mo> </mo><mo> </mo><mo> </mo><mfenced><mrow><mi>y</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>625</mn><mi>x</mi><mo>+</mo><mn>3</mn><mo>.</mo><mn>5</mn></mrow></mfenced></math> <em><strong>A1A1</strong></em></p>
<p><strong><br>Note:</strong> Award <em><strong>A1</strong> </em>for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>625</mn><mi>x</mi></math>, <em><strong>A1</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>.</mo><mn>5</mn></math>.<br>Award a maximum of <em><strong>A0A1</strong></em> if not part of an equation.</p>
<p><strong><br></strong><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"><mi>y</mi><mo>=</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>975</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>9</mn><mo>.</mo><mn>56</mn><mi>x</mi><mo>-</mo><mn>16</mn><mo>.</mo><mn>7</mn></math> <em><strong>(M1)A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>y</mi><mo>=</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>974630</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>9</mn><mo>.</mo><mn>55919</mn><mi>x</mi><mo>-</mo><mn>16</mn><mo>.</mo><mn>6569</mn><mo>…</mo></mrow></mfenced></math></p>
<p><strong><br></strong><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>gradient of curve is positive at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>4</mn></math> <em><strong>R1</strong></em></p>
<p><em><br></em><strong>Note:</strong> Accept a sensible rationale that refers to the gradient.</p>
<p><strong><br></strong><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">b.ii.</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"><mi>y</mi><mo>=</mo><mi>a</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>b</mi><mi>x</mi><mo>+</mo><mi>c</mi></math></p>
<p>differentiating or using <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mrow><mo>-</mo><mi>b</mi></mrow><mrow><mn>2</mn><mi>a</mi></mrow></mfrac></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn><mi>a</mi><mo>+</mo><mi>b</mi><mo>=</mo><mn>0</mn></math></p>
<p>substituting in the coordinates<br><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn><mo>.</mo><msup><mn>5</mn><mn>2</mn></msup><mi>a</mi><mo>+</mo><mn>7</mn><mo>.</mo><mn>5</mn><mi>b</mi><mo>+</mo><mi>c</mi><mo>=</mo><mn>0</mn></math> <em><strong>(A1)<br></strong></em><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>4</mn><mn>2</mn></msup><mi>a</mi><mo>+</mo><mn>4</mn><mi>b</mi><mo>+</mo><mi>c</mi><mo>=</mo><mn>6</mn></math> <em><strong>(A1)</strong></em></p>
<p>solve to get<br><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>-</mo><mfrac><mn>24</mn><mn>49</mn></mfrac><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mfrac><mn>192</mn><mn>49</mn></mfrac><mi>x</mi><mo>-</mo><mfrac><mn>90</mn><mn>49</mn></mfrac></math> <strong>OR </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>490</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>3</mn><mo>.</mo><mn>92</mn><mi>x</mi><mo>-</mo><mn>1</mn><mo>.</mo><mn>84</mn></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Use of quadratic regression with points using the symmetry of the graph is a valid method.</p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>a</mi><msup><mfenced><mrow><mi>x</mi><mo>-</mo><mn>4</mn></mrow></mfenced><mn>2</mn></msup><mo>+</mo><mn>6</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>=</mo><mi>a</mi><msup><mfenced><mrow><mn>7</mn><mo>.</mo><mn>5</mn><mo>-</mo><mn>4</mn></mrow></mfenced><mn>2</mn></msup><mo>+</mo><mn>6</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mo>-</mo><mfrac><mn>24</mn><mn>49</mn></mfrac></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>-</mo><mfrac><mn>24</mn><mn>49</mn></mfrac><msup><mfenced><mrow><mi>x</mi><mo>-</mo><mn>4</mn></mrow></mfenced><mn>2</mn></msup><mo>+</mo><mn>6</mn></math> <strong>OR </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>490</mn><msup><mfenced><mrow><mi>x</mi><mo>-</mo><mn>4</mn></mrow></mfenced><mn>2</mn></msup><mo>+</mo><mn>6</mn></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><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">π</mi><msubsup><mo>∫</mo><mn>0</mn><mn>4</mn></msubsup><msup><mfenced><mrow><mfrac><mn>5</mn><mn>8</mn></mfrac><mi>x</mi><mo>+</mo><mn>3</mn><mo>.</mo><mn>5</mn></mrow></mfenced><mn>2</mn></msup><mo>d</mo><mi>x</mi><mo>+</mo><mi mathvariant="normal">π</mi><msubsup><mo>∫</mo><mn>4</mn><mrow><mn>7</mn><mo>.</mo><mn>5</mn></mrow></msubsup><msup><mfenced><mrow><mo>-</mo><mfrac><mn>24</mn><mn>49</mn></mfrac><msup><mfenced><mrow><mi>x</mi><mo>-</mo><mn>4</mn></mrow></mfenced><mn>2</mn></msup><mo>+</mo><mn>6</mn></mrow></mfenced><mn>2</mn></msup><mo>d</mo><mi>x</mi></math> <em><strong>(M1)(M1) (M1)A1</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>(M1)(M1)(M1)A0</strong></em> if <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">π</mi></math> is omitted but response is otherwise correct. Award <em><strong>(M1)</strong></em> for an integral that indicates volume,<em><strong> (M1)</strong></em> for their part (a) within their volume integral, <em><strong>(M1)</strong></em> for their part (b)(i) within their volume integral, <em><strong>A1</strong></em> for their correct two integrals with all correct limits.</p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>501</mn><mo> </mo><msup><mtext>cm</mtext><mn>3</mn></msup><mo> </mo><mo> </mo><mfenced><mrow><mn>501</mn><mo>.</mo><mn>189</mn><mo>…</mo></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[1 mark]</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.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.</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>An environmental scientist is asked by a river authority to model the effect of a leak from a power plant on the mercury levels in a local river. The variable <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> measures the concentration of mercury in micrograms per litre.</p>
<p>The situation is modelled using the second order differential equation</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mtext>d</mtext><mn>2</mn></msup><mi>x</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>+</mo><mn>3</mn><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>+</mo><mn>2</mn><mi>x</mi><mo>=</mo><mn>0</mn></math></p>
<p>where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>≥</mo><mn>0</mn></math> is the time measured in days since the leak started. It is known that when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn><mo>,</mo><mo> </mo><mi>x</mi><mo>=</mo><mn>0</mn></math> and <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><mn>1</mn></math>.</p>
</div>
<div class="specification">
<p>If the mercury levels are greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>1</mn></math> micrograms per litre, fishing in the river is considered unsafe and is stopped.</p>
</div>
<div class="specification">
<p>The river authority decides to stop people from fishing in the river for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn><mo>%</mo></math> longer than the time found from the model.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the system of coupled first order equations:</p>
<p style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mi>y</mi></math></p>
<p style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mo>-</mo><mn>2</mn><mi>x</mi><mo>-</mo><mn>3</mn><mi>y</mi></math></p>
<p style="text-align:left;">can be written as the given second order differential equation.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the eigenvalues of the system of coupled first order equations given in part (a).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find the exact solution of the second order differential equation.</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>Sketch the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> against <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>, labelling the maximum point of the graph with its coordinates.</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>Use the model to calculate the total amount of time when fishing should be stopped.</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>Write down one reason, with reference to the context, to support this decision.</p>
<div class="marks">[1]</div>
<div class="question_part_label">f.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>differentiating first equation. <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>x</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></math></p>
<p>substituting in for <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></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><mn>2</mn><mi>x</mi><mo>-</mo><mn>3</mn><mi>y</mi><mo>=</mo><mo>-</mo><mn>2</mn><mi>x</mi><mo>-</mo><mn>3</mn><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></math></p>
<p>therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>x</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>+</mo><mn>3</mn><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>+</mo><mn>2</mn><mi>x</mi><mo>=</mo><mn>0</mn></math> <strong><em>AG</em></strong></p>
