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</div><h2>SL Paper 2</h2><div class="specification">
<p>A particle P moves along a straight line. Its velocity \({v_{\text{P}}}{\text{ m}}\,{{\text{s}}^{ - 1}}\) after \(t\) seconds is given by \({v_{\text{P}}} = \sqrt t \sin \left( {\frac{\pi }{2}t} \right)\), for \(0 \leqslant t \leqslant 8\). The following diagram shows the graph of \({v_{\text{P}}}\).</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-14_om_10.04.21.png" alt="M17/5/MATME/SP2/ENG/TZ1/07"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the first value of \(t\) at which P changes direction.</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>Find the <strong>total </strong>distance travelled by P, for \(0 \leqslant t \leqslant 8\).</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>A second particle Q also moves along a straight line. Its velocity, \({v_{\text{Q}}}{\text{ m}}\,{{\text{s}}^{ - 1}}\) after \(t\) seconds is given by \({v_{\text{Q}}} = \sqrt t \) for \(0 \leqslant t \leqslant 8\). After \(k\) seconds Q has travelled the same total distance as P.</p>
<p>Find \(k\).</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>\(t = 2\) <strong><em>A1</em></strong> <strong><em>N1</em></strong></p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>substitution of limits or function into formula or correct sum <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\int_0^8 {\left| v \right|{\text{d}}t,{\text{ }}\int {\left| {{v_Q}} \right|{\text{d}}t,{\text{ }}\int_0^2 {v{\text{d}}t - \int_2^4 {v{\text{d}}t + \int_4^6 {v{\text{d}}t - \int_6^8 {v{\text{d}}t} } } } } } \)</p>
<p>9.64782</p>
<p>distance \( = 9.65{\text{ (metres)}}\) <strong><em>A1</em></strong> <strong><em>N2</em></strong></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>correct approach <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(s = \int {\sqrt t ,{\text{ }}\int_0^k {\sqrt t } } {\text{d}}t,{\text{ }}\int_0^k {\left| {{v_{\text{Q}}}} \right|{\text{d}}t} \)</p>
<p>correct integration <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\int {\sqrt t = \frac{2}{3}{t^{\frac{3}{2}}} + c,{\text{ }}\left[ {\frac{2}{3}{x^{\frac{3}{2}}}} \right]_0^k,{\text{ }}\frac{2}{3}{k^{\frac{3}{2}}}} \)</p>
<p>equating their expression to the distance travelled by their P <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\frac{2}{3}{k^{\frac{3}{2}}} = 9.65,{\text{ }}\int_0^k {\sqrt t {\text{d}}t = 9.65} \)</p>
<p>5.93855</p>
<p>5.94 (seconds) <strong><em>A1</em></strong> <strong><em>N3</em></strong></p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = \frac{{20x}}{{{{\rm{e}}^{0.3x}}}}\) , for \(0 \le x \le 20\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Sketch the graph of <em>f</em> .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Write down the <em>x</em>-coordinate of the maximum point on the graph of <em>f</em> .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Write down the interval where <em>f</em> is increasing.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Show that \(f'(x) = \frac{{20 - 6x}}{{{{\rm{e}}^{0.3x}}}}\) .</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the interval where the rate of change of <em>f</em> is increasing.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/phoebe.png" alt></span><em><strong><span style="font-family: times new roman,times; font-size: medium;"> A1A1A1 N3</span></strong></em></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Award <em><strong>A1</strong></em> for approximately correct shape with inflexion/change of curvature, </span><span style="font-family: times new roman,times; font-size: medium;"><em><strong>A1</strong></em> for maximum skewed to the left, </span><span style="font-family: times new roman,times; font-size: medium;"><em><strong>A1</strong> </em>for asymptotic behaviour to the right.</span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) \(x = 3.33\) <em><strong>A1 N1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) correct interval, with right end point \(3\frac{1}{3}\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A1A1 N2</span></strong></em></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(0 < x \le 3.33\) , \(0 \le x < 3\frac{1}{3}\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Accept any inequalities in the right direction.</span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></strong></em></p>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">valid approach <em><strong> (M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. quotient rule, product rule</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">2 correct derivatives (must be seen in product or quotient rule) <em><strong>(A1)(A1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(20\) , \(0.3{{\rm{e}}^{0.3x}}\) or \( - 0.3{{\rm{e}}^{ - 0.3x}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct substitution into product or quotient rule <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{{20{{\rm{e}}^{0.3x}} - 20x(0.3){{\rm{e}}^{0.3x}}}}{{{{({{\rm{e}}^{0.3x}})}^2}}}\) , \(20{{\rm{e}}^{ - 0.3x}} + 20x( - 0.3){{\rm{e}}^{ - 0.3x}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct working <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{{20{{\rm{e}}^{0.3x}} - 6x{{\rm{e}}^{0.3x}}}}{{{{\rm{e}}^{0.6x}}}}\) , \(\frac{{{{\rm{e}}^{0.3x}}(20 - 20x(0.3))}}{{{{{\rm{(}}{{\rm{e}}^{0.3x}})}^2}}}\) , \({{\rm{e}}^{ - 0.3x}}(20 + 20x( - 0.3))\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(f'(x) = \frac{{20 - 6x}}{{{{\rm{e}}^{0.3x}}}}\) </span><span style="font-family: times new roman,times; font-size: medium;"> <em><strong>AG N0</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> [5 marks]</span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">consideration of \(f'\) or \(f''\) <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">valid reasoning <em><strong>R1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. sketch of \(f'\) , \(f''\) is positive, \(f'' = 0\) , reference to minimum of \(f'\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct value \(6.6666666 \ldots \) \(\left( {6\frac{2}{3}} \right)\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">(A1)</span></strong></em></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct interval, with <strong>both</strong> endpoints <em><strong>A1 N3</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(6.67 < x \le 20\) , \(6\frac{2}{3} \le x < 20\)</span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks]</span></strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Many candidates earned the first four marks of the question in parts (a) and (b) for correctly using their GDC to graph and find the maximum value. </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Many candidates earned the first four marks of the question in parts (a) and (b) for correctly using their GDC to graph and find the maximum value. </span></p>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Most had a valid approach in part (c) using either the quotient or product rule, but many had difficulty applying the chain rule with a function involving e and simplifying. </span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Part (d) was difficult for most candidates. Although many associated rate of change with derivative, only the best-prepared students had valid reasoning and could find the correct interval with both endpoints. </span></p>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = {{\rm{e}}^{2x}}\cos x\) , \( - 1 \le x \le 2\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Show that \(f'(x) = {{\rm{e}}^{2x}}(2\cos x - \sin x)\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Let the line <em>L</em> be the normal to the curve of <em>f</em> at \(x = 0\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find the equation of <em>L</em> .</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><span style="font-family: times new roman,times; font-size: medium;">The graph of <em>f</em> and the line <em>L</em> intersect at the point (0, 1) and at a second point P.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Find the <em>x</em>-coordinate of P.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Find the area of the region <strong>enclosed</strong> by the graph of <em>f</em> and the line <em>L</em> .</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">c(i) and (ii).</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">correctly finding the derivative of \({{\rm{e}}^{2x}}\) , i.e. \(2{{\rm{e}}^{2x}}\) <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correctly finding the derivative of \(\cos x\) , i.e. \( - \sin x\) <em><strong>A1 </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of using the product rule, seen anywhere <em><strong>M1 </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(f'(x) = 2{{\rm{e}}^{2x}}\cos x - {{\rm{e}}^{2x}}\sin x\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(f'(x) = 2{{\rm{e}}^{2x}}(2\cos x - \sin x)\) <em><strong>AG N0</strong> </em></span></p>
<p><em><span style="font-family: times new roman,times; font-size: medium;"><strong>[3 marks]</strong> </span></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of finding \(f(0) = 1\) , seen anywhere <em><strong>A1 </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">attempt to find the gradient of <em>f</em> <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. substituting \(x = 0\) into \(f'(x)\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">value of the gradient of <em>f</em> <em><strong>A1 </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(f'(0) = 2\) , </span><span style="font-family: times new roman,times; font-size: medium;">equation of tangent is \(y = 2x + 1\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">gradient of normal \( = - \frac{1}{2}\) <em><strong>(A1)</strong></em><br></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(y - 1 = - \frac{1}{2}x\left( {y = - \frac{1}{2}x + 1} \right)\) <em><strong>A1 N3</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[5 marks] </span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) evidence of equating correct functions <em><strong>M1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \({{\rm{e}}^{2x}}\cos x = - \frac{1}{2}x + 1\) , sketch showing intersection of graphs </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(x = 1.56\)<strong><em> </em></strong> <em><strong>A1 N1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) evidence of approach involving subtraction of integrals/areas <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\int {\left[ {f(x) - g(x)} \right]} {\rm{d}}x\) , \(\int {f(x)} {\rm{d}}x - {\text{area under trapezium}}\) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">fully correct integral expression <em><strong>A2 </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\int_0^{1.56} {\left[ {{{\rm{e}}^{2x}}\cos x - \left( { - \frac{1}{2}x + 1} \right)} \right]} {\rm{d}}x\) , \(\int_0^{1.56} {{{\rm{e}}^{2x}}\cos x} {\rm{d}}x - 0.951 \ldots \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{area}} = 3.28\) <em><strong>A1 N2</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[6 marks] </span></strong></em></p>
<div class="question_part_label">c(i) and (ii).</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">A good number of candidates demonstrated the ability to apply the product and chain rules to obtain the given derivative. </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Where candidates recognized that the gradient of the tangent is the derivative, many went on to correctly find the equation of the normal. </span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Few candidates showed the setup of the equation in part (c) before writing their answer from the GDC. Although a good number of candidates correctly expressed the integral to find the area between the curves, surprisingly few found a correct answer. Although this is a GDC paper, some candidates attempted to integrate this function analytically. </span></p>
<div class="question_part_label">c(i) and (ii).</div>
</div>
<br><hr><br><div class="specification">
<p class="p1"><span class="s1">A particle P </span>moves along a straight line so that its velocity, \(v\,{\text{m}}{{\text{s}}^{ - 1}}\), after \(t\) seconds, is given by \(v = \cos 3t - 2\sin t - 0.5\)<span class="s1">, for \(0 \leqslant t \leqslant 5\). The initial displacement of P from a fixed point O is 4 </span>metres.</p>
</div>
<div class="specification">
<p class="p1">The following sketch shows the graph of \(v\).</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2017-02-01_om_15.51.25.png" alt="M16/5/MATME/SP2/ENG/TZ1/09.b+c+d+e"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the displacement of <span class="s1">P </span>from <span class="s1">O </span>after <span class="s1">5 </span>seconds.</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find when <span class="s1">P </span>is first at rest.</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 class="p1">Write down the number of times <span class="s1">P </span>changes direction.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the acceleration of <span class="s1">P </span>after 3 seconds.</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 class="p1">Find the maximum speed of <span class="s1">P</span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1"><strong>METHOD 1</strong></p>
<p class="p1">recognizing \(s = \int v \) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p2">recognizing displacement of P in first 5 <span class="s1">seconds (seen anywhere) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></span></p>
<p class="p2">(accept missing \({\text{d}}t\)<span class="s1">)</span></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(\int_0^5 {v{\text{d}}t,{\text{ }} - 3.71591} \)</p>
<p class="p1">valid approach to find total displacement <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(4 + ( - 3.7159),{\text{ }}s = 4 + \int_0^5 v \)</p>
<p class="p2">0.284086</p>
<p class="p2">0.284 (m) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A2 <span class="Apple-converted-space"> </span>N3</em></strong></span></p>
<p class="p1"><strong>METHOD 2</strong></p>
<p class="p1">recognizing \(s = \int v \) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1">correct integration <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(\frac{1}{3}\sin 3t + 2\cos t - \frac{t}{2} + c\) (do not penalize missing “\(c\)”)</p>
<p class="p1">attempt to find \(c\) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(4 = \frac{1}{3}\sin (0) + 2\cos (0)--\frac{0}{2} + c,{\text{ }}4 = \frac{1}{3}\sin 3t + 2\cos t - \frac{t}{2} + c,{\text{ }}2 + c = 4\)</p>
<p class="p1">attempt to substitute \(t = 5\) into their expression with \(c\) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(s(5),{\text{ }}\frac{1}{3}\sin (15) + 2\cos (5)5--\frac{5}{2} + 2\)</p>
<p class="p2">0.284086</p>
<p class="p2">0.284 (m) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1 <span class="Apple-converted-space"> </span>N3</em></strong></span></p>
<p class="p1"><strong><em>[5 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">recognizing that at rest, \(v = 0\) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p2">\(t = 0.179900\)</p>
<p class="p1"><span class="s1">\(t = 0.180{\text{ (secs)}}\) <span class="Apple-converted-space"> </span></span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">recognizing when change of direction occurs <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p2"><em>eg</em>\(\,\,\,\,\,\)\(v\) <span class="s1">crosses </span>\(t\) axis</p>
<p class="p1"><span class="s2">2 </span>(times) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">acceleration is \({v'}\) (seen anywhere) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(v'(3)\)</p>
<p class="p2">0.743631</p>
<p class="p1"><span class="s1">\(0.744{\text{ }}({\text{m}}{{\text{s}}^{ - 2}})\) <span class="Apple-converted-space"> </span></span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">valid approach involving max or min of \(v\) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><span class="s1"><em>eg</em>\(\,\,\,\,\,\)\(v\prime = 0,{\text{ }}a = 0\)</span>, graph</p>
<p class="p1">one correct co-ordinate for min <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(1.14102,{\text{ }}-3.27876\)</p>
<p class="p1"><span class="s1">\(3.28{\text{ }}({\text{m}}{{\text{s}}^{ - 1}})\) <span class="Apple-converted-space"> </span></span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">This question was not well done throughout. Analytical approaches were almost always unsuccessful as a result of poor integration and differentiation skills and many of the errors were a result of having the GDC in degree mode. In (a), most candidates recognized the need to integrate \(v\) to find the displacement, although a significant number differentiated \(v\). Of those that integrated, many assumed incorrectly that the initial displacement was the value of the constant of integration. Some candidates integrated \(\left| v \right|\) and obtained no marks for an invalid approach. In the case where a correct definite integral was given, it was disappointing to see many candidates try to evaluate it analytically rather than using their GDC.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">This question was not well done throughout. Analytical approaches were almost always unsuccessful as a result of poor integration and differentiation skills and many of the errors were a result of having the GDC in degree mode. In part (b), many candidates did not read the question carefully and gave the two occasions, in the given domain, where the particle was at rest.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">This question was not well done throughout. Analytical approaches were almost always unsuccessful as a result of poor integration and differentiation skills and many of the errors were a result of having the GDC in degree mode. In part (c), many candidates did not appreciate that velocity is a vector and that the particle would change direction when its velocity changes sign. Consequently, many candidates gave the incorrect answer of four changes in directions, rather than the correct two direction changes.</p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">This question was not well done throughout. Analytical approaches were almost always unsuccessful as a result of poor integration and differentiation skills and many of the errors were a result of having the GDC in degree mode. Part (d), was done very poorly, with candidates struggling to differentiate sine and cosine correctly and to evaluate their derivative. As with question 3, many candidates worked with the incorrect angle setting on their calculator.</p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">This question was not well done throughout. Analytical approaches were almost always unsuccessful as a result of poor integration and differentiation skills and many of the errors were a result of having the GDC in degree mode. Few candidates attempted part (e). Of those that did, many attempted to find the largest local maximum of the graph rather than least local minimum as they did not recognise speed as \(\left| v \right|\).</p>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Let \(f(x) = \sqrt[3]{{{x^4}}} - \frac{1}{2}\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find \(f'(x)\).</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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find \(\int {f(x){\text{d}}x} \).</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">expressing \(f\) as \({x^{\frac{4}{3}}}\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(f'(x) = \frac{4}{3}{x^{\frac{1}{3}}}{\text{ }}\left( { = \frac{4}{3}\sqrt[3]{x}} \right)\) <strong><em>A1 N2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">attempt to integrate \({\sqrt[3]{{{x^4}}}}\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(\frac{{{x^{\frac{4}{3} + 1}}}}{{\frac{4}{3} + 1}}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(\int {f(x){\text{d}}x = \frac{3}{7}{x^{\frac{7}{3}}} - \frac{x}{2} + c} \) <strong><em>A1A1A1 N4</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><strong><em><span style="font-family: 'times new roman', times; font-size: medium;">[4 marks]</span><br></em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The following diagram shows the graphs of \(f(x) = \ln (3x - 2) + 1\) and \(g(x) = - 4\cos (0.5x) + 2\) , for \(1 \le x \le 10\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/laurie.png" alt></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Let <em>A</em> be the area of the region <strong>enclosed</strong> by the curves of <em>f</em> and <em>g</em>. </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(i) Find an expression for <em>A</em>. </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> (ii) Calculate the value of <em>A</em>.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) Find \(f'(x)\) . </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> (ii) Find \(g'(x)\) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">There are two values of <em>x</em> for which the gradient of <em>f</em> is equal to the gradient </span><span style="font-family: times new roman,times; font-size: medium;">of <em>g</em>. Find both these values of <em>x</em>.</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) intersection points \(x = 3.77\) , \(x = 8.30\) (may be seen as the limits) <em><strong>(A1)(A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">approach involving subtraction and integrals <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">fully correct expression <em><strong>A2</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\int_{3.77}^{8.30} {(( - 4\cos (0.5x) + 2) - (\ln (3x - 2) + 1)){\rm{d}}x} \) , \(\int_{3.77}^{8.30} {g(x){\rm{d}}x - } \int_{3.77}^{8.30} {f(x){\rm{d}}x} \)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) \(A = 6.46\) <em><strong>A1 N1</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[6 marks]</span></strong></em></p>
<div class="question_part_label">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) \(f'(x) = \frac{3}{{3x - 2}}\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A1A1 N2</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Award <em><strong>A1</strong></em> for numerator (3), <em><strong>A1</strong></em> for denominator (\({3x - 2}\)) , but penalize </span><span style="font-family: times new roman,times; font-size: medium;">1 mark for additional terms.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) \(g'(x) = 2\sin (0.5x)\) <em><strong>A1A1 N2</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Award <em><strong>A1</strong></em> for 2, <em><strong>A1</strong></em> for \(\sin (0.5x)\) , but penalize 1 mark for additional terms.</span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks]</span></strong></em></p>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of using derivatives for gradients <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct approach <em><strong>(A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(f'(x) = g'(x)\) , points of intersection</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(x = 1.43\) , \(x = 6.10\) <em><strong>A1A1 N2N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks]</span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Many candidates did not make good use of the GDC in this problem. Most had the correct </span><span style="font-family: times new roman,times; font-size: medium;">expression but incorrect limits. Some tried to integrate to find the area without using their </span><span style="font-family: times new roman,times; font-size: medium;">GDC. This became extremely complicated and time consuming.</span></p>
<div class="question_part_label">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">In part (b), the chain rule was </span><span style="font-family: times new roman,times; font-size: medium;">not used by some.</span></p>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Most candidates realized the relationship between the gradient and the </span><span style="font-family: times new roman,times; font-size: medium;">first derivative and set the two derivatives equal to one another. Once again many did not </span><span style="font-family: times new roman,times; font-size: medium;">realize that the intersection could be easily found on their GDC.</span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Consider the function \(f(x) = {x^2} - 4x + 1\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Sketch the graph of <em>f</em> , for \( - 1 \le x \le 5\) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">This function can also be written as \(f(x) = {(x - p)^2} - 3\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Write down the value of <em>p </em>.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The graph of <em>g</em> is obtained by reflecting the graph of <em>f</em> in the <em>x</em>-axis, followed by a </span><span style="font-family: times new roman,times; font-size: medium;">translation of \(\left( {\begin{array}{*{20}{c}}<br>0\\<br>6<br>\end{array}} \right)\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Show that \(g(x) = - {x^2} + 4x + 5\) .</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The graph of <em>g </em>is obtained by reflecting the graph of <em>f </em>in the <em>x</em>-axis, followed by a translation of \(\left( {\begin{array}{*{20}{c}}<br>0\\<br>6<br>\end{array}} \right)\) .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The graphs of <em>f</em> and <em>g</em> intersect at two points.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Write down the <em>x</em>-coordinates of these two points.</span></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><span style="font-family: times new roman,times; font-size: medium;">The graph of \(g\) is obtained by reflecting the graph of \(f\) in the <em>x</em>-axis, followed by a translation of \(\left( {\begin{array}{*{20}{c}}<br> 0 \\ <br> 6 <br>\end{array}} \right)\) .<br></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Let <em>R</em> be the region enclosed by the graphs of <em>f</em> and <em>g</em> .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find the area of <em>R</em> .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/N12P2Q9.jpg" alt> <em><strong>A1A1A1A1 N4</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note:</strong> The shape <strong>must</strong> be an approximately correct upwards parabola. </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Only</strong> if the shape is approximately correct, award the following: </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em><strong>A1</strong></em> for vertex \(x \approx 2\) , <em><strong>A1</strong></em> for <em>x</em>-intercepts between 0 and 1, and 3 and 4, </span><span style="font-family: times new roman,times; font-size: medium;"><em><strong>A1</strong></em> for correct <em>y</em>-intercept \((0{\text{, }}1)\), <em><strong>A1</strong></em> for correct domain \([ - 1{\text{, }}5]\).</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Scale not required on the axes, but approximate positions need to be clear. </span></p>
<p><em><strong> <span style="font-family: times new roman,times; font-size: medium;">[4 marks] </span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">\(p = 2\) <em><strong>A1 N1 </strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[1 mark] </span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">correct vertical reflection, correct vertical translation <strong><em>(A1)(A1) </em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \( - f(x)\) , \( - ({(x - 2)^2} - 3)\) , \( - y\) , \( - f(x) + 6\) , \(y + 6\) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">transformations in correct order <em><strong>(A1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \( - ({x^2} - 4x + 1) + 6\) , \( - ({(x - 2)^2} - 3) + 6\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">simplification which clearly leads to given answer <em><strong>A1 </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \( - {x^2} + 4x - 1 + 6\) , \( - ({x^2} - 4x + 4 - 3) + 6\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(g(x) = - {x^2} + 4x + 5\) <em><strong>AG N0</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: If working shown, award <em><strong>A1A1A0A0</strong></em> if transformations correct, but done in reverse order, e.g. \( - ({x^2} - 4x + 1 + 6)\).</span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks] </span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">valid approach <em><strong> (M1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. sketch, \(f = g\) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\( - 0.449489 \ldots \) , \(4.449489 \ldots \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\((2 \pm \sqrt 6 )\) (exact), \( - 0.449{\text{ }}[ - 0.450{\text{, }} - 0.449]\) ; \(4.45{\text{ }}[4.44{\text{, }}4.45]\) <em><strong>A1A1 N3</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks] </span></strong></em></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">attempt to substitute limits or functions into area formula (accept absence of \({\rm{d}}x\) ) <em><strong>(M1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\int_a^b {(( - {x^2}} + 4x + 5) - ({x^2} - 4x + 1)){\rm{d}}x\) , \(\int_{4.45}^{ - 0.449} {(f - g)} \) , \(\int {( - 2{x^2}} + 8x + 4){\rm{d}}x\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">approach involving subtraction of integrals/areas (accept absence of \({\rm{d}}x\) ) <strong><em> (M1) </em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\int_a^b {( - {x^2}} + 4x + 5) - \int_a^b {({x^2}} - 4x + 1)\) , \(\int {(f - g){\rm{d}}x} \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{area}} = 39.19183 \ldots \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\rm{area}} = 39.2\) \([39.1{\text{, }}39.2]\) <em><strong>A1 N3</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks] </span></strong></em></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">A good number of students provided a clear sketch of the quadratic function within the given domain. Some lost marks as they did not clearly indicate the approximate positions of the most important points of the parabola either by labelling or providing a suitable scale. </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">There were few difficulties in part (b).</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">In part (c), candidates often used an insufficient number of steps to show the required result or had difficulty setting out their work logically. </span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Part (d) was generally done well though many candidates gave at least one answer to fewer than three significant figures, potentially resulting in more lost marks. </span></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">In part (e), many candidates were unable to connect the points of intersection found in part (d) with the limits of integration. Mistakes were also made here either using a GDC incorrectly or not subtracting the correct functions. Other candidates tried to divide the region into four areas and made obvious errors in the process. Very few candidates subtracted \(f(x)\) from \(g(x)\) to get a simple function before integrating and there were numerous, fruitless analytical attempts to find the required integral.</span></p>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = \cos 2x\) and \(g(x) = \ln (3x - 5)\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find \(f'(x)\) .</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><span style="font-family: times new roman,times; font-size: medium;">Find \(g'(x)\) .</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><span style="font-family: times new roman,times; font-size: medium;">Let \(h(x) = f(x) \times g(x)\) . Find \(h'(x)\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(a) \(f'(x) = - \sin 2x \times 2( = - 2\sin 2x)\) <em><strong>A1A1 N2</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Award <em><strong>A1</strong></em> for 2, <em><strong>A1</strong></em> for \( - \sin 2x\) . </span></p>
<p><span style="font-family: times new roman,times;"><em><strong><span style="font-size: medium;">[2 marks]</span></strong></em></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">\(g'(x) = 3 \times \frac{1}{{3x - 5}}\) \(\left( { = \frac{3}{{3x - 5}}} \right)\) <em><strong>A1A1 N2</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Award <em><strong>A1</strong></em> for 3, <em><strong>A1</strong></em> for \(\frac{1}{{3x - 5}}\) . </span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of using product rule <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(h'(x) = (\cos 2x)\left( {\frac{3}{{3x - 5}}} \right) + \ln (3x - 5)( - 2\sin 2x)\) <em><strong>A1 N2 </strong></em></span></p>
<p><span style="font-family: times new roman,times;"><em><strong><span style="font-size: medium;">[2 marks]</span></strong></em></span></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Almost all candidates earned at least some of the marks on this question. Some weaker students showed partial knowledge of the chain rule, forgetting to account for the coefficient of \(x\) in their derivatives. A few did not know how to use the product rule, even though it is in the information booklet. </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Almost all candidates earned at least some of the marks on this question. Some weaker students showed partial knowledge of the chain rule, forgetting to account for the coefficient of \(x\) in their derivatives. A few did not know how to use the product rule, even though it is in the information booklet. </span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Almost all candidates earned at least some of the marks on this question. Some weaker students showed partial knowledge of the chain rule, forgetting to account for the coefficient of <em>x</em> in their derivatives. A few did not know how to use the product rule, even though it is in the information booklet. </span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = x{(x - 5)^2}\) , for \(0 \le x \le 6\) . The following diagram shows the graph of <em>f</em> .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/sully.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Let R be the region enclosed by the <em>x</em>-axis and the curve of <em>f</em> .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the area of <em>R</em>.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Find the volume of the solid formed when <em>R</em> is rotated through \({360^ \circ }\) about the </span><span style="font-family: times new roman,times; font-size: medium;"><em>x</em>-axis.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The diagram below shows a part of the graph of a quadratic function </span><span style="font-family: times new roman,times; font-size: medium;">\(g(x) = x(a - x)\) . The graph of <em>g</em> crosses the <em>x</em>-axis when \(x = a\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/555.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The area of the shaded region is equal to the area of <em>R</em>. Find the value of <em>a</em>.</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">finding the limits \(x = 0\) , \(x = 5\) <em><strong>(A1)</strong> </em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">integral expression <em><strong>A1</strong> </em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\int_0^5 {f(x){\rm{d}}x} \) </span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">area = 52.1 <em> <strong>A1 N2</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks] </span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of using formula \(v = \int {\pi {y^2}{\rm{d}}x} \) <em><strong>(M1)</strong> </em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct expression <em><strong>A1</strong> </em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. volume \( = \pi \int_0^5 {{x^2}{{(x - 5)}^4}{\rm{d}}x} \)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">volume = 2340 <em><strong>A2 N2</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks] </span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">area is \(\int_0^a {x(a - x){\rm{d}}x} \) <em><strong>A1 </strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\( = \left[ {\frac{{a{x^2}}}{2} - \frac{{{x^3}}}{3}} \right]_0^a\) <em><strong>A1A1</strong> </em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">substituting limits <em><strong> (M1)</strong> </em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{{{a^3}}}{2} - \frac{{{a^3}}}{3}\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">setting expression equal to area of <em>R</em> <em><strong>(M1) </strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct equation <em><strong>A1</strong> </em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{{{a^3}}}{2} - \frac{{{a^3}}}{3} = 52.1\) , \({a^3} = 6 \times 52.1\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(a = 6.79\) <em><strong> A1 N3</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[7 marks] <br></span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Many candidates set up a completely correct equation for the area enclosed by the <em>x</em>-axis and the curve. Also, many of them tried an analytic approach which sometimes returned incorrect answers. Using the wrong limits \(0\) and \(6\) was a common error. </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">The formula for the volume of revolution given in the data booklet was seen many times in part (b). Some candidates wrote the integrand incorrectly, either missing the \(\pi \) or not squaring. A good number of students could write a completely correct integral expression for the volume of revolution but fewer could evaluate it correctly as many started an analytical approach instead of using their GDC. </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Many candidates did not use a GDC at all in this question. Pages of calculations were produced in an effort to find the area and the volume of revolution. This probably caused a shortage of time for later questions. </span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Many candidates did not use a GDC at all in this question. Pages of calculations were produced in an effort to find the area and the volume of revolution. This probably caused a shortage of time for later questions. </span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The velocity <em>v</em> ms<sup>−1</sup> of an object after <em>t</em> seconds is given by \(v(t) = 15\sqrt t - 3t\) , </span><span style="font-family: times new roman,times; font-size: medium;">for \(0 \le t \le 25\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">On the grid below, sketch the graph of <em>v</em> , clearly indicating the maximum point.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br></span><img src="data:image/png;base64,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" alt></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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Write down an expression for <em>d</em> .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Hence, write down the value of <em>d</em> .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/HSM3.png" alt></span><em><strong><span style="font-family: times new roman,times; font-size: medium;"> A1A1A1 N3</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong> Note</strong>: Award <em><strong>A1</strong></em> for approximately correct shape, <em><strong>A1</strong></em> for right endpoint at \((25{\text{, }}0)\) and <em><strong>A1</strong></em> for maximum point in circle.</span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) recognizing that <em>d</em> is the area under the curve <em><strong> (M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\int {v(t)} \)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct expression in terms of <em>t</em>, with correct limits <em><strong>A2 N3</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(d = \int_0^9 {(15\sqrt t } - 3t){\rm{d}}t\) , \(d = \int_0^9 v {\rm{d}}t\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) \(d = 148.5\) (m) (accept 149 to 3 sf) <em><strong>A1 N1</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks]</span></strong></em></p>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">The graph in part (a) was well done. It was pleasing to see many candidates considering the domain as they sketched their graph. </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Part (b) (i) asked for an expression which bewildered a great many candidates. However, few had difficulty obtaining the correct answer in (b) (ii). </span></p>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<br><hr><br><div class="question">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">A gradient function is given by </span><span style="font-family: times new roman,times; font-size: medium;">\(\frac{{{\rm{d}}y}}{{{\rm{d}}x}} = 10{{\rm{e}}^{2x}} - 5\) . When \(x = 0\) , \(y = 8\) . Find the value </span><span style="font-family: times new roman,times; font-size: medium;">of <em>y</em> when \(x = 1\) .</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><strong><span style="font-family: times new roman,times; font-size: medium;">METHOD 1</span></strong></p>
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of anti-differentiation <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\int {(10{{\rm{e}}^{2x}} - 5){\rm{d}}x} \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(y = 5{{\rm{e}}^{2x}} - 5x + C\) <em><strong>A2A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Award <em><strong>A2</strong></em> for \(5{{\rm{e}}^{2x}}\) , <em><strong>A1</strong></em> for <span lang="EN-US">\( - 5x\)</span> . If “<em>C</em>” is omitted, award no further marks.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">substituting \((0{\text{, }}8)\) <em><strong> (M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(8 = 5 + C\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(C = 3\) \((y = 5{{\rm{e}}^{2x}} - 5x + 3)\) <em><strong>(A1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">substituting \(x = 1\) <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(y = 34.9\) \((5{{\rm{e}}^2} - 2)\) <em><strong>A1 N4</strong></em></span></p>
<p><strong> <span style="font-family: times new roman,times; font-size: medium;">METHOD 2</span></strong></p>
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of definite integral function expression <em><strong>(M2)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\int_a^x {f'(t){\rm{d}}t = } f(x) - f(a)\) , \(\int_0^x {(10{{\rm{e}}^{2x}} - 5)} \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">initial condition in definite integral function expression <em><strong> (A2)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\int_0^x {(10{{\rm{e}}^{2t}} - 5)} {\rm{d}}t = y - 8\) , \(\int_0^x {(10{{\rm{e}}^{2x}} - 5)} {\rm{d}}x + 8\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct definite integral expression for <em>y</em> when \(x = 1\) <em><strong>(A2)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\int_0^1 {(10{{\rm{e}}^{2x}} - 5){\rm{d}}x + 8} \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: times new roman,times; font-size: medium;">\(y = 34.9\) \((5{{\rm{e}}^2} - 2)\) <em><strong>A1 N4</strong></em></span></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> [8 marks]</span></strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">Although a pleasing number of candidates recognized the requirement of integration, many did not correctly apply the reverse of the chain rule to integration. While some candidates did not write the constant of integration, many did, earning additional follow-through marks even with an incorrect integral. Weaker candidates sometimes substituted \(x = 1\) into \(\frac{{{\rm{d}}y}}{{{\rm{d}}x}}\) or attempted some work with a tangent line equation, earning no marks. </span></p>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">A particle moves in a straight line. Its velocity, \(v{\text{ m}}{{\text{s}}^{ - 1}}\), at time \(t\) seconds, is given by</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\[v = {\left( {{t^2} - 4} \right)^3},{\text{ for }}0 \leqslant t \leqslant 3.\]</span></p>
<div> </div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find the velocity of the particle when \(t = 1\).</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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find the value of \(t\) for which the particle is at rest.</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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find the total distance the particle travels during the first three seconds.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Show that the acceleration of the particle is given by \(a = 6t{({t^2} - 4)^2}\).</span></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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find all possible values of \(t\) for which the velocity and acceleration are both positive or</span><span style="font-family: 'times new roman', times; font-size: medium;"> both negative.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">substituting \(t = 1\) into \(v\)<em> </em><strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(v(1),{\text{ }}{\left( {{1^2} - 4} \right)^3}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">velocity \( = - 27{\text{ }}\left( {{\text{m}}{{\text{s}}^{ - 1}}} \right)\) <strong><em>A1 N2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">valid reasoning <strong><em>(R1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(v = 0,{\text{ }}{\left( {{t^2} - 4} \right)^3} = 0\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">correct working <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \({t^2} - 4 = 0,{\text{ }}t = \pm 2\), sketch</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(t = 2\) <strong><em>A1 N2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">correct integral expression for distance <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(\int_0^3 {\left| v \right|,{\text{ }}\int {\left| {{{\left( {{t^2} - 4} \right)}^3}} \right|,{\text{ }} - \int_0^2 {v{\text{d}}t + \int_2^3 {v{\text{d}}t} } } } \),</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(\int_0^2 {{{\left( {4 - {t^2}} \right)}^3}{\text{d}}t + \int_2^3 {{{\left( {{t^2} - 4} \right)}^3}{\text{d}}t} }\) (do not accept \(\int_0^3 {v{\text{d}}t} \))</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(86.2571\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{distance}} = 86.3{\text{ (m)}}\) <strong><em>A2 N3</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">evidence of differentiating velocity <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(v'(t)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(a = 3{\left( {{t^2} - 4} \right)^2}(2t)\) <strong><em>A2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(a = 6t{\left( {{t^2} - 4} \right)^2}\) <strong><em>AG N0</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 1</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">valid approach <strong><em>M1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> graphs of \(v\) and \(a\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">correct working <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg </em>areas of same sign indicated on graph</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(2 < t \leqslant 3\)<span style="font: 20.5px 'Times New Roman';"> </span>(accept \(t > 2\)) <strong><em>A2 N2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 2</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">recognizing that \(a \geqslant 0\) (accept \(a\)<em> is </em>always positive) (seen anywhere) <strong><em>R1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">recognizing that \(v\) is positive when \(t > 2\) (seen anywhere) <strong><em>(R1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(2 < t \leqslant 3\)<span style="font: 20.5px 'Times New Roman';"> </span>(accept \(t > 2\)) <strong><em>A2 N2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The following diagram shows the graph of \(f(x) = {{\rm{e}}^{ - {x^2}}}\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/berlin.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The points A, B, C, D and E lie on the graph of <em>f</em> . Two of these are points of inflexion.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Identify the <strong>two</strong> points of inflexion.</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Find \(f'(x)\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Show that \(f''(x) = (4{x^2} - 2){{\rm{e}}^{ - {x^2}}}\) .</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the <em>x</em>-coordinate of each point of inflexion.</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><span style="font-family: times new roman,times; font-size: medium;">Use the second derivative to show that one of these points is a point of inflexion.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">B, D <strong><em>A1A1 N2</em></strong></span></p>
<p><strong><em><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) \(f'(x) = - 2x{{\rm{e}}^{ - {x^2}}}\) <em><strong>A1A1 N2</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Award <em><strong>A1</strong></em> for \({{\rm{e}}^{ - {x^2}}}\) and <strong><em>A1</em></strong> for \( - 2x\) .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) finding the derivative of \( - 2x\) , i.e. \( - 2\) <em><strong>(A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of choosing the product rule <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \( - 2{{\rm{e}}^{ - {x^2}}}\) \( - 2x \times - 2x{{\rm{e}}^{ - {x^2}}}\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\( - 2{{\rm{e}}^{ - {x^2}}} + 4{x^2}{{\rm{e}}^{ - {x^2}}}\) <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(f''(x) = (4{x^2} - 2){{\rm{e}}^{ - {x^2}}}\) <em><strong>AG N0</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[5 marks]</span></strong></em></p>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">valid reasoning <em><strong>R1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(f''(x) = 0\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">attempting to solve the equation <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \((4{x^2} - 2) = 0\) , sketch of \(f''(x)\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(p = 0.707\) \(\left( { = \frac{1}{{\sqrt 2 }}} \right)\) , \(q = - 0.707\) \(\left( { = - \frac{1}{{\sqrt 2 }}} \right)\) </span><strong><em><span style="font-family: times new roman,times; font-size: medium;">A1A1 N3</span></em></strong></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks]</span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of using second derivative to test values on either side of POI <em><strong>M1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. finding values, reference to graph of \(f''\) , sign table</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct working <em><strong>A1A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. finding any two correct values either side of POI,</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">checking sign of \(f''\) on either side of POI</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">reference to sign change of \(f''(x)\) <em><strong>R1 N0</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks]</span></strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Most candidates were able to recognize the points of inflexion in part (a).</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Most candidates were able to recognize the points of inflexion in part (a) and had little difficulty with the first and second derivatives in part (b). A few did not recognize the application of the product rule in part (b). </span></p>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Obtaining the <em>x</em>-coordinates of the inflexion points in (c) usually did not cause many problems. </span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Only the better-prepared candidates understood how to set up a second derivative test in part (d). Many of those did not show, or clearly indicate, the values of <em>x</em> used to test for a point of inflexion, but merely gave an indication of the sign. Some candidates simply resorted to showing that \(f''\left( { \pm \frac{1}{{\sqrt 2 }}} \right) = 0\) , completely missing the point of the question. The necessary condition for a point of inflexion, i.e. \(f''(x) = 0\) <strong>and</strong> the change of sign for \(f''(x)\) , seemed not to be known by the vast majority of candidates. </span></p>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>Let \(f(x) = - 0.5{x^4} + 3{x^2} + 2x\). The following diagram shows part of the graph of \(f\).</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-15_om_06.09.00.png" alt="M17/5/MATME/SP2/ENG/TZ2/08"></p>
<p> </p>
<p>There are \(x\)-intercepts at \(x = 0\) and at \(x = p\). There is a maximum at A where \(x = a\), and a point of inflexion at B where \(x = b\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of \(p\).</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 the coordinates of A.</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>Write down the rate of change of \(f\) at A.</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 coordinates of B.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the the rate of change of \(f\) at B.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Let \(R\) be the region enclosed by the graph of \(f\) , the \(x\)-axis, the line \(x = b\) and the line \(x = a\). The region \(R\) is rotated 360° about the \(x\)-axis. Find the volume of the solid formed.</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>evidence of valid approach <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(f(x) = 0,{\text{ }}y = 0\)</p>
<p>2.73205</p>
<p>\(p = 2.73\) <strong><em>A1</em></strong> <strong><em>N2</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>1.87938, 8.11721</p>
<p>\((1.88,{\text{ }}8.12)\) <strong><em>A2</em></strong> <strong><em>N2</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>rate of change is 0 (do not accept decimals) <strong><em>A1</em></strong> <strong><em>N1</em></strong></p>
<p><strong><em>[1 marks]</em></strong></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1 (using GDC)</strong></p>
<p>valid approach <strong><em>M1</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(f’’ = 0\), max/min on \(f’,{\text{ }}x = - 1\)</p>
<p>sketch of either \(f’\) or \(f’’\), with max/min or root (respectively) <strong><em>(A1)</em></strong></p>
<p>\(x = 1\) <strong><em>A1</em></strong> <strong><em>N1</em></strong></p>
<p>Substituting <strong>their</strong> \(x\) value into \(f\) <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(f(1)\)</p>
<p>\(y = 4.5\) <strong><em>A1</em></strong> <strong><em>N1</em></strong></p>
<p><strong>METHOD 2 (analytical)</strong></p>
<p>\(f’’ = - 6{x^2} + 6\) <strong><em>A1</em></strong></p>
<p>setting \(f’’ = 0\) <strong><em>(M1)</em></strong></p>
<p>\(x = 1\) <strong><em>A1</em></strong> <strong><em>N1</em></strong></p>
<p>substituting <strong>their</strong> \(x\) value into \(f\) <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(f(1)\)</p>
<p>\(y = 4.5\) <strong><em>A1</em></strong> <strong><em>N1</em></strong></p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>recognizing rate of change is \(f’\) <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(y’,{\text{ }}f’(1)\)</p>
<p>rate of change is 6 <strong><em>A1</em></strong> <strong><em>N2</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to substitute either limits or the function into formula <strong><em>(M1)</em></strong></p>
<p>involving \({f^2}\) (accept absence of \(\pi \) and/or \({\text{d}}x\))</p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\pi \int {{{( - 0.5{x^4} + 3{x^2} + 2x)}^2}{\text{d}}x,{\text{ }}\int_1^{1.88} {{f^2}} } \)</p>
<p>128.890</p>
<p>\({\text{volume}} = 129\) <strong><em>A2</em></strong> <strong><em>N3</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.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="question">
<p><strong>Note:</strong> <strong>In this question, distance is in metres and time is in seconds.</strong></p>
<p> </p>
<p>A particle moves along a horizontal line starting at a fixed point A. The velocity \(v\) of the particle, at time \(t\), is given by \(v(t) = \frac{{2{t^2} - 4t}}{{{t^2} - 2t + 2}}\), for \(0 \leqslant t \leqslant 5\). The following diagram shows the graph of \(v\)</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-15_om_08.18.11.png" alt="M17/5/MATME/SP2/ENG/TZ2/07"></p>
<p>There are \(t\)-intercepts at \((0,{\text{ }}0)\) and \((2,{\text{ }}0)\).</p>
<p>Find the maximum distance of the particle from A during the time \(0 \leqslant t \leqslant 5\) and justify your answer.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><strong>METHOD 1 (displacement)</strong></p>
<p>recognizing \(s = \int {v{\text{d}}t} \) <strong><em>(M1)</em></strong></p>
<p>consideration of displacement at \(t = 2\) <strong>and</strong> \(t = 5\) (seen anywhere) <strong><em>M1</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\int_0^2 v \) and \(\int_0^5 v \)</p>
<p> </p>
<p><strong>Note:</strong> Must have both for any further marks.</p>
<p> </p>
<p>correct displacement at \(t = 2\) and \(t = 5\) (seen anywhere) <strong><em>A1A1</em></strong></p>
<p>\( - 2.28318\) (accept 2.28318), 1.55513</p>
<p>valid reasoning comparing correct displacements <strong><em>R1</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\left| { - 2.28} \right| > \left| {1.56} \right|\), more left than right</p>
<p>2.28 (m) <strong><em>A1</em></strong> <strong><em>N1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Do not award the final <strong><em>A1 </em></strong>without the <strong><em>R1</em></strong><em>.</em></p>
<p> </p>
<p><strong>METHOD 2 (distance travelled)</strong></p>
<p>recognizing distance \( = \int {\left| v \right|{\text{d}}t} \) <strong><em>(M1)</em></strong></p>
<p>consideration of distance travelled from \(t = 0\) to 2 <strong>and</strong> \(t = 2\) to 5 (seen anywhere) <strong><em>M1</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\int_0^2 v \) and \(\int_2^5 v \)</p>
<p> </p>
<p><strong>Note:</strong> Must have both for any further marks</p>
<p> </p>
<p>correct distances travelled (seen anywhere) <strong><em>A1A1</em></strong></p>
<p>2.28318, (accept \( - 2.28318\)), 3.83832</p>
<p>valid reasoning comparing correct distance values <strong><em>R1</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(3.84 - 2.28 < 2.28,{\text{ }}3.84 < 2 \times 2.28\)</p>
<p>2.28 (m) <strong><em>A1</em></strong> <strong><em>N1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Do not award the final <strong><em>A1 </em></strong>without the <strong><em>R1</em></strong>.</p>
<p> </p>
<p><strong><em>[6 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = x\cos x\) , for \(0 \le x \le 6\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find \(f'(x)\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">On the grid below, sketch the graph of \(y = f'(x)\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/bbc.png" alt></span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of choosing the product rule <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(x \times ( - \sin x) + 1 \times \cos x\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(f'(x) = \cos x - x\sin x\) <em><strong>A1A1 N3</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/bbc2.png" alt></span><em><span style="font-family: times new roman,times; font-size: medium;"><strong> A1A1A1A1 N4</strong> </span></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Award <em><strong>A1</strong></em> for correct domain, \(0 \le x \le 6\) with endpoints in circles, <em><strong>A1</strong></em> for approximately correct shape, <em><strong>A1</strong></em> for local minimum in circle, <em><strong>A1</strong></em> for local maximum in circle. </span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks] </span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">This problem was well done by most candidates. There were some candidates that struggled </span><span style="font-family: times new roman,times; font-size: medium;">to apply the product rule in part (a) and often wrote nonsense like \( - x\sin x = - \sin {x^2}\) .</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">In part </span><span style="font-family: times new roman,times; font-size: medium;">(b), few candidates were able to sketch the function within the required domain and a large </span><span style="font-family: times new roman,times; font-size: medium;">number of candidates did not have their calculator in the correct mode.</span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">The acceleration, \(a{\text{ m}}{{\text{s}}^{ - 2}}\), of a particle at time <em>t</em> seconds is given by \[a = \frac{1}{t} + 3\sin 2t {\text{, for }} t \ge 1.\]<br></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The particle is at rest when \(t = 1\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find the velocity of the particle when \(t = 5\) .</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of integrating the acceleration function <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\int {\left( {\frac{1}{t} + 3\sin 2t} \right)} {\rm{d}}t\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct expression \(\ln t - \frac{3}{2}\cos 2t + c\) <em><strong>A1A1 </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of substituting (1, 0) <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(0 = \ln 1 - \frac{3}{2}\cos 2 + c\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(c = - 0.624\) \(\left( { = \frac{3}{2}\cos 2 - \ln {\text{1 or }}\frac{{\rm{3}}}{{\rm{2}}}\cos 2} \right)\) <em><strong> (A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(v = \ln t - \frac{3}{2}\cos 2t - 0.624\) \(\left( { = \ln t - \frac{3}{2}\cos 2t + \frac{3}{2}\cos {\text{2 or ln}}t - \frac{3}{2}\cos 2t + \frac{3}{2}\cos 2 - \ln 1} \right)\) <em><strong>(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(v(5) = 2.24\) (accept the exact answer \(\ln 5 - 1.5\cos 10 + 1.5\cos 2\) ) <em><strong>A1 N3</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[7 marks] </span></strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">This problem was not well done. A large number of students failed to recognize that they </span><span style="font-family: times new roman,times; font-size: medium;">needed to integrate the acceleration function. Even among those who integrated the function, </span><span style="font-family: times new roman,times; font-size: medium;">there were many who integrated incorrectly. A great number of candidates were not able to </span><span style="font-family: times new roman,times; font-size: medium;">handle the given initial condition to find the integration constant but incorrectly substituted </span><span style="font-family: times new roman,times; font-size: medium;">\(t = 5\) directly into their expression.</span></p>
</div>
<br><hr><br><div class="specification">
<p class="p1">A particle <span class="s1">P </span>starts from a point <span class="s1">A </span>and moves along a horizontal straight line. Its velocity \(v{\text{ cm}}\,{{\text{s}}^{ - 1}}\) after \(t\) <span class="s2">seconds is given by</span></p>
<p class="p2">\[v(t) = \left\{ {\begin{array}{*{20}{l}} { - 2t + 2,}&{{\text{for }}0 \leqslant t \leqslant 1} \\ {3\sqrt t + \frac{4}{{{t^2}}} - 7,}&{{\text{for }}1 \leqslant t \leqslant 12} \end{array}} \right.\]</p>
<p class="p1">The following diagram shows the graph of \(v\).</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2017-03-03_om_16.40.29.png" alt="N16/5/MATME/SP2/ENG/TZ0/09"></p>
</div>
<div class="specification">
<p class="p1"><span class="s1">P </span>is at rest when \(t = 1\) and \(t = p\).</p>
</div>
<div class="specification">
<p class="p1">When \(t = q\), the acceleration of <span class="s1">P </span>is zero.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the initial velocity of \(P\).</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 class="p1">Find the value of \(p\).</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 class="p1">(i) <span class="Apple-converted-space"> </span>Find the value of \(q\).</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Hence, find the <strong>speed </strong>of <span class="s1">P </span>when \(t = q\).</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 class="p1">(i) <span class="Apple-converted-space"> </span>Find the total distance travelled by <span class="s1">P </span>between \(t = 1\) and \(t = p\).</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Hence or otherwise, find the displacement of <span class="s1">P </span>from <span class="s1">A </span>when \(t = p\).</p>
<div class="marks">[6]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">valid attempt to substitute \(t = 0\) into the correct function <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p2"><em>eg</em>\(\,\,\,\,\,\)\( - 2(0) + 2\)</p>
<p class="p2"><span class="s2">2 <span class="Apple-converted-space"> </span></span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p2"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">recognizing \(v = 0\) <span class="s1">when P </span>is at rest <span class="Apple-converted-space"> </span><span class="s2"><strong><em>(M1)</em></strong></span></p>
<p class="p2">5.21834</p>
<p class="p3"><span class="s1">\(p = 5.22{\text{ }}({\text{seconds}})\) <span class="Apple-converted-space"> </span></span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p3"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>recognizing that \(a = v'\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p2"><span class="s2"><em>eg</em>\(\,\,\,\,\,\)\(v' = 0\)</span>, minimum on graph</p>
<p class="p2">1.95343</p>
<p class="p3"><span class="s3">\(q = 1.95\) <span class="Apple-converted-space"> </span></span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>valid approach to find <strong>their </strong>minimum <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p2"><span class="s2"><em>eg</em>\(\,\,\,\,\,\)\(v(q),{\text{ }} - 1.75879\)</span>, reference to min on graph</p>
<p class="p2">1.75879</p>
<p class="p1">speed \( = 1.76{\text{ }}(c\,{\text{m}}\,{{\text{s}}^{ - 1}})\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></span></p>
<p class="p3"><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>substitution of <strong>correct</strong> \(v(t)\) into distance formula, <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(A1)</em></strong></span></p>
<p class="p2"><em>eg</em>\(\,\,\,\,\,\)\(\int_1^p {\left| {3\sqrt t + \frac{4}{{{t^2}}} - 7} \right|{\text{d}}t,{\text{ }}\left| {\int {3\sqrt t + \frac{4}{{{t^2}}} - 7{\text{d}}t} } \right|} \)</p>
<p class="p3">4.45368</p>
<p class="p1">distance \( = 4.45{\text{ }}({\text{cm}})\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></span></p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>displacement from \(t = 1\) to \(t = p\) (seen anywhere) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(A1)</em></strong></span></p>
<p class="p2"><em>eg</em>\(\,\,\,\,\,\)\( - 4.45368,{\text{ }}\int_1^p {\left( {3\sqrt t + \frac{4}{{{t^2}}} - 7} \right){\text{d}}t} \)</p>
<p class="p2">displacement from \(t = 0\) to \(t = 1\) <span class="s1"><strong><em>(A1)</em></strong></span></p>
<p class="p2"><em>eg</em>\(\,\,\,\,\,\)\(\int_0^1 {( - 2t + 2){\text{d}}t,{\text{ }}0.5 \times 1 \times 2,{\text{ 1}}} \)</p>
<p class="p2">valid approach to find displacement for \(0 \leqslant t \leqslant p\) <span class="s1"><strong><em>M1</em></strong></span></p>
<p class="p2"><em>eg</em>\(\,\,\,\,\,\)\(\int_0^1 {( - 2t + 2){\text{d}}t + \int_1^p {\left( {3\sqrt t + \frac{4}{{{t^2}}} - 7} \right){\text{d}}t,{\text{ }}\int_0^1 {( - 2t + 2){\text{d}}t - 4.45} } } \)</p>
<p class="p2">\( - 3.45368\)</p>
<p class="p1">displacement \( = - 3.45{\text{ }}({\text{cm}})\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></span></p>
<p class="p2"><strong><em>[6 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.</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><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = {{\rm{e}}^x}(1 - {x^2})\) .</span></p>
</div>
<div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Part of the graph of \(y = f(x)\), for \( - 6 \le x \le 2\) , is shown below. The <em>x</em>-coordinates of the local minimum and maximum points are <em>r</em> and <em>s</em> respectively. </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/aching.png" alt></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Show that \(f'(x) = {{\rm{e}}^x}(1 - 2x - {x^2})\) . </span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Write down the <strong>equation</strong> of the horizontal asymptote.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Write down the value of <em>r</em> and of <em>s</em>.</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><span style="font-family: times new roman,times; font-size: medium;">Let <em>L</em> be the normal to the curve of <em>f</em> at \({\text{P}}(0{\text{, }}1)\) . Show that <em>L</em> has equation \(x + y = 1\) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Let <em>R</em> be the region enclosed by the curve \(y = f(x)\) and the line <em>L</em>. </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(i) Find an expression for the area of <em>R</em>. </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> (ii) Calculate the area of <em>R</em>.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">e(i) and (ii).</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of using the product rule <em><strong>M1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(f'(x) = {{\rm{e}}^x}(1 - {x^2}) + {{\rm{e}}^x}( - 2x)\) <em><strong>A1A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Award <em><strong>A1</strong></em> for \({{\rm{e}}^x}(1 - {x^2})\) , <em><strong>A1</strong></em> for \({{\rm{e}}^x}( - 2x)\) .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(f'(x) = {{\rm{e}}^x}(1 - 2x - {x^2})\) <em><strong>AG N0</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(y = 0\) <em><strong>A1 N1</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[1 mark]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">at the local maximum or minimum point</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(f'(x) = 0\) \(({{\rm{e}}^x}(1 - 2x - {x^2}) = 0)\) <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\( \Rightarrow 1 - 2x - {x^2} = 0\) <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(r = - 2.41\) \(s = 0.414\) <em><strong>A1A1 N2N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks]</span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(f'(0) = 1\) <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">gradient of the normal \(= - 1\) <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of substituting into an equation for a straight line <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct substitution <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(y - 1 = - 1(x - 0)\) , \(y - 1 = - x\) , \(y = - x + 1\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(x + y = 1\) <em><strong>AG N0</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks]</span></strong></em></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) intersection points at \(x = 0\) and \(x = 1\) (may be seen as the limits) <em><strong>(A1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">approach involving subtraction and integrals <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">fully correct expression <em><strong>A2 N4</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\int_0^1 {\left( {{{\rm{e}}^x}(1 - {x^2}) - (1 - x)} \right)} {\rm{d}}x\) , \(\int_0^1 {f(x){\rm{d}}x - \int_0^1 {(1 - x){\rm{d}}x} } \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) area \(R = 0.5\) <em><strong>A1 N1</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> [5 marks]</span></strong></em></p>
<div class="question_part_label">e(i) and (ii).</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Many candidates clearly applied the product rule to correctly show the given derivative. Some </span><span style="font-family: times new roman,times; font-size: medium;">candidates missed the multiplicative nature of the function and attempted to apply a chain rule </span><span style="font-family: times new roman,times; font-size: medium;">instead.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">For part (b), the equation of the horizontal asymptote was commonly written as \(x = 0\) .</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Although part (c) was a “write down” question where no working is required, a good number of </span><span style="font-family: times new roman,times; font-size: medium;">candidates used an algebraic method of solving for <em>r</em> and <em>s</em> which sometimes returned </span><span style="font-family: times new roman,times; font-size: medium;">incorrect answers. Those who used their GDC usually found correct values, although not </span><span style="font-family: times new roman,times; font-size: medium;">always to three significant figures.</span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">In part (d), many candidates showed some skill showing the equation of a normal, although </span><span style="font-family: times new roman,times; font-size: medium;">some tried to work with the gradient of the tangent.</span></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Surprisingly few candidates set up a completely correct expression for the area between </span><span style="font-family: times new roman,times; font-size: medium;">curves that considered both integration and the correct subtraction of functions. Using limits </span><span style="font-family: times new roman,times; font-size: medium;">of \( - 6\) and 2 was a common error, as was integrating on \(f(x)\) alone. Where candidates did </span><span style="font-family: times new roman,times; font-size: medium;">write a correct expression, many attempted to perform analytic techniques to calculate the </span><span style="font-family: times new roman,times; font-size: medium;">area instead of using their GDC.</span></p>
<div class="question_part_label">e(i) and (ii).</div>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">A particle’s displacement, in metres, is given by \(s(t) = 2t\cos t\) , for \(0 \le t \le 6\) , </span><span style="font-family: times new roman,times; font-size: medium;">where <em>t</em> is the time in seconds.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">On the grid below, sketch the graph of \(s\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/N12P2Q7.jpg" alt></span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the maximum velocity of the particle.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/N12P2Q7ms.jpg" alt> <em><strong>A1A1A1A1 N4</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong> Note:</strong> Award <em><strong>A1</strong></em> for approximately correct shape (do not accept line segments). </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong> Only</strong> if this <em><strong>A1</strong></em> is awarded, award the following: </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em><strong> A1</strong></em> for maximum and minimum within circles, </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em><strong> A1</strong></em> for <em>x</em>-intercepts between 1 and 2 <strong>and</strong> between 4 and 5, </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em><strong> A1</strong></em> for left endpoint at \((0{\text{, }}0)\) and right endpoint within circle. </span></p>
<p><strong><em><span style="font-family: times new roman,times; font-size: medium;">[4 marks] </span></em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">appropriate approach <em><strong>(M1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. recognizing that \(v = s'\) , finding derivative, \(a = s''\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">valid method to find maximum <em><strong>(M1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. sketch of \(v\) , \(v'(t) = 0\) , \(t = 5.08698 \ldots \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(v = 10.20025 \ldots \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(v = 10.2\) \([10.2{\text{, }}10.3]\) <em><strong>A1 N2</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks] </span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Most candidates sketched an approximately correct shape for the displacement of a particle in the given domain, but many lost marks for carelessness in graphing the local extrema or the right endpoint. </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">In part (b), most candidates knew to differentiate displacement to find velocity, but few knew how to then find the maximum. Occasionally, a candidate would give the time value of the maximum. Others attempted to incorrectly set the first derivative equal to zero and solve analytically rather than take the maximum value from the graph of the velocity function. </span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = a{x^3} + b{x^2} + c\) , where <em>a</em> , <em>b</em> and <em>c</em> are real numbers. The graph of <em>f</em> passes </span><span style="font-family: times new roman,times; font-size: medium;">through the point (2, 9) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Show that \(8a + 4b + c = 9\) .</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><span style="font-family: times new roman,times; font-size: medium;">The graph of <em>f</em> has a local minimum at \((1{\text{, }}4)\) .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Find two other equations in <em>a</em> , <em>b</em> and <em>c</em> , giving your answers in a similar form to </span><span style="font-family: times new roman,times; font-size: medium;">part (a).</span></p>