<p><br><strong>Note:</strong> The <strong>AG</strong> line must be seen to award the final <em><strong>M1</strong></em> mark.</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>the relevant matrix is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>0</mn><mo> </mo><mo> </mo></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn><mo> </mo><mo> </mo></mtd><mtd><mo>-</mo><mn>3</mn></mtd></mtr></mtable></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><br><strong>Note:</strong> <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mo>-</mo><mn>3</mn><mo> </mo><mo> </mo></mtd><mtd><mo>-</mo><mn>2</mn></mtd></mtr><mtr><mtd><mn>1</mn><mo> </mo><mo> </mo></mtd><mtd><mn>0</mn></mtd></mtr></mtable></mfenced></math> is also possible.</p>
<p><br>(this has characteristic equation) <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mi>λ</mi><mfenced><mrow><mo>-</mo><mn>3</mn><mo>-</mo><mi>λ</mi></mrow></mfenced><mo>+</mo><mn>2</mn><mo>=</mo><mn>0</mn></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi><mo>=</mo><mo>-</mo><mn>1</mn><mo>,</mo><mo> </mo><mo>-</mo><mn>2</mn></math> <strong><em>A1</em></strong></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><strong>EITHER </strong></p>
<p>the general solution is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mi>A</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup><mo>+</mo><mi>B</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mn>2</mn><mi>t</mi></mrow></msup></math> <em><strong>M1</strong></em></p>
<p><br><strong>Note:</strong> Must have constants, but condone sign error for the <em><strong>M1</strong></em>.</p>
<p><br>so <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><mi>A</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup><mo>-</mo><mn>2</mn><mi>B</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mn>2</mn><mi>t</mi></mrow></msup></math> <em><strong>M1A1</strong></em></p>
<p> </p>
<p><strong>OR</strong></p>
<p>attempt to find eigenvectors <em><strong>(M1)</strong></em></p>
<p>respective eigenvectors are <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr></mtable></mfenced></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr></mtable></mfenced></math> (or any multiple)</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mi>x</mi></mtd></mtr><mtr><mtd><mi>y</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mi>A</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mi>B</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mn>2</mn><mi>t</mi></mrow></msup><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr></mtable></mfenced></math> <em><strong>(M1)A1</strong></em></p>
<p> </p>
<p><strong>THEN</strong></p>
<p>the initial conditions become:</p>
<p style="padding-left:30px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>=</mo><mi>A</mi><mo>+</mo><mi>B</mi></math></p>
<p style="padding-left:30px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>=</mo><mo>-</mo><mi>A</mi><mo>-</mo><mn>2</mn><mi>B</mi></math> <em><strong>M1</strong></em></p>
<p>this is solved by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo> </mo><mi>B</mi><mo>=</mo><mo>-</mo><mn>1</mn></math></p>
<p>so the solution is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup><mo>-</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mn>2</mn><mi>t</mi></mrow></msup></math> <strong><em>A1</em></strong></p>
<p> </p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="padding-left:60px;"><img src="data:image/png;base64,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"> <strong><em>A1A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A1</strong> </em>for correct shape (needs to go through origin, have asymptote at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>0</mn></math> and a single maximum; condone <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo><</mo><mn>0</mn></math>). Award <em><strong>A1</strong></em> for correct coordinates of maximum.</p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>intersecting graph with <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>1</mn></math> <em><strong>(M1)</strong></em></p>
<p style="padding-left:60px;"><img src="data:image/png;base64,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"></p>
<p>so the time fishing is stopped between <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>.</mo><mn>1830</mn><mo>…</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>11957</mn><mo>…</mo></math> <strong><em>(A1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>2</mn><mo>.</mo><mn>06</mn><mo> </mo><mfenced><mrow><mn>343</mn><mo>…</mo></mrow></mfenced></math> days <strong><em>A1</em></strong></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><em>Any reasonable answer. For example:</em></p>
<p>There are greater downsides to allowing fishing when the levels may be dangerous than preventing fishing when the levels are safe.</p>
<p>The concentration of mercury may not be uniform across the river due to natural variation / randomness.</p>
<p>The situation at the power plant might get worse.</p>
<p>Mercury levels are low in water but still may be high in fish. <strong><em>R1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>R1</strong> </em>for a reasonable answer that refers to this specific context (and not a generic response that could apply to <em>any</em> model).</p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">f.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>Many candidates did not get this far, but the attempts at the question that were seen were generally good. The greater difficulties were seen in parts (e) and (f), but this could be a problem with time running out.</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.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">f.</div>
</div>
<br><hr><br><div class="specification">
<p>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 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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 <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = - 1 + \ln \left( {\sqrt {{x^2} - 1} } \right)">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo>+</mo>
<mi>ln</mi>
<mo><!-- --></mo>
<mrow>
<mo>(</mo>
<mrow>
<msqrt>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</msqrt>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
</div>
<div class="specification">
<p>The function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> is defined by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = - 1 + \ln \left( {\sqrt {{x^2} - 1} } \right),{\text{ }}x \in D">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo>+</mo>
<mi>ln</mi>
<mo><!-- --></mo>
<mrow>
<mo>(</mo>
<mrow>
<msqrt>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</msqrt>
</mrow>
<mo>)</mo>
</mrow>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>x</mi>
<mo>∈<!-- ∈ --></mo>
<mi>D</mi>
</math></span></p>
</div>
<div class="specification">
<p>The function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g">
<mi>g</mi>
</math></span> is defined by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g(x) = - 1 + \ln \left( {\sqrt {{x^2} - 1} } \right),{\text{ }}x \in \left] {1,{\text{ }}\infty } \right[">
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo>+</mo>
<mi>ln</mi>
<mo><!-- --></mo>
<mrow>
<mo>(</mo>
<mrow>
<msqrt>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</msqrt>
</mrow>
<mo>)</mo>
</mrow>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>x</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mo>]</mo>
<mrow>
<mn>1</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi mathvariant="normal">∞<!-- ∞ --></mi>
</mrow>
<mo>[</mo>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the largest possible domain <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="D">
<mi>D</mi>
</math></span> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> to be a function.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f(x)">
<mi>y</mi>
<mo>=</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span> showing clearly the equations of asymptotes and the coordinates of any intercepts with the axes.</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>Explain why <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> is an even function.</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>Explain why the inverse function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f^{ - 1}}">
<mrow>
<msup>
<mi>f</mi>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</math></span> does not exist.</p>
<div class="marks">[1]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the inverse function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{g^{ - 1}}">
<mrow>
<msup>
<mi>g</mi>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</math></span> and state its domain.</p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g'(x)">
<msup>
<mi>g</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span>.</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>Hence, show that there are no solutions to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g'(x) = 0">
<msup>
<mi>g</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0</mn>
</math></span>;</p>
<div class="marks">[2]</div>
<div class="question_part_label">g.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, show that there are no solutions to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="({g^{ - 1}})'(x) = 0">
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mi>g</mi>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<msup>
<mo stretchy="false">)</mo>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0</mn>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">g.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p 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="{x^2} - 1 > 0">