<p> </p>
<div class="marks">[7]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the value of <em>a</em> , of <em>b</em> and of <em>c</em> .</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><span style="font-family: times new roman,times; font-size: medium;">attempt to substitute coordinates in <em>f</em> <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(f(2) = 9\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct substitution <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(a \times {2^3} + b \times {2^2} + c = 9\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(8a + 4b + c = 9\) <em><strong>AG N0</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks] </span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">recognizing that \((1{\text{, }}4)\) is on the graph of <em>f</em> <strong><em>(M1) </em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(f(1) = 4\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct equation <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(a + b + c = 4\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">recognizing that \(f' = 0\) at minimum (seen anywhere) <strong><em>(M1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(f'(1) = 0\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(f'(x) = 3a{x^2} + 2bx\) (seen anywhere) <strong><em>A1A1 </em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct substitution into derivative <strong><em>(A1) </em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(3a \times {1^2} + 2b \times 1 = 0\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct simplified equation <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(3a + 2b = 0\)</span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[7 marks] </span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">valid method for solving system of equations <strong><em>(M1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. inverse of a matrix, substitution </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(a = 2\) , \(b = - 3\) , \(c = 5\) <em><strong>A1A1A1 N4</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks] </span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Part (a) was generally well done, with a few candidates failing to show a detailed substitution. Some substituted 2 in place of <em>x</em>, but didn't make it clear that they had substituted in <em>y</em> as well. </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">A great majority could find the two equations in part (b). However there were a significant number of candidates who failed to identify that the gradient of the tangent is zero at a minimum point, thus getting the incorrect equation \(3a + 2b = 4\) .</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">A considerable number of candidates only had 2 equations, so that they either had a hard time trying to come up with a third equation (incorrectly combining some of the information given in the question) to solve part (c) or they completely failed to solve it. </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Despite obtaining three correct equations many used long elimination methods that caused algebraic errors. Pages of calculations leading nowhere were seen. </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Those who used matrix methods were almost completely successful. </span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(f'(x) = - 24{x^3} + 9{x^2} + 3x + 1\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">There are two points of inflexion on the graph of <em>f</em> . Write down the <em>x</em>-coordinates </span><span style="font-family: times new roman,times; font-size: medium;">of these points.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(g(x) = f''(x)\) . Explain why the graph of <em>g</em> has no points of inflexion.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">valid approach <em><strong>R1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(f''(x) = 0\) , the max and min of \(f'\) gives the points of inflexion on <em>f</em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\( - 0.114{\text{, }}0.364\) (accept (\( - 0.114{\text{, }}0.811\)) and (\(0.364{\text{, }}2.13)\)) <em><strong>A1A1 N1N1</strong></em></span></p>
<p><span style="font-family: times new roman,times;"><em><strong><span style="font-size: medium;">[3 marks]</span></strong></em></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><strong><span style="font-family: times new roman,times; font-size: medium;">METHOD 1</span></strong></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">graph of <em>g</em> is a quadratic function <em><strong>R1 N1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">a quadratic function does not have any points of inflexion <em><strong>R1 N1</strong></em></span></p>
<p align="LEFT"><strong><span style="font-family: times new roman,times; font-size: medium;">METHOD 2</span></strong></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">graph of <em>g</em> is concave down over entire domain <strong><em>R1 N1</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">therefore no change in concavity <em><strong>R1 N1</strong></em></span></p>
<p align="LEFT"><strong><span style="font-family: times new roman,times; font-size: medium;">METHOD 3</span></strong></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(g''(x) = - 144\) <em><strong>R1 N1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">therefore no points of inflexion as \(g''(x) \ne 0\) <em><strong> R1 N1</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">There were mixed results in part (a). Students were required to understand the relationships between a function and its derivative and often obtained the correct solutions with incorrect or missing reasoning. </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">In part (b), the question was worth two marks and candidates were required to make two valid points in their explanation. There were many approaches to take here and candidates often confused their reasoning or just kept writing hoping that somewhere along the way they would say something correct to pick up the points. Many confused \(f'\) and \(g'\) . </span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Let \(f(x) = {x^2}\) and \(g(x) = 3\ln (x + 1)\), for \(x > - 1\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Solve \(f(x) = g(x)\).</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 class="p1">Find the area of the region enclosed by the graphs of \(f\) and \(g\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">valid approach <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg </em>sketch</p>
<p class="p2">0, 1.73843</p>
<p class="p1"><span class="s1">\(x = 0,{\text{ }}x = 1.74{\text{ }}\left( {{\text{accept }}(0,{\text{ }}0){\text{ and }}(1.74,{\text{ }}3.02)} \right)\) <span class="Apple-converted-space"> </span></span><strong><em>A1A1 <span class="Apple-converted-space"> </span>N3</em></strong></p>
<p class="p1"><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">integrating and subtracting functions (in any order) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(\int {g - f} \)</p>
<p class="p1">correct substitution of their limits <strong>or </strong><span class="s1">function (accept missing \({\text{d}}x\)</span>)</p>
<p class="p1"><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(\int_0^{1.74} {g - f,{\text{ }}\int {3\ln (x + 1) - {x^2}} } \)</p>
<p class="p3"><span class="s2"><strong>Note: <span class="Apple-converted-space"> </span></strong>Do not award <strong><em>A1 </em></strong></span>if there is an error in the substitution.</p>
<p class="p3">1.30940</p>
<p class="p1"><span class="s1">1.31 <span class="Apple-converted-space"> </span></span><strong><em>A1 <span class="Apple-converted-space"> </span>N3</em></strong></p>
<p class="p1"><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">Candidates often did not make the connection between parts (a) and (b). The extraordinary number of failed analytical approaches in part (a) and correct use of the GDC to find the limits in part (b) suggests that candidates are equating the command term “solve” to mean use an algebraic approach to solve equations or inequalities, instead of their GDC. Many candidates appeared to interpret part (a) as something they should do by hand and often did not recognize that their answer to part (a) were the limits in part (b). Quite a few candidates failed to interpret a GDC solution of \(x = 5 \times {10^{ - 14}}\) correctly as \(x = 0\) and others found the solution \(x = 1.74\) as the only solution, ignoring the second intersection point until part (b).</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Candidates often did not make the connection between parts (a) and (b). The extraordinary number of failed analytical approaches in part (a) and correct use of the GDC to find the limits in part (b) suggests that candidates are equating the command term “solve” to mean use an algebraic approach to solve equations or inequalities, instead of their GDC. Many candidates appeared to interpret part (a) as something they should do by hand and often did not recognize that their answer to part (a) were the limits in part (b). Quite a few candidates failed to interpret a GDC solution of \(x = 5 \times {10^{ - 14}}\) correctly as \(x = 0\) and others found the solution \(x = 1.74\) as the only solution, ignoring the second intersection point until part (b).</p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Consider the graph of the semicircle given by \(f(x) = \sqrt {6x - {x^2}} \), for \(0 \leqslant x \leqslant 6\). A rectangle \(\rm{PQRS}\) is drawn with upper vertices \(\rm{R}\) and \(\rm{S}\) on the graph of \(f\), and \(\rm{PQ}\) on the \(x\)-axis, as shown in the following diagram.</span></p>
<p style="font: normal normal normal 21px/normal 'Times New Roman'; text-align: center; margin: 0px;"><img src="images/maths_7.png" alt></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">Let \({\text{OP}} = x\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Find \({\text{PQ}}\), giving your answer in terms of \(x\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Hence, write down an expression for the area of the rectangle, giving your answer in terms of \(x\).</span></p>
<div class="marks">[[N/A]]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find the rate of change of area when \(x = 2\).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b(i).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The area is decreasing for \(a < x < b\). Find the value of \(a\) and of \(b\).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b(ii).</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(i) valid approach (may be seen on diagram) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg </em>\({\text{Q}}\) to \(6\) is \(x\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{PQ}} = 6 - 2x\) <strong><em>A1 N2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) \(A = (6 - 2x)\sqrt {6x - {x^2}} \) <strong><em>A1 N1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">recognising \(\frac{{{\text{d}}A}}{{{\text{d}}x}}\) at \(x = 2\) needed (must be the derivative of area) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(\frac{{{\text{d}}A}}{{{\text{d}}x}} = - \frac{{7\sqrt 2 }}{2},{\text{ }} - 4.95\) <strong><em>A1 N2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">b(i).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(a = 0.879{\text{ }}b = 3\) <strong><em>A1A1 N2<br></em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><strong><em><span style="font-family: 'times new roman', times; font-size: medium;">[4 marks]</span><br></em></strong></p>
<div class="question_part_label">b(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>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The diagram below shows a plan for a window in the shape of a trapezium.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/tpc.png" alt></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Three sides of the window are \(2{\text{ m}}\) long. The angle between the sloping sides of the </span><span style="font-family: times new roman,times; font-size: medium;">window and the base is \(\theta \) , where \(0 < \theta < \frac{\pi }{2}\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Show that the area of the window is given by \(y = 4\sin \theta + 2\sin 2\theta \) .</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Zoe wants a window to have an area of \(5{\text{ }}{{\text{m}}^2}\). Find the two possible values of \(\theta \) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">John wants two windows which have the same area <em>A</em> but different values of \(\theta \) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find all possible values for <em>A</em> .</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of finding height, <em>h</em> <strong><em>(A1)</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\sin \theta = \frac{h}{2}\) </span><span style="font-family: times new roman,times; font-size: medium;">, \(2\sin \theta \)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of finding base of triangle, <em>b</em> <strong><em>(A1)</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\cos \theta = \frac{b}{2}\) </span><span style="font-family: times new roman,times; font-size: medium;">, \(2\cos \theta \)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">attempt to substitute valid values into a formula for the area of the window <strong><em>(M1)</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. two triangles plus rectangle, trapezium area formula</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct expression (must be in terms of \(\theta \) ) <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(2\left( {\frac{1}{2} \times 2\cos \theta \times 2\sin \theta } \right) + 2 \times 2\sin \theta \) , \(\frac{1}{2}(2\sin \theta )(2 + 2 + 4\cos \theta )\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">attempt to replace \(2\sin \theta \cos \theta \) by \(\sin 2\theta \) <strong><em>M1</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(4\sin \theta + 2(2\sin \theta \cos \theta )\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(y = 4\sin \theta + 2\sin 2\theta \) <em><strong>AG N0</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[5 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct equation <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(y = 5\) , \(4\sin \theta + 2\sin 2\theta = 5\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of attempt to solve <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. a sketch, \(4\sin \theta + 2\sin \theta - 5 = 0\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(\theta = 0.856\) \(({49.0^ \circ })\) , \(\theta = 1.25\) \(({71.4^ \circ })\) <em><strong>A1A1 N3</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">recognition that lower area value occurs at \(\theta = \frac{\pi }{2}\) </span><strong><em><span style="font-family: times new roman,times; font-size: medium;">(M1)</span></em></strong></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">finding value of area at \(\theta = \frac{\pi }{2}\) </span><strong><em><span style="font-family: times new roman,times; font-size: medium;">(M1)</span></em></strong></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(4\sin \left( {\frac{\pi }{2}} \right) + 2\sin \left( {2 \times \frac{\pi }{2}} \right)\) , </span><span style="font-family: times new roman,times; font-size: medium;">draw square</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(A = 4\) <em><strong>(A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">recognition that maximum value of <em>y</em> is needed <strong><em>(M1)</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(A = 5.19615 \ldots \) <strong><em>(A1)</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(4 < A < 5.20\) (accept \(4 < A < 5.19\) ) <strong><em>A2 N5</em></strong></span></p>
<p><strong><em><span style="font-family: times new roman,times; font-size: medium;">[7 marks]</span></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;">
<p><span style="font-family: times new roman,times; font-size: medium;">As the final question of the paper, this question was understandably challenging for the majority of the candidates. Part (a) was generally attempted, but often with a lack of method or correct reasoning. Many candidates had difficulty presenting their ideas in a clear and organized manner. Some tried a "working backwards" approach, earning no marks. </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">In part (b), most candidates understood what was required and set up an equation, but many did not make use of the GDC and instead attempted to solve this equation algebraically which did not result in the correct solution. A common error was finding a second solution outside the domain. </span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">A pleasing number of stronger candidates made progress on part (c), recognizing the need for the end point of the domain and/or the maximum value of the area function (found graphically, analytically, or on occasion, geometrically). However, it was evident from candidate work and teacher comments that some candidates did not understand the wording of the question. This has been taken into consideration for future paper writing. </span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: TimesNewRomanPSMT;">Consider </span><span style="font-family: TimesNewRomanPS-ItalicMT;">\(f(x) = x\ln (4 - {x^2})\)</span><span style="font-family: TimesNewRomanPSMT;"> , for </span><span style="font-family: Times New Roman;" lang="JA">\( - 2 < x < 2\)</span><span style="font-family: TimesNewRomanPSMT;"> . The graph of </span><em><span style="font-family: TimesNewRomanPS-ItalicMT;">f </span></em><span style="font-family: TimesNewRomanPSMT;">is given below.</span></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: TimesNewRomanPSMT;"><br><img src="images/witch.png" alt></span></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Let P and Q be points on the curve of <em>f</em> where the tangent to the graph of <em>f</em> is </span><span style="font-family: times new roman,times; font-size: medium;">parallel to the <em>x</em>-axis.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Find the <em>x</em>-coordinate of P and of Q.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) Consider \(f(x) = k\) . Write down all values of <em>k</em> for which there are </span><span style="font-family: times new roman,times; font-size: medium;">exactly two solutions.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Let \(g(x) = {x^3}\ln (4 - {x^2})\) , for \( - 2 < x < 2\) .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Show that \(g'(x) = \frac{{ - 2{x^4}}}{{4 - {x^2}}} + 3{x^2}\ln (4 - {x^2})\)</span><span style="font-family: times new roman,times; font-size: medium;"> .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: times new roman,times; font-size: medium;">Let \(g(x) = {x^3}\ln (4 - {x^2})\) , for \( - 2 < x < 2\) .</span></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Sketch the graph of \(g'\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: times new roman,times; font-size: medium;">Let \(g(x) = {x^3}\ln (4 - {x^2})\) , for \( - 2 < x < 2\) .</span></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Consider \(g'(x) = w\) . Write down all values of <em>w</em> for which there are exactly </span><span style="font-family: times new roman,times; font-size: medium;">two solutions.</span></p>
<p align="LEFT"> </p>
<p align="LEFT"> </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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) \( - 1.15{\text{, }}1.15\) <em><strong>A1A1 N2</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) recognizing that it occurs at P and Q <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(x = - 1.15\) , \(x = 1.15\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(k = - 1.13\) , \(k = 1.13\) <em><strong>A1A1 N3</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[5 marks]</span></strong></em></p>
<div class="question_part_label">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of choosing the product rule <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(uv' + vu'\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">derivative of \({x^3}\) is \(3{x^2}\) <em><strong>(A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">derivative of \(\ln (4 - {x^2})\) is \(\frac{{ - 2x}}{{4 - {x^2}}}\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">(A1)</span></strong></em></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct substitution <em><strong> A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \({x^3} \times \frac{{ - 2x}}{{4 - {x^2}}} + \ln (4 - {x^2}) \times 3{x^2}\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(g'(x) = \frac{{ - 2{x^4}}}{{4 - {x^2}}} + 3{x^2}\ln (4 - {x^2})\) <em><strong>AG N0</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/hill.png" alt></span><em><strong><span style="font-family: times new roman,times; font-size: medium;"> A1A1 N2</span></strong></em></p>
<p><em><strong> <span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(w = 2.69\) , \(w < 0\) <em><strong>A1A2 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Many candidates correctly found the <em>x</em>-coordinates of P and Q in (a)(i) with their GDC. In </span><span style="font-family: times new roman,times; font-size: medium;">(a)(ii) some candidates incorrectly interpreted the words “exactly two solutions” as an </span><span style="font-family: times new roman,times; font-size: medium;">indication that the discriminant of a quadratic was required. Many failed to realise that the </span><span style="font-family: times new roman,times; font-size: medium;">values of <em>k</em> they were looking for in this question were the <em>y</em>-coordinates of the points found in </span><span style="font-family: times new roman,times; font-size: medium;">(a)(i).</span></p>
<div class="question_part_label">a(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Many candidates were unclear in their application of the product formula in the verifying the </span><span style="font-family: times new roman,times; font-size: medium;">given derivative of <em>g</em>. Showing that the derivative was the given expression often received full </span><span style="font-family: times new roman,times; font-size: medium;">marks though it was not easy to tell in some cases if that demonstration came through </span><span style="font-family: times new roman,times; font-size: medium;">understanding of the product and chain rules or from reasoning backwards from the given </span><span style="font-family: times new roman,times; font-size: medium;">result.</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Some candidates drew their graphs of the derivative in (c) on their examination papers despite </span><span style="font-family: times new roman,times; font-size: medium;">clear instructions to do their work on separate sheets. Most who tried to plot the graph in (c) </span><span style="font-family: times new roman,times; font-size: medium;">did so successfully.</span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Correct solutions to 10(d) were not often seen.</span></p>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">A particle starts from point \(A\) and moves along a straight line. Its velocity, \(v\;{\text{m}}{{\text{s}}^{ - 1}}\), after \(t\) seconds is given by \(v(t) = {{\text{e}}^{\frac{1}{2}\cos t}} - 1\), for \(0 \le t \le 4\). The particle is at rest when \(t = \frac{\pi }{2}\).</p>
<p class="p1">The following diagram shows the graph of \(v\).</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2015-12-14_om_09.53.11.png" alt></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the distance travelled by the particle for \(0 \le t \le\ \frac{\pi }{2}\).</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 class="p1">Explain why the particle passes through \(A\) again.</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 class="p1">correct substitution of function and/or limits into formula <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1">(accept absence of d\(t\), but do not accept any errors)</p>
<p class="p1"><em>eg</em>\(\;\;\;\)\(\int_0^{\frac{\pi }{2}} {v,{\text{ }}\int {\left| {{{\text{e}}^{\frac{1}{2}\cos t}} - 1} \right|{\text{d}}t,{\text{ }}\int {\left( {{{\text{e}}^{\frac{1}{2}\cos t}} - 1} \right)} } } \)</p>
<p class="p1">\(0.613747\)</p>
<p class="p1">distance is \(0.614{\text{ }}[0.613,{\text{ }}0.614]{\text{ (m)}}\) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1"><strong>METHOD 1</strong></p>
<p class="p1">valid attempt to find the distance travelled between \(t = \frac{\pi }{2}\) and \(t = 4\) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\)\(\int_{\frac{\pi }{2}}^4 {\left( {{{\text{e}}^{\frac{1}{2}\cos t}} - 1} \right),{\text{ }}\int_0^4 {\left| {{{\text{e}}^{\frac{1}{2}\cos t}} - 1} \right|{\text{d}}t - 0.614} } \)</p>
<p class="p1">distance is \(0.719565\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">valid reason, referring to change of direction (may be seen in explanation) <span class="Apple-converted-space"> </span><strong><em>R1</em></strong></p>
<p class="p1">valid explanation comparing <strong>their </strong>distances <span class="Apple-converted-space"> </span><strong><em>R1</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\)\(0.719565 > 0.614\), distance moving back is more than distance moving forward</p>
<p class="p2"> </p>
<p class="p1"><strong>Note: <span class="Apple-converted-space"> </span></strong>Do not award the final <strong><em>R1 </em></strong>unless the <strong><em>A1 </em></strong>is awarded.</p>
<p class="p2"> </p>
<p class="p3">particle passes through \(A\) again <span class="Apple-converted-space"> </span><strong><em>AG <span class="Apple-converted-space"> </span>N0</em></strong></p>
<p class="p1"><strong>METHOD 2</strong></p>
<p class="p1">valid attempt to find displacement <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\)\(\int_{\frac{\pi }{2}}^4 {\left( {{{\text{e}}^{\frac{1}{2}\cos t}} - 1} \right),{\text{ }}\int_0^4 {\left( {{{\text{e}}^{\frac{1}{2}\cos t}} - 1} \right)} } \)</p>
<p class="p1">correct displacement <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\)\(-0.719565,{\text{ }}-0.105817\)</p>
<p class="p1">recognizing that displacement from \(0\) to \(\frac{\pi }{2}\) is positive <span class="Apple-converted-space"> </span><strong><em>R1</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\)displacement = distance from \(0\) to \(\frac{\pi }{2}\)</p>
<p class="p1">valid explanation referring to positive and negative displacement <span class="Apple-converted-space"> </span><strong><em>R1</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\)\(0.719565 > 0.614\), overall displacement is negative, since displacement after \(\frac{\pi }{2}\) is negative, then particle gone backwards more than forwards</p>
<p class="p2"> </p>
<p class="p1"><strong>Note: <span class="Apple-converted-space"> </span></strong>Do not award the final <strong><em>R1 </em></strong>unless the <strong><em>A1 </em></strong>and the first <strong><em>R1 </em></strong>are awarded.</p>
<p class="p2"> </p>
<p class="p1">particle passes through A\(A\) again <span class="Apple-converted-space"> </span><span class="s1"><strong><em>AG <span class="Apple-converted-space"> </span>N0</em></strong></span></p>
<p class="p1"><span class="s1"><strong><em>[4 marks]</em></strong></span></p>
<p class="p1"><span class="s1"><strong>Note: Special Case. </strong></span></p>
<p class="p1"><span class="s1">If all working shown, and candidates seem to have misread the question, using [equation], award marks as follows:<br></span></p>
<p class="p1"><span class="s1">(a) correct substitution of function and/or limits into formula (accept absence of dt, but do not accept any errors) <em><strong>A0MR</strong></em></span></p>
<p class="p1"><span class="s1">eg [equation]</span></p>
<p class="p1"><span class="s1">\(2.184544\)</span></p>
<p class="p1"><span class="s1">distance is \(2.18\) [\(2.18\), \(2.19\)] (m) <em><strong>A1 N0</strong></em></span></p>
<p class="p1"><span class="s1">(b) <strong>METHOD 1</strong></span></p>
<p class="p1"><span class="s1">valid attempt to find the distance travelled between [equation] <em><strong>M1</strong></em></span></p>
<p class="p1"><span class="s1">eg [equation]</span></p>
<p class="p1"><span class="s1">distance is \(1.709638\) <em><strong>A1</strong></em></span></p>
<p class="p1"><span class="s1">reference to change of direction (may be seen in explanation) <strong><em> R1</em></strong></span></p>
<p class="p1"><span class="s1">reasoning/stating particle passes/does not pass through \(A\) again <em><strong>R0</strong></em></span></p>
<p class="p1"><strong><span class="s1">METHOD 2</span></strong></p>
<p class="p1"><span class="s1">valid attempt to find displacement <em><strong>M1</strong></em></span></p>
<p class="p1"><span class="s1">eg [equation]</span></p>
<p class="p1"><span class="s1">correct displacement <em><strong>A1</strong></em></span></p>
<p class="p1"><span class="s1">eg \(1.709638,{\rm{ }}3.894182\)</span></p>
<p class="p1"><span class="s1">recognising that displacement from [ \(0\) to (pi/2] is positive <em><strong>R0</strong></em></span></p>
<p class="p1"><span class="s1">reasoning/stating particle passes/does not pass through \(A\) again <em><strong>R0</strong></em></span></p>
<p class="p1"><span class="s1">With method 2, there is no valid reasoning about whether the particle passes through \(A\) again or not, so they cannot gain the <em><strong>R</strong></em> marks.</span></p>
<p class="p1"><em><strong><span class="s1">Total [6 marks]</span></strong></em></p>
<p class="p1"> </p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">For part (a), a large number of candidates chose the correct formula to find the distance but many got an incorrect value. A considerable number of candidates misread the function as \(v(t) = {{\text{e}}^{\frac{1}{2}\cos t}}\), losing a mark for this part.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Only a few candidates gained full marks in part (b). Although many mentioned the change of direction, very few supported their answer with a calculation of the distance travelled back or the displacement, thus showing poor understanding of the command term “explain”.</p>
<p class="p1">The periodic nature of the function confused many candidates, who used this fact to assure that the particle would pass through A again.</p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Let \(f(x) = \ln x\) and \(g(x) = 3 + \ln \left( {\frac{x}{2}} \right)\), for \(x > 0\).</p>
<p>The graph of \(g\) can be obtained from the graph of \(f\) by two transformations:</p>
<p>\[\begin{array}{*{20}{l}} {{\text{a horizontal stretch of scale factor }}q{\text{ followed by}}} \\ {{\text{a translation of }}\left( {\begin{array}{*{20}{c}} h \\ k \end{array}} \right).} \end{array}\]</p>
</div>
<div class="specification">
<p>Let \(h(x) = g(x) \times \cos (0.1x)\), for \(0 < x < 4\). The following diagram shows the graph of \(h\) and the line \(y = x\).</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-14_om_10.34.27.png" alt="M17/5/MATME/SP2/ENG/TZ1/10.b.c"></p>
<p>The graph of \(h\) intersects the graph of \({h^{ - 1}}\) at two points. These points have \(x\) coordinates 0.111 and 3.31 correct to three significant figures.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the value of \(q\);</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>Write down the value of \(h\);</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>Write down the value of \(k\).</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>Find \(\int_{0.111}^{3.31} {\left( {h(x) - x} \right){\text{d}}x} \).</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>Hence, find the area of the region enclosed by the graphs of \(h\) and \({h^{ - 1}}\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Let \(d\) be the vertical distance from a point on the graph of \(h\) to the line \(y = x\). There is a point \({\text{P}}(a,{\text{ }}b)\) on the graph of \(h\) where \(d\) is a maximum.</p>
<p>Find the coordinates of P, where \(0.111 < a < 3.31\).</p>
<div class="marks">[7]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>\(q = 2\) <strong><em>A1</em></strong> <strong><em>N1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Accept \(q = 1\), \(h = 0\), and \(k = 3 - \ln (2)\), 2.31 as candidate may have rewritten \(g(x)\) as equal to \(3 + \ln (x) - \ln (2)\).</p>
<p> </p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\(h = 0\) <strong><em>A1</em></strong> <strong><em>N1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Accept \(q = 1\), \(h = 0\), and \(k = 3 - \ln (2)\), 2.31 as candidate may have rewritten \(g(x)\) as equal to \(3 + \ln (x) - \ln (2)\).</p>
<p> </p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\(k = 3\) <strong><em>A1</em></strong> <strong><em>N1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Accept \(q = 1\), \(h = 0\), and \(k = 3 - \ln (2)\), 2.31 as candidate may have rewritten \(g(x)\) as equal to \(3 + \ln (x) - \ln (2)\).</p>
<p> </p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>2.72409</p>
<p>2.72 <strong><em>A2</em></strong> <strong><em>N2</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>recognizing area between \(y = x\) and \(h\) equals 2.72 <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)<img src="images/Schermafbeelding_2017-08-14_om_17.00.04.png" alt="M17/5/MATME/SP2/ENG/TZ1/10.b.ii/M"></p>
<p>recognizing graphs of \(h\) and \({h^{ - 1}}\) are reflections of each other in \(y = x\) <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)area between \(y = x\) and \(h\) equals between \(y = x\) and \({h^{ - 1}}\)</p>
<p>\(2 \times 2.72\int_{0.111}^{3.31} {\left( {x - {h^{ - 1}}(x)} \right){\text{d}}x = 2.72} \)</p>
<p>5.44819</p>
<p>5.45 <strong><em>A1</em></strong> <strong><em>N3</em></strong></p>
<p><strong><em>[??? marks]</em></strong></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>valid attempt to find \(d\) <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)difference in \(y\)-coordinates, \(d = h(x) - x\)</p>
<p>correct expression for \(d\) <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\left( {\ln \frac{1}{2}x + 3} \right)(\cos 0.1x) - x\)</p>
<p>valid approach to find when \(d\) is a maximum <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)max on sketch of \(d\), attempt to solve \(d’ = 0\)</p>
<p>0.973679</p>
<p>\(x = 0.974\) <strong><em>A2 N4 </em></strong></p>
<p>substituting <strong>their</strong> \(x\) value into \(h(x)\) <strong><em>(M1)</em></strong></p>
<p>2.26938</p>
<p>\(y = 2.27\) <strong><em>A1</em></strong> <strong><em>N2</em></strong></p>
<p><strong><em>[7 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = x\ln (4 - {x^2})\) , for \( - 2 < x < 2\) . The graph of <em>f</em> is shown below.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/troy.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The graph of <em>f</em> crosses the <em>x</em>-axis at \(x = a\) , \(x = 0\) and \(x = b\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the value of <em>a</em> and of <em>b</em> .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">The graph of <em>f</em> has a maximum value when \(x = c\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find the value of <em>c</em> .</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The region under the graph of <em>f</em> from \(x = 0\) to \(x = c\) is rotated \({360^ \circ }\) about </span><span style="font-family: times new roman,times; font-size: medium;">the <em>x</em>-axis. Find the volume of the solid formed.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Let <em>R</em> be the region enclosed by the curve, the <em>x</em>-axis and the line \(x = c\) , </span><span style="font-family: times new roman,times; font-size: medium;">between \(x = a\) and \(x = c\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find the area of <em>R</em> .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of valid approach <em><strong> (M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(f(x) = 0\) , graph</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(a = - 1.73\) , \(b = 1.73\) \((a = - \sqrt 3 {\text{, }}b = \sqrt 3 )\) <em><strong>A1A1 N3</strong></em></span></p>