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>1</mn>
<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="x < - 1">
<mi>x</mi>
<mo><</mo>
<mo>−</mo>
<mn>1</mn>
</math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x > 1">
<mi>x</mi>
<mo>></mo>
<mn>1</mn>
</math></span> <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><img src="images/Schermafbeelding_2017-08-09_om_15.40.09.png" alt="M17/5/MATHL/HP2/ENG/TZ1/12.b/M"></p>
<p>shape <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 1">
<mi>x</mi>
<mo>=</mo>
<mn>1</mn>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = - 1">
<mi>x</mi>
<mo>=</mo>
<mo>−</mo>
<mn>1</mn>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>-intercepts <strong><em>A1</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</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="f">
<mi>f</mi>
</math></span> is symmetrical about the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span>-axis <strong><em>R1</em></strong></p>
<p><strong>OR</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f( - x) = f(x)">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mo>−</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span> <strong><em>R1</em></strong></p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">c.</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="f">
<mi>f</mi>
</math></span> is not one-to-one function <strong><em>R1</em></strong></p>
<p><strong>OR</strong></p>
<p>horizontal line cuts twice <strong><em>R1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Accept any equivalent correct statement.</p>
<p> </p>
<p><strong><em>[1 mark]</em></strong></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="x = - 1 + \ln \left( {\sqrt {{y^2} - 1} } \right)">
<mi>x</mi>
<mo>=</mo>
<mo>−</mo>
<mn>1</mn>
<mo>+</mo>
<mi>ln</mi>
<mo></mo>
<mrow>
<mo>(</mo>
<mrow>
<msqrt>
<mrow>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>1</mn>
</msqrt>
</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="{{\text{e}}^{2x + 2}} = {y^2} - 1">
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>1</mn>
</math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{g^{ - 1}}(x) = \sqrt {{{\text{e}}^{2x + 2}} + 1} ,{\text{ }}x \in \mathbb{R}">
<mrow>
<msup>
<mi>g</mi>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msqrt>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mo>+</mo>
<mn>1</mn>
</msqrt>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>x</mi>
<mo>∈</mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
</math></span> <strong><em>A1A1</em></strong></p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g'(x) = \frac{1}{{\sqrt {{x^2} - 1} }} \times \frac{{2x}}{{2\sqrt {{x^2} - 1} }}">
<msup>
<mi>g</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>1</mn>
</msqrt>
</mrow>
</mfrac>
<mo>×</mo>
<mfrac>
<mrow>
<mn>2</mn>
<mi>x</mi>
</mrow>
<mrow>
<mn>2</mn>
<msqrt>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>1</mn>
</msqrt>
</mrow>
</mfrac>
</math></span> <strong><em>M1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g'(x) = \frac{x}{{{x^2} - 1}}">
<msup>
<mi>g</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mi>x</mi>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mfrac>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[3 marks]</em></strong></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="g'(x) = \frac{x}{{{x^2} - 1}} = 0 \Rightarrow x = 0">
<msup>
<mi>g</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mi>x</mi>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mfrac>
<mo>=</mo>
<mn>0</mn>
<mo stretchy="false">⇒</mo>
<mi>x</mi>
<mo>=</mo>
<mn>0</mn>
</math></span> <strong><em>M1</em></strong></p>
<p>which is not in the domain of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g">
<mi>g</mi>
</math></span> (hence no solutions to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g'(x) = 0">
<msup>
<mi>g</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0</mn>
</math></span>) <strong><em>R1</em></strong></p>
<p> </p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">g.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="({g^{ - 1}})'(x) = \frac{{{{\text{e}}^{2x + 2}}}}{{\sqrt {{{\text{e}}^{2x + 2}} + 1} }}">
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mi>g</mi>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<msup>
<mo stretchy="false">)</mo>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
</mrow>
</msup>
</mrow>
</mrow>
<mrow>
<msqrt>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mo>+</mo>
<mn>1</mn>
</msqrt>
</mrow>
</mfrac>
</math></span> <strong><em>M1</em></strong></p>
<p>as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{{\text{e}}^{2x + 2}} > 0 \Rightarrow ({g^{ - 1}})'(x) > 0">
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mo>></mo>
<mn>0</mn>
<mo stretchy="false">⇒</mo>
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mi>g</mi>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<msup>
<mo stretchy="false">)</mo>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>></mo>
<mn>0</mn>
</math></span> so no solutions to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="({g^{ - 1}})'(x) = 0">
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mi>g</mi>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<msup>
<mo stretchy="false">)</mo>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0</mn>
</math></span> <strong><em>R1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Accept: equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{{\text{e}}^{2x + 2}} = 0">
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span> has no solutions.</p>
<p> </p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">g.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">g.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">g.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>At an archery tournament, a particular competition sees a ball launched into the air while an archer attempts to hit it with an arrow.</p>
<p>The path of the ball is modelled by the equation</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mi>x</mi></mtd></mtr><mtr><mtd><mi>y</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mtable><mtr><mtd><mn>5</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mi>t</mi><mfenced><mtable><mtr><mtd><msub><mi>u</mi><mi>x</mi></msub></mtd></mtr><mtr><mtd><msub><mi>u</mi><mi>y</mi></msub><mo>-</mo><mn>5</mn><mi>t</mi></mtd></mtr></mtable></mfenced></math></p>
<p>where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> is the horizontal displacement from the archer and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> is the vertical displacement from the ground, both measured in metres, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> is the time, in seconds, since the ball was launched.</p>
<ul>
<li><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mi>x</mi></msub></math> is the horizontal component of the initial velocity</li>
<li><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mi>y</mi></msub></math> is the vertical component of the initial velocity.</li>
</ul>
<p>In this question both the ball and the arrow are modelled as single points. The ball is launched with an initial velocity such that <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mi>x</mi></msub><mo>=</mo><mn>8</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mi>y</mi></msub><mo>=</mo><mn>10</mn></math>.</p>
</div>
<div class="specification">
<p>An archer releases an arrow from the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>0</mn><mo>,</mo><mo> </mo><mn>2</mn><mo>)</mo></math>. The arrow is modelled as travelling in a straight line, in the same plane as the ball, with speed <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>60</mn><mo> </mo><msup><mtext>m s</mtext><mrow><mo>-</mo><mn>1</mn></mrow></msup></math> and an angle of elevation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn><mo>°</mo></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the initial speed of the ball.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the angle of elevation of the ball as it is launched.</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 maximum height reached by the ball.</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>Assuming that the ground is horizontal and the ball is not hit by the arrow, find the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> coordinate of the point where the ball lands.</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 the path of the ball, find an expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the two positions where the path of the arrow intersects the path of the ball.</p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the time when the arrow should be released to hit the ball before the ball reaches its maximum height.</p>
<div class="marks">[4]</div>