<p><span style="font-family: times new roman,times;"><em><strong><span style="font-size: medium;">[3 marks]</span></strong></em></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">attempt to find max <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. setting \(f'(x) = 0\) , graph</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(c = 1.15\) (accept (1.15, 1.13)) <em><strong>A1 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">attempt to substitute either limits or the function into formula <em><strong>M1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(V = \pi {\int_0^c {\left[ {f(x)} \right]} ^2}{\rm{d}}x\) , \(\pi {\int {\left[ {x\ln (4 - {x^2})} \right]} ^2}\) , \(\pi \int_0^{1.149 \ldots } {{y^2}{\rm{d}}x} \)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(V = 2.16\) <em><strong>A2 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">valid approach recognizing 2 regions <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. finding 2 areas</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct working <em><strong>(A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\int_0^{ - 1.73 \ldots } {f(x){\rm{d}}x + } \int_0^{1.149 \ldots } {f(x){\rm{d}}x} \) , \( - \int_{ - 1.73 \ldots }^0 {f(x){\rm{d}}x + } \int_0^{1.149 \ldots } {f(x){\rm{d}}x} \)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">area \( = 2.07\) (accept 2.06) <em><strong>A2 N3</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks]</span></strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">This question was well done by many candidates. If there were problems, it was often with incorrect or inappropriate GDC use. For example, some candidates used the trace feature to answer parts (a) and (b), which at best, only provides an approximation. </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">This question was well done by many candidates. If there were problems, it was often with incorrect or inappropriate GDC use. For example, some candidates used the trace feature to answer parts (a) and (b), which at best, only provides an approximation. </span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Most candidates were able to set up correct expressions for parts (c) and (d) and if they had used their calculators, could find the correct answers. Some candidates omitted the important parts of the volume formula. Analytical approaches to (c) and (d) were always futile and no marks were gained. </span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Most candidates were able to set up correct expressions for parts (c) and (d) and if they had used their calculators, could find the correct answers. Some candidates omitted the important parts of the volume formula. Analytical approaches to (c) and (d) were always futile and no marks were gained. </span></p>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">A farmer wishes to create a rectangular enclosure, ABCD, of area 525 m<sup>2</sup>, as shown below.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/friends.png" alt></span></p>
</div>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">The fencing used for side AB costs \(\$ 11\) per metre. The fencing for the other three sides </span><span style="font-family: times new roman,times; font-size: medium;">costs \(\$ 3\) per metre. The farmer creates an enclosure so that the cost is a minimum. Find this minimum cost.</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>METHOD 1</strong> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct expression for <strong>second</strong> side, using area = 525 <em><strong>(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. let \({\rm{AB}} = x\) , \({\rm{AD}} = \frac{{525}}{x}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">attempt to set up cost function using $3 for three sides and $11 for one side <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(3({\rm{AD}} + {\rm{BC}} + {\rm{CD}}) + 11{\rm{AB}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct expression for cost <em><strong>A2</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{{525}}{x} \times 3 + \frac{{525}}{x} \times 3 + 11x + 3x\) , \(\frac{{525}}{{{\rm{AB}}}} \times 3 + \frac{{525}}{{{\rm{AB}}}} \times 3 + 11{\rm{AB}} + 3{\rm{AB}}\) , \(\frac{{3150}}{x} + 14x\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>EITHER</strong> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">sketch of cost function <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">identifying minimum point <em><strong>(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. marking point on graph, \(x = 15\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">minimum cost is 420 (dollars) <em><strong>A1 N4</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>OR</strong> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct derivative (may be seen in equation below) <em><strong>(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(C'(x) = \frac{{ - 1575}}{{{x^2}}} + \frac{{ - 1575}}{{{x^2}}} + 14\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">setting their derivative equal to 0 (seen anywhere) <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{{ - 3150}}{{{x^2}}} + 14 = 0\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">minimum cost is 420 (dollars) <em><strong>A1 N4</strong></em></span></p>
<p><strong> <span style="font-family: times new roman,times; font-size: medium;">METHOD 2</span></strong></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct expression for <strong>second</strong> side, using area = 525 <em><strong>(A1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. let \({\rm{AD}} = x\) , \({\rm{AB}} = \frac{{525}}{x}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">attempt to set up cost function using \(\$ 3\) for three sides and \(\$ 11\) for one side <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(3({\rm{AD}} + {\rm{BC}} + {\rm{CD}}) + 11{\rm{AB}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct expression for cost <em><strong>A2</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(3\left( {x + x + \frac{{525}}{x}} \right) + \frac{{525}}{x} \times 11\) , \(3\left( {{\rm{AD}} + {\rm{AD}} + \frac{{525}}{{{\rm{AD}}}}} \right) + \frac{{525}}{{{\rm{AD}}}} \times 11\) , \(6x + \frac{{7350}}{x}\)</span></p>
<p><strong> <span style="font-family: times new roman,times; font-size: medium;">EITHER</span></strong></p>
<p><span style="font-family: times new roman,times; font-size: medium;">sketch of cost function <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">identifying minimum point <em><strong>(A1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. marking point on graph, \(x = 35\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">minimum cost is 420 (dollars) <em><strong>A1 N4</strong></em></span></p>
<p><strong> <span style="font-family: times new roman,times; font-size: medium;">OR</span></strong></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct derivative (may be seen in equation below) <em><strong>(A1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(C'(x) = 6 - \frac{{7350}}{{{x^2}}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">setting their derivative equal to 0 (seen anywhere) <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(6 - \frac{{7350}}{{{x^2}}} = 0\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">minimum cost is 420 (dollars) <em><strong>A1 N4</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> [7 marks]</span></strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">Although this question was a rather straight-forward optimisation question, the lack of structure caused many candidates difficulty. Some were able to calculate cost values but were unable to create an algebraic cost function. Those who were able to create a cost function in two variables often could not use the area relationship to obtain a function in a single variable and so could make no further progress. Of those few who created a correct cost function, most set the derivative to zero to find that the minimum cost occurred at \(x = 15\) , leading to \(\$ 420\). Although this is a correct approach earning full marks, candidates seem not to recognise that the result can be obtained from the GDC, without the use of calculus. </span></p>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = {{\rm{e}}^{\frac{x}{4}}}\)</span><span style="font-family: times new roman,times; font-size: medium;"> and \(g(x) = mx\) , where \(m \ge 0\) , and \( - 5 \le x \le 5\) . Let \(R\) be the region </span><span style="font-family: times new roman,times; font-size: medium;">enclosed by the \(y\)-axis, the graph of \(f\) , and the graph of \(g\) .</span></p>
</div>
<div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(m = 1\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) Sketch the graphs of \(f\) and \(g\) on the same axes.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) Find the area of \(R\) .</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the area of \(R\) .</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Consider all values of \(m\) such that the graphs of \(f\) and \(g\) intersect. Find the </span><span style="font-family: times new roman,times; font-size: medium;">value of \(m\) that gives the greatest value for the area of \(R\) .</span></p>
<div class="marks">[8]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><em><span style="font-family: times new roman,times; font-size: medium;"><strong> (i)</strong></span></em></p>
<p><em><span style="font-family: times new roman,times; font-size: medium;"><strong> A1A1 N2</strong> </span></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Notes</strong>: Award <em><strong>A1</strong></em> for the graph of \(f\) positive, increasing and concave up. </span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"> Award <strong><em>A1</em></strong> for graph of \(g\) increasing and linear with \(y\)-intercept of \(0\). </span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"> Penalize one mark if domain is not [\( - 5\), \(5\)] and/or if \(f\) and \(g\) do not intersect in the first quadrant. </span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> [2 marks]</span></strong></em></p>
<p> </p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"><br>(ii)<br>attempt to find intersection of the graphs of \(f\) and \(g\) (M1)<br><br>eg \({{\rm{e}}^{\frac{x}{4}}} = x\)<br><br>\(x = 1.42961 \ldots \) A1<br><br>valid attempt to find area of \(R\) (M1)<br><br>eg \(\int {(x - {{\rm{e}}^{\frac{x}{4}}}} ){\rm{d}}x\) , \(\int_0^1 {(g - f)} \) , \(\int {(f - g)} \)<br><br>area \( = 0.697\) A2 N3<br><br><br>[5 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">attempt to find intersection of the graphs of \(f\) and \(g\)<strong> <em>(M1) </em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \({{\rm{e}}^{\frac{x}{4}}} = x\)</span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\(x = 1.42961 \ldots \) </span><strong><span style="font-family: times new roman,times; font-size: medium;"><em>A1</em> </span></strong></p>
<p><span style="font-family: times new roman,times; font-size: medium;">valid attempt to find area of \(R\) <strong><em>(M1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(\int {(x - {{\rm{e}}^{\frac{x}{4}}}} ){\rm{d}}x\) , \(\int_0^1 {(g - f)} \) , \(\int {(f - g)} \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">area \( = 0.697\) <em><strong>A2 N3 </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em><strong>[5 marks]<br></strong></em></span></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">recognize that area of \(R\) is a maximum at point of tangency <strong><em>(R1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(m = f'(x)\) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">equating functions <em><strong> (M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(f(x) = g(x)\) , \({{\rm{e}}^{\frac{x}{4}}} = mx\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(f'(x) = \frac{1}{4}{{\rm{e}}^{\frac{x}{4}}}\) <strong><em> (A1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">equating gradients <strong><em>(A1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(f'(x) = g'(x)\) , \(\frac{1}{4}{{\rm{e}}^{\frac{x}{4}}} = m\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">attempt to solve system of two equations for \(x\) <strong><em>(M1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(\frac{1}{4}{{\rm{e}}^{\frac{x}{4}}} \times x = {{\rm{e}}^{\frac{x}{4}}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(x = 4\) <strong><em>(A1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">attempt to find \(m\) <em><strong>(M1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(f'(4)\) , \(\frac{1}{4}{{\rm{e}}^{\frac{4}{4}}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(m = \frac{1}{4}e\) (exact), \(0.680\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A1 N3 </span></strong></em></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[8 marks] </span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">There was a flaw with the domain noted in this question. While not an error in itself, it meant that part (b) no longer assessed what was intended. The markscheme included a variety of solutions based on candidate work seen, and examiners were instructed to notify the IB assessment centre of any candidates adversely affected, and these were looked at during the grade award meeting.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">While some candidates sketched accurate graphs on the given domain, the majority did not. Besides the common domain error, some exponential curves were graphed with several concavity changes.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">There was a flaw with the domain noted in this question. While not an error in itself, it meant that part (b) no longer assessed what was intended. The markscheme included a variety of solutions based on candidate work seen, and examiners were instructed to notify the IB assessment centre of any candidates adversely affected, and these were looked at during the grade award meeting.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">In part (a)(ii), most candidates found the intersection correctly. Those who used their GDC to evaluate the integral numerically were usually successful, unlike those who attempted to solve with antiderivatives. A common error was to find the area of the region enclosed by \(f\) and \(g\) (although it involved a point of intersection outside of the given domain), rather than the area of the region enclosed by \(f\) and \(g\) and the \(y\)-axis.<br></span></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">There was a flaw with the domain noted in this question. While not an error in itself, it meant that part (b) no longer assessed what was intended. The markscheme included a variety of solutions based on candidate work seen, and examiners were instructed to notify the IB assessment centre of any candidates adversely affected, and these were looked at during the grade award meeting.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">While some candidates were able to show some good reasoning in part (b), fewer were able to find the value of \(m\) which maximized the area of the region. In addition to the answer obtained from the restricted domain, full marks were awarded for the answer obtained by using the point of tangency.<br></span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Let \(f(x) = \frac{{3x}}{{x - q}}\), where \(x \ne q\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Write down the equations of the vertical and horizontal asymptotes of the graph of \(f\).</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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The vertical and horizontal asymptotes to the graph of \(f\) intersect at the point \({\text{Q}}(1,3)\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find the value of \(q\).</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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The vertical and horizontal asymptotes to the graph of \(f\) intersect at the point \({\text{Q}}(1,3)\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The point \({\text{P}}(x,{\text{ }}y)\) lies on the graph of \(f\). Show that \({\text{PQ}} = \sqrt {{{(x - 1)}^2} + {{\left( {\frac{3}{{x - 1}}} \right)}^2}} \).</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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The vertical and horizontal asymptotes to the graph of \(f\) intersect at the point \({\text{Q}}(1,3)\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Hence find the coordinates of the points on the graph of \(f\) that are closest to \((1,3)\).</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(x = q,{\text{ }}y = 3\) (must be equations) <strong><em>A1A1 N2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 17.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">recognizing connection between point of intersection and asymptote <strong><em>(R1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 17.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(x = 1\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 17.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(q = 1\) <strong><em>A1 N2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 17.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">correct substitution into distance formula <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(\sqrt {{{(x - 1)}^2} + {{(y - 3)}^2}} \)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">attempt to substitute \(y = \frac{{3x}}{{x - 1}}\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(\sqrt {{{(x - 1)}^2} + {{\left( {\frac{{3x}}{{x - 1}} - 3} \right)}^2}} \)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">correct simplification of \(\left( {\frac{{3x}}{{x - 1}} - 3} \right)\) <em><strong>(A1)</strong></em></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(\frac{{3x - 3x(x - 1)}}{{x - 1}}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">correct expression clearly leading to the required answer <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(\frac{{3x - 3x + 3}}{{x - 1}},{\text{ }}\sqrt {{{(x - 1)}^2} + {{\left( {\frac{{3x - 3x + 3}}{{x - 1}}} \right)}^2}} \)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{PQ}} = \sqrt {{{(x - 1)}^2} + {{\left( {\frac{3}{{x - 1}}} \right)}^2}} \) <strong><em>AG N0</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[4 marks]</em></strong></span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">recognizing that closest is when \({\text{PQ}}\) is a minimum <strong><em>(R1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> sketch of \({\text{PQ}}\), \(({\text{PQ}})'(x) = 0\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(x = - 0.73205{\text{ }}x = 2.73205\) (seen anywhere) <strong><em>A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">attempt to find <em>y</em>-coordinates <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(f( - 0.73205)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\((-0.73205, 1.267949) , (2.73205, 4.73205)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\((-0.732, 1.27) , (2.73, 4.73) \) <strong><em>A1A1 N4</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[6 marks]</em></strong></span></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The velocity \(v{\text{ m}}{{\text{s}}^{ - 1}}\) of a particle after \(t\) seconds is given by</p>
<p class="p1">\(v(t) = {(0.3t + 0.1)^t} - 4\), for \(0 \le t \le 5\)</p>
<p class="p1">The following diagram shows the graph of \(v\).</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2016-01-25_om_06.42.03.png" alt></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the value of \(t\) when the particle is at rest.</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 class="p1">Find the value of \(t\) when the acceleration of the particle is \(0\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">recognizing particle at rest when \(v = 0\) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;{(0.3t + 0.1)^t} - 4 = 0\), \(x\)-intercept on graph of \(v\)</p>
<p class="p1">\(t = 4.27631\)</p>
<p class="p1">\(t = 4.28{\text{ }} {\text{ (seconds)}}\) <span class="Apple-converted-space"> </span><strong><em>A2 <span class="Apple-converted-space"> </span>N3</em></strong></p>
<p class="p1"><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">valid approach to find \(t\) when \(a\) is \(0\) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;v'(t) = 0\), \(v\) minimum</p>
<p class="p1">\(t = 1.19236\)</p>
<p class="p1">\(t = 1.19{\text{ }}{\text{ (seconds)}}\) <span class="Apple-converted-space"> </span><strong><em>A2 <span class="Apple-converted-space"> </span>N3</em></strong></p>
<p class="p1"><strong><em>[3 marks]</em></strong></p>
<p class="p1"><strong><em>Total [6 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = A{{\rm{e}}^{kx}} + 3\) . Part of the graph of <em>f</em> is shown below.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/ryan.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The <em>y</em>-intercept is at (0, 13) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Show that \(A = 10\) .</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><span style="font-family: times new roman,times; font-size: medium;">Given that \(f(15) = 3.49\) (correct to 3 significant figures), find the value of <em>k</em>.</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Using your value of <em>k</em> , find \(f'(x)\) .</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) Hence, explain why <em>f</em> is a decreasing function.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(iii) Write down the equation of the horizontal asymptote of the graph <em>f</em> .</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">c(i), (ii) and (iii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Let \(g(x) = - {x^2} + 12x - 24\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find the area enclosed by the graphs of <em>f</em> and <em>g</em> .</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">substituting (0, 13) into function <em><strong>M1 </strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(13 = A{{\rm{e}}^0} + 3\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(13 = A + 3\) <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(A = 10\) <em><strong>AG N0</strong> </em></span></p>
<p><em><span style="font-family: times new roman,times; font-size: medium;"><strong>[2 marks]</strong> </span></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">substituting into \(f(15) = 3.49\) <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(3.49 = 10{{\rm{e}}^{15k}} + 3\) , \(0.049 = {{\rm{e}}^{15k}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of solving equation <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. sketch, using \(\ln \) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(k = - 0.201\) (accept \(\frac{{\ln 0.049}}{{15}}\) ) <em><strong>A1 N2</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks] </span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) \(f(x) = 10{{\rm{e}}^{ - 0.201x}} + 3\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(f(x) = 10{{\rm{e}}^{ - 0.201x}} \times - 0.201\) \(( = - 2.01{{\rm{e}}^{ - 0.201x}})\) <em><strong>A1A1A1 N3</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Award <em><strong>A1</strong></em> for \(10{{\rm{e}}^{ - 0.201x}}\) , <em><strong>A1</strong></em> for \( \times - 0.201\) , <em><strong>A1</strong></em> for the derivative of 3 is zero. </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) valid reason with reference to derivative <em><strong>R1 N1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(f'(x) < 0\) , derivative always negative </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(iii) \(y = 3\) <em><strong> A1 N1</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[5 marks] </span></strong></em></p>
<div class="question_part_label">c(i), (ii) and (iii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">finding limits \(3.8953 \ldots \), \(8.6940 \ldots \) (seen anywhere) <em><strong>A1A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of integrating and subtracting functions <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct expression <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\int_{3.90}^{8.69} {g(x) - f(x){\rm{d}}x} \) , \(\int_{3.90}^{8.69} {\left[ {\left( { - {x^2} + 12x - 24} \right) - \left( {10{{\rm{e}}^{ - 0.201x}} + 3} \right)} \right]} {\rm{d}}x\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">area \(= 19.5\) <em><strong>A2 N4</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[6 marks] </span></strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">This question was quite well done by a great number of candidates indicating that calculus is </span><span style="font-family: times new roman,times; font-size: medium;">a topic that is covered well by most centres. Parts (a) and (b) proved very accessible to many </span><span style="font-family: times new roman,times; font-size: medium;">candidates.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">This question was quite well done by a great number of candidates indicating that calculus is </span><span style="font-family: times new roman,times; font-size: medium;">a topic that is covered well by most centres. Parts (a) and (b) proved very accessible to many </span><span style="font-family: times new roman,times; font-size: medium;">candidates.</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The chain rule in part (c) was also carried out well. Few however, recognized the </span><span style="font-family: times new roman,times; font-size: medium;">command term “hence” and that </span><span style="font-family: times new roman,times; font-size: medium;">\(f'(x) < 0\) guarantees a decreasing function. A common </span><span style="font-family: times new roman,times; font-size: medium;">answer for the equation of the asymptote was to give </span><span style="font-family: times new roman,times; font-size: medium;">\(y = 0\) or </span><span style="font-family: times new roman,times; font-size: medium;">\(x = 3\) .</span></p>
<div class="question_part_label">c(i), (ii) and (iii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">In part (d), it was again </span><span style="font-family: times new roman,times; font-size: medium;">surprising and somewhat disappointing to see how few candidates were able to use their GDC </span><span style="font-family: times new roman,times; font-size: medium;">effectively to find the area between curves, often not finding correct limits, and often trying to </span><span style="font-family: times new roman,times; font-size: medium;">evaluate the definite integral without the GDC, which led nowhere.</span></p>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Let \(f(t) = 2{t^2} + 7\) , where \(t > 0\) . The function <em>v</em> is obtained when the graph of <em>f</em> is </span><span style="font-family: times new roman,times; font-size: medium;">transformed by</span></p>
<p style="margin-left: 60px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">a stretch by a scale factor of \(\frac{1}{3}\) </span><span style="font-family: times new roman,times; font-size: medium;">parallel to the <em>y</em>-axis,</span></p>
<p style="margin-left: 60px;"><span style="font-family: times new roman,times; font-size: medium;">followed by a translation by the vector \(\left( {\begin{array}{*{20}{c}}<br>2\\<br>{ - 4}<br>\end{array}} \right)\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find \(v(t)\) , giving your answer in the form \(a{(t - b)^2} + c\) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">A particle moves along a straight line so that its velocity in ms<sup>−1</sup> , at </span><span style="font-family: times new roman,times; font-size: medium;">time <em>t </em>seconds, is given by<em> v</em> . Find the distance the particle travels between </span><span style="font-family: times new roman,times; font-size: medium;">\(t = 5.0\) and \(t = 6.8\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">applies vertical stretch parallel to the <em>y</em>-axis factor of \(\frac{1}{3}\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">(M1)</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. multiply by \(\frac{1}{3}\) , \(\frac{1}{3}f(t)\) , \(\frac{1}{3} \times 2\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">applies horizontal shift 2 units to the right <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(f(t - 2)\) , \(t - 2\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">applies a vertical shift 4 units down <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. subtracting 4, \(f(t) - 4\) , \(\frac{7}{3} - 4\)</span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\(v(t) = \frac{2}{3}{(t - 2)^2} - \frac{5}{3}\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A1 N4</span></strong></em></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> [4 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">recognizing that distance travelled is area under the curve <em><strong>M1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\int {v,\frac{2}{9}} {(t - 2)^3} - \frac{5}{3}t\) </span><span style="font-family: times new roman,times; font-size: medium;">, sketch</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/belle.png" alt></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">distance = 15.576 (accept 15.6) <em><strong>A2 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">While a number of candidates had an understanding of each transformation, most had difficulty applying them in the correct order, and few obtained the completely correct answer in part (a). Many earned method marks for discerning three distinct transformations. Few candidates knew to integrate to find the distance travelled. Many instead substituted time values into the velocity function or its derivative and subtracted. A number of those who did recognize the need for integration attempted an analytic approach rather than using the GDC, which often proved unsuccessful.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">While a number of candidates had an understanding of each transformation, most had difficulty applying them in the correct order, and few obtained the completely correct answer in part (a). Many earned method marks for discerning three distinct transformations. Few candidates knew to integrate to find the distance travelled. Many instead substituted time values into the velocity function or its derivative and subtracted. A number of those who did recognize the need for integration attempted an analytic approach rather than using the GDC, which often proved unsuccessful.</span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Let \(f(x) = 2\ln (x - 3)\), for \(x > 3\). The following diagram shows part of the graph of \(f\).</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2016-01-22_om_17.05.00.png" alt></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the equation of the vertical asymptote to the graph of \(f\).</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 class="p1">Find the \(x\)-intercept of the graph of \(f\).</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 class="p1">The region enclosed by the graph of \(f\), the \(x\)-axis and the line \(x = 10\) <span class="s1">is rotated \(360\)° </span>about the \(x\)-axis. Find the volume of the solid formed.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">valid approach <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\)horizontal translation \(3\) units to the right</p>
<p class="p1">\(x = 3\) (must be an equation) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">valid approach <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;f(x) = 0,{\text{ }}{e^0} = x - 3\)</p>
<p class="p1">\(4,{\text{ }}x = 4,{\text{ }}(4,{\text{ }}0)\) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">attempt to substitute either <strong>their correct </strong>limits or the function into formula involving \({f^2}\) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\int_4^{10} {{f^2},{\text{ }}\pi \int {{{\left( {2\ln (x - 3)} \right)}^2}{\text{d}}x} } \)</p>
<p class="p1">\(141.537\)</p>
<p class="p1">volume = \(142\) <span class="Apple-converted-space"> </span><strong><em>A2 <span class="Apple-converted-space"> </span>N3</em></strong></p>
<p class="p1"><strong><em>[3 marks]</em></strong></p>
<p class="p1"><strong><em>Total [7 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>Let \(f(x) = - {x^4} + 2{x^3} - 1\), for \(0 \le x \le 2\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of \(f\) on the following grid.</p>
<p style="text-align: center;"><img src="image_1.html" alt></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 class="p1">Solve \(f(x) = 0\).</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 class="p1">The region enclosed by the graph of \(f\) and the \(x\)-axis is rotated \(360°\) about the <em>\(x\)</em>-axis.</p>
<p class="p1">Find the volume of the solid formed.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><img src="image_2.html" alt> <strong><em>A1A1A1 N3</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>A1 </em></strong>for both endpoints in circles,</p>
<p><strong><em>A1 </em></strong>for approximately correct shape (concave up to concave down).</p>
<p>Only if this <strong><em>A1 </em></strong>for shape is awarded, award <strong><em>A1 </em></strong>for maximum point in circle.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">\(x = 1\;\;\;x = 1.83928\)</p>
<p class="p1">\(x = 1{\text{ (exact)}}\;\;\;x = 1.84{\text{ }}[1.83,{\text{ }}1.84]\) <span class="Apple-converted-space"> </span><strong><em>A1A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">attempt to substitute either (<strong><em>FT</em></strong> ) limits or function into formula with \({f^2}\) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\;\;\;\)\(V = \pi \int_1^{1.84} {{f^2},{\text{ }}\int {{{( - {x^4} + 2{x^3} - 1)}^2}{\text{d}}x} } \)</p>
<p class="p1">\(0.636581\)</p>
<p class="p1">\(V = 0.637{\text{ }}[0.636,{\text{ }}0.637]\) <span class="Apple-converted-space"> </span><strong><em>A2 <span class="Apple-converted-space"> </span>N3</em></strong></p>
<p class="p1"><strong><em>[3 marks]</em></strong></p>
<p class="p1"><strong><em>Total [8 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;">
<p class="p1">Despite being a straightforward question, and although most candidates had a roughly correct shape for their graph, their sketches were either out of scale or missed one of the endpoints. In part (b), a few did not give both answers despite going on to use 1.84 in part (c).</p>
<p class="p1">Part (c) proved difficult for most candidates, as only a small number could write the correct expression for the volume: some included the correct limits but did not square the function, whilst others squared the function but did not write the correct limits in the integral. Many did not find a volume, or found an incorrect volume. The latter included finding the integral from 0 to 2, or dividing the region into three parts, showing a lack of understanding of “enclosed”.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Despite being a straightforward question, and although most candidates had a roughly correct shape for their graph, their sketches were either out of scale or missed one of the endpoints. In part (b), a few did not give both answers despite going on to use 1.84 in part (c).</p>
<p class="p1">Part (c) proved difficult for most candidates, as only a small number could write the correct expression for the volume: some included the correct limits but did not square the function, whilst others squared the function but did not write the correct limits in the integral. Many did not find a volume, or found an incorrect volume. The latter included finding the integral from 0 to 2, or dividing the region into three parts, showing a lack of understanding of “enclosed”.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">Despite being a straightforward question, and although most candidates had a roughly correct shape for their graph, their sketches were either out of scale or missed one of the endpoints. In part (b), a few did not give both answers despite going on to use 1.84 in part (c).</p>
<p class="p1">Part (c) proved difficult for most candidates, as only a small number could write the correct expression for the volume: some included the correct limits but did not square the function, whilst others squared the function but did not write the correct limits in the integral. Many did not find a volume, or found an incorrect volume. The latter included finding the integral from 0 to 2, or dividing the region into three parts, showing a lack of understanding of “enclosed”.</p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The diagram below shows part of the graph of the gradient function, \(y = f'(x)\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/bed.png" alt></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">On the grid below, sketch a graph of \(y = f''(x)\) , clearly indicating the <em>x</em>-intercept.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><img src="images/hsm2.png" alt></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><span style="font-family: times new roman,times; font-size: medium;">Complete the table, for the graph of \(y = f(x)\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><img src="images/abacus.png" alt></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><span style="font-family: times new roman,times; font-size: medium;">Justify your answer to part (b) (ii).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;"><img src="images/fin.png" alt></span><em><strong><span style="font-family: times new roman,times; font-size: medium;"> A1A1 N2</span></strong></em></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Award <em><strong>A1</strong></em> for negative gradient throughout, <em><strong>A1</strong></em> for <em>x</em>-intercept of <em>q</em>. </span><span style="font-family: times new roman,times; font-size: medium;">It need not be linear.</span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times;"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAYYAAABJCAIAAAC2KxZnAAAN/UlEQVR4nO2dvYvjSBrG69+oWKkzR9dRJadkLrjEyQYVOOmgswkEDhYabmHgBIaFgYWBgmZgoLmmYOAYWMyJhWO4waBsGIy45WgaIxg6aIQ4jGmKuqBKUkmW1PJHu0u+9xc1balUsl89qs/nRRIAAMAa0EtXAAAAoAAkCQAAiwBJAgDAIkCSAACwCJAkAAAsokaSEAAAwFHoJElKlQ4sfQBgExDhNgCSBAAaiHAbAEkCAA1EuA2AJAGA5jQjPI0C5hF0zuNHKaWMOUXED9OXrlYjIEkAoDnFCE9DnyCEUC5Je5IEP07DQxTUCEgSAGhONMKVKh1EklYRGzk+SBIAHIUTjfBDSZJIF1cUI5AkADgSB4rwO04dhBAe86WQIv48pQNUP3wj0uhXnw4QQgi5Hl/oI/ToD0IIIeJdhbEojg+Y5yKEEMLEex/GaylFGv1241PsXPJfLwnCxJ+nUop4fqWOJBceHWhJSqPgxqdYV0bEIfcpdt4E37hHMEKYTGbqYsXpCGH6dh6vlB5lqBKSaDZV/8R0OouSQ3x7IEkAkHG4CBdpOCVoxMLfpt77RSrqD1ryMR5Q9jWVD6H/CqGhF9xLccvHA0yvFqmQ6VdGBwi98sOH8vG6wYLINHyY+46SicH45p+ziYvHfJnMfeKQySwW2ZHonMdpxEaFoIgFc5XGDOj0cyxWERshNGLRSsr7wBsiMg3TxzScEpS1jGJO87/leskvHPp2Hq9FPJsQjMg0bLjTrQBJAgDNQSP8jlMH4Qu+XDcccB94Q+SySLdKZhNy5gX3ImKu0iYppZYhhFwWKcnAkyBRj/16yS8wwi5bCH2t/KPKkfeBN8w6bo8xPzeabHecOrnGPIa+gxzK77QsekGS/3dTksSCuWdZPZWcFdXeB5AkANAcNMJXERshyuOmz1UjpXpARTKywxw/fCzJh5RSRMzVAlH56I5TxyjZHEvqJkn6AnH48YPqV9ZIUsxpdR+Ice4egCQBgObwklQ0VSrPsEP/9qlOklRzwxx4ylRjnWuTNAuskSSlYln7aydJWsfztxQPqH8dfJvVt5KebX0TSBIAaA4Y4WL5ceK9prilL3MfeMONnp1Iggk2e0BiwVyMvSDRxxcaZ3TxKq2kSsnbS1ISeFh1CZs7bkngYT2UnlX1e/T7AUa4QZIAQHOwCE+/Mu+XMP0eeGcuWzwuP/54tdgY+FVD4EiPZEsp4mDKwlQPb7+dx2sp13FwSTJx0cPb08+xkFIsAzWSLczOXaVkY+wZIYRGLHooDfqUTlRqiF22EDGnevZNz6npcaWYU4SwN1uGH1j4fckvMHK9q3kspJRJxN/xaLX/lweSBACaA0S4esjJhEdJ9pBnIlLDOg7fe1ov8mdbyjSa+XrCvTy5bi4aGFD/1ygVqhWTdQdzVVI9L13CO+8MU5+H0Vyv5EYIOfT62jjxJ359bnQq/zVXk/uYTr/M3rlYr1EQy2DiovyORBxe1S5W2AuQJADQQITbAEgSAGggwm0AJAkANBDhNgCSBAAaiHAbAEkCAA1EuA2AJAGABiLcBrpKUnXtOAAAwPPQSZKa1AsATgaIcBsASQIADUS4DYAkAYAGItwG9pCkNJr552N+p/fyKf+E8bk/i+zNfrAj+Q2+dEW2wM46F1si9P4smwBJsoEdJUnEswkZbmzeEWnEPTKk7OtRVamwfRiM+e1mnGsfLIQOZelyWtwHP/5yFOVSDocDyr4mEXNN4w47AEmygZ0kSdzy8bDkS2CSzn0yrJWG50RZ56E6t03lIoqQfc+ADYiIucdpTBU+G5YCkmQDO0iSSIIJbnu8nzzgOUhD/xUhw82GkljysXNGzp4/tUIfUe7Ox/hmkgUbY23tbCkgSTawvSRV33V5ggQjrE0LqCORhj71Z9zDlYbSKmI/EP8DK9nuGaYKmE7nsSgy8CHk+OEqN3w45/G6dIM6qwPxg1A7SJAJj+4zLwjlMlPO55f3KymP5ToOuU8Hjj/7prM7uB5fJDqPBULkMogrbs1JFFz79IyyT1wljdB1zj9lGwYXoqHOc12Cuop2my/uu6xLtfkwmktroWSoeKAch89AjyUptyjBkyB5iPiE2J3StoXtJSnmtGTQm+U5KEV0xf33CKShT/3wvmpLngSec8GXvxvuecrF5pUfPsh07pO89bSOgzznzCpiI22jVbpBZcqnZOHtPF6rUSpMxm/4ItUO7erqpge70miEKI/zh1PJirjl4wHChL75GKVCl1bq2qjaFlfUOqJtvdZLfoHxmC2SrBmCiT9PS3VeZXkpsjpHzEX5C6Pq6FxcuDYfRvrf1tKasb7XJnstSVKqkMPezZern9nNG7ctE4HVbC1Jj6HvVAV4M6zvA2943I6SkqRUP/zaeHgVsRH2gqRUQ/UYq35lGvrEqKcadRpQ74KUmlrm6XVWoZnPseFeXM7np15iSqNLmWdUaVl3xjzMoOLTXqhA5TlXAqdLa65z6SpNktSSD6OltGaSwMO2zy30W5KU8yydsPDhpauyF88kSZVH/QhkkmSmxEoCD28+n4p1HH66UT0v8//6qa5YJj8lSTWG6ltJ0hOPdzV1RBJ4qjFVKkqWzeT3lKSWfBg7SJJq61k9kCR7Lknqd7C8HdqF3SSp5YlVvKAk5amvfubvRtkvVKqhSkCKqX8ThEGlnvnwSml8xDJJyhza14VM6A+M0vaTpKqds1ntHSQpDX1i/4xnnyVJtWr72lkz2XssScq6sH6psSQlE9msf/FaNmtopvQrS6dYBpPRmN8+qjGdYi3fDpKkGizPIklFaoq8uaQPNHtbe7aSWvJhbC9JfRhIkr2WpJ58w13Yf8Yte6Oa78AXmHG7V9kg1BWr48SlGt5x6iByGcQrvZJY/z9ZsLGjT9HZGnSK9NLpq+YcD+rxVDeureDH/FaIWJurI4QoX0bMLRrY5bxd6vEuMnBpdJvcTMqsh7rUuJiegFPrV/UaiJo6Z1cpDbdliVKXc8ZKvfHGfBhtpTUQc3qgVKjPSo8lqSffcBf2XpdkJkjQjYKjr0sy55iLptC5aiKVKogcyv+dTdgP6HQWvBshhIl3/Q82RgjpJ80scERpcf6fKB0WRZVyPPhfvpRTRRR9wAn/NvMd12N//8Ivi8Io40XGCOJ/CepSTeTfMSavX+t1Bvl8vJRS6rQ2ag5M7+Yp3bNZZ+JzVtyaWpQQXJJ8Oq9EXT4Mc7VETWm1vMg6tV3orSSp95+9qyu2YgdJsnP19ilTHUvqGWpyugd9it5K0kmxkyRJNQxMNhNUifjzlJJj73E7dfotScW8p+2AJNnArpIkTSeAnFN1AnhZjDSkL12VLUjnPnH9+SKYvGpsUFsGSJIN7CFJwDEo1otvDDHZjV7h1ZLr1Togwm2gqyQhAACAo9BJkprUCwBOBohwGwBJAgANRLgNgCQBgAYi3Ab2kKQa720ppVq4mG2zsnQCbh2H77XH0B+IO379mri7msu8uMX1HaeOfcPe+QLOeuthOwFJsoEdJanBe7vCsay4Sw5h6EmTMBExF41YtBLxpz/qU/rqd9UZkQTT6ZHuMVmwMcZjtviemcP0A5AkG9hJktpXb1c40mJutfGqy+KdVcRGRrumso32RBEL5u7cEtzyUsUG3Z4BkmQDO0jStluWjrPFqbuyVIxT/h8kSRlOHuUe1c6+9l24tgKSZAPbS1K793YaBTc+xeXoP4YxQElZtD208yb4xj2CUb6nv9LFK7xr8wobe031LtbOttz69Bq/6sb6lNjBY1udMihZbFevovTIvOXN626WbHiE8wlBCCF3ErQnXiut6jyuNc1h6Lkk5aGLif+R92Rf4SbbS1Kb93Zu81xnO/m8MWooS14lvXRY1SrfZtXWShJLPnaUK/YymLhIWXR3suWWssmvOvnWWh/9Pe7gsZ1EzM11pu2uW1qCT5SceYQvmIu7tH3622uT/ZYkFW+YeDxKV4YHfP/YWpKeMrqtjf4jWHF3dFaTrZK0ModjRcnYqIstd4tfdUt9jHvYwWO7ZAjXdJVmSepactkks5F+u2T0V5JE2XTwMfSd0/oVDi5JR/C9PYgk3XHDG6naAXnalrvFr3oXSerksb2nJHUtuaMkmY6d/aO3kqTel3m7exWx0Yn9Cvt4b/dekhqr+aQtd5tf9U6S1MVjez9JEl1L7iZJymuun0MYsr+SVPE7Tuc+cXrmG2Gw91iSlB0k6bhjSdUqdZek+8AbljNTruPoVg0ydbDlbvGr3rnj9pTH9p6tpK4ld5KkXg8kyf5KUumX6kFW4Xb2n3FrtaZWHGPGrbwuqdRgqfgNVYSjVGGx5GOcz5SJNPo45ZHoasvd7FfdVp+CHTy2SwNejVdRkjT0gihkHwzBlU+WnAW6ypTZHuiPMT+3Pw1JC32VJPVeGfPlYzyfTjzvh17YCjex97qkRpvn/G3//OuSqqu3DZyf+PV5UaW//sU80vGDL4WTtKqwsddEz4jnc9vtttzad7zGr9r8iir1KTeUtvXYLn33ZmU2riLi2YTg8qoC2bFkx7++LopuaSj1xtC2CdRTSZLZOg/iXYX/WfRqxfwmtb9CH1dvnwL9N7Ttca9N9liSDMSCuWen9yvs7r1dAay4t6LPkiSSYIJ7O8ujOAVJ6o/TeRO7SpKs9d6uYK0TgJ300WNbpOGUOP78bjYhf/ZPMRt9r3gI/Ve9skOuYQ9JAg5JXz22dbrz+lGqntHrCBf5gnv0VKZPu+kqSQgAAOAodJIkAACAlwIkCQAAiwBJAgDAIkCSAACwiP8BLAFG+hJf0ocAAAAASUVORK5CYII=" alt><span style="font-size: medium;"> </span></span><span style="font-size: medium; font-family: times new roman,times;"> <em><strong>A1A1 N1N1</strong></em></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><strong><span style="font-family: times new roman,times; font-size: medium;">METHOD 1</span></strong></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Second derivative is zero, second derivative changes sign. <em><strong>R1R1 N2</strong></em></span></p>