<div class="question_part_label">f.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><msup><mn>10</mn><mn>2</mn></msup><mo>+</mo><msup><mn>8</mn><mn>2</mn></msup></msqrt></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>12</mn><mo>.</mo><mn>8</mn><mo> </mo><mo> </mo><mo> </mo><mfenced><mrow><mn>12</mn><mo>.</mo><mn>8062</mn><mo>…</mo><mo>,</mo><mo> </mo><msqrt><mn>164</mn></msqrt></mrow></mfenced><mo> </mo><mfenced><mrow><mtext>m</mtext><mo> </mo><msup><mtext>s</mtext><mrow><mo>-</mo><mn>1</mn></mrow></msup></mrow></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><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>tan</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mfrac><mn>10</mn><mn>8</mn></mfrac></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>0</mn><mo>.</mo><mn>896</mn></math> <strong>OR </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>51</mn><mo>.</mo><mn>3</mn></math> (<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>896055</mn><mo>…</mo></math> <strong>OR </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>51</mn><mo>.</mo><mn>3401</mn><mo>…</mo><mo>°</mo></math>) <em><strong>A1</strong></em></p>
<p><strong><br>Note:</strong> Accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>897</mn></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>51</mn><mo>.</mo><mn>4</mn></math> from use of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>arcsin</mtext><mfenced><mfrac><mn>10</mn><mrow><mn>12</mn><mo>.</mo><mn>8</mn></mrow></mfrac></mfenced></math>.</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><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>t</mi><mfenced><mrow><mn>10</mn><mo>-</mo><mn>5</mn><mi>t</mi></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><br><strong>Note:</strong> The <em><strong>M1</strong> </em>might be implied by a correct graph or use of the correct equation.</p>
<p> </p>
<p><strong>METHOD 1 – graphical Method</strong></p>
<p>sketch graph <em><strong>(M1)</strong></em></p>
<p><br><strong>Note:</strong> The <em><strong>M1</strong> </em>might be implied by correct graph or correct maximum (eg <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>1</mn></math>).</p>
<p><br>max occurs when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>5</mn><mo> </mo><mtext>m</mtext></math> <em><strong>A1</strong></em><br><br></p>
<p><strong>METHOD 2 – calculus</strong><br><br>differentiating and equating to zero <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><mn>10</mn><mo>-</mo><mn>10</mn><mi>t</mi><mo>=</mo><mn>0</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>1</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mfenced><mrow><mo>=</mo><mn>1</mn><mfenced><mrow><mn>10</mn><mo>-</mo><mn>5</mn></mrow></mfenced></mrow></mfenced><mo>=</mo><mn>5</mn><mo> </mo><mtext>m</mtext></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>METHOD 3 – symmetry</strong></p>
<p>line of symmetry is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>1</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mfenced><mrow><mo>=</mo><mn>1</mn><mfenced><mrow><mn>10</mn><mo>-</mo><mn>5</mn></mrow></mfenced></mrow></mfenced><mo>=</mo><mn>5</mn><mo> </mo><mtext>m</mtext></math> <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>attempt to solve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mfenced><mrow><mn>10</mn><mo>-</mo><mn>5</mn><mi>t</mi></mrow></mfenced><mo>=</mo><mn>0</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>2</mn></math> (or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn></math>) <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo> </mo><mfenced><mrow><mo>=</mo><mn>5</mn><mo>+</mo><mn>8</mn><mo>×</mo><mn>2</mn></mrow></mfenced><mo>=</mo><mo> </mo><mn>21</mn><mo> </mo><mtext>m</mtext></math> <em><strong>A1</strong></em><br><br></p>
<p><strong>Note:</strong> Do not award the final <em><strong>A1</strong> </em>if <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>5</mn></math> is also seen.</p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mfrac><mrow><mi>x</mi><mo>-</mo><mn>5</mn></mrow><mn>8</mn></mfrac></math> <em><strong>M1A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfenced><mfrac><mrow><mi>x</mi><mo>-</mo><mn>5</mn></mrow><mn>8</mn></mfrac></mfenced><mfenced><mrow><mn>10</mn><mo>-</mo><mn>5</mn><mo>×</mo><mfrac><mrow><mi>x</mi><mo>-</mo><mn>5</mn></mrow><mn>8</mn></mfrac></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><br><strong>METHOD 2</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>k</mi><mfenced><mrow><mi>x</mi><mo>-</mo><mn>5</mn></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>-</mo><mn>21</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p>when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>13</mn><mo>,</mo><mo> </mo><mi>y</mi><mo>=</mo><mn>5</mn></math> so <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mfrac><mn>5</mn><mrow><mfenced><mrow><mn>13</mn><mo>-</mo><mn>5</mn></mrow></mfenced><mfenced><mrow><mn>13</mn><mo>-</mo><mn>21</mn></mrow></mfenced></mrow></mfrac><mo>=</mo><mo>-</mo><mfrac><mn>5</mn><mn>64</mn></mfrac></math> <em><strong>M1A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>y</mi><mo>=</mo><mo>-</mo><mfrac><mn>5</mn><mn>64</mn></mfrac><mfenced><mrow><mi>x</mi><mo>-</mo><mn>5</mn></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>-</mo><mn>21</mn></mrow></mfenced></mrow></mfenced></math></p>
<p> </p>
<p><strong>METHOD 3</strong></p>
<p>if <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>a</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>b</mi><mi>x</mi><mo>+</mo><mi>c</mi></math></p>
<p> <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>=</mo><mn>25</mn><mi>a</mi><mo>+</mo><mn>5</mn><mi>b</mi><mo>+</mo><mi>c</mi></math><br> <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mo>=</mo><mn>169</mn><mi>a</mi><mo>+</mo><mn>13</mn><mi>b</mi><mo>+</mo><mi>c</mi></math><br> <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>=</mo><mn>441</mn><mi>a</mi><mo>+</mo><mn>21</mn><mi>b</mi><mo>+</mo><mi>c</mi></math> <em><strong>M1A1</strong></em></p>
<p>solving simultaneously, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mo>-</mo><mfrac><mn>5</mn><mn>64</mn></mfrac><mo>,</mo><mo> </mo><mi>b</mi><mo>=</mo><mfrac><mn>130</mn><mn>64</mn></mfrac><mo>,</mo><mo> </mo><mi>c</mi><mo>=</mo><mo>-</mo><mfrac><mn>525</mn><mn>64</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p>(<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>-</mo><mfrac><mn>5</mn><mn>64</mn></mfrac><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mfrac><mn>130</mn><mn>64</mn></mfrac><mi>x</mi><mo>-</mo><mfrac><mn>525</mn><mn>64</mn></mfrac></math>)</p>
<p> </p>
<p><strong>METHOD 4</strong><br><br>use quadratic regression on <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>5</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo><mo>,</mo><mo> </mo><mo>(</mo><mn>13</mn><mo>,</mo><mo> </mo><mn>5</mn><mo>)</mo><mo>,</mo><mo> </mo><mo>(</mo><mn>21</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math> <em><strong>M1A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>-</mo><mfrac><mn>5</mn><mn>64</mn></mfrac><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mfrac><mn>130</mn><mn>64</mn></mfrac><mi>x</mi><mo>-</mo><mfrac><mn>525</mn><mn>64</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Question asks for expression; condone omission of "<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo></math>".</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>trajectory of arrow is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>x</mi><mo> </mo><mi>tan</mi><mo> </mo><mn>10</mn><mo>+</mo><mn>2</mn></math> <em><strong>(A1)</strong></em></p>
<p>intersecting <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>x</mi><mo> </mo><mi>tan</mi><mo> </mo><mn>10</mn><mo>+</mo><mn>2</mn></math> and their answer to (d) <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>8</mn><mo>.</mo><mn>66</mn><mo>,</mo><mo> </mo><mn>3</mn><mo>.</mo><mn>53</mn></mrow></mfenced><mo> </mo><mo> </mo><mfenced><mfenced><mrow><mn>8</mn><mo>.</mo><mn>65705</mn><mo>…</mo><mo>,</mo><mo> </mo><mn>3</mn><mo>.</mo><mn>52647</mn><mo>…</mo></mrow></mfenced></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>15</mn><mo>.</mo><mn>1</mn><mo>,</mo><mo> </mo><mn>4</mn><mo>.</mo><mn>66</mn></mrow></mfenced><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mfenced><mfenced><mrow><mn>15</mn><mo>.</mo><mn>0859</mn><mo>…</mo><mo>,</mo><mo> </mo><mn>4</mn><mo>.</mo><mn>66006</mn><mo>…</mo></mrow></mfenced></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>when <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>x</mi><mtext>target</mtext></msub><mo>=</mo><mn>8</mn><mo>.</mo><mn>65705</mn><mo>…</mo><mo>,</mo><mo> </mo><mo> </mo><msub><mi>t</mi><mtext>target</mtext></msub><mo>=</mo><mfrac><mrow><mn>8</mn><mo>.</mo><mn>65705</mn><mo>…</mo><mo>-</mo><mn>5</mn></mrow><mn>8</mn></mfrac><mo>=</mo><mn>0</mn><mo>.</mo><mn>457132</mn><mo>…</mo><mo> </mo><mtext>s</mtext></math> <em><strong>(A1)</strong></em></p>
<p>attempt to find the distance from point of release to intersection <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mn>8</mn><mo>.</mo><mn>65705</mn><msup><mo>…</mo><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mn>3</mn><mo>.</mo><mn>52647</mn><mo>…</mo><mo>-</mo><mn>2</mn></mrow></mfenced><mn>2</mn></msup></msqrt><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mn>8</mn><mo>.</mo><mn>79060</mn><mo>…</mo><mo> </mo><mtext>m</mtext></mrow></mfenced></math></p>
<p>time for arrow to get there is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>8</mn><mo>.</mo><mn>79060</mn><mo>…</mo></mrow><mn>60</mn></mfrac><mo>=</mo><mn>0</mn><mo>.</mo><mn>146510</mn><mo>…</mo><mtext>s</mtext></math> <em><strong>(A1)</strong></em></p>