<p align="LEFT"><strong><span style="font-family: times new roman,times; font-size: medium;">METHOD 2</span></strong></p>
<p><span style="font-family: times new roman,times; font-size: medium;">There is a maximum on the graph of the first derivative. <em><strong>R2 N2</strong></em></span></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Several candidates had a correct sketch in part (a).</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The majority of the errors occurred in </span><span style="font-family: times new roman,times; font-size: medium;">parts (b) and (c). In part (b), some seemed to just guess while others left it blank.</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">In part (c), </span><span style="font-family: times new roman,times; font-size: medium;">justification lacked completeness. For example, many stated that the second derivative must </span><span style="font-family: times new roman,times; font-size: medium;">equal zero but said nothing of its change in sign.</span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">A particle moves along a straight line such that its velocity, \(v{\text{ m}}{{\text{s}}^{ - 1}}\), is given by \(v(t) = 10t{{\text{e}}^{ - 1.7t}}\), for \(t \geqslant 0\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 17.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">On the grid below, sketch the graph of \(v\), for \(0 \leqslant t \leqslant 4\).</span></p>
<p style="font: normal normal normal 17px/normal 'Times New Roman'; text-align: center; margin: 0px;"><img src="images/maths_5a.png" alt></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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find the distance travelled by the particle in the first three seconds.</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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find the velocity of the particle when its acceleration is zero.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times;"><span style="font-size: medium;"><img src="images/maths_5a_markscheme.png" alt> <strong><em>A1A2 N3</em></strong></span></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman'; min-height: 23.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Notes: </strong>Award <strong><em>A1 </em></strong>for approximately correct domain \(0 \leqslant t \leqslant 4\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The shape must be approximately correct, with maximum skewed left. <strong>Only</strong> if the shape is approximately correct, award <strong><em>A2 </em></strong>for all the following approximately correct features, in circle of tolerance where drawn (accept seeing correct coordinates for the maximum, even if point outside circle):</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Maximum point, passes through origin, asymptotic to \(t\)-axis (but must not touch the axis).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">If only two of these features are correct, award <strong><em>A1</em></strong>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman'; min-height: 23.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">valid approach (including \(0\) and \(3\)) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(\int_0^3 {10t{{\text{e}}^{ - 1.7t}}{\text{d}}t,{\text{ }}\int_0^3 {f(x)} } \), area from \(0\) to \(3\) (may be shaded in diagram)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{distance}} = 3.33{\text{ (m)}}\) <strong><em>A1 N2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">recognizing acceleration is derivative of velocity <strong><em>(R1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(a = \frac{{{\text{d}}v}}{{{\text{d}}t}}\), attempt to find \(\frac{{{\text{d}}v}}{{{\text{d}}t}}\), reference to maximum on the graph of \(v\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">valid approach to find \(v\) when \(a = 0\) (may be seen on graph) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(\frac{{{\text{d}}v}}{{{\text{d}}t}} = 0,{\text{ }}10{{\text{e}}^{ - 1.7t}} - 17t{{\text{e}}^{ - 1.7t}} = 0,{\text{ }}t = 0.588\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{velocity}} = 2.16{\text{ (m}}{{\text{s}}^{ - 1}})\) <strong><em>A1 N3</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note: </strong>Award <strong><em>R1M1A0 </em></strong>for \((0.588, 216)\) if velocity is not identified as final answer</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The population of deer in an enclosed game reserve is modelled by the function \(P(t) = 210\sin (0.5t - 2.6) + 990\), where \(t\) is in months, and \(t = 1\) corresponds to 1 January 2014.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find the number of deer in the reserve on 1 May 2014.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find the rate of change of the deer population on 1 May 2014.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b(i).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Interpret the answer to part (i) with reference to the deer population size on 1 May 2014.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">b(ii).</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(t = 5\) <span style="font: 21.0px 'Times New Roman';"><strong><em>(A1)</em></strong></span></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">correct substitution into formula <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg </em>\(210\sin (0.5 \times 5 - 2.6) + 990,{\text{ }}P(5)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(969.034982 \ldots \)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">969 (deer) (must be an integer) <strong><em>A1 N3</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">evidence of considering derivative <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(P'\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(104.475\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(104\) (deer per month) <strong><em>A1 N2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">b(i).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(the deer population size is) <strong>increasing</strong> <strong><em>A1 N1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[1 mark]</em></strong></span></p>
<div class="question_part_label">b(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>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The graph of \(y = (x - 1)\sin x\) , for \(0 \le x \le \frac{{5\pi }}{2}\)</span><span style="font-family: times new roman,times; font-size: medium;"> , is shown below.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/exhausted.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The graph has \(x\)-intercepts at \(0\), \(1\), \( \pi\) and \(k\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find <em>k</em> .</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The shaded region is rotated \(360^\circ \) about the <em>x</em>-axis. Let <em>V</em> be the volume of the </span><span style="font-family: times new roman,times; font-size: medium;">solid formed.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Write down an expression for <em>V</em> .</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The shaded region is rotated \(360^\circ \) about the <em>x</em>-axis. Let <em>V</em> be the volume of the </span><span style="font-family: times new roman,times; font-size: medium;">solid formed.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find <em>V</em> .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of valid approach <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(y = 0\) , \(\sin x = 0\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(2\pi = 6.283185 \ldots \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(k = 6.28\) <em><strong>A1 N2</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks] </span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">attempt to substitute either limits or the function into formula <strong><em>(M1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(accept absence of \({\rm{d}}x\) ) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(V = \pi \int_\pi ^k {{{(f(x))}^2}{\rm{d}}x} \) , \(\pi \int {{{((x - 1)\sin x)}^2}} \) , \(\pi \int_\pi ^{6.28 \ldots } {{y^2}{\rm{d}}x} \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct expression <em><strong>A2 N3</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\pi \int_\pi ^{6.28} {{{(x - 1)}^2}{{\sin }^2}x{\rm{d}}x} \) , \(\pi \int_\pi ^{2\pi } {{{((x - 1)\sin x)}^2}{\rm{d}}x} \) </span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> [3 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(V = 69.60192562 \ldots \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(V = 69.6\) <em><strong>A2 N2</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks] </span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Candidates showed marked improvement in writing fully correct expressions for a volume of revolution. Common errors of course included the omission of d<em>x</em> , using the given domain as the upper and lower bounds of integration, forgetting to square their function and/or the omission of \(\pi \) . There were still many who were unable to use their calculator successfully to find the required volume. </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Candidates showed marked improvement in writing fully correct expressions for a volume of revolution. Common errors of course included the omission of d<em>x</em> , using the given domain as the upper and lower bounds of integration, forgetting to square their function and/or the omission of \(\pi \) . There were still many who were unable to use their calculator successfully to find the required volume. </span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Candidates showed marked improvement in writing fully correct expressions for a volume of revolution. Common errors of course included the omission of d<em>x</em>, using the given domain as the upper and lower bounds of integration, forgetting to square their function and/or the omission of \(\pi \) . There were still many who were unable to use their calculator successfully to find the required volume. </span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p>Let \(f(x) = {({x^2} + 3)^7}\). Find the term in \({x^5}\) in the expansion of the derivative, \(f’(x)\).</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><strong>METHOD 1 </strong></p>
<p>derivative of \(f(x)\) <strong><em>A2</em></strong></p>
<p>\(7{({x^2} + 3)^6}(x2)\)</p>
<p>recognizing need to find \({x^4}\) term in \({({x^2} + 3)^6}\) (seen anywhere) <strong><em>R1</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(14x{\text{ (term in }}{x^4})\)</p>
<p>valid approach to find the terms in \({({x^2} + 3)^6}\) <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\left( {\begin{array}{*{20}{c}} 6 \\ r \end{array}} \right){({x^2})^{6 - r}}{(3)^r},{\text{ }}{({x^2})^6}{(3)^0} + {({x^2})^5}{(3)^1} + \ldots \), Pascal’s triangle to 6th row</p>
<p>identifying correct term (may be indicated in expansion) <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\({\text{5th term, }}r = 2,{\text{ }}\left( {\begin{array}{*{20}{c}} 6 \\ 4 \end{array}} \right),{\text{ }}{({x^2})^2}{(3)^4}\)</p>
<p>correct working (may be seen in expansion) <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\left( {\begin{array}{*{20}{c}} 6 \\ 4 \end{array}} \right){({x^2})^2}{(3)^4},{\text{ }}15 \times {3^4},{\text{ }}14x \times 15 \times 81{({x^2})^2}\)</p>
<p>\(17010{x^5}\) <strong><em>A1</em></strong> <strong><em>N3</em></strong></p>
<p><strong>METHOD 2</strong></p>
<p>recognition of need to find \({x^6}\) in \({({x^2} + 3)^7}\) (seen anywhere) <strong><em>R1 </em></strong></p>
<p>valid approach to find the terms in \({({x^2} + 3)^7}\) <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\left( {\begin{array}{*{20}{c}} 7 \\ r \end{array}} \right){({x^2})^{7 - r}}{(3)^r},{\text{ }}{({x^2})^7}{(3)^0} + {({x^2})^6}{(3)^1} + \ldots \), Pascal’s triangle to 7th row</p>
<p>identifying correct term (may be indicated in expansion) <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)6th term, \(r = 3,{\text{ }}\left( {\begin{array}{*{20}{c}} 7 \\ 3 \end{array}} \right),{\text{ (}}{{\text{x}}^2}{)^3}{(3)^4}\)</p>
<p>correct working (may be seen in expansion) <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\left( {\begin{array}{*{20}{c}} 7 \\ 4 \end{array}} \right){{\text{(}}{{\text{x}}^2})^3}{(3)^4},{\text{ }}35 \times {3^4}\)</p>
<p>correct term <strong><em>(A1)</em></strong></p>
<p>\(2835{x^6}\)</p>
<p>differentiating their term in \({x^6}\) <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\((2835{x^6})',{\text{ (6)(2835}}{{\text{x}}^5})\)</p>
<p>\(17010{x^5}\) <strong><em>A1</em></strong> <strong><em>N3</em></strong></p>
<p><strong><em>[7 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(h(x) = \frac{{2x - 1}}{{x + 1}}\) , \(x \ne - 1\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find \({h^{ - 1}}(x)\) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Sketch the graph of <em>h</em> for \( - 4 \le x \le 4\) and \( - 5 \le y \le 8\) , including any </span><span style="font-family: times new roman,times; font-size: medium;">asymptotes.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) Write down the equations of the asymptotes.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(iii) Write down the <em>x</em>-intercept of the graph of <em>h</em> .</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">b(i), (ii) and (iii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Let <em>R</em> be the region in the first quadrant enclosed by the graph of <em>h</em> , the <em>x</em>-axis </span><span style="font-family: times new roman,times; font-size: medium;">and the line \(x = 3\).</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) Find the area of <em>R</em>.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) Write down an expression for the volume obtained when <em>R</em> is revolved </span><span style="font-family: times new roman,times; font-size: medium;">through \({360^ \circ }\) about the <em>x</em>-axis.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">c(i) and (ii).</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">\(y = \frac{{2x - 1}}{{x + 1}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">interchanging <em>x</em> and <em>y</em> (seen anywhere) <em><strong>M1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(x = \frac{{2y - 1}}{{y + 1}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct working <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(xy + x = 2y - 1\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">collecting terms <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(x + 1 = 2y - xy\) , \(x + 1 = y(2 - x)\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({h^{ - 1}}(x) = \frac{{x + 1}}{{2 - x}}\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A1 N2</span></strong></em></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> [4 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/piper.png" alt></span> <em><strong><span style="font-family: times new roman,times; font-size: medium;">A1A1A1A1 N4</span></strong></em></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Award <em><strong>A1</strong></em> for approximately correct intercepts, </span><span style="font-family: times new roman,times; font-size: medium;"><em><strong>A1</strong></em> for correct shape, <em><strong>A1</strong></em> for asymptotes, </span><span style="font-family: times new roman,times; font-size: medium;"><em><strong>A1</strong></em> for approximately correct domain and range.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) \(x = - 1\) , \(y = 2\) <em><strong>A1A1 N2</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(iii) \(\frac{1}{2}\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A1 N1</span></strong></em></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[7 marks]</span></strong></em></p>
<div class="question_part_label">b(i), (ii) and (iii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) \({\text{area}} = 2.06\) <em><strong>A2 N2</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) attempt to substitute into volume formula (do not accept \(\pi \int_a^b {{y^2}{\rm{d}}x} \) ) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">M1</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;">volume \( = \pi {\int_{\frac{1}{2}}^3 {\left( {\frac{{2x - 1}}{{x + 1}}} \right)} ^2}{\rm{d}}x\) <em><strong>A2 N3</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> [5 marks]</span></strong></em></p>
<div class="question_part_label">c(i) and (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), (ii) and (iii).</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c(i) and (ii).</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Let \(f(x) = \frac{1}{{x - 1}} + 2\), for \(x > 1\).</p>
</div>
<div class="specification">
<p class="p1">Let \(g(x) = a{e^{ - x}} + b\), for \(x \geqslant 1\). The graphs of \(f\) and \(g\) have the same horizontal asymptote.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Write down the equation of the horizontal asymptote of the graph of \(f\).</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 class="p1">Find \(f'(x)\).</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 class="p1">Write down the value of \(b\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Given that \(g'(1) = - e\), find the value of \(a\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">There is a value of \(x\)<span class="s1">, for \(1 < x < 4\)</span>, for which the graphs of \(f\) and \(g\) have the same gradient. Find this gradient.</p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">\(y = 2\) (correct equation only) <span class="Apple-converted-space"> </span><strong><em>A2 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">valid approach <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\({(x - 1)^{ - 1}} + 2,{\text{ }}f'(x) = \frac{{0(x - 1) - 1}}{{{{(x - 1)}^2}}}\)</p>
<p class="p1"><span class="Apple-converted-space">\( - {(x - 1)^{ - 2}},{\text{ }}f'(x) = \frac{{ - 1}}{{{{(x - 1)}^2}}}\) </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">correct equation for the asymptote of \(g\)</p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(y = b\) <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><span class="s1">\(b = 2\) <span class="Apple-converted-space"> </span></span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">correct derivative of <span class="s1"><em>g </em></span>(seen anywhere) <span class="Apple-converted-space"> </span><strong><em>(A2)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(g'(x) = - a{{\text{e}}^{ - x}}\)</p>
<p class="p1">correct equation <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\( - {\text{e}} = - a{{\text{e}}^{ - 1}}\)</p>
<p class="p2">7.38905</p>
<p class="p1"><span class="s1">\(a = {{\text{e}}^2}{\text{ }}({\text{exact}}),{\text{ }}7.39\) <span class="Apple-converted-space"> </span></span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">attempt to equate <strong>their </strong>derivatives <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(f'(x) = g'(x),{\text{ }}\frac{{ - 1}}{{{{(x - 1)}^2}}} = - a{{\text{e}}^{ - x}}\)</p>
<p class="p1">valid attempt to solve <strong>their </strong>equation <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)correct value outside the domain of \(f\) <span class="s1">such as 0.522 or 4.51,</span></p>
<p class="p2"><img src="images/Schermafbeelding_2017-02-03_om_09.34.38.png" alt="M16/5/MATME/SP2/ENG/TZ2/09.e/M"></p>
<p class="p1">correct solution (may be seen in sketch) <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p2"><em>eg</em>\(\,\,\,\,\,\)\(x = 2,{\text{ }}(2,{\text{ }} - 1)\)</p>
<p class="p2">gradient is \( - 1\) <span class="Apple-converted-space"> </span><span class="s2"><strong><em>A1 <span class="Apple-converted-space"> </span>N3</em></strong></span></p>
<p class="p1"><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">Part (a) was in general well answered. Many candidates lost the marks for writing 2 or \(y \ne 2\) instead of \(y = 2\).</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">In part (b) some candidates got confused and found \({f^{ - 1}}(x)\) instead of \(f'(x)\)<span class="s1"><em>. </em></span>When calculating the derivative, two types of approaches were seen. Most of the ones who rewrote the function as \(f(x) = {(x - 1)^{ - 1}} + 2\), applied the chain rule correctly. Those who tried to apply the quotient rule made various mistakes: incorrect derivative of a constant, incorrect multiplication by zero, wrong subtraction order in the numerator, omitted the negative sign in the answer.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">In (c), most candidates were coherent and obtained the same value as the one written in part (a).</p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">In part (d) many candidates did not manage to differentiate the function g correctly. Of those who could, the equation was generally well solved algebraically.</p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">For part (e), not many candidates wrote a correct equation with their derivatives. There was mixed performance for this question, as those who knew they needed to use their GDC managed to obtain an answer, while many got tangled in unsuccessful attempts to solve the equation algebraically. Many candidates tried to solve quite complex equations ‘manually’ instead of trying to graph the expressions on their calculators and finding the value of \(x\) at the point of intersection. Of those students who tried to solve graphically only a small percentage actually sketched the two curves that they were considering. This sketch is particularly useful to examiners to see how the student is thinking, or what steps s/he is taking to solve the equations.</p>
<p class="p1">Only a few realized that the question asked for the gradient, which was represented by the \(y\)-coordinate of the point of intersection, rather than the \(x\)-coordinate.</p>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Let \(f(x) = 0.225{x^3} - 2.7x\), for \( - 3 \leqslant x \leqslant 3\). There is a local minimum point at <span class="s1">A</span>.</p>
</div>
<div class="specification">
<p class="p1">On the following grid,</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the coordinates of <span class="s1">A</span><span class="s2">.</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 class="p1">(i) <span class="Apple-converted-space"> </span>sketch the graph of \(f\), clearly indicating the point <span class="s1">A</span><span class="s2">;</span></p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>sketch the tangent to the graph of \(f\) at <span class="s1">A</span>.</p>
<p class="p1" style="text-align: left;"><img src="images/Schermafbeelding_2017-03-03_om_17.19.46.png" alt="N16/5/MATME/SP2/ENG/TZ0/02.b"></p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1"><span class="s1">\({\text{A }}(2,{\text{ }}-3.6)\) <span class="Apple-converted-space"> </span></span><strong><em>A1A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">(i) (ii) <span class="Apple-converted-space"> <img src="images/Schermafbeelding_2017-03-03_om_17.24.16.png" alt="N16/5/MATME/SP2/ENG/TZ0/02.b/M"></span> <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p2"><strong><em>A1A1A1 <span class="Apple-converted-space"> </span>N4</em></strong></p>
<p class="p2"><strong><em>A1 <span class="Apple-converted-space"> </span>N1</em></strong></p>
<p class="p3"> </p>
<p class="p1"><strong>Notes:</strong> (i) Award <span class="s1"><strong><em>A1</em></strong> </span>for correct cubic shape with correct curvature.</p>
<p class="p1">Only if this <span class="s1"><strong><em>A1 </em></strong></span>is awarded, award the following:</p>
<p class="p1"><span class="s1"><strong><em>A1 </em></strong></span>for passing through <strong>their </strong><span class="s2">point A </span>and the origin,</p>
<p class="p1"><span class="s1"><strong><em>A1 </em></strong></span>for endpoints,</p>
<p class="p1"><span class="s1"><strong><em>A1 </em></strong></span>for maximum.</p>
<p class="p1">(ii) Award <span class="s1"><strong><em>A1 </em></strong></span>for horizontal line through <strong>their </strong><span class="s2">A</span>.</p>
<p class="p3"> </p>
<p class="p2"><strong><em>[5 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Let \(f(x) = 6 - \ln ({x^2} + 2)\), for \(x \in \mathbb{R}\). The graph of \(f\) passes through the point \((p,{\text{ }}4)\), where \(p > 0\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of \(p\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The following diagram shows part of the graph of \(f\).</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2018-02-12_om_13.30.18.png" alt="N17/5/MATME/SP2/ENG/TZ0/05.b"></p>
<p>The region enclosed by the graph of \(f\), the \(x\)-axis and the lines \(x = - p\) and \(x = p\) is rotated 360° about the \(x\)-axis. Find the volume of the solid formed.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>valid approach <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(f(p) = 4\), intersection with \(y = 4,{\text{ }} \pm 2.32\)</p>
<p>2.32143</p>
<p>\(p = \sqrt {{{\text{e}}^2} - 2} \) (exact), 2.32 <strong><em>A1 N2</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>attempt to substitute <strong>either their</strong> limits <strong>or</strong> the function into volume formula (must involve \({f^2}\), accept reversed limits and absence of \(\pi \) and/or \({\text{d}}x\), but do not accept any other errors) <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\int_{ - 2.32}^{2.32} {{f^2},{\text{ }}\pi \int {{{\left( {6 - \ln ({x^2} + 2)} \right)}^2}{\text{d}}x,{\text{ 105.675}}} } \)</p>
<p>331.989</p>
<p>\({\text{volume}} = 332\) <strong><em>A2 N3</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">A particle moves in a straight line with velocity \(v = 12t - 2{t^3} - 1\) , for \(t \ge 0\) , where <em>v</em> is </span><span style="font-family: times new roman,times; font-size: medium;">in centimetres per second and <em>t</em> is in seconds.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the acceleration of the particle after 2.7 seconds.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the displacement of the particle after 1.3 seconds.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">recognizing that acceleration is the derivative of velocity (seen anywhere) <em><strong>(R1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(a = \frac{{{{\rm{d}}^2}s}}{{{\rm{d}}{t^2}}},v',12 - 6{t^2}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correctly substituting 2.7 into their expression for <em>a</em> (not into <em>v</em>) <strong><em>(A1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(s''(2.7)\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\text{acceleration}} = - 31.74\) (exact), \( - 31.7\) <strong><em>A1 N3</em> </strong></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks] </span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">recognizing that displacement is the integral of velocity <em><strong>R1 </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(s = \int v \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correctly substituting 1.3 <strong><em>(A1) </em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\int_0^{1.3} {v{\rm{d}}t} \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\text{displacement}} = 7.41195\) (exact), \(7.41\) (cm) <em><strong>A1 N2</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks] </span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">This question was well answered by many candidates, although there were some who did not recognize the relationship between velocity, acceleration and displacement. Many of them substituted into the original expression given for the velocity, losing most of the marks. Very few appear to have used their GDC for the integration. </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">This question was well answered by many candidates, although there were some who did not recognize the relationship between velocity, acceleration and displacement. Many of them substituted into the original expression given for the velocity, losing most of the marks. Very few appear to have used their GDC for the integration. </span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p class="p1">Let \(f(x) = \frac{{\ln (4x)}}{x}\) for \(0 < x \le 5\).</p>
<p class="p1">Points \({\text{P}}(0.25,{\text{ }}0)\) and \(Q\) are on the curve of \(f\). The tangent to the curve of \(f\) at \(P\) is perpendicular to the tangent at \(Q\). Find the coordinates of \(Q\).</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p>recognizing that the gradient of tangent is the derivative <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\;\;\;f'\)</p>
<p>finding the gradient of \(f\) at \(P\) <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\;\;\;f'(0.25) = 16\)</p>
<p>evidence of taking negative reciprocal of <strong>their </strong>gradient at \(P\) <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\;\;\;\frac{{ - 1}}{m},{\text{ }} - \frac{1}{{f'(0.25)}}\)</p>
<p>equating derivatives <strong><em>M1</em></strong></p>
<p><em>eg</em>\(\;\;\;f'(x) = \frac{{ - 1}}{{16}},{\text{ }}f' = - \frac{1}{m},{\text{ }}\frac{{x\left( {\frac{1}{x}} \right) - \ln (4x)}}{{{x^2}}} = 16\)</p>
<p>finding the \(x\)-coordinate of \(Q\), \(x = 0.700750\)</p>
<p>\(x = 0.701\) <strong><em>A1 N3</em></strong></p>
<p>attempt to substitute <strong>their</strong> \(x\) into \(f\) to find the \(y\)-coordinate of \(Q\) <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\;\;\;f(0.7)\)</p>
<p>\(y = 1.47083\)</p>
<p>\(y = 1.47\) <strong><em>A1 N2</em></strong></p>
<p><strong><em>[7 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p class="p1">Few candidates were completely successful with this question. Students using an analytical approach were aware of the relationship between the gradients of the tangent and normal, but were often unable to find a correct derivative initially. The solution was made significantly easier when the GDC was used effectively and the few candidates who used this approach, were generally successful. Attempting to find the equation of the tangent and/or the normal were common, ineffective approaches.</p>
</div>
<br><hr><br><div class="specification">
<p>Let \(f\left( x \right) = 12\,\,{\text{cos}}\,x - 5\,\,{\text{sin}}\,x,\,\, - \pi \leqslant x \leqslant 2\pi \), be a periodic function with \(f\left( x \right) = f\left( {x + 2\pi } \right)\)</p>
<p>The following diagram shows the graph of \(f\).</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<p style="text-align: left;">There is a maximum point at A. The minimum value of \(f\) is −13 .</p>
</div>
<div class="specification">
<p>A ball on a spring is attached to a fixed point O. The ball is then pulled down and released, so that it moves back and forth vertically.</p>
<p style="text-align: center;"><img 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"></p>
<p>The distance, <em>d</em> centimetres, of the centre of the ball from O at time <em>t</em> seconds, is given by</p>
<p style="padding-left: 90px;">\(d\left( t \right) = f\left( t \right) + 17,\,\,0 \leqslant t \leqslant 5.\)</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the coordinates of A.</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>For the graph of \(f\), write down the amplitude.</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>For the graph of \(f\), write down the period.</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>Hence, write \(f\left( x \right)\) in the form \(p\,\,{\text{cos}}\,\left( {x + r} \right)\).</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 maximum speed of the ball.</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 first time when the ball’s speed is changing at a rate of 2 cm s<sup>−2</sup>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>−0.394791,13</p>
<p>A(−0.395, 13) <em><strong>A1A1 N2</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>13 <em><strong>A1 N1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\({2\pi }\), 6.28 <em><strong>A1 N1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>valid approach <em><strong>(M1)</strong></em></p>
<p><em>eg</em> recognizing that amplitude is <em>p</em> or shift is <em>r</em></p>
<p>\(f\left( x \right) = 13\,\,{\text{cos}}\,\left( {x + 0.395} \right)\) (accept <em>p</em> = 13, <em>r</em> = 0.395) <em><strong>A1A1 N3</strong></em></p>
<p><strong>Note:</strong> Accept any value of <em>r</em> of the form \(0.395 + 2\pi k,\,\,k \in \mathbb{Z}\)</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>recognizing need for <em>d </em>′(<em>t</em>) <em><strong>(M1)</strong></em></p>
<p><em>eg</em> −12 sin(<em>t</em>) − 5 cos(<em>t</em>)</p>
<p>correct approach (accept any variable for <em>t</em>) <em><strong>(A1)</strong></em></p>
<p><em>eg </em> −13 sin(<em>t + </em>0.395), sketch of <em>d</em>′, (1.18, −13), <em>t</em> = 4.32</p>
<p>maximum speed = 13 (cms<sup>−1</sup>) <em><strong>A1 N2</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>recognizing that acceleration is needed <em><strong>(M1)</strong></em></p>
<p><em>eg </em>a(<em>t</em>), <em>d "</em>(t)</p>
<p>correct equation (accept any variable for <em>t</em>) <em><strong> (A1)</strong></em></p>
<p><em>eg </em>\(a\left( t \right) = - 2,\,\,\left| {\frac{{\text{d}}}{{{\text{d}}t}}\left( {d'\left( t \right)} \right)} \right| = 2,\,\, - 12\,\,{\text{cos}}\,\left( t \right) + 5\,\,{\text{sin}}\,\left( t \right) = - 2\)</p>
<p>valid attempt to solve <strong>their</strong> equation <em><strong>(M1)</strong></em></p>
<p><em>eg </em> sketch, 1.33</p>
<p>1.02154</p>
<p>1.02 <em><strong>A2 N3</strong></em></p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[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.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p>Let \(f\left( x \right) = \,\,{\text{sin}}\,\left( {{e^x}} \right)\) for 0 ≤ \(x\) ≤ 1.5. The following diagram shows the graph of \(f\).</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the <em>x</em>-intercept of the graph of \(f\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The region enclosed by the graph of \(f\), the<em> y</em>-axis and the <em>x</em>-axis is rotated 360° about the <em>x</em>-axis.</p>
<p>Find the volume of the solid formed.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>valid approach <em><strong>(M1)</strong></em><br><em>eg </em> \(f\left( x \right) = 0,\,\,\,\,{e^x} = 180\) or 0…</p>
<p>1.14472</p>
<p>\(x = {\text{ln}}\,\pi \) (exact), 1.14 <em><strong>A1 N2</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>attempt to substitute either their <strong>limits</strong> or the function into formula involving \({f^2}\). <em><strong>(M1)</strong></em></p>
<p><em>eg</em> \({\int_0^{1.14} {{f^2},\,\,\pi \int {\left( {{\text{sin}}\,\left( {{e^x}} \right)} \right)} } ^2}dx,\,\,0.795135\)</p>
<p>2.49799</p>