<p>so the arrow should be released when</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>311</mn><mo> </mo><mfenced><mtext>s</mtext></mfenced><mo> </mo><mo> </mo><mfenced><mrow><mn>0</mn><mo>.</mo><mn>310622</mn><mo>…</mo><mo> </mo><mfenced><mtext>s</mtext></mfenced></mrow></mfenced></math> <em><strong>A1</strong></em> </p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">f.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>This question was found to be the most difficult on the paper. There were a good number of good solutions to parts (a) and part (b), frequently with answers just written down with no working. Part (c) caused some difficulties with confusing variables. The most significant difficulties started with part (d) and became greater to the end of the question. Few candidates were able to work through the final two parts.</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>
<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>Sketch the graphs <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = {\text{si}}{{\text{n}}^3}\,x + {\text{ln}}\,x">
<mi>y</mi>
<mo>=</mo>
<mrow>
<mtext>si</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mn>3</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>+</mo>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 1 + {\text{cos}}\,x">
<mi>y</mi>
<mo>=</mo>
<mn>1</mn>
<mo>+</mo>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</math></span> on the following axes for 0 < <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> ≤ 9.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence solve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{si}}{{\text{n}}^3}\,x + {\text{ln}}\,x - {\text{cos}}\,x - 1 < 0">
<mrow>
<mtext>si</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mn>3</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>+</mo>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>−</mo>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>−</mo>
<mn>1</mn>
<mo><</mo>
<mn>0</mn>
</math></span> in the range 0 < <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> ≤ 9.</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><img 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"> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for each correct curve, showing all local max & mins.</p>
<p><strong>Note:</strong> Award<em><strong> A0A0</strong></em> for the curves drawn in degrees.</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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> = 1.35, 4.35, 6.64 <em><strong>(M1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1</strong></em> for attempt to find points of intersections between two curves.</p>
<p>0 < <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> < 1.35 <em><strong> A1</strong></em></p>
<p><strong>Note:</strong> Accept <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> < 1.35.</p>
<p>4.35 < <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> < 6.64 <em><strong>A1A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for correct endpoints, <em><strong>A1</strong></em> for correct inequalities.</p>
<p><strong>Note:</strong> Award <em><strong>M1FTA1FTA0FTA0FT</strong></em> for 0 < <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> < 7.31.</p>
<p><strong>Note:</strong> Accept <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> < 7.31.</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>Consider the function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = \frac{{\sqrt x }}{{\sin x}},{\text{ }}0 < x < \pi ">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mrow>
<msqrt>
<mi>x</mi>
</msqrt>
</mrow>
<mrow>
<mi>sin</mi>
<mo><!-- --></mo>
<mi>x</mi>
</mrow>
</mfrac>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>0</mn>
<mo><</mo>
<mi>x</mi>
<mo><</mo>
<mi>π<!-- π --></mi>
</math></span>.</p>
</div>
<div class="specification">
<p>Consider the region bounded by the curve <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>, the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>-axis and the lines <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \frac{\pi }{6},{\text{ }}x = \frac{\pi }{3}">
<mi>x</mi>
<mo>=</mo>
<mfrac>
<mi>π<!-- π --></mi>
<mn>6</mn>
</mfrac>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>x</mi>
<mo>=</mo>
<mfrac>
<mi>π<!-- π --></mi>
<mn>3</mn>
</mfrac>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>-coordinate of the minimum point on the curve <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> satisfies the equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\tan x = 2x">
<mi>tan</mi>
<mo></mo>
<mi>x</mi>
<mo>=</mo>
<mn>2</mn>
<mi>x</mi>
</math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> for which <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x)">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span> is a decreasing function.</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>Sketch the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f(x)">
<mi>y</mi>
<mo>=</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span> showing clearly the minimum point and any asymptotic behaviour.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the coordinates of the point on the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> where the normal to the graph is parallel to the line <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = - x">
<mi>y</mi>
<mo>=</mo>
<mo>−</mo>
<mi>x</mi>
</math></span>.</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>This region is now rotated through <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2\pi ">
<mn>2</mn>
<mi>π</mi>
</math></span> radians about the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>-axis. Find the volume of revolution.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>attempt to use quotient rule or product rule <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f’(x) = \frac{{\sin x\left( {\frac{1}{2}{x^{ - \frac{1}{2}}}} \right) - \sqrt x \cos x}}{{{{\sin }^2}x}}{\text{ }}\left( { = \frac{1}{{2\sqrt x \sin x}} - \frac{{\sqrt x \cos x}}{{{{\sin }^2}x}}} \right)">
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mrow>
<mi>sin</mi>
<mo></mo>
<mi>x</mi>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mrow>
<msup>
<mi>x</mi>
<mrow>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>−</mo>
<msqrt>
<mi>x</mi>
</msqrt>
<mi>cos</mi>
<mo></mo>
<mi>x</mi>
</mrow>
<mrow>
<mrow>
<msup>
<mrow>
<mi>sin</mi>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mn>2</mn>
<msqrt>
<mi>x</mi>
</msqrt>
<mi>sin</mi>
<mo></mo>
<mi>x</mi>
</mrow>
</mfrac>
<mo>−</mo>
<mfrac>
<mrow>
<msqrt>
<mi>x</mi>
</msqrt>
<mi>cos</mi>
<mo></mo>
<mi>x</mi>
</mrow>
<mrow>
<mrow>
<msup>
<mrow>
<mi>sin</mi>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>A1A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>A1 </em></strong>for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{{2\sqrt x \sin x}}">
<mfrac>
<mn>1</mn>
<mrow>
<mn>2</mn>
<msqrt>
<mi>x</mi>
</msqrt>
<mi>sin</mi>
<mo></mo>
<mi>x</mi>
</mrow>
</mfrac>
</math></span> or equivalent and <strong><em>A1 </em></strong>for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - \frac{{\sqrt x \cos x}}{{{{\sin }^2}x}}">
<mo>−</mo>
<mfrac>
<mrow>
<msqrt>
<mi>x</mi>
</msqrt>
<mi>cos</mi>
<mo></mo>
<mi>x</mi>
</mrow>
<mrow>
<mrow>
<msup>
<mrow>
<mi>sin</mi>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
</math></span> or equivalent.</p>
<p> </p>
<p>setting <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f’(x) = 0">
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<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="\frac{{\sin x}}{{2\sqrt x }} - \sqrt x \cos x = 0">
<mfrac>
<mrow>
<mi>sin</mi>
<mo></mo>
<mi>x</mi>
</mrow>
<mrow>
<mn>2</mn>
<msqrt>
<mi>x</mi>
</msqrt>
</mrow>
</mfrac>
<mo>−</mo>
<msqrt>
<mi>x</mi>
</msqrt>
<mi>cos</mi>
<mo></mo>
<mi>x</mi>
<mo>=</mo>
<mn>0</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{\sin x}}{{2\sqrt x }} = \sqrt x \cos x">
<mfrac>
<mrow>
<mi>sin</mi>
<mo></mo>
<mi>x</mi>
</mrow>
<mrow>
<mn>2</mn>
<msqrt>
<mi>x</mi>
</msqrt>
</mrow>
</mfrac>
<mo>=</mo>
<msqrt>
<mi>x</mi>
</msqrt>
<mi>cos</mi>
<mo></mo>
<mi>x</mi>
</math></span> or equivalent <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\tan x = 2x">
<mi>tan</mi>
<mo></mo>
<mi>x</mi>
<mo>=</mo>
<mn>2</mn>
<mi>x</mi>
</math></span> <strong><em>AG</em></strong></p>
<p><strong><em>[5 marks]</em></strong></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 1.17">
<mi>x</mi>
<mo>=</mo>
<mn>1.17</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0 < x \leqslant 1.17">
<mn>0</mn>
<mo><</mo>
<mi>x</mi>
<mo>⩽</mo>
<mn>1.17</mn>
</math></span> <strong><em>A1A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>A1 </em></strong>for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0 < x">
<mn>0</mn>
<mo><</mo>
<mi>x</mi>
</math></span> and <strong><em>A1 </em></strong>for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x \leqslant 1.17">
<mi>x</mi>
<mo>⩽</mo>
<mn>1.17</mn>
</math></span>. Accept <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x < 1.17">
<mi>x</mi>
<mo><</mo>
<mn>1.17</mn>
</math></span>.</p>
<p> </p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img src="images/Schermafbeelding_2018-02-08_om_16.19.25.png" alt="N17/5/MATHL/HP2/ENG/TZ0/10.b/M"></p>
<p>concave up curve over correct domain with one minimum point above the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>-axis. <strong><em>A1</em></strong></p>
<p>approaches <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> asymptotically <strong><em>A1</em></strong></p>
<p>approaches <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \pi ">
<mi>x</mi>
<mo>=</mo>
<mi>π</mi>
</math></span> asymptotically <strong><em>A1</em></strong></p>
<p> </p>
<p>Note: For the final <strong><em>A1 </em></strong>an asymptote must be seen, and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\pi ">
<mi>π</mi>
</math></span> must be seen on the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>-axis or in an equation.</p>
<p> </p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f’(x){\text{ }}\left( { = \frac{{\sin x\left( {\frac{1}{2}{x^{ - \frac{1}{2}}}} \right) - \sqrt x \cos x}}{{{{\sin }^2}x}}} \right) = 1">
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mi>sin</mi>
<mo></mo>
<mi>x</mi>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mrow>
<msup>
<mi>x</mi>
<mrow>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>−</mo>
<msqrt>
<mi>x</mi>
</msqrt>
<mi>cos</mi>
<mo></mo>
<mi>x</mi>
</mrow>
<mrow>
<mrow>
<msup>
<mrow>
<mi>sin</mi>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>1</mn>
</math></span> <strong><em>(A1)</em></strong></p>
<p>attempt 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 = 1.96">
<mi>x</mi>
<mo>=</mo>
<mn>1.96</mn>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f(1.96 \ldots )">
<mi>y</mi>
<mo>=</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mn>1.96</mn>
<mo>…</mo>
<mo stretchy="false">)</mo>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 1.51">
<mo>=</mo>
<mn>1.51</mn>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[4 marks]</em></strong></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="V = \pi \int_{\frac{\pi }{6}}^{\frac{\pi }{3}} {\frac{{x{\text{d}}x}}{{{{\sin }^2}x}}} ">
<mi>V</mi>
<mo>=</mo>
<mi>π</mi>
<msubsup>
<mo>∫</mo>
<mrow>
<mfrac>
<mi>π</mi>
<mn>6</mn>
</mfrac>
</mrow>
<mrow>
<mfrac>
<mi>π</mi>
<mn>3</mn>
</mfrac>
</mrow>
</msubsup>
<mrow>
<mfrac>
<mrow>
<mi>x</mi>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
<mrow>
<mrow>
<msup>
<mrow>
<mi>sin</mi>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
</mrow>
</math></span> <strong><em>(M1)(A1)</em></strong></p>
<p> </p>
<p><strong>Note:</strong> <strong><em>M1 </em></strong>is for an integral of the correct squared function (with or without limits and/or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\pi ">
<mi>π</mi>
</math></span>).</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 2.68{\text{ }}( = 0.852\pi )">
<mo>=</mo>
<mn>2.68</mn>
<mrow>
<mtext> </mtext>
</mrow>
<mo stretchy="false">(</mo>
<mo>=</mo>
<mn>0.852</mn>
<mi>π</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">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.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>Beth goes for a run. She uses a fitness app to record her distance, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="s">
<mi>s</mi>
</math></span> km, and time, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> minutes. A graph of her distance against time is shown.</p>
<p><img src="data:image/png;base64,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"></p>
<p>Beth runs at a constant speed of 2.3 ms<sup>–1</sup> for the first 8 minutes.</p>
</div>
<div class="specification">
<p>Between 8 and 20 minutes, her distance can be modeled by a cubic function, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="s = a{t^3} + b{t^2} + ct + d">
<mi>s</mi>
<mo>=</mo>
<mi>a</mi>
<mrow>
<msup>
<mi>t</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>+</mo>
<mi>b</mi>
<mrow>
<msup>
<mi>t</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mi>c</mi>
<mi>t</mi>
<mo>+</mo>
<mi>d</mi>
</math></span>. She reads the following data from her app.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<div class="specification">
<p>Hence find</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate her distance after 8 minutes. Give your answer in km, correct to 3 decimal places.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="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>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>the distance she runs in 20 minutes.</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>her maximum speed, in ms<sup>–1</sup>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.ii.</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="\frac{{2.3 \times 8 \times 60}}{{1000}} = 1.104">
<mfrac>
<mrow>
<mn>2.3</mn>
<mo>×</mo>
<mn>8</mn>
<mo>×</mo>
<mn>60</mn>
</mrow>
<mrow>
<mn>1000</mn>
</mrow>
</mfrac>
<mo>=</mo>
<mn>1.104</mn>
</math></span> <em><strong>M1A1</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>either using a cubic regression or solving a system of 4 equations <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a = - 0.00364{\text{,}}\,\,b = 0.150{\text{,}}\,\,c = - 1.67{\text{,}}\,\,d = 6.72">
<mi>a</mi>
<mo>=</mo>
<mo>−</mo>
<mn>0.00364</mn>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>b</mi>
<mo>=</mo>
<mn>0.150</mn>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>c</mi>
<mo>=</mo>
<mo>−</mo>
<mn>1.67</mn>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mo>=</mo>
<mn>6.72</mn>
</math></span> <em><strong>A1A1A1A1</strong></em></p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="s\left( {20} \right) = 4.21">
<mi>s</mi>
<mrow>
<mo>(</mo>
<mrow>
<mn>20</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>4.21</mn>
</math></span> km (Note: Condone <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="s\left( {20} \right) = 4.2">
<mi>s</mi>
<mrow>
<mo>(</mo>
<mrow>
<mn>20</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>4.2</mn>
</math></span> km obtained from using rounded values.) <em><strong>M1A1</strong></em></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>EITHER finding maximum of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{ds}}{{dt}}">
<mfrac>
<mrow>
<mi>d</mi>
<mi>s</mi>
</mrow>
<mrow>
<mi>d</mi>
<mi>t</mi>
</mrow>
</mfrac>
</math></span> OR solving <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{d^2}s}}{{d{t^2}}} = 0">
<mfrac>
<mrow>
<mrow>
<msup>
<mi>d</mi>
<mn>2</mn>
</msup>
</mrow>
<mi>s</mi>
</mrow>
<mrow>
<mi>d</mi>
<mrow>
<msup>
<mi>t</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
<mo>=</mo>
<mn>0</mn>
</math></span> <em><strong>M1</strong></em></p>
<p>maximum speed = 0.390… km per minute <em><strong>A1</strong></em></p>
<p>maximum speed = 6.51 ms<sup>–1</sup> <em><strong>M1A1</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">c.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.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.ii.</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>Jorge is carefully observing the rise in sales of a new app he has created.</p>
<p>The number of sales in the first four months is shown in the table below.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<p style="text-align: left;">Jorge believes that the increase is exponential and proposes to model the number of sales <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math> in month <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> with the equation</p>
<p style="text-align: left; padding-left: 30px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mo>=</mo><mi>A</mi><msup><mtext>e</mtext><mrow><mi>r</mi><mi>t</mi></mrow></msup><mo>,</mo><mo> </mo><mi>A</mi><mo>,</mo><mo> </mo><mi>r</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math></p>
</div>
<div class="specification">
<p>Jorge plans to adapt Euler’s method to find an approximate value for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math>.</p>
<p>With a step length of one month the solution to the differential equation can be approximated using Euler’s method where</p>
<p style="padding-left: 30px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>≈</mo><mi>N</mi><mfenced><mi>n</mi></mfenced><mo>+</mo><mn>1</mn><mo>×</mo><mi>N</mi><mo>'</mo><mfenced><mi>n</mi></mfenced><mo>,</mo><mo> </mo><mi>n</mi><mo>∈</mo><mi mathvariant="normal">ℕ</mi></math></p>
</div>
<div class="specification">
<p>Jorge decides to take the mean of these values as the approximation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math> for his model. He also decides the graph of the model should pass through the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>2</mn><mo>,</mo><mo> </mo><mn>52</mn><mo>)</mo></math>.</p>
</div>
<div class="specification">
<p>The sum of the square residuals for these points for the least squares regression model is approximately <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mo>.</mo><mn>555</mn></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that Jorge’s model satisfies the differential equation</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>N</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mi>r</mi><mi>N</mi></math></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>≈</mo><mfrac><mrow><mi>N</mi><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>-</mo><mi>N</mi><mfenced><mi>n</mi></mfenced></mrow><mrow><mi>N</mi><mfenced><mi>n</mi></mfenced></mrow></mfrac></math></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find three approximations for the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</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>Find the equation for Jorge’s model.</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 sum of the square residuals for Jorge’s model using the values <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo> </mo><mn>2</mn><mo>,</mo><mo> </mo><mn>3</mn><mo>,</mo><mo> </mo><mn>4</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>Comment how well Jorge’s model fits the data.</p>