<p>volume = 2.50 <em><strong> A2 N3</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Let \(f(x) = (x - 1)(x - 4)\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find the \(x\)-intercepts of the graph of \(f\).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The region enclosed by the graph of \(f\) and the \(x\)-axis is rotated \(360^\circ\) about the \(x\)-axis.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find the volume of the solid formed.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">valid approach <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(f(x) = 0\), sketch of parabola showing two \(x\)-intercepts</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(x = 1,{\text{ }}x = 4{\text{ }}\left( {{\text{accept (1, 0), (4, 0)}}} \right)\) <strong><em>A1A1 N3</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">attempt to substitute either limits or the function into formula involving \({f^2}\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(\int_1^4 {{{\left( {f(x)} \right)}^2}{\text{d}}x,{\text{ }}\pi \int {{{\left( {(x - 1)(x - 4)} \right)}^2}} } \)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\({\text{volume}} = 8.1\pi {\text{ (exact), 25.4}}\) <strong><em>A2 N3</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.5px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks] </em></strong></span></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><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = \cos ({{\rm{e}}^x})\) , for \( - 2 \le x \le 2\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find \(f'(x)\) .</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><span style="font-family: times new roman,times; font-size: medium;">On the grid below, sketch the graph of \(f'(x)\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/M12P2TZ2Q2.png" alt></span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">\(f'(x) = - {{\rm{e}}^x}\sin ({{\rm{e}}^x})\) <em><strong>A1A1 N2</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks] </span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/Jon.png" alt> <em><strong>A1A1A1A1 N4</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Award <em><strong>A1</strong></em> for shape that must have the correct domain (from \( - 2\) to \( + 2\) ) and correct range (from \( - 6\) to \(4\) ), <em><strong>A1</strong></em> for minimum in circle, <em><strong>A1</strong></em> for maximum in circle and <em><strong>A1</strong></em> for intercepts in circles. </span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Many students failed in applying the chain rule to find the correct derivative, and some inappropriately used the product rule. However, many of those obtained full follow through marks in part (b) for the sketch of the function they found in part (a). </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Many students failed in applying the chain rule to find the correct derivative, and some inappropriately used the product rule. However, many of those obtained full follow through marks in part (b) for the sketch of the function they found in part (a). </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> Most candidates sketched an approximately correct shape in the given domain, though there were some that did not realize they had to set their GDC to radians, producing a meaningless sketch.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> It is very important to stress to students that although they are asked to produce a sketch, it is still necessary to show its key features such as domain and range, stationary points and intercepts.</span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The following diagram shows a waterwheel with a bucket. The wheel rotates at a </span><span style="font-family: times new roman,times; font-size: medium;">constant rate in an anticlockwise (counter-clockwise) direction.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/bucket.png" alt></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The diameter of the wheel is 8 metres. The centre of the wheel, A, is 2 metres </span><span style="font-family: times new roman,times; font-size: medium;">above the water level. After <em>t</em> seconds, the height of the bucket above the water level </span><span style="font-family: times new roman,times; font-size: medium;">is given by \(h = a\sin bt + 2\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Show that \(a = 4\) .</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The wheel turns at a rate of one rotation every 30 seconds.</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Show that \(b = \frac{\pi }{{15}}\)</span><span style="font-family: times new roman,times; font-size: medium;"> .</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">In the first rotation, there are two values of <em>t</em> when the bucket is <strong>descending</strong> at a rate </span><span style="font-family: times new roman,times; font-size: medium;">of \(0.5{\text{ m}}{{\text{s}}^{ - 1}}\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find these values of <em>t</em> .</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">In the first rotation, there are two values of <em>t</em> when the bucket is <strong>descending</strong> at a rate </span><span style="font-family: times new roman,times; font-size: medium;">of \(0.5{\text{ m}}{{\text{s}}^{ - 1}}\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Determine whether the bucket is underwater at the second value of <em>t</em> .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><strong><span style="font-family: times new roman,times; font-size: medium;">METHOD 1</span></strong></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of recognizing the amplitude is the radius <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. amplitude is half the diameter</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(a = \frac{8}{2}\) <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(a = 4\) <em><strong>AG N0</strong></em></span></p>
<p align="LEFT"><strong><span style="font-family: times new roman,times; font-size: medium;">METHOD 2</span></strong></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of recognizing the maximum height <em><strong> (M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(h = 6\) , \(a\sin bt + 2 = 6\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct reasoning</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(a\sin bt = 4\) and \(\sin bt\) has amplitude of 1 <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(a = 4\) <em><strong>AG N0</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><strong><span style="font-family: times new roman,times; font-size: medium;">METHOD 1</span></strong></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">period = 30 <em><strong>(A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(b = \frac{{2\pi }}{{30}}\) <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(b = \frac{\pi }{{15}}\) </span><span style="font-family: times new roman,times; font-size: medium;"> <em><strong>AG N0</strong></em></span></p>
<p align="LEFT"><strong><span style="font-family: times new roman,times; font-size: medium;">METHOD 2</span></strong></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct equation <em><strong> (A1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(2 = 4\sin 30b + 2\) , \(\sin 30b = 0\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(30b = 2\pi \) <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(b = \frac{\pi }{{15}}\) <em><strong>AG N0</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">recognizing \(h'(t) = - 0.5\) (seen anywhere) <em><strong>R1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">attempting to solve <em><strong> (M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. sketch of \(h'\) , finding \(h'\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct work involving \(h'\) <em><strong>A2</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. sketch of \(h'\) showing intersection, \( - 0.5 = \frac{{4\pi }}{{15}}\cos \left( {\frac{\pi }{{15}}t} \right)\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(t = 10.6\) , \(t = 19.4\) <em><strong>A1A1 N3</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[6 marks]</span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><strong><span style="font-family: times new roman,times; font-size: medium;">METHOD 1</span></strong></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">valid reasoning for <strong>their</strong> conclusion (seen anywhere) <strong><em>R1</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(h(t) < 0\) so underwater; \(h(t) > 0\) so not underwater</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of substituting into <em>h</em> <strong><em>(M1)</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(h(19.4)\) , \(4\sin \frac{{19.4\pi }}{{15}} + 2\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct calculation <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(h(19.4) = - 1.19\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct statement <em><strong>A1 N0</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. the bucket is underwater, yes</span></p>
<p align="LEFT"><strong><span style="font-family: times new roman,times; font-size: medium;">METHOD 2</span></strong></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">valid reasoning for <strong>their</strong> conclusion (seen anywhere) <strong><em>R1</em></strong></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(h(t) < 0\) so underwater; \(h(t) > 0\) so not underwater</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of valid approach <em><strong> (M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. solving \(h(t) = 0\) , graph showing region below <em>x</em>-axis</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct roots <em><strong>A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(17.5\), \(27.5\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">correct statement <em><strong>A1 N0</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. the bucket is underwater, yes</span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks]</span></strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Parts (a) and (b) were generally well done. </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Parts (a) and (b) were generally well done, however there were several instances of candidates working backwards from the given answer in part (b). </span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Parts (c) and (d) proved to be quite challenging for a large proportion of candidates. Many did not attempt these parts. The most common error was a misinterpretation of the word "descending" where numerous candidates took \(h'(t)\) to be 0.5 instead of \( - 0.5\) but incorrect derivatives for <em>h</em> were also widespread. The process required to solve for <em>t</em> from the equation \( - 0.5 = \frac{{4\pi }}{{15}}\cos \left( {\frac{\pi }{{15}}t} \right)\) overwhelmed those who attempted algebraic methods. Few could obtain both correct solutions, more had one correct while others included unreasonable values including \(t < 0\) . </span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">In part (d), not many understood that the condition for underwater was \(h(t) < 0\) and had trouble interpreting the meaning of "second value". Many candidates, however, did recover to gain some marks in follow through. </span></p>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Let \(f(x) = \frac{{g(x)}}{{h(x)}}\), where \(g(2) = 18,{\text{ }}h(2) = 6,{\text{ }}g'(2) = 5\), and \(h'(2) = 2\). Find the equation of the normal to the graph of \(f\) at \(x = 2\).</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">recognizing need to find \(f(2)\) or \(f'(2)\) <strong><em>(R1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(f(2) = \frac{{18}}{6}\) (seen anywhere) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">correct substitution into the quotient rule <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(\frac{{6(5) - 18(2)}}{{{6^2}}}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(f'(2) = - \frac{6}{{36}}\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">gradient of normal is 6 <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">attempt to use the point and gradient to find equation of straight line <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(y - f(2) = - \frac{1}{{f'(2)}}(x - 2)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">correct equation in any form <strong><em>A1 N4</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(y - 3 = 6(x - 2),{\text{ }}y = 6x - 9\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[7 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = {{\rm{e}}^x}\sin 2x + 10\) , for \(0 \le x \le 4\) . Part of the graph of <em>f</em> is given below.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/apple.png" alt></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">There is an <em>x</em>-intercept at the point A, a local maximum point at M, where \(x = p\) and </span><span style="font-size: medium;"><span style="font-family: times new roman,times;">a local minimum point at N, where \(x = q\)</span><span style="font-family: times new roman,times;"> .</span></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Write down the <em>x</em>-coordinate of A.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Find the value of</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) <em>p</em> ;</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) <em>q</em> .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Find \(\int_p^q {f(x){\rm{d}}x} \)</span><span style="font-family: times new roman,times; font-size: medium;"> . Explain why this is not the area of the shaded region.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">\(2.31\) <em><strong>A1 N1</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[1 mark]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) 1.02 <em><strong>A1 N1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) 2.59 <em><strong>A1 N1</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(\int_p^q {f(x){\rm{d}}x} = 9.96\) <em><strong>A1 N1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">split into two regions, make the area below the <em>x</em>-axis positive <em><strong>R1R1 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Parts (a) and (b) were generally well answered, the main problem being the accuracy.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Parts (a) and (b) were generally well answered, the main problem being the accuracy.</span></p>
<div class="question_part_label">b(i) and (ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Many students lacked the calculator skills to successfully complete (6)(c) in that they could </span><span style="font-family: times new roman,times; font-size: medium;">not find the value of the definite integral. Some tried to find it by hand. </span><span style="font-family: times new roman,times; font-size: medium;">When trying to explain why the integral was not the area, most knew the region under the <em>x</em>-axis </span><span style="font-family: times new roman,times; font-size: medium;">was the cause of the integral not giving the total area, but the explanations were not </span><span style="font-family: times new roman,times; font-size: medium;">sufficiently clear. It was often stated that the area below the axis was negative rather than the </span><span style="font-family: times new roman,times; font-size: medium;">integral was negative.</span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = 5\cos \frac{\pi }{4}x\) and \(g(x) = - 0.5{x^2} + 5x - 8\) for \(0 \le x \le 9\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">On the same diagram, sketch the graphs of <em>f</em> and <em>g</em> .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Consider the graph of \(f\) . Write down</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) the <em>x</em>-intercept that lies between \(x = 0\) and \(x = 3\) ;</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) the period;</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(iii) the amplitude.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Consider the graph of <em>g</em> . Write down</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) the two <em>x</em>-intercepts;</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) the equation of the axis of symmetry.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Let <em>R</em> be the region enclosed by the graphs of <em>f</em> and <em>g</em> . Find the area of <em>R</em>.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times;"><span style="font-size: medium;"><img src="images/luke.png" alt></span><em><span style="font-size: medium;"><strong> A1A1A1 N3</strong> </span></em></span></p>
<p><span style="font-family: times new roman,times;"><span style="font-size: medium;"><strong>Note</strong>: Award <em><strong>A1</strong></em> for <em>f</em> being of sinusoidal shape, with 2 maxima and one minimum, </span><span style="font-size: medium;"><em><strong>A1</strong></em> for <em>g</em> being a parabola opening down, </span><span style="font-size: medium;"><em><strong>A1</strong></em> for <strong>two</strong> intersection points in approximately correct position. </span></span></p>
<p><span style="font-family: times new roman,times;"><em><strong><span style="font-size: medium;">[3 marks] </span></strong></em></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) \((2{\text{, }}0)\) (accept \(x = 2\) ) <em><strong>A1 N1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) \({\text{period}} = 8\) <em><strong>A2 N2</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(iii) \({\text{amplitude}} = 5\) <em><strong>A1 N1 </strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks] </span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">(i) \((2{\text{, }}0)\) , \((8{\text{, }}0)\) (accept \(x = 2\) , \(x = 8\) ) <em><strong>A1A1 N1N1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">(ii) \(x = 5\) (must be an equation) <em><strong>A1 N1</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks] </span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong><span style="font-family: times new roman,times; font-size: medium;">METHOD 1</span></strong></p>
<p><span style="font-family: times new roman,times; font-size: medium;">intersect when \(x = 2\) and \(x = 6.79\) (may be seen as limits of integration) <em><strong>A1A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of approach <em><strong>(M1)</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\int {g - f} \) , \(\int {f(x){\rm{d}}x - \int {g(x){\rm{d}}x}}\) , \(\int_2^{6.79} {\left( {( - 0.5{x^2} + 5x - 8) - \left( {5\cos \frac{\pi }{4}x} \right)} \right)}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\text{area}} = 27.6\) <em><strong>A2 N3</strong></em></span></p>
<p><strong> <span style="font-family: times new roman,times; font-size: medium;">METHOD 2</span></strong></p>
<p><span style="font-family: times new roman,times; font-size: medium;">intersect when \(x = 2\) and \(x = 6.79\) (seen anywhere) <em><strong>A1A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of approach using a sketch of <em>g</em> and <em>f</em> , or \(g - f\) . <em><strong>(M1)</strong></em></span></p>
<p><br><img src="images/going_out.png" alt></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. area = \(A + B - C\) , \(12.7324 + 16.0938 - 1.18129 \ldots \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({\text{area}} = 27.6\) <em><strong>A2 N3</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> [5 marks]</span></strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Graph sketches were much improved over previous sessions. Most candidates graphed the two functions correctly, but many ignored the domain restrictions. </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Many candidates found parts (b) and (c) accessible, although quite a few did not know how to find the period of the cosine function.</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Many candidates found parts (b) and (c) accessible, although quite a few did not know how to find the period of the cosine function.</span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Part (d) proved elusive to many candidates. Some used creative approaches that split the area into parts above and below the <em>x</em>-axis; while this leads to a correct result, few were able to achieve it. Many candidates were unable to use their GDCs effectively to find points of intersection and the subsequent area. </span></p>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = x\cos (x - \sin x)\) , \(0 \le x \le 3\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Sketch the graph of <em>f</em> on the following set of axes.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><img src="images/marvin.png" alt></span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The graph of <em>f</em> intersects the <em>x</em>-axis when \(x = a\) , \(a \ne 0\) . Write down the </span><span style="font-family: times new roman,times; font-size: medium;">value of <em>a</em>.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The graph of <em>f</em> is revolved \(360^\circ \) about the <em>x</em>-axis from \(x = 0\) to \(x = a\) . </span><span style="font-family: times new roman,times; font-size: medium;">Find the volume of the solid formed.</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><span style="font-family: times new roman,times; font-size: medium;"><img src="images/marvin2.png" alt></span><em><strong><span style="font-family: times new roman,times; font-size: medium;"> A1A2 N3</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Notes</strong>: Award <em><strong>A1</strong></em> for correct domain, \(0 \le x \le 3\) . Award <em><strong>A2</strong></em> for approximately correct shape, with local maximum in circle 1 and right endpoint in circle 2.</span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks] </span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">\(a = 2.31\) <em><strong>A1 N1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em><strong>[1 mark]</strong></em></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of using \(V = \pi {\int {\left[ {f(x)} \right]} ^2}{\rm{d}}x\) <em><strong>(M1) </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">fully correct integral expression <em><strong>A2</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(V = \pi {\int_0^{2.31} {\left[ {x\cos (x - \sin x)} \right]} ^2}{\rm{d}}x\) , \(V = \pi {\int_0^{2.31} {\left[ {f(x)} \right]} ^2}{\rm{d}}x\) <em><strong>A1 N2</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(V = 5.90\)</span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks]</span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Many candidates sketched a clear and smooth freehand curve with the local maximum, <em>x</em>-intercept and endpoints in approximately correct positions. Commonly, candidates sketched a graph across \([ - 3{\text{, }}3]\) , which neglects the given domain of the function. There were some candidates who sketched a straight line through the origin, presumably from being in the degree mode of their GDC. </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Many candidates sketched a clear and smooth freehand curve with the local maximum, <em>x</em>-intercept and endpoints in approximately correct positions. Commonly, candidates sketched a graph across \([ - 3{\text{, }}3]\) , which neglects the given domain of the function. There were some candidates who sketched a straight line through the origin, presumably from being in the degree mode of their GDC. </span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">A good number of candidates could set up the correct integral expression for volume, but surprisingly few were able to use their GDC to find the correct value. Some attempted to analytically integrate the square of this unusual function, expending valuable time in this effort. A small but significant number of candidates wrote a final answer as \(1.88\pi \) , which accrued the accuracy penalty.</span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Let <em>f</em>(<em>x</em>) = ln <em>x</em> − 5<em>x</em> , for <em>x</em> > 0 .</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <em>f '</em>(<em>x</em>).</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 <em>f "</em>(<em>x</em>).</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Solve<em> f '</em>(<em>x</em>)<em> = f "</em>(<em>x</em>).</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>\(f'\left( x \right) = \frac{1}{x} - 5\) <em><strong>A1A1 N2</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><em>f "</em>(<em>x</em>) = −<em>x</em><sup>−2 </sup> <em><strong>A1 N1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1 (using GDC)</strong></p>
<p>valid approach <em><strong>(M1)</strong></em></p>
<p><em>eg </em><img src="data:image/png;base64,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"></p>
<p>0.558257</p>
<p><em>x</em> = 0.558 <em><strong>A1 N2</strong></em></p>
<p><strong>Note:</strong> Do not award <em><strong>A1</strong></em> if additional answers given.</p>
<p> </p>
<p><strong>METHOD 2 (analytical)</strong></p>
<p>attempt to solve their equation <em>f '(x) = f "</em>(<em>x</em>) (do not accept \(\frac{1}{x} - 5 = - \frac{1}{{{x^2}}}\)) <em><strong>(M1)</strong></em></p>
<p><em>eg </em>\(5{x^2} - x - 1 = 0,\,\,\frac{{1 \pm \sqrt {21} }}{{10}},\,\,\frac{1}{x} = \frac{{ - 1 \pm \sqrt {21} }}{2},\,\, - 0.358\)</p>
<p>0.558257</p>
<p><em>x</em> = 0.558 <em><strong>A1 N2</strong></em></p>
<p><strong>Note:</strong> Do not award <em><strong>A1</strong></em> if additional answers given.</p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = {x^3} - 4x + 1\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Expand \({(x + h)^3}\) .</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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Use the formula \(f'(x) = \mathop {\lim }\limits_{h \to 0} \frac{{f(x + h) - f(x)}}{h}\) </span><span style="font-family: times new roman,times; font-size: medium;">to show that </span><span style="font-family: times new roman,times; font-size: medium;">the derivative of \(f(x)\) is \(3{x^2} - 4\) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The tangent to the curve of f at the point \({\text{P}}(1{\text{, }} - 2)\) is parallel to the tangent at </span><span style="font-family: times new roman,times; font-size: medium;">a point Q. Find the coordinates of Q.</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><span style="font-family: times new roman,times; font-size: medium;">The graph of <em>f</em> is decreasing for \(p < x < q\) . Find the value of <em>p</em> and of <em>q</em>.</span></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><span style="font-family: times new roman,times; font-size: medium;">Write down the range of values for the gradient of \(f\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">attempt to expand <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\({(x + h)^3} = {x^3} + 3{x^2}h + 3x{h^2} + {h^3}\) <em><strong>A1 N2</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks] </span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of substituting \(x + h\) <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct substitution <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(f'(x) = \mathop {\lim }\limits_{h \to 0} \frac{{{{(x + h)}^3} - 4(x + h) + 1 - ({x^3} - 4x + 1)}}{h}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">simplifying <em><strong>A1 </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{{({x^3} + 3{x^2}h + 3x{h^2} + {h^3} - 4x - 4h + 1 - {x^3} + 4x - 1)}}{h}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">factoring out <em>h</em> <em><strong>A1 </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\frac{{h(3{x^2} + 3xh + {h^2} - 4)}}{h}\)</span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\(f'(x) = 3{x^2} - 4\) </span><em><span style="font-family: times new roman,times; font-size: medium;"><strong>AG N0</strong> </span></em></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;"> [4 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">\(f'(1) = - 1\) <em><strong>(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">setting up an appropriate equation <em><strong>M1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(3{x^2} - 4 = - 1\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">at Q, \(x = - 1,y = 4\) (Q is \(( - 1{\text{, }}4)\)) <em><strong>A1 A1</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks] </span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">recognizing that <em>f</em> is decreasing when \(f'(x) < 0\) <em><strong>R1</strong></em> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct values for <em>p</em> and <em>q</em> (but do not accept \(p = 1.15{\text{, }}q = - 1.15\) ) <em><strong>A1A1 N1N1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. \(p = - 1.15{\text{, }}q = 1.15\) ; \( \pm \frac{2}{{\sqrt 3 }}\) ; an interval such as \( - 1.15 \le x \le 1.15\)</span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks] </span></strong></em></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">\(f'(x) \ge - 4\) , \(y \ge - 4\) , \(\left[ { - 4,\infty } \right[\) <em><strong>A2 N2</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks] </span></strong></em></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">In part (a), the basic expansion was not done well. Rather than use the binomial theorem, many candidates opted to expand by multiplication which resulted in algebraic errors. </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">In part (b), it was clear that many candidates had difficulty with differentiation from first principles. Those that successfully set the answer up, often got lost in the simplification. </span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Part (c) was poorly done with many candidates assuming that the tangents were horizontal and then incorrectly estimating the maximum of <em>f</em> as the required point. Many candidates unnecessarily found the equation of the tangent and could not make any further progress. </span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">In part (d) many correct solutions were seen but only a very few earned the reasoning mark. </span></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Part (e) was often not attempted and if it was, candidates were not clear on what was expected. </span></p>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Consider a function \(f\), for \(0 \le x \le 10\). The following diagram shows the graph of \(f'\), the derivative of \(f\).</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2016-01-14_om_07.44.17.png" alt></p>
<p class="p1">The graph of \(f'\) passes through \((2,{\text{ }} - 2)\) and \((5,{\text{ }}1)\), and has \(x\)-intercepts at \(0\), \(4\) and \(6\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">The graph of \(f\) has a local maximum point when \(x = p\). State the value of \(p\), and justify your answer.</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 class="p1">Write down \(f'(2)\).</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Let \(g(x) = \ln \left( {f(x)} \right)\) and \(f(2) = 3\).</p>
<p class="p1">Find \(g'(2)\).</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 class="p1">Verify that \(\ln 3 + \int_2^a {g'(x){\text{d}}x = g(a)} \), where \(0 \le a \le 10\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">The following diagram shows the graph of \(g'\), the derivative of \(g\).</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2016-01-14_om_07.59.38.png" alt></p>
<p class="p1">The shaded region \(A\) is enclosed by the curve, the <em>\(x\)</em>-axis and the line \(x = 2\), and has area \({\text{0.66 unit}}{{\text{s}}^{\text{2}}}\).</p>
<p class="p1">The shaded region \(B\) is enclosed by the curve, the \(x\)-axis and the line \(x = 5\), and has area \({\text{0.21 unit}}{{\text{s}}^{\text{2}}}\)<span class="s1">.</span></p>
<p class="p2">Find \(g(5)\)<span class="s2">.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">\(p = 6\) <span class="Apple-converted-space"> </span><strong><em>A1 <span class="Apple-converted-space"> </span>N1</em></strong></p>
<p class="p1">recognizing that turning points occur when \(f'(x) = 0\) <span class="Apple-converted-space"> </span><strong><em>R1 <span class="Apple-converted-space"> </span>N1</em></strong></p>
<p class="p1">eg\(\;\;\;\)correct sign diagram</p>
<p class="p1">\(f'\) changes from positive to negative at \(x = 6\) <span class="Apple-converted-space"> </span><strong><em>R1 <span class="Apple-converted-space"> </span>N1</em></strong></p>
<p class="p1"><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\(f'(2) = - 2\) <strong><em>A1 N1</em></strong></p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to apply chain rule <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\;\;\;\ln (x)' \times f'(x)\)</p>
<p>correct expression for \(g'(x)\) <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\;\;\;g'(x) = \frac{1}{{f(x)}} \times f'(x)\)</p>
<p>substituting \(x = 2\) into <strong>their</strong> \(g'\) <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\;\;\;\frac{{f'(2)}}{{f(2)}}\)</p>
<p>\( - 0.666667\)</p>
<p>\(g'(2) = - \frac{2}{3}{\text{ (exact), }} - 0.667\) <strong><em>A1 N3</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 class="p1">evidence of integrating \(g'(x)\) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1">eg\(\;\;\;g(x)|_2^a,{\text{ }}g(x)|_a^2\)</p>
<p class="p1">applying the fundamental theorem of calculus (seen anywhere) <span class="Apple-converted-space"> </span><strong><em>R1</em></strong></p>
<p class="p1">eg\(\;\;\;\int_2^a {g'(x) = g(a) - g(2)} \)</p>
<p class="p1">correct substitution into integral <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1">eg\(\;\;\;\ln 3 + g(a) - g(2),{\text{ }}\ln 3 + g(a) - \ln \left( {f(2)} \right)\)</p>
<p class="p1">\(\ln 3 + g(a) - \ln 3\) <span class="Apple-converted-space"> </span><strong><em>A1</em></strong></p>
<p class="p1">\(\ln 3 + \int_2^a {g'(x) = g(a)} \) <span class="Apple-converted-space"> </span><strong><em>AG <span class="Apple-converted-space"> </span>N0</em></strong></p>
<p class="p1"><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>substituting \(a = 5\) into the formula for \(g(a)\) <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\;\;\;\int_2^5 {g'(x){\text{d}}x,{\text{ }}g(5) = \ln 3 + \int_2^5 {g'(x){\text{d}}x\;\;\;} } \left( {{\text{do not accept only }}g(5)} \right)\)</p>
<p>attempt to substitute areas <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\;\;\;\ln 3 + 0.66 - 0.21,{\text{ }}\ln 3 + 0.66 + 0.21\)</p>
<p>correct working</p>
<p><em>eg</em>\(\;\;\;g(5) = \ln 3 + ( - 0.66 + 0.21)\) <strong><em>(A1)</em></strong></p>
<p>\(0.648612\)</p>
<p>\(g(5) = \ln 3 - 0.45{\text{ (exact), }}0.649\) <strong><em>A1 N3</em></strong></p>
<p><strong>METHOD 2</strong></p>
<p>attempt to set up an equation for one shaded region <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\;\;\;\int_4^5 {g'(x){\text{d}}x = 0.21,{\text{ }}\int_2^4 {g'(x){\text{d}}x = - 0.66,{\text{ }}\int_2^5 {g'(x){\text{d}}x = - 0.45} } } \)</p>
<p>two correct equations <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\;\;\;g(5) - g(4) = 0.21,{\text{ }}g(2) - g(4) = 0.66\)</p>
<p>combining equations to eliminate \(g(4)\) <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\;\;\;g(5) - [\ln 3 - 0.66] = 0.21\)</p>
<p>\(0.648612\)</p>
<p>\(g(5) = \ln 3 - 0.45{\text{ (exact), }}0.649\) <strong><em>A1 N3</em></strong></p>
<p><strong>METHOD 3</strong></p>
<p>attempt to set up a definite integral <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\;\;\;\int_2^5 {g'(x){\text{d}}x = - 0.66 + 0.21,{\text{ }}\int_2^5 {g'(x){\text{d}}x = - 0.45} } \)</p>
<p>correct working <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\;\;\;g(5) - g(2) = - 0.45\)</p>
<p>correct substitution <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\;\;\;g(5) - \ln 3 = - 0.45\)</p>
<p>\(0.648612\)</p>
<p>\(g(5) = \ln 3 - 0.45{\text{ (exact), }}0.649\) <strong><em>A1 N3</em></strong></p>
<p><strong><em>[4 marks]</em></strong></p>