<div class="marks">[1]</div>
<div class="question_part_label">f.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Give two possible sources of error in the construction of his model.</p>
<div class="marks">[2]</div>
<div class="question_part_label">f.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 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><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>N</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mi>r</mi><mi>A</mi><msup><mtext>e</mtext><mrow><mi>r</mi><mi>t</mi></mrow></msup></math> <strong>(M1)A1</strong></p>
<p> </p>
<p><strong>Note: M1</strong> is for an attempt to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>N</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></math></p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi>r</mi><mi>N</mi></math> <strong>AG</strong></p>
<p> </p>
<p><strong>Note:</strong> Accept solution of the differential equation by separating variables</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><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>≈</mo><mi>N</mi><mfenced><mi>n</mi></mfenced><mo>+</mo><mn>1</mn><mo>×</mo><mi>N</mi><mo>'</mo><mfenced><mi>n</mi></mfenced><mo>⇒</mo><mi>N</mi><mo>'</mo><mfenced><mi>n</mi></mfenced><mo>≈</mo><mi>N</mi><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>-</mo><mi>N</mi><mfenced><mi>n</mi></mfenced></math> <strong>M1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>r</mi><mi>N</mi><mfenced><mi>n</mi></mfenced><mo>≈</mo><mi>N</mi><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>-</mo><mi>N</mi><mfenced><mi>n</mi></mfenced></math> <strong>M1A1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>r</mi><mo>≈</mo><mfrac><mrow><mi>N</mi><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>-</mo><mi>N</mi><mfenced><mi>n</mi></mfenced></mrow><mrow><mi>N</mi><mfenced><mi>n</mi></mfenced></mrow></mfrac></math> <strong>AG</strong></p>
<p> </p>
<p><strong>Note:</strong> Do not penalize the use of the <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo></math> sign.</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>Correct method <strong>(M1)</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>≈</mo><mfrac><mrow><mn>52</mn><mo>-</mo><mn>40</mn></mrow><mn>40</mn></mfrac><mo>=</mo><mn>0</mn><mo>.</mo><mn>3</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>≈</mo><mfrac><mrow><mn>70</mn><mo>-</mo><mn>52</mn></mrow><mn>52</mn></mfrac><mo>=</mo><mn>0</mn><mo>.</mo><mn>346</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>≈</mo><mfrac><mrow><mn>98</mn><mo>-</mo><mn>70</mn></mrow><mn>70</mn></mfrac><mo>=</mo><mn>0</mn><mo>.</mo><mn>4</mn></math> <strong>A2</strong></p>
<p> </p>
<p><strong>Note: A1</strong> for a single error <strong>A0</strong> for two or more errors.</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>r</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>349</mn><mo> </mo><mfenced><mrow><mn>0</mn><mo>.</mo><mn>34871</mn><mo>…</mo></mrow></mfenced></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>68</mn><mn>195</mn></mfrac></math> <strong>A1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>52</mn><mo>=</mo><mi>A</mi><msup><mtext>e</mtext><mrow><mn>0</mn><mo>.</mo><mn>34871</mn><mo>…</mo><mo>×</mo><mn>2</mn></mrow></msup></math> <strong>(M1)</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mn>25</mn><mo>.</mo><mn>8887</mn><mo>…</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mo>=</mo><mn>25</mn><mo>.</mo><mn>9</mn><msup><mtext>e</mtext><mrow><mn>0</mn><mo>.</mo><mn>349</mn><mi>t</mi></mrow></msup></math> <strong>A1</strong></p>
<p> </p>
<p><strong>[3 marks]</strong></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mn>36</mn><mo>.</mo><mn>6904</mn><mo>…</mo><mo>-</mo><mn>40</mn></mrow></mfenced><mn>2</mn></msup><mo>+</mo><mn>0</mn><mo>+</mo><msup><mfenced><mrow><mn>73</mn><mo>.</mo><mn>6951</mn><mo>…</mo><mo>-</mo><mn>70</mn></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mn>104</mn><mo>.</mo><mn>4435</mn><mo>…</mo><mo>-</mo><mn>98</mn></mrow></mfenced><mn>2</mn></msup></math> <strong>(M1)</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>66</mn><mo>.</mo><mn>1</mn><mo> </mo><mfenced><mrow><mn>66</mn><mo>.</mo><mn>126</mn><mo>…</mo></mrow></mfenced></math> <strong>A1</strong></p>
<p> </p>
<p><strong>[2 marks]</strong></p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>The sum of the square residuals is approximately <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn></math> times as large as the minimum possible, so Jorge’s model is unlikely to fit the data exactly <strong>R1</strong></p>
<p> </p>
<p><strong>[1 mark]</strong></p>
<div class="question_part_label">f.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>For example</p>
<p>Selecting a single point for the curve to pass through</p>
<p>Approximating the gradient of the curve by the gradient of a chord <strong>R1R1</strong></p>
<p> </p>
<p><strong>[2 marks]</strong></p>
<div class="question_part_label">f.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.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">f.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">f.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the curve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><msqrt><mi>x</mi></msqrt></math>.</p>
</div>
<div class="specification">
<p>The shape of a piece of metal can be modelled by the region bounded by the functions <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math>, the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis and the line segment <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>[AB]</mtext></math>, as shown in the following diagram. The units on the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> axes are measured in metres.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<p>The piecewise function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> is defined by</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mo>{</mo><mtable><mtr><mtd><msqrt><mi>x</mi></msqrt><mo> </mo><mo> </mo></mtd><mtd><mn>0</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mn>0</mn><mo>.</mo><mn>16</mn></mtd></mtr><mtr><mtd><mn>1</mn><mo>.</mo><mn>25</mn><mi>x</mi><mo>+</mo><mn>0</mn><mo>.</mo><mn>2</mn><mo> </mo><mo> </mo></mtd><mtd><mn>0</mn><mo>.</mo><mn>16</mn><mo><</mo><mi>x</mi><mo>≤</mo><mn>0</mn><mo>.</mo><mn>5</mn></mtd></mtr></mtable></math></p>
<p>The graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math> is obtained from the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> by:</p>
<ul>
<li>a stretch scale factor of <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac></math> in the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> direction,</li>
<li>followed by a stretch scale factor <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac></math> in the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> direction,</li>
<li>followed by a translation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>2</mn></math> units to the right.</li>
</ul>
<p>Point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> lies on the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> and has coordinates <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>0</mn><mo>.</mo><mn>5</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>.</mo><mn>825</mn><mo>)</mo></math>. Point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math> is the image of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> under the given transformations and has coordinates <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>p</mi><mo>,</mo><mo> </mo><mi>q</mi><mo>)</mo></math>.</p>
</div>
<div class="specification">
<p>The piecewise function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math> is given by</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mo>{</mo><mtable><mtr><mtd><mi>h</mi><mfenced><mi>x</mi></mfenced><mo> </mo><mo> </mo></mtd><mtd><mn>0</mn><mo>.</mo><mn>2</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mi>a</mi></mtd></mtr><mtr><mtd><mn>1</mn><mo>.</mo><mn>25</mn><mi>x</mi><mo>+</mo><mi>b</mi><mo> </mo><mo> </mo></mtd><mtd><mi>a</mi><mo><</mo><mi>x</mi><mo>≤</mo><mi>p</mi></mtd></mtr></mtable></math></p>
</div>
<div class="specification">