<p><strong><em>Total [16 marks]</em></strong></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">In part (a), many candidates did not get full marks in justifying that \(p = 6\) was where the maximum occurs. The derivative changing from positive to negative was not sufficient since there are cases where the derivative changes signs at a value where there is no turning point. Part (c) was very poorly done as most candidates did not recognize the use of the chain rule to find the derivative of \(\ln \left( {f(x)} \right)\), a fairly basic application for Mathematics SL. In part (d), candidates appeared to have difficulty with the command term “verify”, and even if they were successful, did not make the connection to part (e) where they attempted a variety of interesting ways to find \(g(5)\) - the most common approach was to set up two incorrect integrals involving areas \(A\) and \(B\). Many students did not realize that integrating a function over an interval where the function is negative gives the opposite of the area between the function and the <em>x</em>-axis.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">In part (a), many candidates did not get full marks in justifying that \(p = 6\) was where the maximum occurs. The derivative changing from positive to negative was not sufficient since there are cases where the derivative changes signs at a value where there is no turning point. Part (c) was very poorly done as most candidates did not recognize the use of the chain rule to find the derivative of \(\ln \left( {f(x)} \right)\), a fairly basic application for Mathematics SL. In part (d), candidates appeared to have difficulty with the command term “verify”, and even if they were successful, did not make the connection to part (e) where they attempted a variety of interesting ways to find \(g(5)\) - the most common approach was to set up two incorrect integrals involving areas A and B. Many students did not realize that integrating a function over an interval where the function is negative gives the opposite of the area between the function and the <em>x</em>-axis.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">In part (a), many candidates did not get full marks in justifying that \(p = 6\) was where the maximum occurs. The derivative changing from positive to negative was not sufficient since there are cases where the derivative changes signs at a value where there is no turning point. Part (c) was very poorly done as most candidates did not recognize the use of the chain rule to find the derivative of \(\ln \left( {f(x)} \right)\), a fairly basic application for Mathematics SL. In part (d), candidates appeared to have difficulty with the command term “verify”, and even if they were successful, did not make the connection to part (e) where they attempted a variety of interesting ways to find \(g(5)\) - the most common approach was to set up two incorrect integrals involving areas A and B. Many students did not realize that integrating a function over an interval where the function is negative gives the opposite of the area between the function and the <em>x</em>-axis.</p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">In part (a), many candidates did not get full marks in justifying that \(p = 6\) was where the maximum occurs. The derivative changing from positive to negative was not sufficient since there are cases where the derivative changes signs at a value where there is no turning point. Part (c) was very poorly done as most candidates did not recognize the use of the chain rule to find the derivative of \(\ln \left( {f(x)} \right)\), a fairly basic application for Mathematics SL. In part (d), candidates appeared to have difficulty with the command term “verify”, and even if they were successful, did not make the connection to part (e) where they attempted a variety of interesting ways to find \(g(5)\) - the most common approach was to set up two incorrect integrals involving areas A and B. Many students did not realize that integrating a function over an interval where the function is negative gives the opposite of the area between the function and the <em>x</em>-axis.</p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">In part (a), many candidates did not get full marks in justifying that \(p = 6\) was where the maximum occurs. The derivative changing from positive to negative was not sufficient since there are cases where the derivative changes signs at a value where there is no turning point. Part (c) was very poorly done as most candidates did not recognize the use of the chain rule to find the derivative of \(\ln \left( {f(x)} \right)\), a fairly basic application for Mathematics SL. In part (d), candidates appeared to have difficulty with the command term “verify”, and even if they were successful, did not make the connection to part (e) where they attempted a variety of interesting ways to find \(g(5)\) - the most common approach was to set up two incorrect integrals involving areas A and B. Many students did not realize that integrating a function over an interval where the function is negative gives the opposite of the area between the function and the <em>x</em>-axis.</p>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p><strong>Note: In this question, distance is in metres and time is in seconds.</strong></p>
<p>A particle P moves in a straight line for five seconds. Its acceleration at time \(t\) is given by \(a = 3{t^2} - 14t + 8\), for \(0 \leqslant t \leqslant 5\).</p>
</div>
<div class="specification">
<p>When \(t = 0\), the velocity of P is \(3{\text{ m}}\,{{\text{s}}^{ - 1}}\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the values of \(t\) when \(a = 0\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence or otherwise, find all possible values of \(t\) for which the velocity of P is decreasing.</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 the velocity of P at time \(t\).</p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the total distance travelled by P when its velocity is increasing.</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>\(t = \frac{2}{3}{\text{ (exact), }}0.667,{\text{ }}t = 4\) <strong> <em>A1A1 N2</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>recognizing that \(v\) is decreasing when \(a\) is negative <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(a < 0,{\text{ }}3{t^2} - 14t + 8 \leqslant 0\), sketch of \(a\)</p>
<p>correct interval <strong><em>A1 N2</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\frac{2}{3} < t < 4\)</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>valid approach (do not accept a definite integral) <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(v\int a \)</p>
<p>correct integration (accept missing \(c\)) <strong><em>(A1)(A1)(A1)</em></strong></p>
<p>\({t^3} - 7{t^2} + 8t + c\)</p>
<p>substituting \(t = 0,{\text{ }}v = 3\) , (must have \(c\)) <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(3 = {0^3} - 7({0^2}) + 8(0) + c,{\text{ }}c = 3\)</p>
<p>\(v = {t^3} - 7{t^2} + 8t + 3\) <strong><em>A1 N6</em></strong></p>
<p><strong><em>[6 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>recognizing that \(v\) increases outside the interval found in part (b) <strong><em>(M1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(0 < t < \frac{2}{3},{\text{ }}4 < t < 5\), diagram</p>
<p>one correct substitution into distance formula <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)\(\int_0^{\frac{2}{3}} {\left| v \right|,{\text{ }}\int_4^5 {\left| v \right|} ,{\text{ }}\int_{\frac{2}{3}}^4 {\left| v \right|} ,{\text{ }}\int_0^5 {\left| v \right|} } \)</p>
<p>one correct pair <strong><em>(A1)</em></strong></p>
<p><em>eg</em>\(\,\,\,\,\,\)3.13580 and 11.0833, 20.9906 and 35.2097</p>
<p>14.2191 <strong><em>A1 N2</em></strong></p>
<p>\(d = 14.2{\text{ (m)}}\)</p>
<p><strong><em>[4 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.</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 class="p1">The following diagram shows the graph of \(f(x) = a\sin bx + c\), for \(0 \leqslant x \leqslant 12\).</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2017-03-03_om_16.53.31.png" alt="N16/5/MATME/SP2/ENG/TZ0/10"></p>
<p class="p1" style="text-align: center;">The graph of \(f\) has a minimum point at \((3,{\text{ }}5)\) and a maximum point at \((9,{\text{ }}17)\).</p>
</div>
<div class="specification">
<p class="p1">The graph of \(g\) is obtained from the graph of \(f\) by a translation of \(\left( {\begin{array}{*{20}{c}} k \\ 0 \end{array}} \right)\). The maximum point on the graph of \(g\) has coordinates \((11.5,{\text{ }}17)\).</p>
</div>
<div class="specification">
<p class="p1">The graph of \(g\) changes from concave-up to concave-down when \(x = w\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Find the value of \(c\).</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Show that \(b = \frac{\pi }{6}\).</p>
<p class="p1">(iii) <span class="Apple-converted-space"> </span>Find the value of \(a\).</p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Write down the value of \(k\).</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Find \(g(x)\).</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 class="p1">(i) <span class="Apple-converted-space"> </span>Find \(w\).</p>
<p class="p2">(ii) <span class="Apple-converted-space"> </span>Hence or otherwise, find the maximum positive rate of change of \(g\).</p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>valid approach <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p2"><em>eg</em>\(\,\,\,\,\,\)\(\frac{{5 + 17}}{2}\)</p>
<p class="p2"><span class="Apple-converted-space">\(c = 11\) </span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>valid approach <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p3"><em>eg</em>\(\,\,\,\,\,\)period is 12, per \( = \frac{{2\pi }}{b},{\text{ }}9 - 3\)</p>
<p class="p3"><span class="Apple-converted-space">\(b = \frac{{2\pi }}{{12}}\) </span><span class="s1"><strong><em>A1</em></strong></span></p>
<p class="p2"><span class="s2">\(b = \frac{\pi }{6}\) <span class="Apple-converted-space"> </span></span><strong><em>AG <span class="Apple-converted-space"> </span>N0</em></strong></p>
<p class="p1">(iii) <span class="Apple-converted-space"> </span><strong>METHOD 1</strong></p>
<p class="p1">valid approach <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p1"><span class="s1"><em>eg</em>\(\,\,\,\,\,\)\(5 = a\sin \left( {\frac{\pi }{6} \times 3} \right) + 11\)</span>, substitution of points</p>
<p class="p2"><span class="s3">\(a = - 6\) <span class="Apple-converted-space"> </span></span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p1"><strong>METHOD 2</strong></p>
<p class="p1">valid approach <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p3"><span class="s1"><em>eg</em>\(\,\,\,\,\,\)\(\frac{{17 - 5}}{2}\)</span>, amplitude is 6</p>
<p class="p2"><span class="s2">\(a = - 6\) <span class="Apple-converted-space"> </span></span><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></p>
<p class="p2"><strong><em>[6 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> \(k = 2.5\)</span> <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1 <span class="Apple-converted-space"> </span>N1</em></strong></span></p>
<p class="p2">(ii) <span class="Apple-converted-space"> \(g(x) = - 6\sin \left( {\frac{\pi }{6}(x - 2.5)} \right) + 11\)</span> <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A2 <span class="Apple-converted-space"> </span>N2</em></strong></span></p>
<p class="p3"><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span><strong>METHOD 1 </strong>Using \(g\)</p>
<p class="p1">recognizing that a point of inflexion is required <span class="Apple-converted-space"> </span><span class="s1"><strong><em>M1</em></strong></span></p>
<p class="p1"><span class="s1"><em>eg</em>\(\,\,\,\,\,\)s</span>ketch, recognizing change in concavity</p>
<p class="p1">evidence of valid approach <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p1"><span class="s2"><em>eg</em>\(\,\,\,\,\,\)\(g''(x) = 0\)</span>, sketch, coordinates of max/min on \({g'}\)</p>
<p class="p1">\(w = 8.5\) (exact) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></span></p>
<p class="p1"><strong>METHOD 2 </strong>Using \(f\)</p>
<p class="p1">recognizing that a point of inflexion is required <span class="Apple-converted-space"> </span><span class="s1"><strong><em>M1</em></strong></span></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)sketch, recognizing change in concavity</p>
<p class="p1">evidence of valid approach involving translation <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p2"><span class="s2"><em>eg</em>\(\,\,\,\,\,\)\(x = w - k\)</span>, sketch, \(6 + 2.5\)</p>
<p class="p1"><span class="s3">\(w = 8.5\) </span>(exact) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></span></p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>valid approach involving the derivative of \(g\) or \(f\) (seen anywhere) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p1"><span class="s2"><em>eg</em>\(\,\,\,\,\,\)\(g'(w),{\text{ }} - \pi \cos \left( {\frac{\pi }{6}x} \right)\)</span>, max on derivative, sketch of derivative</p>
<p class="p1">attempt to find max value on derivative <span class="Apple-converted-space"> </span><span class="s1"><strong><em>M1</em></strong></span></p>
<p class="p2"><span class="s1"><em>eg</em>\(\,\,\,\,\,\)\( - \pi \cos \left( {\frac{\pi }{6}(8.5 - 2.5)} \right),{\text{ }}f'(6)\), </span>dot on max of sketch</p>
<p class="p2">3.14159</p>
<p class="p1">max rate of change \( = \pi \) <span class="s3">(exact), 3.14 <span class="Apple-converted-space"> </span></span><span class="s1"><strong><em>A1 <span class="Apple-converted-space"> </span>N2</em></strong></span></p>
<p class="p3"><strong><em>[6 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 class="p1">Let \(f(x) = x{{\text{e}}^{ - x}}\) and \(g(x) = - 3f(x) + 1\).</p>
<p class="p1">The graphs of \(f\) and \(g\) intersect at \(x = p\) and \(x = q\), where \(p < q\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the value of \(p\) <span class="s1">and of \(q\).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Hence, find the area of the region enclosed by the graphs of \(f\) <span class="s1">and \(g\).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">valid attempt to find the intersection <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p1"><span class="s2"><em>eg</em>\(\,\,\,\,\,\)\(f = g\)</span>, sketch, one correct answer</p>
<p class="p1">\(p = 0.357402,{\text{ }}q = 2.15329\)</p>
<p class="p2"><span class="s3">\(p = 0.357,{\text{ }}q = 2.15\) <span class="Apple-converted-space"> </span></span><strong><em>A1A1 <span class="Apple-converted-space"> </span>N3</em></strong></p>
<p class="p2"><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">attempt to set up an integral involving subtraction (in any order) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p2"><em>eg</em>\(\,\,\,\,\,\)\(\int_p^q {\left[ {f(x) - g(x)} \right]{\text{d}}x,{\text{ }}} \int_p^q {f(x){\text{d}}x - } \int_p^q {g(x){\text{d}}x} \)</p>
<p class="p3">0.537667</p>
<p class="p2"><span class="s2">\({\text{area}} = 0.538\) <span class="Apple-converted-space"> </span></span><strong><em>A2 <span class="Apple-converted-space"> </span>N3</em></strong></p>
<p class="p2"><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1"><strong>All lengths in this question are in metres.</strong></p>
<p class="p1">Let \(f(x) = - 0.8{x^2} + 0.5\), for \( - 0.5 \leqslant x \leqslant 0.5\). Mark uses \(f(x)\) as a model to create a barrel. The region enclosed by the graph of \(f\), the \(x\)-axis, the line \(x = - 0.5\) and the line \(x = 0.5\) is rotated <span class="s1">360°</span> about the \(x\)-axis. This is shown in the following diagram.</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2017-03-03_om_15.49.19.png" alt="N16/5/MATME/SP2/ENG/TZ0/06"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Use the model to find the volume of the barrel.</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 class="p1"><span class="s1">The empty barrel is being filled with water. The volume \(V{\text{ }}{{\text{m}}^3}\) </span>of water in the barrel after \(t\) minutes is given by \(V = 0.8(1 - {{\text{e}}^{ - 0.1t}})\). How long will it take for the barrel to be half-full?</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="p1">attempt to substitute correct limits or the function into the formula involving</p>
<p class="p1">\({y^2}\)</p>
<p class="p2"><em>eg</em>\(\,\,\,\,\,\)\(\pi \int_{ - 0.5}^{0.5} {{y^2}{\text{d}}x,{\text{ }}\pi \int {{{( - 0.8{x^2} + 0.5)}^2}{\text{d}}x} } \)</p>
<p class="p3">0.601091</p>
<p class="p1">volume \( = 0.601{\text{ }}({{\text{m}}^3})\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A2 <span class="Apple-converted-space"> </span>N3</em></strong></span></p>
<p class="p2"><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">attempt to equate half <strong>their </strong>volume to \(V\) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>(M1)</em></strong></span></p>
<p class="p2"><span class="s1"><em>eg</em>\(\,\,\,\,\,\)\(0.30055 = 0.8(1 - {{\text{e}}^{ - 0.1t}})\)</span>, graph</p>
<p class="p2">4.71104</p>
<p class="p1"><span class="s2">4.71 </span>(minutes) <span class="Apple-converted-space"> </span><span class="s1"><strong><em>A2 <span class="Apple-converted-space"> </span>N3</em></strong></span></p>
<p class="p3"><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Consider the curve with equation \(f(x) = p{x^2} + qx\) , where <em>p</em> and <em>q</em> are constants. </span><span style="font-family: times new roman,times; font-size: medium;">The point \({\text{A}}(1{\text{, }}3)\) lies on the curve. The tangent to the curve at A has gradient \(8\). </span><span style="font-family: times new roman,times; font-size: medium;">Find the value of <em>p</em> and of <em>q</em> .</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">substituting \(x = 1\) , \(y = 3\) into \(f(x)\) <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(3 = p + q\) <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">finding derivative <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(f'(x) = 2px + q\) <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct substitution, \(2p + q = 8\) <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(p = 5\) , \(q = - 2\) <em><strong>A1A1 N2N2</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[7 marks] </span></strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">A good number of candidates were able to obtain an equation by substituting the point \(1{\text{, }}3\) into the function’s equation. Not as many knew how to find the other equation by using the derivative. Some candidates thought they needed to find the equation of the tangent line rather than recognising that the information about the tangent provided the gradient of the function at the point. While they were usually able to find this equation correctly, it was irrelevant to the question asked. </span></p>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Consider the curve \(y = \ln (3x - 1)\) . Let P be the point on the curve where \(x = 2\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Write down the gradient of the curve at P.</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><span style="font-family: times new roman,times; font-size: medium;">The normal to the curve at P cuts the <em>x</em>-axis at R. Find the coordinates of R.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">gradient is \(0.6\) <em><strong>A2 N2</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">at R, \(y = 0\) (seen anywhere) <em><strong>A1</strong> </em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">at \(x = 2\) , \(y = \ln 5\) \(( = 1.609 \ldots )\) <em><strong> (A1)</strong> </em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">gradient of normal \( = - 1.6666 \ldots \) <em><strong>(A1)</strong> </em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of finding correct equation of normal <em><strong>A1</strong> </em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(y = \ln 5 = - \frac{5}{3}(x - 2)\) , \(y = - 1.67x + c\)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(x = 2.97\) (accept 2.96) <em><strong>A1</strong> </em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">coordinates of R are (2.97,0) <em><strong> N3</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[5 marks] </span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Although the command term "write down" was used in part (a), many candidates still opted for an analytic method for finding the derivative value. Although this value was often incorrect, many candidates knew how to find the equation of the normal and earned follow through marks in part (b). </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Although the command term "write down" was used in part (a), many candidates still opted for an analytic method for finding the derivative value. Although this value was often incorrect, many candidates knew how to find the equation of the normal and earned follow through marks in part (b). </span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p class="p1">A particle moves in a straight line. Its velocity \(v{\text{ m}}\,{{\text{s}}^{ - 1}}\) after \(t\) seconds is given by</p>
<p class="p1">\[v = 6t - 6,{\text{ for }}0 \leqslant t \leqslant 2.\]</p>
<p class="p1">After \(p\) <span class="s1">seconds, the particle is 2 m </span>from its initial position. Find the possible values of \(p\).</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p class="p1">correct approach <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(s = \int {v,{\text{ }}\int_0^p {6t - 6{\text{d}}t} } \)</p>
<p class="p1">correct integration <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(\int {6t - 6{\text{d}}t = 3{t^2} - 6t + C,{\text{ }}\left[ {3{t^2} - 6t} \right]_0^p} \)</p>
<p class="p1">recognizing that there are two possibilities <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)2 correct answers, \(s = \pm 2,{\text{ }}c \pm 2\)</p>
<p class="p1">two correct equations in \(p\) <span class="Apple-converted-space"> </span><strong><em>A1A1</em></strong></p>
<p class="p1"><em>eg</em>\(\,\,\,\,\,\)\(3{p^2} - 6p = 2,{\text{ }}3{p^2} - 6p = - 2\)</p>
<p class="p1">0.42265, 1.57735</p>
<p class="p1"><span class="Apple-converted-space">\(p = 0.423{\text{ or }}p = 1.58\) </span><strong><em>A1A1 <span class="Apple-converted-space"> </span>N3</em></strong></p>
<p class="p1"><strong><em>[7 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p class="p1">Most candidates realized that they needed to calculate the integral of the velocity, and did it correctly. However, only a few realized that there were two possible positions for the particle, as it could move in two directions. In general, the only equation candidates wrote was \(3{p^2} - 6p = 2\), that gave solutions outside the given domain. Candidates failed to differentiate between displacement and distance travelled.</p>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = 4x - {{\rm{e}}^{x - 2}} - 3\) , for \(0 \le x \le 5\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the <em>x</em>-intercepts of the graph of <em>f</em> .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">On the grid below, sketch the graph of <em>f</em> .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/chops.png" alt></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><span style="font-family: times new roman,times; font-size: medium;">Write down the gradient of the graph of <em>f</em> at \(x = 3\) .</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">intercepts when \(f(x) = 0\) <em><strong>M1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(0.827, 0) (4.78, 0) (accept \(x = 0.827\), \(x = 4.78\) ) <em><strong>A1A1 N3</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/sock.png" alt></span><em><strong><span style="font-family: times new roman,times; font-size: medium;"> A1A1A1 N3</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Award <em><strong>A1</strong></em> for maximum point in circle, <em><strong>A1</strong></em> for <em>x</em>-intercepts in circles, </span><span style="font-family: times new roman,times; font-size: medium;"><em><strong>A1</strong></em> for correct shape (<em>y</em> approximately greater than \( - 3.14\)).</span></p>
<p><em><strong> <span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">gradient is 1.28 <em><strong>A1 N1</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[1 mark]</span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Ramiro and Lautaro are travelling from Buenos Aires to El Moro.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Ramiro travels in a vehicle whose velocity in \({\text{m}}{{\text{s}}^{ - 1}}\) is given by \({V_R} = 40 - {t^2}\), where \(t\) is in seconds.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Lautaro travels in a vehicle whose displacement from Buenos Aires in metres is given by \({S_L} = 2{t^2} + 60\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">When \(t = 0\), both vehicles are at the same point.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find Ramiro’s displacement from Buenos Aires when \(t = 10\).</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 1</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\({S_L}(0) = 60\) (seen anywhere) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">recognizing need to integrate \({V_R}\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \({S_R}(t)\int {{V_R}{\text{d}}t} \)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">correct expression <strong><em>A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(40t - \frac{1}{3}{t^3} + C\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Award <strong><em>A1 </em></strong>for \(40t\), and <strong><em>A1 </em></strong>for \( - \frac{1}{3}{t^3}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">equate displacements to find <em>C </em><strong><em>(R1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(40(0) - \frac{1}{3}{(0)^3} + C = 60,{\text{ }}{S_L}(0) = {S_R}(0)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(C = 60\) <em><strong>A1</strong></em></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">attempt to find displacement <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \({S_R}(10),{\text{ }}40(10) - \frac{1}{3}{(10)^3} + 60\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(126.666\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(126\frac{2}{3}{\text{ (exact), 127 (m)}}\) <strong><em>A1 N5</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 2</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">recognizing need to integrate \({V_R}\) <em><strong>(M1)</strong></em></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \({S_R}(t) = \int {{V_R}{\text{d}}t} \)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">valid approach involving a definite integral <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(\int_a^b {{V_R}{\text{d}}t} \)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">correct expression with limits <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(\int_0^{10} {\left( {40 - {t^2}} \right){\text{d}}t,{\text{ }}} \int_0^{10} {{V_R}{\text{d}}t,{\text{ }}\left[ {40t - \frac{1}{3}{t^3}} \right]} _0^{10}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(66.6666\) <strong><em>A2</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\({S_L}(0) = 60\) (seen anywhere) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">valid approach to find total displacement <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg </em>\(60 + 66.666\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(126.666\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(126\frac{2}{3}\) (exact), \(127\) (m) <strong><em>A1 N5</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em> </em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>METHOD 3</strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\({S_L}(0) = 60\) (seen anywhere) <strong><em>(A1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">recognizing need to integrate \({V_R}\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \({S_R}(t) = \int {{V_R}{\text{d}}t} \)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">correct expression <strong><em>A1A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(40t - \frac{1}{3}{t^3} + C\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong>Note:</strong> Award <strong><em>A1 </em></strong>for \(40t\), and <strong><em>A1 </em></strong>for \( - \frac{1}{3}{t^3}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman'; min-height: 25.0px;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">correct expression for Ramiro displacement <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \({S_R}(10) - {S_R}(0),{\text{ }}\left[ {40t - \frac{1}{3}{t^3} + C} \right]_0^{10}\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(66.6666\) <strong><em>A1</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">valid approach to find total displacement <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(60 + 66.6666\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\(126\frac{2}{3}\) (exact), 127 (m) <strong><em>A1 N5</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[8 marks]</em></strong></span></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Let \(f(x) = 5 - {x^2}\). Part of the graph of \(f\)is shown in the following diagram.</span></p>
<p style="margin: 0px; font-style: normal; font-variant: normal; font-weight: normal; font-size: 21px; line-height: normal; font-family: 'Times New Roman'; text-align: center;"><img src="images/maths_2.png" alt></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The graph crosses the \(x\)-axis at the points \(\rm{A}\) and \(\rm{B}\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find the \(x\)-coordinate of \({\text{A}}\) and of \({\text{B}}\).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The region enclosed by the graph of \(f\) and the \(x\)-axis is revolved \(360^\circ \) about the \(x\)-axis.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">Find the volume of the solid formed.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">recognizing \(f(x) = 0\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(f = 0,{\text{ }}{x^2} = 5\)</span></p>
<p style="font: normal normal normal 21px/normal 'Times New Roman'; text-align: left; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;">\(x = \pm 2.23606\)</span></p>
<p style="font: normal normal normal 21px/normal 'Times New Roman'; text-align: left; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;">\(x = \pm \sqrt 5 {\text{ (exact), }}x = \pm 2.24\) <strong><em>A1A1 N3</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">attempt to substitute either limits or the function into formula</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">involving \({f^2}\) <strong><em>(M1)</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><em>eg</em> \(\pi \int {{{\left( {5 - {x^2}} \right)}^2}{\text{d}}x,{\text{ }}\pi \int_{ - 2.24}^{2.24} {\left( {{x^4} - 10{x^2} + 25} \right)} ,{\text{ }}2\pi \int_0^{\sqrt 5 } {{f^2}} } \)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">\(187.328\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">volume \(= 187\) <strong><em>A2 N3</em></strong></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"><strong><em>[3 marks]</em></strong></span></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><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = 3\sin x + 4\cos x\) , for \( - 2\pi \le x \le 2\pi \) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Sketch the graph of <em>f</em> .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Write down</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) the amplitude;</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) the period;</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(iii) the <em>x</em>-intercept that lies between \( - \frac{\pi }{2}\) </span><span style="font-family: times new roman,times; font-size: medium;">and 0.</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><span style="font-family: times new roman,times; font-size: medium;">Hence write \(f(x)\) in the form \(p\sin (qx + r)\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Write down one value of <em>x</em> such that \(f'(x) = 0\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Write down the two values of <em>k</em> for which the equation \(f(x) = k\) has exactly </span><span style="font-family: times new roman,times; font-size: medium;">two solutions.</span></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 align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Let \(g(x) = \ln (x + 1)\) , for \(0 \le x \le \pi \) . There is a value of <em>x</em>, between \(0\) and \(1\), </span><span style="font-family: times new roman,times; font-size: medium;">for which the gradient of <em>f</em> is equal to the gradient of <em>g</em>. Find this value of <em>x</em>.</span></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> </p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/mike.png" alt></span><em><span style="font-family: times new roman,times; font-size: medium;"><strong> A1A1A1 N3</strong> </span></em></p>
<p> </p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note:</strong> Award <em><strong>A1</strong></em> for approximately sinusoidal shape, </span><span style="font-family: times new roman,times; font-size: medium;"><em><strong>A1</strong></em> for end points approximately correct \(( - 2\pi {\text{, }}4)\) \((2\pi {\text{, }}4)\), </span><span style="font-family: times new roman,times; font-size: medium;"><em><strong>A1</strong></em> for approximately correct position of graph, (<em>y</em>-intercept \((0{\text{, }}4)\), maximum to right of <em>y</em>-axis). </span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(i) 5 <em><strong>A1 N1</strong> </em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(ii) \(2\pi \) (6.28) <em><strong>A1 N1</strong> </em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">(iii) \( - 0.927\) <em><strong>A1 N1</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks] </span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(f(x) = 5\sin (x + 0.927)\) (accept \(p = 5\) , \(q = 1\) , \(r = 0.927\) ) <em><strong>A1A1A1 N3</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks] </span></strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of correct approach <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. max/min, sketch of \(f'(x)\) indicating roots </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/wee.png" alt></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">one 3 s.f. value which rounds to one of \( - 5.6\), \( - 2.5\), \(0.64\), \(3.8\) <em><strong>A1 N2 </strong></em></span></p>
<p> </p>
<p><em><span style="font-family: times new roman,times; font-size: medium;"><strong>[2 marks]</strong> </span></em></p>
<p> </p>
<p> </p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(k = - 5\) , \(k = 5\) <em><strong>A1A1 N2</strong> </em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks] </span></strong></em></p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"><strong>METHOD 1</strong> </span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">graphical approach (but must involve derivative functions) <em><strong>M1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">e.g. </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/visit.png" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">each curve <em><strong>A1A1</strong> </em></span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\(x = 0.511\) </span><em><strong><span style="font-family: times new roman,times; font-size: medium;">A2 N2</span></strong></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>METHOD 2</strong> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(g'(x) = \frac{1}{{x + 1}}\) <em><strong>A1 </strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(f'(x) = 3\cos x - 4\sin x\) \((5\cos (x + 0.927))\) <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of attempt to solve \(g'(x) = f'(x)\) <em><strong> M1</strong> </em></span></p>
<p><span style="font-family: Times New Roman; font-size: medium;">\(x = 0.511\) </span><em><span style="font-family: times new roman,times; font-size: medium;"><strong>A2 N2</strong> </span></em></p>
<p><em> <span style="font-family: times new roman,times; font-size: medium;"><strong>[5 marks]</strong> </span></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><span style="font-family: times new roman,times; font-size: medium;">Some graphs in part (a) were almost too detailed for just a sketch but more often, the important features were far from clear. Some graphs lacked scales on the axes. </span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">A number of candidates had difficulty finding the period in part (b)(ii).</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">A number of candidates had difficulty writing the correct value of <em>q</em> in part (c). </span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">The most common approach in part (d) was to differentiate and set \(f'(x) = 0\) . Fewer students found the values of <em>x</em> given by the maximum or minimum values on their graphs. </span></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Part (e) proved challenging for many candidates, although if candidates answered this part, they generally did so correctly. </span></p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">In part (f), many candidates were able to get as far as equating the two derivatives but fewer used their GDC to solve the resulting equation. Again, many had trouble demonstrating their method of solution. </span></p>
<div class="question_part_label">f.</div>
</div>
<br><hr><br><div class="question">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = \cos ({x^2})\) and \(g(x) = {{\rm{e}}^x}\) , for \( - 1.5 \le x \le 0.5\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Find the area of the region enclosed by the graphs of <em>f</em> and <em>g</em> .</span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of finding intersection points <em><strong>(M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(f(x) = g(x)\) , \(\cos {x^2} = {{\rm{e}}^x}\) , sketch showing intersection</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\(x = - 1.11\) , \(x = 0\) (may be seen as limits in the integral) <em><strong>A1A1</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">evidence of approach involving integration and subtraction (in any order) <em><strong> (M1)</strong></em></span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">e.g. \(\int_{ - 1.11}^0 {\cos {x^2} - {{\rm{e}}^x}} \) , \(\int {(\cos {x^2} - {{\rm{e}}^x}){\rm{d}}x} \) , \(\int {g - f} \)</span></p>
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">\({\text{area}} = 0.282\) <em><strong>A2 N3</strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[6 marks]</span></strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p><span style="font-family: times new roman,times; font-size: medium;">This question was poorly done by a great many candidates. Most seemed not to understand what was meant by the phrase "region enclosed by" as several candidates assumed that the limits of the integral were those given in the domain. Few realized what area was required, or that intersection points were needed. Candidates who used their GDCs to first draw a suitable sketch could normally recognize the required region and could find the intersection points correctly. However, it was disappointing to see the number of candidates who could not then use their GDC to find the required area or who attempted unsuccessful analytical approaches. </span></p>
</div>
<br><hr><br><div class="specification">
<p>A particle P moves along a straight line. The velocity <em>v</em> m s<sup>−1</sup> of P after <em>t</em> seconds is given by <em>v</em> (<em>t</em>) = 7 cos <em>t</em> − 5<em>t </em><sup>cos <em>t</em></sup>, for 0 ≤ <em>t</em> ≤ 7.</p>
<p>The following diagram shows the graph of <em>v</em>.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the initial velocity of P.</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 maximum speed of P.</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>Write down the number of times that the acceleration of P is 0 m s<sup>−2</sup> .</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 acceleration of P when it changes direction.</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the total distance travelled by P.</p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>initial velocity when <em>t</em> = 0 <em><strong>(M1)</strong></em></p>
<p>eg <em>v</em>(0)</p>
<p><em>v</em> = 17 (m s<sup>−1</sup>) <em><strong>A1 N2</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>recognizing maximum speed when \(\left| v \right|\) is greatest <em><strong>(M1)</strong></em></p>
<p><em>eg</em> minimum, maximum, <em>v'</em> = 0</p>
<p>one correct coordinate for minimum <em><strong>(A1)</strong></em></p>