<p>The area enclosed by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo></math>, the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis and the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mi>p</mi></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>0627292</mn><mo> </mo><msup><mtext>m</mtext><mn>2</mn></msup></math> correct to six significant figures.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac></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 show that the equation of the tangent to the curve at the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>0</mn><mo>.</mo><mn>16</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>.</mo><mn>4</mn></mrow></mfenced></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>1</mn><mo>.</mo><mn>25</mn><mi>x</mi><mo>+</mo><mn>0</mn><mo>.</mo><mn>2</mn></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math> and the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</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"><mo> </mo><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo></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>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></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 value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the area enclosed by <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>, the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis and the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>5</mn></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the area of the shaded region on the diagram.</p>
<div class="marks">[4]</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><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><msup><mi>x</mi><mfrac><mn>1</mn><mn>2</mn></mfrac></msup></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>x</mi><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup></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>gradient at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>16</mn></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>×</mo><mfrac><mn>1</mn><msqrt><mn>0</mn><mo>.</mo><mn>16</mn></msqrt></mfrac></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>1</mn><mo>.</mo><mn>25</mn></math></p>
<p><br><strong>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>-</mo><mn>0</mn><mo>.</mo><mn>4</mn><mo>=</mo><mn>1</mn><mo>.</mo><mn>25</mn><mfenced><mrow><mi>x</mi><mo>-</mo><mn>0</mn><mo>.</mo><mn>16</mn></mrow></mfenced></math> <em><strong>M1</strong></em></p>
<p><br><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>4</mn><mo>=</mo><mn>1</mn><mo>.</mo><mn>25</mn><mfenced><mrow><mn>0</mn><mo>.</mo><mn>16</mn></mrow></mfenced><mo>+</mo><mi>b</mi></math> <em><strong>M1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Do not allow working backwards from the given answer.</p>
<p> </p>
<p><strong>THEN</strong></p>
<p>hence <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>1</mn><mo>.</mo><mn>25</mn><mi>x</mi><mo>+</mo><mn>0</mn><mo>.</mo><mn>2</mn></math> <em><strong>AG</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><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>45</mn><mo>,</mo><mo> </mo><mo> </mo><mi>q</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>4125</mn></math> (or <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>413</mn></math>) (accept " <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>0</mn><mo>.</mo><mn>45</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>.</mo><mn>4125</mn><mo>)</mo></math> ") <em><strong>A1A1</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"><mfenced><mrow><mi>h</mi><mfenced><mi>x</mi></mfenced><mo>=</mo></mrow></mfenced><mo> </mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msqrt><mn>2</mn><mfenced><mrow><mi>x</mi><mo>-</mo><mn>0</mn><mo>.</mo><mn>2</mn></mrow></mfenced></msqrt></math> <em><strong>A2</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>A1</strong> </em>if only two correct transformations are seen. </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"><mfenced><mrow><mi>a</mi><mo>=</mo></mrow></mfenced><mo> </mo><mn>0</mn><mo>.</mo><mn>28</mn></math> <em><strong>A1</strong></em></p>
<p><br><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p>Correct substitution of their part (b) (or <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>0</mn><mo>.</mo><mn>28</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>.</mo><mn>2</mn></mrow></mfenced></math>) into the given expression <strong><em>(M1)</em></strong></p>
<p><br><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac><mfenced><mrow><mn>1</mn><mo>.</mo><mn>25</mn><mo>×</mo><mn>2</mn><mfenced><mrow><mi>x</mi><mo>-</mo><mn>0</mn><mo>.</mo><mn>2</mn></mrow></mfenced><mo>+</mo><mn>0</mn><mo>.</mo><mn>2</mn></mrow></mfenced></math> <strong><em>(M1)</em></strong></p>
<p><br><strong>Note:</strong> Award <em><strong>M1</strong> </em>for transforming the equivalent expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> correctly.</p>
<p><br><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>b</mi><mo>=</mo></mrow></mfenced><mo> </mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>15</mn></math> <em><strong>A1</strong></em></p>
<p><br><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>recognizing need to add two integrals <strong><em>(M1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mn>0</mn><mrow><mn>0</mn><mo>.</mo><mn>16</mn></mrow></msubsup><msqrt><mi>x</mi></msqrt><mo>d</mo><mi>x</mi><mo>+</mo><msubsup><mo>∫</mo><mrow><mn>0</mn><mo>.</mo><mn>16</mn></mrow><mrow><mn>0</mn><mo>.</mo><mn>5</mn></mrow></msubsup><mfenced><mrow><mn>1</mn><mo>.</mo><mn>25</mn><mi>x</mi><mo>+</mo><mn>0</mn><mo>.</mo><mn>2</mn></mrow></mfenced><mo>d</mo><mi>x</mi></math> <strong><em>(A1)</em></strong></p>
<p><br><strong>Note:</strong> The second integral could be replaced by the formula for the area of a trapezoid <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>×</mo><mn>0</mn><mo>.</mo><mn>34</mn><mfenced><mrow><mn>0</mn><mo>.</mo><mn>4</mn><mo>+</mo><mn>0</mn><mo>.</mo><mn>825</mn></mrow></mfenced></math>.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>251</mn><mo> </mo><msup><mtext>m</mtext><mn>2</mn></msup><mo> </mo><mo> </mo><mfenced><mrow><mn>0</mn><mo>.</mo><mn>250916</mn><mo>…</mo></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><br><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p>area of a trapezoid <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>×</mo><mn>0</mn><mo>.</mo><mn>05</mn><mfenced><mrow><mn>0</mn><mo>.</mo><mn>4125</mn><mo>+</mo><mn>0</mn><mo>.</mo><mn>825</mn></mrow></mfenced><mo>=</mo><mn>0</mn><mo>.</mo><mn>0309375</mn></math> <strong><em>(M1)(A1)</em></strong></p>
<p><br><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mrow><mn>0</mn><mo>.</mo><mn>45</mn></mrow><mrow><mn>0</mn><mo>.</mo><mn>5</mn></mrow></msubsup><mfenced><mrow><mn>8</mn><mo>.</mo><mn>25</mn><mi>x</mi><mo>-</mo><mn>3</mn><mo>.</mo><mn>3</mn></mrow></mfenced><mo>d</mo><mi>x</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>0309375</mn></math> <strong><em>(M1)(A1)</em></strong></p>
<p><strong><br>Note:</strong> If the rounded answer of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>413</mn></math> from part (b) is used, the integral is <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mrow><mn>0</mn><mo>.</mo><mn>45</mn></mrow><mrow><mn>0</mn><mo>.</mo><mn>5</mn></mrow></msubsup><mfenced><mrow><mn>8</mn><mo>.</mo><mn>24</mn><mi>x</mi><mo>-</mo><mn>3</mn><mo>.</mo><mn>295</mn></mrow></mfenced><mo>d</mo><mi>x</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>03095</mn></math> which would be awarded <strong><em>(M1)(A1)</em></strong>.</p>
<p> </p>
<p><strong>THEN</strong></p>
<p>shaded area <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>0</mn><mo>.</mo><mn>250916</mn><mo>…</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>0627292</mn><mo>-</mo><mn>0</mn><mo>.</mo><mn>0309375</mn></math> <strong><em>(M1)</em></strong></p>
<p><br><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for the subtraction of both <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>0627292</mn><mo>…</mo></math> and their area for the trapezoid from their answer to (a)(i).</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>0</mn><mo>.</mo><mn>157</mn><mo> </mo><msup><mtext>m</mtext><mn>2</mn></msup><mo> </mo><mo> </mo><mfenced><mrow><mn>0</mn><mo>.</mo><mn>15725</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[4 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;">
<p>The differentiation using the power rule was well done. In part (ii) some candidates felt it was sufficient to refer to the equation being the same as the one generated by their calculator. Generally, for ‘show that’ questions an algebraic derivation is expected.</p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>The candidates were successful at applying transformations to points but very few were able to apply these transformations to derive the correct function <em>h</em>. In most cases it was due to not appreciating the effect the horizontal transformations have on <em>x</em>.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>The candidates were successful at applying transformations to points but very few were able to apply these transformations to derive the correct function <em>h</em>. In most cases it was due to not appreciating the effect the horizontal transformations have on <em>x</em>.</p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Part (i) was frequently done well using the inbuilt functionality of the GDC. Part (ii) was less structured, and candidates needed to create a clear diagram so they could easily see which areas needed to be subtracted. Most of those who were successful used the formula for the trapezoid for the area they needed to find, though others were also successful through finding the equation of the line AB.</p>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.ii.</div>
</div>
<br><hr><br>