<p><em>eg</em> 6.37896, −24.6571</p>
<p>24.7 (ms<sup>−1</sup>) <em><strong>A1 N2</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>recognizing a = <em>v </em>′ <em><strong>(M1)</strong></em></p>
<p>eg \(a = \frac{{{\text{d}}v}}{{{\text{d}}t}}\), correct derivative of first term</p>
<p>identifying when a = 0 <em><strong>(M1)</strong></em></p>
<p><em>eg</em> turning points of <em>v</em>, <em>t</em>-intercepts of <em>v </em>′</p>
<p>3 <em><strong>A1 N3</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>recognizing P changes direction when <em>v </em>= 0 <em><strong>(M1) </strong></em></p>
<p><em>t</em> = 0.863851 <em><strong>(A1) </strong></em></p>
<p>−9.24689</p>
<p><em>a</em> = −9.25 (ms<sup>−2</sup>) <em><strong> A2 N3</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>correct substitution of limits or function into formula <em><strong>(A1)</strong></em><br><em>eg</em> \(\int_0^7 {\left| {\,v\,} \right|,\,\int_0^{0.8638} {v{\text{d}}t - \int_{0.8638}^7 {v{\text{d}}t} } ,\,\,\int {\left| {\,7\,{\text{cos}}\,x - 5{x^{{\text{cos}}\,x}}\,} \right|} \,dx,\,\,3.32 = 60.6} \)</p>
<p>63.8874</p>
<p>63.9 (metres) <em><strong>A2 N3</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[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>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The velocity of a particle in ms<sup>−1</sup> is given by \(v = {{\rm{e}}^{\sin t}} - 1\) , for \(0 \le t \le 5\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">On the grid below, sketch the graph of \(v\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><br><img src="images/ronan.png" alt></span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the total distance travelled by the particle in the first five seconds.</span></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><span style="font-family: times new roman,times; font-size: medium;">Write down the positive \(t\)-intercept.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.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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" alt><em><span style="font-family: times new roman,times; font-size: medium;"><strong> A1A1A1 N3</strong> </span></em></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Note</strong>: Award <em><strong>A1</strong></em> for approximately correct shape crossing <em>x</em>-axis with \(3 < x < 3.5\) . </span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"><strong> Only</strong> if this <em><strong>A1</strong></em> is awarded, award the following: </span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"><em><strong> A1</strong></em> for maximum in circle, <em><strong>A1</strong></em> for endpoints in circle. </span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[3 marks] </span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">\(t = \pi \) (exact), \(3.14\) <strong><em>A1 N1</em> </strong></span></p>
<p><em><span style="font-family: times new roman,times; font-size: medium;"><strong>[1 mark]</strong> </span></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">recognizing distance is area under velocity curve <strong><em>(M1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(s = \int v \) , shading on diagram, attempt to integrate </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">valid approach to find the total area <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \({\text{area A}} + {\text{area B}}\) , \(\int {v{\rm{d}}t - \int {v{\rm{d}}t} } \) , \(\int_0^{3.14} {v{\rm{d}}t + } \int_{3.14}^5 {v{\rm{d}}t} \) , \(\int {\left| v \right|} \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct working with integration and limits (accept \({\rm{d}}x\) or missing \({\rm{d}}t\) ) <strong><em>(A1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(\int_0^{3.14} {v{\rm{d}}t + } \int_5^{3.14} {v{\rm{d}}t} \) , \(3.067 \ldots + 0.878 \ldots \) , \(\int_0^5 {\left| {{{\rm{e}}^{\sin t}} - 1} \right|} \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">distance \( = 3.95\) (m) <em><strong>A1 N3 </strong></em></span></p>
<p><em><span style="font-family: times new roman,times; font-size: medium;"><strong>[4 marks]</strong> </span></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">There was a minor error on this question, where the units for velocity were given as ms<sup>-2</sup> rather than ms<sup>-1</sup> . Examiners were instructed to notify the IB assessment centre of any candidates adversely affected, and these were considered at the grade award meeting.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Candidates continue to produce sloppy graphs resulting in loss of marks. Although the shape was often correctly drawn, students were careless when considering the domain and other key features such as the root and the location of the maximum point.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">The fact that most candidates with poorly drawn graphs correctly found the root in (b)(i), clearly emphasized the disconnect between geometric and algebraic approaches to problems.</span></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">In (b)(ii), most appreciated that the definite integral would give the distance travelled but few could write a valid expression and normally just integrated from \(t = 0\) to \(t = 5\) without considering the part of the graph below the \(t\)-axis. Again, analytic approaches to evaluating their integral predominated over simpler GDC approaches and some candidates had their calculator set in degree mode rather than radian mode.</span></p>
<div class="question_part_label">b.ii.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Let \(f(x) = \frac{{100}}{{(1 + 50{{\rm{e}}^{ - 0.2x}})}}\) . Part of the graph of \(f\) is shown below.</span></p>
<p><br><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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" alt></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Write down \(f(0)\) .</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Solve \(f(x) = 95\) .</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><span style="font-family: times new roman,times; font-size: medium;">Find the range of \(f\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Show that \(f'(x) = \frac{{1000{{\rm{e}}^{ - 0.2x}}}}{{{{(1 + 50{{\rm{e}}^{ - 0.2x}})}^2}}}\) .</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Find the maximum rate of change of \(f\) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">\(f(0) = \frac{{100}}{{51}}\) (exact), \(1.96\) <em><strong>A1 N1 </strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[1 mark] </span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">setting up equation <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(95 = \frac{{100}}{{1 + 50{{\rm{e}}^{ - 0.2x}}}}\) , sketch of graph with horizontal line at \(y = 95\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(x = 34.3\) <em><strong>A1 N2 </strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[2 marks] </span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">upper bound of \(y\) is \(100\) <strong><em>(A1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">lower bound of \(y\) is \(0\) <strong><em>(A1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">range is \(0 < y < 100\) <strong><em>A1 N3 </em></strong></span></p>
<p><strong><em><span style="font-family: times new roman,times; font-size: medium;">[3 marks] </span></em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong><span style="font-family: times new roman,times; font-size: medium;">METHOD 1 </span></strong></p>
<p><span style="font-family: times new roman,times; font-size: medium;">setting function ready to apply the chain rule <em><strong>(M1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(100{(1 + 50{{\rm{e}}^{ - 0.2x}})^{ - 1}}\) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of correct differentiation (must be substituted into chain rule) <strong><em>(A1)(A1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(u' = - 100{(1 + 50{{\rm{e}}^{ - 0.2x}})^{ - 2}}\) , \(v' = (50{{\rm{e}}^{ - 0.2x}})( - 0.2)\) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct chain rule derivative <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(f'(x) = - 100{(1 + 50{{\rm{e}}^{ - 0.2x}})^{ - 2}}(50{{\rm{e}}^{ - 0.2x}})( - 0.2)\) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct working clearly leading to the required answer <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(f'(x) = 1000{{\rm{e}}^{ - 0.2x}}{(1 + 50{{\rm{e}}^{ - 0.2x}})^{ - 2}}\) </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(f'(x) = \frac{{1000{{\rm{e}}^{ - 0.2x}}}}{{{{(1 + 50{{\rm{e}}^{ - 0.2x}})}^2}}}\) <strong><em> AG N0</em> </strong></span></p>
<p><strong><span style="font-family: times new roman,times; font-size: medium;">METHOD 2 </span></strong></p>
<p><span style="font-family: times new roman,times; font-size: medium;">attempt to apply the quotient rule (accept reversed numerator terms) <strong><em> (M1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(\frac{{vu' - uv'}}{{{v^2}}}\) , \(\frac{{uv' - vu'}}{{{v^2}}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">evidence of correct differentiation inside the quotient rule <strong><em>(A1)(A1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(f'(x) = \frac{{(1 + 50{{\rm{e}}^{ - 0.2x}})(0) - 100(50{{\rm{e}}^{ - 0.2x}} \times - 0.2)}}{{{{(1 + 50{{\rm{e}}^{ - 0.2x}})}^2}}}\) , \(\frac{{100( - 10){{\rm{e}}^{ - 0.2x}} - 0}}{{{{(1 + 50{{\rm{e}}^{ - 0.2x}})}^2}}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">any correct expression for derivative (\(0\) may not be explicitly seen) <strong><em>(A1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(\frac{{ - 100(50{{\rm{e}}^{ - 0.2x}} \times - 0.2)}}{{{{(1 + 50{{\rm{e}}^{ - 0.2x}})}^2}}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">correct working clearly leading to the required answer <strong><em>A1</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(f'(x) = \frac{{0 - 100( - 10){{\rm{e}}^{ - 0.2x}}}}{{{{(1 + 50{{\rm{e}}^{ - 0.2x}})}^2}}}\) , \(\frac{{ - 100( - 10){{\rm{e}}^{ - 0.2x}}}}{{{{(1 + 50{{\rm{e}}^{ - 0.2x}})}^2}}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">\(f'(x) = \frac{{{\rm{1000}}{{\rm{e}}^{ - 0.2x}}}}{{{{(1 + 50{{\rm{e}}^{ - 0.2x}})}^2}}}\) <strong><em>AG N0 </em></strong></span></p>
<p><strong><em><span style="font-family: times new roman,times; font-size: medium;">[5 marks] </span></em></strong></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>METHOD 1</strong> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">sketch of \(f'(x)\) <em><strong>(A1)</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg</em> </span></p>
<p><img 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" alt></p>
<p><span style="font-family: times new roman,times; font-size: medium;">recognizing maximum on \(f'(x)\) <strong><em>(M1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg</em> dot on max of sketch </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">finding maximum on graph of \(f'(x)\) <em><strong>A1</strong></em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg</em> (\(19.6\), \(5\)) , \(x = 19.560 \ldots \)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">maximum rate of increase is \(5\) <em><strong>A1 N2</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>METHOD</strong></span><span style="font-family: times new roman,times; font-size: medium;"><strong> 2</strong> </span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">recognizing \(f''(x) = 0\) <strong><em>(M1) </em></strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">finding any correct expression for \(f''(x) = 0\) <strong><em>(A1)</em> </strong></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><em>eg </em> \(\frac{{{{(1 + 50{{\rm{e}}^{ - 0.2x}})}^2}( - 200{{\rm{e}}^{ - 0.2x}}) - (1000{{\rm{e}}^{ - 0.2x}})(2(1 + 50{{\rm{e}}^{ - 0.2x}})( - 10{{\rm{e}}^{ - 0.2x}}))}}{{{{(1 + 50{{\rm{e}}^{ - 0.2x}})}^4}}}\)</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">finding \(x = 19.560 \ldots \) <em><strong>A1</strong> </em></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">maximum rate of increase is \(5\) <em><strong>A1 N2 </strong></em></span></p>
<p><em><strong><span style="font-family: times new roman,times; font-size: medium;">[4 marks] </span></strong></em></p>
<div class="question_part_label">e.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Candidates had little difficulty with parts (a), (b) and (c).</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Candidates had little difficulty with parts (a), (b) and (c). Successful analytical approaches were often used in part (b) but again, this was not the most efficient or expected method.</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Candidates had little difficulty with parts (a), (b) and (c). In part (c), candidates gained marks by correctly identifying upper and lower bounds but often did not express them properly using an appropriate notation.</span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">In part (d), the majority of candidates opted to use the quotient rule and did so with some degree of competency, but failed to recognize the command term “show that” and consequently did not show enough to gain full marks. Approaches involving the chain rule were also successful but with the same point regarding sufficiency of work.</span></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Part (e) was poorly done as most were unable to interpret what was required. There were a few responses involving the use of the “trace” feature of the GDC which often led to inaccurate answers and a number of candidates incorrectly reported \(x = 19.6\) as their final answer. Some found the maximum value of \(f\) rather than \(f'\).</span></p>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p>Let <em>g</em>(<em>x</em>) = −(<em>x</em> − 1)<sup>2</sup> + 5.</p>
</div>
<div class="specification">
<p>Let <em>f</em>(<em>x</em>) = x<sup>2</sup>. The following diagram shows part of the graph of <em>f</em>.</p>
<p><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAxcAAAJWCAYAAAAqU3pBAAAgAElEQVR4AezdD3gV1b3o/d9GH5VexJ5gn9sAWqreBHxVQELKLfiIoFF7quGFo209FRSxx0vRFm8BBRsNFYrg1dY/PbYCFTzaUyh5QttTCgciFGgVEoN/DpActSqQ+FyBVzFXWm7JvM/asMPO7Jnsf/NnrZnvPI9mz+w1a36/zxr2zsrMmpWwLMsSFgQQQAABBBBAAAEEEECgSIFeRe7P7ggggAACCCCAAAIIIIBAUoDOBScCAggggAACCCCAAAIIeCJwulstiUTC7S22I4AAAggggAACCCCAQMwFnEZXuHYulJXTDjE3JH0EEEAAAQQQQAABBGIv4HYhgtuiYn9qAIAAAggggAACCCCAgDcCdC68caQWBBBAAAEEEEAAAQRiL0DnIvanAAAIIIAAAggggAACCHgjQOfCG0dqQQABBBBAAAEEEEAg9gJ0LmJ/CgCAAAIIIIAAAggggIA3AnQuvHGkFgQQQAABBBBAAAEEYi9A5yL2pwAACCCAAAIIIIAAAgh4I0DnwhtHakEAAQQQQAABBBBAIPYCdC5ifwoAgAACCCCAAAIIIICANwJ0LrxxpBYEEEAAAQQQQAABBGIvQOci9qcAAAgggAACCCCAAAIIeCNA58IbR2pBAAEEEEAAAQQQQCD2AnQuYn8KAIAAAggggAACCCCAgDcCdC68caQWBBBAAAEEEEAAAQRiL0DnIvanAAAIIIAAAggggAACCHgjQOfCG0dqQQABBBBAAAEEEEAg9gJ0LmJ/CgCAAAIIIIAAAggggIA3AnQuvHGkFgQQQAABBBBAAAEEYi9A5yL2pwAACCCAAAIIIIAAAgh4I0DnwhtHakEAAQQQQAABBBBAIPYCdC5ifwoAgAAC0RLYJ3VTLpJEIiGJxFfl0aaPTqZ3UBrmVEj/O+rkQGe0MiYbBBBAAAF9BOhc6NMWRIIAAgh4IHCeTFzxlhxvWSZV8qr8+2sfyIm+RF8Zcu0NUr7/I/k/HhyFKhBAAAEEEHASoHPhpMI2BBBAwHCBXhcMlWsubJe3D/+fk52LM+S/DhwkF1xzifTnk9/w1iV8BBBAQF8BvmL0bRsiQwABBAoX6PVfpOTCUnn7jffkQ1VL53tS/9hu+ftvDJM+hdfKnggggAACCPQoQOeiRx7eRAABBAwV6PVf5LOf/8zJ4Dulo3mdNFZNlwkDznBMaNeuXaL+Y0EAAQQQQKAYAToXxeixLwIIIKCtQG/57Oc/K7L9Hdm3b5MsXlMq357wBXH60D98+LB861vfkuHDh8v+/fu1zYjAEEAAAQT0F3D6ntE/aiJEAAEEEMgicJac87mzRT7dLk//6DUZe/f1MsDlE/+ZZ57pqqu2trbrNS8QQAABBBDIV8DlqybfaiiPAAIIIKCXwOlydsnnRKRErrzzWzKu1P12qHnz5snDDz+cDH/p0qWyYcMGvVIhGgQQQAABYwToXBjTVASKAAII5Ctwntz/L/PltsF9HXc8evRo8naoWbNmSVVVVbLMggUL5IEHHhB1qxQLAggggAAC+QrQuchXjPIIIICA9gKd0tH0nCyT22XuuAGO4yxUCqtXr5adO3fKPffc05XRzJkzk68XLVrUtY0XCCCAAAII5CqQsCzLciqsZnd1ecupONsQQAABBEIVUB2KH8sNN7wnt/78v0nL7pHy/ZmVro+dbW1tlfLyclmzZo1MnDgxGXnqc1/dFnXttddKc3OzDBs2LNSsODgCCCCAgJ4Cqe8Me3R0LuwirCOAAAJGCnTKkYYHZPD4Jrn1uR/Id2+tlNIerk1PmDAhmWV9fX1XtulfFLNnz5bNmzfLli1bpHfv3l1leIEAAggggIASSP/OSBehc5GuwWsEEEAgBgJ1dXUyadIkaWlpkbKysq6M078o1JiLfv36yZNPPikzZszoKsMLBBBAAAEElED6d0a6SA9/10ovxmsEEEAAgagIdHR0JG+HSu9Y2HMrKSmR9evXy2uvvWZ/i3UEEEAAAQRcBbhy4UrDGwgggEC8BNz+ChUvBbJFAAEEEMhFwO07gysXuehRBgEEEEAAAQQQQAABBLIK0LnISkQBBBBAAAEEEEAAAQQQyEWAzkUuSpRBAAEEEEAAAQQQQACBrAJ0LrISUQABBBBAAAEEEEAAAQRyEaBzkYsSZRBAAAHNBTrbm2Ttrx6VKf0Tkug/VxqOdGoeMeEhgAACCERRgM5FFFuVnBBAIEYCx6R9x9Ny+4jbpO7d82TK5o/Falso4/ry8R6jk4BUEUAAAW0ETtcmEgJBAAEEEMhToFM6mp6WWybskhvrN8h3KkuFLkWehBRHAAEEEPBUgM6Fp5xUhgACCAQo0NEoz3zvBbngqTV0LAJk51AIIIAAAu4C/JHL3YZ3EEAAAY0F/iKtqx6VWTJeRneultvVWIvEpTLl0fXS2sF4C40bjtAQQACBSAvQuYh085IcAghEVqDzXdn2y1dlbOUlMnT0PbKi7bh8sud7Io/dLv/0TKN0RDZxEkMAAQQQ0FmA26J0bh1i6ybw4IMPJtdra2sl9VptsK932+nkir2MfZ19nAQybQtxu++++2Tu3LnSt29f54M4tGEhx4ndPn/9s/zljc/Ksf/6uvz6mbfk10nH++XsisWyedbtcsu7E2R4v8yPeLuTfT3931aqwexl7Oupcuk/7WXs6+llU6/tZezrqXLpP+1l7OvpZVOv7WXs66ly6T/tZezr6WVTr+1l7Oupcuk/7WXs60ePHk0W7927d9du9jL29a6CaS/sZbKtq12zlbG/H8V90giTL51y7qmMKs+CQOQFLJdFRFzeYTMC4QjU1NSEc+ACjkqsp9DWrFljzZo169SGIl9hewLweMsyq0pGWLM3fdhN1Gm7+jzP9b9ulWm8EtfzYMWKFZ7+e7I3cVxd7Q5+rJtk60f+1Bk9Abe+ArdFRb77SIIIhCtwwQUXyJIlS8INIoJH7/X5QTKstE12vXtQMkdYXCDlA/t0ZW1ZlvprUY//1dTUdJXnhb4CW7dulVGjRukbIJEhgEDsBRKqH+WkkEgkkl9ETu+xDQEEEMhV4PDhw9KvXz/Zt2+fDBw4MNfdKJdV4KA0zLlOvnlwrux8dqIMSP6pqFOONDwggx+5SDavmyplef75iM/9rOihF1Bt1NzcLMOGDQs9FgJAAIF4C7h9Z+T51RNvRLIPV8DpXvBwI3I/OrGesikpKZGRI0fK7t27T20s4hW2KbxzZew9c+X6dQ/K3OfeTA7g7mz/kyxb8Z7cu3Bi3h0Lk1yVgEnxehVra2trsvHPP//81Eng+U+vYvU8MIcKTYpVhW9avA7kbEIgJwE6FzkxUQgBBIoRmDBhgqR+MSqmHvbtLtBrwAT58eZH5dIt35CzEwk57ZZfS8k9j8q9Iz7bvSBrkRB49913pbq6WlSHnQUBBBDQVYDOha4tQ1wIREhg8ODBsnHjxghlpEsqvaRP2bXyvRVvnBhP8dIjMmUEs3Tr0jpex9HY2CiVlZVeV0t9CCCAgKcCjLnwlJPKEEDASUBdtSgvL5dPP/1U0h+h6VSWbeEJuN0/G15EHDldQF0BnD59ulRVVaVv5jUCCCAQioDbdwZXLkJpDg5aiIBJ96sSa/cWLisrS25Qg7qLXbAtVtB5f5NcVQYmxetFrOrBCGvXrpVBgwY5N6BHW72I1aNQslZjUqwqGdPizdoAFEDARYDOhQsMmxFAwFuBadOmyZtvvultpdSGQEwE3n///WSmqY56TNImTQQQMFCAzoWBjUbICJgoMHToUNm7d6+JoRMzAqELvPPOO6I66CwIIICA7gJ0LnRvIeJDICIC6i+uO3bsiEg2pIFAsAIvv/yyqA46i7kCtbW15gZP5AjkIUDnIg8siiKAQOEC6l5xdc/40aNHC6+EPRGIqYB6KAK3RJnd+Iy5MLv9iD53AToXuVtREgEEihBI/WLkxaDuIsJgVwSME1Ad8iAGcxsHQ8AIIKClAJ0LLZuFoBCIpgCDuqPZrmTlr0CqQ57qoPt7NGpHAAEEihOgc1GcH3sjgEAeAl/84hcZ1J2HF0URUALqKWsM5uZcQAABUwToXJjSUsSJQAQE1EzdDOqOQEOSQqAC6ilrDOYOlJyDIYBAEQJ0LorAY1cEEMhP4JJLLmFQd35klEYg2SHnlihOBAQQMEWAzoUpLUWcCERAIPULUktLSwSyIQUE/BcIamZu/zPhCAggEBcBOhdxaWnyREATAXXvuJoQjAUBBLILMDN3diNKIICAXgIJy7Isp5ASiYS4vOVUnG0IIIBATgIrV65MDlBdvHhxTuUpFJwAn/vBWed6JP695CpFOQQQCFrA7TuDKxdBtwTHK1jApAmIiNW9mS+88EJZsmSJe4Es72CbBajAt01yVSmaFG8xsaonRamxSkEtxcQaVIyp45gUq4rZtHhTzvxEIF8BOhf5ilEeAQSKEhgyZEhy//379xdVDzsjEAcB1RG/7LLL4pAqOSKAQEQE6FxEpCFJAwFTBEpKSmTkyJGye/duU0ImTgRCEWhtbU0et7y8PJTjc1AEEECgEAHGXBSixj4IIFCUwMKFC6Vv374yY8aMouphZ28F3O6f9fYo1JarQF1dnagxF/X19bnuQjkEEEAgMAG37wyuXATWBByoWAGT7lcl1p5bW02mt3Hjxp4LubyLrQtMkZtNclWpmhRvobGqyfMqKyuLbNn8di801vyO4k1pk2JVGZsWrzetRC1xFKBzEcdWJ2cEQhZgMr2QG4DDGyGgZrOvqKgwIlaCRAABBFICdC5SEvxEAIHABM4777zksfbt2xfYMTkQAiYJHD16NDmb/aBBg0wKm1gRQAABoXPBSYAAAoEL9O7dW6qrq5PzXQR+cA6IgAECqY53alZ7A0ImRAQQQCApwIBuTgQEEAhF4KmnnpIjR47I3LlzQzk+B80UcBucl1mSLX4LqMHc69atk2effdbvQ1F/QAJqzEVtbW1AR+MwCPgv4PadwZUL/+05gkcCJg2GI9bsja7+IlvIU3CwzW5bSAmTXFV+JsVbSKwvv/yyDB06tJCmLGqfQmIt6oBF7GxSrEWkya4IGCdA58K4JiNgBKIhcPHFF8vOnTvl8OHD0UiILBDwUGDz5s0yfPhwD2ukKgQQQCAYAToXwThzFAQQsAkMHDgwuWXPnj22d1hFIN4CavZ61fFOzWYfbw2yRwAB0wQYc2FaixEvAhESmD17tqjH0k6ePDlCWZmbitv9s+ZmZGbk27dvl5kzZ4p6FC0LAgggoKuA23cGVy50bTHiyhAw6f5aYs1oPscNo0aNyvuJUdg6Uha90SRXlaxJ8eYb69tvvy1jx44tuk0LqSDfWAs5hlf7mBSrytm0eL1qJ+qJnwCdi/i1ORkjoI3ABRdcIEuWLNEmHgJBQAeBN998U1THmwUBBBAwUYDOhYmtRswIRETg/PPPT2ai7jFnQQCBEwKqw6063iwIIICAiQJ0LkxsNWJGICICJSUlMnLkSNm9e3dEMgojjYPSMKdC1L2vJ/7rL9cu3yudYYTCMYsWaG1tTdZRXl5edF1UoJcAc1zo1R5E458AnQv/bKkZAQRyEJgwYYI0NjbmUJIiTgKdB/4gLzzflPbWGPnamEHCh3saiUEv1S1RavZ6NYs9S7QEGHMRrfYkG3cBvn/cbXgHAQQCEKioqOCpOAU7fyTNv1gr5/7Lh2JZ1sn/VsnUsrMKrpEdwxVQk+ddffXV4QbB0RFAAIEiBOhcFIHHrgggULyAmkxv7dq1TKZXCOWRV2XVYytl8Q+WyPK6zdLawc1QhTDqtA+T5+nUGsSCAAKFCNC5KESNfRBAwDOB1GR677//vmd1xqOiv0jrr56Rxe0isnmx3DHpKim/4QGpaz0Sj/QjmKWarV5NnveFL3whgtmREgIIxEWAzkVcWpo8EdBYYNasWfLOO+9oHKGOoZ0lZVNXiWX9VdoafyPLZleJbP6hTPqnZdLEFQwdGyxrTKnZ6lMd7qw7UAABBBDQUIDOhYaNQkgIxE1AzdKt7jVnKUTgDCkd8VWZ+shvpG3TQzJ28wuyasfhQipin5AF1OR5qqPNggACCJgskLDUKECHxW1Kb4eibEIgEIHUkzbU4/xSr9WB7etOwdjL2NfZx0kg09Yvt6qqKhkzZozU1NQkA/HrOPZ67etOCvYy9nW99rlbJo++TOrlapl+9RflzLTg5s+fn7bm/jLVBukl7Dnb19PLpl7by9jXU+XSf9rL2NfTy6Ze28vY11Pl0n/ay9jX08umXtvL2NdT5dJ/2svY19PLqte/+c1v5Lvf/a6oTkZqybaPKmcvY19P1ZX+014m23oux7HXEcV90g0LyU8ZsSAQFQHXvoLqXDgtIurBIywI6CNQU1OjTzBZIiHWLEC2tw8dOqT+yGHt27fP9k7mKraZJqe2HLValt1iVS3bYx0/tTGnV8rVpM/9KJ4Hyr+5uTmn9vKrUBRd/bLKt16TbPPNjfLxFHD7zuDKRVS6j+SBgOECar6L6dOni7qKwVKowEFpmLtU5L7ZMq5v/ne9uv4VqtBw2C9nATV5npo479NPP2WOi5zVKIgAAmEKuH1n5P/tE2YWHDvWAum3QukOQaz5t5B6tn8uk+lhe9K2s12a1v5OmtqPndjQ2S47Hn9MNl99q4wtoGNhkqtK2KR4c4lVTZ43bdq00DsWucSa/79uf/YwKVbTzll/Woxa4yJA5yIuLU2eCGguUFZWxmR6ebVRp3z8yo+lov+ZkkhcKlMe+4N0/P19Mn/cAGbnzstRj8LqgQZDhw7VIxiiQAABBIoQOL2IfdkVAQQQ8Exg0KBBXZPplZSUeFZvZCvqNUDGLVwv1sLIZhirxNTkeY8//nisciZZBBCIpgBjLqLZrmSFgJEC6v7N5uZmGTZsmJHxmx602/2zpuele/xq8rx+/frJvn37hDkudG8t4kMAgZSA23cGt0WlhPipvYBJ99cSa2Gnk3rG/+uvv97jztj2yFPwmya5qiRNijdbrDpNnpct1oJPMB92NClW085ZH5qLKmMkQOciRo1NqgjoLqAm01MDW1kQiJMAk+fFqbXJFYHoC9C5iH4bkyECxghcdtllsmTJEmPiJVAEvBDYunWrjBo1youqqAMBBBAIXYDORehNQAAIIJASUM/5V4t65j8LAnEQOHr0qCxdulTUVTsWBBBAIAoCDOiOQiuSAwIRElCT6U2ePFkmTpwYoazMSMVtcJ4Z0ZsZ5a5du2T48OFiWWqCdJYoC6gxIrW1tVFOkdxiJuD2ncGVi5idCCana9LgPWIt/ExTk+nt3bvXtQJsXWmKesMkV5WoSfH2FOs777yTnDyvqMbzcOeeYvXwMJ5UZVKsniRMJQgYIkDnwpCGIkwE4iKgJtOrr6+PS7rkGXMBJs+L+QlA+ghEUIDORQQblZQQMFng4osvlp07d4p69j8LAlEXUJPnqduiWBBAAIGoCDDmIiotSR4IREhA3ce5bds2GT16dISy0j8Vt/tn9Y/czAj3798v5513nhw6dEiYld7MNiRqBOIs4PadwZWLOJ8VhuVu0v21xFrcyaUm01MzdTst2DqpFL/NJFeVrUnxusW6e/duqa6u1qpj4RZr8WeY9zWYFKtp56z3rUWNcRKgcxGn1iZXBAwRUM/837hxoyHREiYChQk0NjZKZWVlYTuzFwIIIKCpAJ0LTRuGsBCIs4B65v/atWsZdxHnkyAGuasHF1RUVMQgU1JEAIE4CdC5iFNrkysChgioJ0ap5f333zckYsJEID8B9cAC9eAC9QADlngIMMdFPNqZLEXoXHAWIICAlgJq3MXrr7+uZWwEhUCxAnv27ElWMXDgwGKrYn9DBEwbI2IIK2FqKEDnQsNGISQEEBBRt0a9+eabUCAQSYG3335bVAeaBQEEEIiaAJ2LqLUo+SAQEYHLLrtMlixZEpFsSAOB7gJbt24V9eACFgQQQCBqAnQuotai5INARATKy8uTmbS2tkYkI9JA4ITA0aNHZenSpcmrc5gggAACUROgcxG1FiUfBCIi0Lt3b5k2bRq3RkWkPUnjlEBLS0tyJfXgglPv8AoBBBAwX4DOhfltSAYIRFbgiiuukJdffjmy+ZFYPAXUgwoYbxHPtidrBOIgQOciDq1MjggYKnDhhRcy7sLQtiNsdwH1oAL1wAIWBBBAIIoCdC6i2KrkhEBEBIYMGZLMZP/+/RHJiDQQkGSHWT2wgAUBBBCIokDCsizLKbFEIiEubzkVZxsCCCDgi0BlZaU8/PDDUlVV5Uv9VHpKgM/9UxZ+vVIPKFAPK/j0009FjStiQQABBEwVcPvO4MqFqS0aw7hNmoCIWL07QSdMmCCNjY1dFWLbReHpC5NcVeImxZseq7olqrq6WtuORXqsnp5gPlRmUqymnbM+NBdVxkiAzkWMGptUETBRoKKiQurr600MnZgRyBBQDyi4+uqrM7azAQEEEIiKAJ2LqLQkeSAQUYGLL75Ydu7cKYy7iGgDxywtNTHk8OHDY5Y16SKAQJwEGHMRp9YmVwQMFWDcRTAN53b/bDBHj/5RUuMtDh06JCUlJdFPmAwRQCDSAm7fGVy5iHSzRys5k+6vJVZvzz017iI1Uze23tqmajPJVcVsUrypWFPjLXTuWKRiTZ0XOv80KVbTzlmd253Y9Begc6F/GxEhArEXUOMuNm7cGHsHAMwW2Lt3L+MtzG5CokcAgRwE6FzkgEQRBBAIV2DQoEGydu1aOXz4cLiBcHQEihBQDyYoKysrogZ2RQABBPQXoHOhfxsRIQKxF0j9QrZnz57YW7gDHJMDdTOk/7XLpbXTvRTvhCOgHkigHkygHlDAggACCERZgAHdUW5dckMgQgILFy6Uvn37yowZMyKUlXepdB6okztHTpLlly6TlnVTpayAPx25Dc7zLsr41rRhwwb5yU9+wmOV43sKJMcJ1dbWxliA1KMm4PadUcDXT9RoyMcUAZMG7xGr92dVatwFtg62HTvksbl/kHOvucThzdw2meSqMjIpXhWrmgjShPktTHPN7eymFAIIBClA5yJIbY6FAAIFC6jbSdS4i6NHjxZcRzR3/EiannlR5J7pcu3nz4xmihHISo23YH6LCDQkKSCAQFYBOhdZiSiAAAI6CAwcOFBGjhwpBw8e1CEcTWLolI6mlfKE3CJ3jWDeBE0aJSOMI0eOJMdbDBkyJOM9NiCAAAJRE6BzEbUWJR8EIiyg5rt47733Ipxhnql1NMozT4jcc1eF9MlzV4oHJ9DW1ibV1dVMnBccuZZHYryFls1CUD4I0LnwAZUqEUDAHwE17kLNFcCiBD6Spmcb5IKFd8mIPnyU63xOqCdFmTDeQmfDKMRm0niWKHiTQ3gCPC0qPHuOnKdA6oNZ/fUn9VpVYV93qtZexr7OPk4CmbZhu6nbS370ox/Jd7/7XXn88cdjfB5Ycqy9SRLX/UROf225JJLN96lc+ZeXZPya/vLtW4ZLvxMbk+/Mnz/fuYEdttbU1GRstbe7fT1jB4d/l3HeR/nffvvtsnz58m7nrM5uTu1l35ZtXeWXrYz9fd33sbeZPf5c1u11sI6AqQJuT4sSy2URcX3LZQ82I+CvQE1Njb8H8LB2YvUQ01ZV//79rW3bttm26rnq33nwobVp9ghLfU47/1dqVS3bYx3Pg0XFatLnvn+2eaDlUHTfvn1J10OHDuVQOvwiprgqKZNiNTHe8M9GItBdwO07g2vppnYXiRuBmAoMHjxYmpubY5p9Ku1zZdwjjeovQGn/fSibZo8QqVomLcfbZP3UwcIHfMorvJ+7d++W8vJyxluE1wTaHFld1WBBIA4CfPfEoZXJEYEICZx77rmycePGCGVEKlEWUPNbDBgwIMopkluOAum38+a4C8UQMFKAzoWRzUbQCMRX4HOf+1xyvovDhw/HF4HMjRFQ81uUlpYaEy+BIoAAAsUK0LkoVpD9EUAgUIF+/folj7dnz55Aj6v/wU7eKrV+qpTxya5Fc6mnRO3cuVNUh5gFAQQQiIsAX0FxaWnyRCBCArNmzWLcRYTaM6qpqPEWauLHvn37RjVF8kIAAQQyBOhcZJCwAQEEdBcYNWoU4y50byTiEzXeQk38yIIAAgjESYDORZxam1wRiIhAZWUl4y4i0pZRTkONt7jyyiujnCK5IYAAAhkCdC4ySNiAAAK6CwwcODB5uwnjLnRvqfjGlxpvMWTIkPgikDkCCMRSgM5FLJudpBEwX0DdbrJlyxbzEyGDSArs2LFDqqurmd8ikq1LUggg0JNAQs3+51TAdUpvp8JsQwABBAIW2L59u8ycOVPUL3Es3gjwue+No6pl9uzZcv7558uMGTO8q5SaEEAAAY0E3L4zuHKhUSMRSs8CJk1ARKw9t2Ux76Zs1e0m6jGf6vYTXZdUrLrGlx6XSbGquHWO9+jRo7JkyRIZM2ZMkljnWNPPAd1dTY7VNFu7NesI5CNA5yIfLcoigIA2AiUlJcnbTrhyoU2TEMhJgZaWluSr8vJyTBBAAIHYCdC5iF2TkzAC0RG4+uqrZe/evdFJiEwiIfDOO+/ItGnTpHfv3pHIhyQQQACBfAQYc5GPFmURQEArAcZdeNscbvfPenuU6NemxltccsklMnny5OgnS4YIIBBbAbfvDK5cxPaUMC9x7lv2p81MclUC6fHqPu4iPVZ/Ws+7Wk2K1X4eeKfgTU1qvMVll13WVZlJtsTa1WyevzDJ1vPkqTBWAnQuYtXcJItAtARS4y52794drcTIxliB1tbWZOyMtzC2CQkcAQSKFKBzUSQguyOAQLgCatxFY2NjuEFwdAROCrz55puMt+BsQACBWAvQuYh185M8AuYLDB8+XOrr681PhAwiIbBu3Tq54oorIpELSSCAAAKFCDCguxA19kEAAW0EDh8+LP369RP1+M+ysjJt4jIxELfBeSbmEkbMan6Lz3zmM9Lc3CzDhg0LIwSOqaqFwkMAACAASURBVLGAGnNRW1urcYSEhkB+Am7fGVy5yM+R0iEKmDQYjlj9O1HstmrchXrs58svv+zfQQus2R5rgdUEsptJsSoQHeN1m99Cx1jdTipidZNhOwII5CpA5yJXKcohgIC2Auo2lK1bt2obH4HFQ2Dbtm0ya9Ys5reIR3OTJQIIuAjQuXCBYTMCCJgjoB77uXTpUlG3SLEgEJbAxo0bRT1ggAUBBBCIswCdizi3PrkjEBGB1P3te/bsiUhGpGGawP79+2Xt2rVy8cUXmxY68QYkwHiLgKA5TOgCdC5CbwICQAABLwQWLFiQHEjrRV3UgUC+Amqulerqahk4cGC+u1I+JgImjWeJSZOQpk8CdC58gqVaBBAIVqCiokLUbSksCIQhoOZa4ZaoMOQ5JgII6CZA50K3FiEeBBAoSEDdjqJuS2HcRUF87FSkwLx580TNucKCAAIIxF2AzkXczwDyRyAiAup2lJEjRzJbd0Ta06Q0Wltbk+EOGTLEpLCJNWABxlwEDM7hQhOgcxEaPQdGAAGvBSZMmEDnwmtU6ssq8OabbybHW6g5V1gQcBNgzIWbDNujJkDnImotSj4IxFjgyiuvlPr6+hgLkHoYAuvWrZOJEyeGcWiOiQACCGgnQOdCuyYhIAQQKFRA3Zayc+dOSd2mUmg97IdArgJHjx5NzrGi5lphQQABBBAQoXPBWYAAApERULelTJs2TV5++eXI5EQiegu8+uqryQBTc63oHS3RIYAAAv4L0Lnw35gjIIBAgAJXXHGFbN26NcAjcqg4CzQ3N8usWbPiTEDuCCCAQDcBOhfdOFhBAAHTBdTtKUuXLhV1uwoLAn4LqLlVmN/Cb2XqRwABkwToXJjUWsSKAAJZBVK3p6RuV8m6AwUQKFBg//79yblV1BwrLAgggAACJwToXHAmIIBA5AQWLFgg6nYVFgT8FNi9e3fyEbRqjhUWBBBAAIETAgnLsiwnjEQiIS5vORVnGwIIIKCNwPbt22XmzJmyY8cObWLyJ5Bj0r7jWblvwgxZ2V4qY2c/JT/9/gQp61PY34343M+vlWbPni3nn3++zJgxI78dKY0AAghEQMDtO6Owb6AIgJCCeQImTUBErP6dX7nYph5Jq25bCXPJJdbC4+uUjuZ1skEmys/bLDnetlpu/OBBGfuDzXKkgEr9jbWAgLLsEna8akzPkiVLZMyYMVkiFQk71qwBphUg1jQMj1+aZOtx6lQXMwE6FzFrcNJFIA4C6pG01dXV0b5y0fmeNH58udxaWZp8pniv0v8ud0y5QeT5jdJ4pDMOzRxqji0tLcnjl5eXhxoHB0cAAQR0E6BzoVuLEA8CCHgioJ7gE+n5Lnp9UcaOPS9tsqJj8sG770n5vROksi8f7Z6cRD1U8vrrryfnVOndu3cPpXgLAQQQiJ8AYy7i1+ZkjEAsBHbt2iXDhw+XTz/9VCL/C2BHqzSsek5WvHWVLHr4GiktsG/hdv9sLE6YPJOcMGGCTJ48WSZOnJjnnhRHAAEEoiHg9p1R4FdQNFDIwiwBk+5XJVb/zq1cbXV4JG2usRau1SlHGuZK/7PLZfwdP5SVf3pJ/vRWISMuzBoXoLz8t3VvldQjaCsrK90Lpb0TZqxpYeT0klhzYiqokEm2BSXITgicFODKBaeCMQKpD+ba2tpuv1jY150Sspexr7OPk4CI3cm+7rSXvYx9Pch9tm3bJmeeeaaMHDkyeVh7LPb1IGOzH8sei33dXl6tnypzXDra/0NO/89XZHHTQLl5WpUMPvs0p13S9jnx9qk6RObPny81NTUZ+6WX6X7cjKJdG6K8z9tvvy0dHR0ydOjQrnydXphsYI9d5Wfflm09ivvY2zmbgdP79jpYR8BUAbcrF+pxs46LiOtbjuXZiIDfAjU1NX4fwrP6idUzyoyK8rFdv369NXLkyIw6gtqQT6xexHS8ZZlVJSOs2Zs+7Fad+jzP9b9uO2q8ErRtOsWsWbOsJ598Mn1Tj6/DjLXHwBzeJFYHFI82mWTrUcpUE3EBt74Ct0WZ2l0kbgQQyCqgZk7euXOnhP1I2qyBelSg10Vflq9VnZlRm5qzKNt/TlcsMipiQ1JAPYJWjedhQQABBBDIFOC2qEwTtiCAQIQE1MDb6dOnS1VVVYSycknlSIPMubxOvrTlMZk44AyXQu6bXS9xu+8Su3di9aCA2LWuvwmrW3vVbVIsCERFwO07gysXUWnhGOSRGnNhQqrE6l8r5WurHkm7evVq/wLqoeZ8Y+2hqoy3Og/UyR39r5U5K3ZIu5rWovOANCx6Rj6Ye5dUFdCx8DPWjOA92BBWvGocz7Rp0/J6AllYsRbCTKyFqLEPAgikC9C5SNfgNQIIRE5AzaC8dOlSOXz4cKRy63VOmYy+pk0W3/Yl6X9aQvrf/gv5aNKP5edTL5E+kcpUr2Q2btwoN910k15BEQ0CCCCgkQCdC40ag1AQQMB7AfVIWvW0qD179nhfeZg19rlEpq54o2ssRduK78nEESdm6w4zrCgfO/UI2oqKiiinSW4IIIBAUQJ0LoriY2cEEDBBQI272LJliwmhEqPGAjt27JDq6mopKSnROEpC01WA8Ra6tgxxeS1A58JrUepDAAHtBNRfmuvr67WLi4DMEnj55ZdFjeFhQaAQAZPGsxSSH/sgkBKgc5GS4CcCCERWQHUu1CNpW1tbI5sjifkrcPToUVGPoFVjeFgQQAABBNwF6Fy42/AOAghEREDdxqKe8KP+8syCQCECr776anI3NYaHBQEEEEDAXYDOhbsN7yCAQIQErr/+eqmrq4tQRqQSpIAas7NgwYIgD8mxIibAmIuINSjpuArQuXCl4Q0EEIiSQGVlpaxduzY2s3VHqe10yEWN2bnyyit1CIUYDBVgzIWhDUfYeQvQucibjB0QQMBEgYEDByYfSbt7924TwyfmEAXUWB01ZmfIkCEhRsGhEUAAATME6FyY0U5EiQACHghMnjxZGhsbPaiJKuIkoMbqqDE7PII2Tq1OrgggUKgAnYtC5dgPAQSME1BP+pk3b56oJ/+wIJCrgBqrc8UVV+RanHIIIIBArAXoXMS6+UkegXgJpGbrTj35J17Zk20hAqlZuceNG1fI7uyDAAIIxE6AzkXsmpyEEYi3gLo1itm6430O5JN9Q0NDclZuNWaHBQEEEEAguwCdi+xGlEAAgQgJDB8+PHlrVIRSIhUfBbZu3SoTJ0708QhUjQACCERLgM5FtNqTbBBAIIvA5Zdfniyxa9euLCV5O+4Chw8flqVLl8qoUaPiTkH+CCCAQM4CCcuyLKfSiURCXN5yKs42BBBAwBiBO++8U4YOHSozZswwJuYgAuVzv7vyhg0b5IEHHpAdO3Z0f4M1BBBAAAFx+87gygUnhzECJk1ARKz+nVZe2N50002ycuVK/4I8WbMXsfoepIGxqpCDsF29erWoMTrFLkHEWmyMqf2JNSXh/U+TbL3PnhrjJEDnIk6tTa4IIJAUqKioSE6KpiZHY0HASUA9rljdEqUeX8yCAAIIIJC7AJ2L3K0oiQACERFQk6GpSdHU5GgsCDgJpB5XrB5fzIIAAgggkLsAYy5yt6IkAghESEBNjKZujaqvr49QVsWl4nb/bHG1mrn3woULk4HPnTvXzASIGgEEEPBZwO07gysXPsNTvXcCJt2vSqzetbu9Jq9sKysrZe3ataImSfNr8SpWv+JLr9ekWFXcfserOp1XXnllOlHBr/2OteDAHHYkVgcUjzaZZOtRylQTUwE6FzFteNJGIO4CalK06upqUZOksSCQLqAeU7xz505JPbY4/T1eI4AAAgj0LEDnomcf3kUAgQgLqMnR1O1RLAikC/zud7+TBQsWSO/evdM38xoBBBBAIAcBOhc5IFEEAQSiKTBu3Djfb42Kplx0s1JPiZo3b55nt0RFV4rMEEAAAWcBOhfOLmxFAIEYCHBrVAwaOc8UU0+J4paoPOEojgACCJwUoHPBqYAAArEW4NaoWDd/RvJbtmzhlqgMFTYggAACuQvQucjdipIIIBBBAW6NimCjFpESt0QVgceuCCCAgIjQueA0QACBWAukbo3asWNHrB1IXkQ9JUot3BLF2YAAAggULkDnonA79kQAgYgIqFuj1q1bF5FsSKNQgW3btnFLVKF47JdVoLa2NmsZCiAQBQE6F1FoRXJAAIGiBEaNGiVLly6Vw4cPF1UPO5stoGZsr6ioMDsJotdWgEn0tG0aAvNYgM6Fx6BUhwAC5gmUlZXJyJEjpbGx0bzgidgTgdTEeXQuPOGkEgQQiLEAnYsYNz6pI4DAKYHJkyfL6tWrT23gVawE1C1R06ZNk5KSkljlTbIIIICA1wJ0LrwWpT4EEDBSYMKECebdGtXRKv/+6BTpn0hIItFfrprzvDS1HzPSP+yg1S1Rt912W9hhcPwICzDmIsKNS2rdBOhcdONgBQEE4iqQemrUb3/7WzMIOt+Ttc82iNz4pLRZlhxvWy03frBYKm55Wpo6Os3IQZMot2/fLjt37uQpUZq0R1TDYMxFVFuWvOwCdC7sIqwjgEBsBUyaUK/zz/vlnJunyjVlfZPt1at0tHxn3kyp2vyCrNrBwPR8TmImzstHi7IIIIBAzwJ0Lnr24V0EEIiRwFe/+lVZu3at7N+/X/use104WsYOOKNbnL0+P0iGlXbbxEoWgaNHjwoT52VB4m0EEEAgDwE6F3lgURQBBKItoAbzqkG9DQ0N5ib6mVHypfITVzPMTSK4yF999dXkwZg4LzhzjoQAAtEWoHMR7fYlOwQQyFPgpptukrq6ujz30qF4pxxp3Cy77posVbYrGjpEp2sM3BKla8sQFwIImCpA58LUliNuBBDwReDiiy825taobgAdjfKzFefKwrsqpE+3N1hxE+CWKDcZtiOAAAKFCyQsy7Kcdk8kEuLyllNxtiHgu0DqSRvqcX6p1+qg9nWnQOxl7Ovs4ySQaRsXt1/+8pcyePBgGTp0aMb5pafBX6T95Vdk1I9/Le/94n85Nub8+fMdt9s31tTU2DcZYnAi7HzaZ9++ffLzn/9cPv30U1m0aFFG3ukb7PXa19PLpl7by9jXU+XSf9rL2NfTy6Ze28vY11Pl0n/ay2RbV/tmK2N/P4r7pBsWkp8yYkEgKgKufQXVuXBaRFTfggUBfQRqamr0CSZLJMSaBaiIt4OwXbNmjVVdXV1ElCd2DSJWy/qrtb/+n63n9nxcVLwqVpM+972wXbBggaX+83vxIla/Y0zVT6wpCe9/mmTrffbUGEUBt+8MrlxEpftIHggg4JnA4cOHpV+/ftLc3CzDhg3zrF7vKzom7Q3L5V/lBvnOuAFy4j7XI7J3xVr54P/9RxnbN787X13/CuV94KHXaE4bh05FAAgggICjgNt3Rn7fPI5VsxGBYATSb4UK5oiFH4VYC7fLtmcQtuqpUbNmzZJt27ZlC6fH9/2N9Yi01j0kt4z/H3Lv+IFyWnKWbjVT9zky5MX/K/375Pfx7m+sPTIV9Gax8W7evFlGjhwZSOex2FgLAipwJ2ItEC6H3UyyzSEdiiDgKpDft49rNbyBAAIIREugurpa7r77blGDfvVbjsmBurkydtIPZXNGcKVS9bUvy0V8umfIpG9YuXKlzJgxI30TrxFAAAEEPBDg68cDRKpAAIHoCYwePTr5l+2tW7dqmNwZMmDiU9JmWckHb6iHb5z6r03WTx188hYpDUPXIKTW1tbkE8HUpIksCCCAAALeCjDmwltPakMAgQgJPPXUU/Laa6/Js88+G6Gs3FNxu3/WfQ8z34lbu5rZSkSNAAK6C7h9Z3DlQveWI74uAZPuVyXWrmbz/EWQtmPGjJGlS5eKGvxbyBJkrIXEl76PSbGquIuJV90SpSZLDGopJtagYkwdh1hTEt7/NMnW++ypMU4CdC7i1NrkigACeQmoJ0WpQb9q8C9LNAR27dolO3fulIqKimgkRBYIIICAZgJ0LjRrEMJBAAG9BNSgX/WXbpZoCPzud7+TBQsWiHoiGAsCCCCAgPcCdC68N6VGBBCIkIAa9Lt27VpRf/FmMVtA3d42b948+cpXvmJ2IkSPAAIIaCxA50LjxiE0BBAIX0D9hXvatGlFz3kRfiZE0NjYGNjcFmgjgAACcRWgcxHXlidvBBDIWeC2225L3hql55wXOacR+4KrV6+WyZMnx94BAAQQQMBPAToXfupSNwIIRELg8ssvT+bx6quvRiKfOCaxf//+5JO/JkyYEMf0yRkBBBAITIDORWDUHAgBBEwV6N27d/Iv3s8995ypKcQ+7vr6+uTtbQMHDoy9BQAIIICAnwJ0LvzUpW4EEIiMQFVVVfIv3+ov4CxmCajb2dQTv9TtbSwIhCVQW1sb1qE5LgKBCtC5CJSbgyGAgKkCZWVlUl1dLQ0NDaamENu41e1sam6L1O1tsYUg8VAFmEQvVH4OHqAAnYsAsTkUAgiYLaAGAz/11FNmJxHD6NWjhJ988klRt7exIIAAAgj4K0Dnwl9fakcAgQgJXH/99cm/gG/fvj1CWUU7FXUb25IlS0Td1saCAAIIIOC/AJ0L/405AgIIRERA/eVb/QVc/SWcxQwBdRubup1N3dbGgkCYAoy5CFOfYwcpQOciSG2OhQACxguMGTMm+ZdwNdszi/4C6ja26dOn6x8oEUZegDEXkW9iEjwpQOeCUwEBBBDIQ2DYsGHJv4Rv3rw5j70oGoaAun1NDeSuqKgI4/AcEwEEEIilAJ2LWDY7SSOAQDECamD3okWLiqmCfQMQSA3kLikpCeBoHAIBBBBAQAnQueA8QAABBPIUYGB3nmAhFGcgdwjoHBIBBBCgc8E5gAACCOQvwMDu/M2C3oOB3EGLczwEEEDghABXLjgTEEAAgQIE1KNN1SNOGdhdAJ7Pu6gZuRnI7TMy1SOAAAIuAnQuXGDYjAACCPQkkJqx+8UXX+ypGO+FILB169bkUa+44ooQjs4hEUAAgXgL0LmId/uTPQIIFCEwa9Ysufvuu0X9pZxFH4Gf/OQnogbdMyO3Pm1CJAggEB+BhGVZllO6iURCXN5yKs42BBBAIHYCqlNx5ZVXysMPPxyJGaCj8Lnf2toq5eXlsm/fPhk4cGDszkkSRgABBIIScPvO4MpFUC3AcYoWMGkCImIturldK9DJVv1lXP2FXP2l3GnRKVan+NK3mRSritst3l/96leirijp1LFwizXdX5fXxOpfS5hk658CNcdBgM5FHFqZHBFAwDeBW265RdR8CmrCNpZwBdTg+nnz5olqExYEEEAAgXAE6FyE485REUAgIgJqgrYnn3wy2cGISErGpqEG11dXV4uaRZ0FAQQQQCAcAcZchOPOURFAIEICUbnP3+3+WROaKmrjX0wwJ0YEEIi3gNt3Blcu4n1eGJW9SferEqt/p5aOtqnH0tbX13dLXMdYuwWYtmJSrCpse7zr1q1LZqPj42ftsaaxa/eSWP1rEpNs/VOg5jgI0LmIQyuTIwII+C6Qeiwtk+r5Tu14gEWLFsl9993H42cdddiIAAIIBCdA5yI4a46EAAIRFhg9enTyfn8m1Qu+kdVg+p07d8r1118f/ME5IgIIIIBANwE6F904WEEAAQQKF5g+fbqsXLky2En1Olqloe5FeXTKtTKn4WDhwRu855IlS5KD6pk0z+BGJHQEEIiMAJ2LyDQliSCAQNgCqfv9U/f/+x5P515Z/vUHZMWaH8qslYd8P5yOB1BXLdSjgHn8rI6tQ0wIIBBHAToXcWx1ckYAAV8E1F/O1X3/6v5/9fQi35deg2Xqb1fJzx+cKVW+H0zPA6SuWqhHArMggAACCIQvQOci/DYgAgQQiJBA6r7/wK5eRMgu31S4apGvGOURQAAB/wXoXPhvzBEQQCBGAulXL/72t7/FKPPgU1W3Q6kJDLlqEbw9R0QAAQTcBOhcuMmwHQEEEChQIHX1or29vcAa2C2bwKFDh0TdElVVFdcbwrIJ8b5uArW1tbqFRDwI+CJA58IXVipFAIE4C6irF5MnT5Y//vGPcWbwNffm5mZRc4uoCQxZEDBBgEn0TGglYvRCIGFZluVUkduU3k5l2YZAEAKpD2b115/Ua3Vc+7pTLPYy9nX2cRLItMUt08RJTjnNnj07+Zf122+/XZYvX97tnHXbp9Dz2jrULC8+3SiTNr0uZ215MqP6+fPnZ2xz21BTU5Pxlr3d7esZOzj8u/RyH3XV4umnn5Zvf/vb8tRTT3Wz9fI46XnZ67Wvp5dNvbaXsa+nyqX/tJexr6eXTb22l7Gvp8ql/7SXybau9s1Wxv5+FPdJNywkP2XEgkBUBFz7Cqpz4bSIiNNmtiEQmkBNTU1ox873wMSar1ju5U2yvf76663q6urckyuw5PGWZVaVjLBmb/qwwBosS7ma8rk/a9Ys68tf/nLBuQa9o0nnLLH6d3aYZOufAjVHScDtO4PboqLSfSQPBBDQTuCSSy6RtrY2qaur0y42UwNqbW1NXhEaPny4qSkQd0wFuGoR04aPYdp0LmLY6KSMAALBCKQ/OSqQeS+CSSvUoyxdujQ51qJfv36hxsHBEchXIP22x3z3pTwCJgnQuTCptYgVAQSME0g9OcqfeS8OSsOcCjmt/A7ZIE2yePznJHHtcmntNI4pp4BTVy2mTZuWU3kKIYAAAggEL0DnInhzjogAAjES8Pfqxbky7pFGNUDu1H/rp0pZRD/ZU1cteEJUjP4BkSoCCBgnENGvIOPagYARQCDCAv5evYgwXFpqXLVIw+AlAgggoLEAnQuNG4fQEEAgGgL+Xr2IhlG2LNSEecxrkU2J9xFAAIHwBehchN8GRIAAAjEQUFcv+vfvL8uWLYtBtt6muGHDBlG3RN13333eVkxtCCCAAAKeC9C58JyUChFAAIFMAXX14qGHHpK7775b1C0+LLkJqKdsPfDAA7JmzRopKSnJbSdKIYAAAgiEJkDnIjR6DowAAnETGDZsWPLWHvVXeJbcBNSVHnXFJzVuJbe9KIUAAgggEJZAQs0U6HRw1ym9nQqzDQEEEEAgJwF11aK8vFyam5tFdTZ0WnT73D98+LCo+Sy2bdsmo0eP1omKWBBAAIHYC7h9Z3DlIvanhjkAJk1ARKz+nVem26rHqC5YsCB5i5ROE+vp6Lpo0SJRc1o4dSx0jNftrCdWN5nitpvkqjI1Ld7iWoe94yxA5yLOrU/uCCAQisDMmTOlra1N/JlYL5SUPD9o6tGz6glRLAgggAAC5gjQuTCnrYgUAQQiIpD+aFp16w9LdwF1RWf27NnJKzxMmNfdhjUEEEBAdwHGXOjeQsSHAAKRFFC/QH/jG98Q9cvz4sWLtcjR7f7ZoIOrq6sTdUvU73//e54QFTQ+x0MAAQRyFHD7zuDKRY6AFAtfwKT7VYnVv/MlKrbq6oXqVKjJ4Xbt2uUfWI416+K6f/9+mTRpkjz88MM9dix0iTcXXmLNRSn/Mia5quxMizf/FmEPBE4I0LngTEAAAQRCElBXLZ588kn51re+JToN7g6JI3nYJ554IjmIu6qqKswwODYCCCCAQIECdC4KhGM3BBBAwAuBO+64I1nN6tWrvajO6DrUTNzqSg5/4TW6GQkeAQRiLkDnIuYnAOkjgEC4Aur2qJ/97GcyZcqUWM/crQa2p2biHjhwYLiNwtERQAABBAoWoHNRMB07IoAAAt4IqMn01NwX6glJcb09Sg3gHjp0qEycONEbVGpBAAEEEAhFgM5FKOwcFAEEEOgukJr7YtmyZd3fiMEat0PFoJFJEQEEYiNwemwyJVEEEEBAY4HU7VHDhw8X9Z/TrNQah19waOrpUNdee62sWbNGuB2qYEZ2RAABBLQR4MqFNk1BIAggEHcBdXvUihUrRF3FiMPkeuoWsBkzZoiahZvboeJ+9pM/AghERYDORVRakjwQQCASAjfddFNy7MGcOXMikU9PSahbwNra2uS+++7rqRjvIRAJgdra2kjkQRIIZBOgc5FNiPcRQACBAAXU7VHqUayvvfaaPPXUUwEeOdhDbd++Xe6+++7kk7JKSkqCPThHQyAEAR6xHAI6hwxFgDEXobBzUAQQQMBdQI09UI+nVWMv1ER7UZtQrrW1VcaMGZMcZ6FuBWNBAAEEEIiOAFcuotOWZIIAAhESUL90q0HOau4HNeg5KosaZ/HNb34z+ehdxllEpVXJAwEEEDglQOfilAWvEEAAAa0E1C/fau4Hda92VOa/SN0aogatsyAQJwHGXMSpteOdK52LeLc/2SOAgOYCTzzxhHz44Ydyzz33GN/BUGNINm/eLHV1daLGlrAgECeBVMc6TjmTazwF6FzEs93JGgEEDBFQv4SrX8rVAG/VwTB1UTmoAdyqY8F8Fqa2InEjgAAC2QXoXGQ3ogQCCCAQqoD6ZVz9Um7qE6TUDNyqY7Ft2zY6FqGeSRwcAQQQ8F+Ap0X5b8wREEAAgaIF0p8gpSpTk8+ZsKhHzqoZuNevXx+bWcdNaBdiRAABBPwSoHPhlyz1IoAAAh4LqCdIqb/+q8e4qkX3Doa6YqE6Firm0aNHe6xBdQgggAACOgrQudCxVYgJAQQQcBFQv6Tv27dPUo9x1bWDQcfCpQHZjAACCERcgDEXEW9g0kMAgegJpG6RWrlypdx5553aPUVKDd7mikX0zjsyQgABBHIRoHORixJlEEAAAc0E1C1SqUHe6ilSOky0p+bimD17tqhOD7dCaXbCEA4CCCAQkEDCsizL6ViJREJc3nIqzjYEEEAAgRAE1C/0qnOhniT1+OOPFzW2oZjP/dbW1mTHoq2tjcfNhnAecEgEEEAgaAG37wyuXATdEhyvYAGTJiAi1oKbOeuO2HYnUvNgqIn21NgLNdBb3ZJUyGzexbiqKyjl5eVSWVkpW7ZsCeRxs8XE213Q/zVi9cfYJFclUEi86t+zeuIaCwImCdC5MKm1iBUBBBBwEFAd1ZhjcwAAIABJREFUjMmTJ0tzc3PylqRvfOMbgfxCoq5WqDEfixYtSj5qdu7cucy87dA+bEKgUIH3338/+UcDdbvh4cOHC62G/RAIVIDORaDcHAwBBBDwT0CNw/j9738vV199ta+/kKhfctRfVNXVir/7u79L3gZVVVXlX2LUjEBMBRYvXpzsuG/evFmuu+665L+1mFKQtkECjLkwqLEIFQEEEMhVQF1VWLJkiSxdulQWLFggX/nKV0R1Pnpa3O6fTe2jBo3X19cnZ9uurq6Whx56KGudqX35iQAChQuoWx3VmKp58+bJtGnTkrdYqafGsSAQpoDbdwZXLsJsFY6dl0Ah96vmdQAPCxOrh5i2qrC1gbislpWVybPPPpu8Veqjjz6S4cOHy4QJE5K3Te3atStjLzdX1aFQT39S+5533nnJgePqSVCqk5Gts5JxEA83uMXr4SE8q4pYPaPsVpFJrirwYuJVtz6q2w7VrY8ffvhh8t+i+ndZyPiqboisIOCDAFcufEClSn8EUh/MtbW13T6k7etOR7eXsa+zj5OAiN3Jvu60l72MfZ19nAT8t1a/hKgnOalfUtTVDLWoDsc//MM/JDsMan3VqlVy8803yyeffCIff/yx/PGPf0yWU1cp1PLFL35R+vbtm3FeJN+0/c/e7vZ1W/Hkqr2MfZ19nAT8OXec7O3bsq2raLOVsb+v+z72FrDHn8u6vY5819W/5dWrV8uUKVNE/dtcvny5lJSU5FsN5REoWsDtygWdi6JpqSAoAXUSsyCAAAIIIIDAKYGWlhZRVypZEAhawK1zwW1RQbcExytKQM29YsJ/KkkT4lQx1tTUJNvElHhNsjUpVs4D/z5bOA/8sY3jOXvo0CGZNWtW8jNb/VTrdCySHPxPI4HTNYqFUBBAAAEEEEAAAQQcBNR8Muqxz2pZv3698IQ2ByQ2aSHAlQstmoEgEEAAAQQQQACBTAH1UAU1n8ykSZOSD1ZQE1XSsch0Yos+AnQu9GkLIkEAAQS8Eehsl6YVc+SqREIS/afI4zvapdObmqkFAQQCFFBPhFJPaVNPiFJPimKiygDxOVTBAnQuCqZjRwQQQEBHgY+k6bHp8r2Wq+XF45Ycb/qmfDhnujzW9JGOwRITAgj0IKAmq1yxYoX84he/CPXRzz2EyFsIZAgw5iKDhA0IIICAuQKdrXUy97Eh8v2946VU/fmodLzc9/2XZPDcOrlx3VQp409K5jYukcdOYMeOHbHLmYTNF+Brxvw2JAMEEEDgpMBf5K1tv5cNl14kA/ukPt57Sd+Kq+XWN34v2976C1IIIIAAAgj4KpD69vH1IFSOAAIIIBCAQOe7su2X26R02CD5fMan+zb55bZ3GXsRQDNwCAQQQCDOAhlfP3HGIHcE4iqg5rhgiYBAR5u0vCFyaXl/6ROBdEgBAQTcBP4m7XV3iZrELJHoL1c9ukM65Ji073hapvRPSOKiR6Xpb277sh0BfwUYc+GvL7V7KJCaMMnDKn2ryqRYfUPwqWKTbE2K1afm8q1ak2xNilU1mEl/bIhvrKdL6cRnxDp+v9Td+VWZ9Nga2TB0p7yy70p5us2SFb79y6NiBLILcOUiuxElEEAAATME+vSX8ktF3mhpkw4zIiZKBGIj8OCDD3qfa6/z5Op/vEFK2xfLjH8tlXtuu4Srlt4rU2OeAnQu8gSjOAKFCxyTA3UzpP+1y6VVw0kHOtu3y+NTLj1xmf2quVLXeqTwVP3c89gh+fdHp0j/1O0Ac56XpvZjfh6xwLqPyKE/t8ivHp0iF81pkEA0ew2SMV8bkxFv5wfvyq72MfK1MYPE/A/9Tulo3Sx7d/9Rplw0VxqOaPiPKdUCHa3Jc/Xx+fNP3Lqi7bkqIp3tsuPxKXIi1mtlTt1eczqone9J3R0Vcu3yvfqOKTrSIBsfV+eBuo1J/XezLG/14gELJx/YUFoql46++MQT4lLnHz8RCEnA/O+ZkOA4bJgCnXKkYe7JXy5PflBr+gt7ulLngd9KzYynpT19oy6vO16T+g2nydd//oZYx9vklRsPyIyxi/T7xa3zPWl59c8iNz4pbZYlx9tWy40fLJaKW56Wpg6dfsn8i7QunylbXntN7pm1Uj4NrJ3PkovGXCeXPr9RGrt+6e6Ujv1vyRtV18mYi84KLBK/DtTZ+px8vfbnsnv9RlkZHGz+6XS+J2ufbUieqzNrajQ+V1VqH0lz/XaRrz8rM2vmSdsrN8oHM74pP2g4mH/ege9xTA7UL5EZy9sCP3LuBzwmBzbWyRufpO3h4b/Hzk8+koPSLht++Ud5S6ePwbR0eRkvAToX8WrvaGTbuU82vvCbtF/SS6Xqa1+Wi3Q+mzt2yGNz/yDnXnOJhm3wN3m78ah86dZRJ/7q1atUKu+YIrfKBlnfeFireDv/vF/O+n+GyzVlfZNx9SodLd+ZN1OqNr8gq3boFOtZUjZ1mUysHicPV5UGatirbKIsvHeP/GDRJmnvVH+Q3iSLfrBH7l04MRJzXPQqmyq/feGncuXYiwN1zfdg6lw95+apBpyrIp1v/4d8/KUbpbL0DBE5TUorvyZTbhV5fv3rwVxxyxe3q7wlHU3PyNztZ8o1wf4z64ogpxcdu+QXPy2RCXO+nxzPosaJWOs9mnOm8z2p/8EGOfcbVSJvvCX7tfojS046FIqggM6/jkWQm5SKF+iUjuY6+em5P5aP1Qd08r82WT91sMa3e3wkTc+8KHLPdLn282cWT+B5DafLhWNHyYC0T4PkbTTl/yg3V5Z4frRiKux14Wj5wtmndaui1+cHyTCdf7HoFm0QK5+VEff+RB753L/IiNMSctotG6X80Z/IvSM+G8TBOcZJAXWujh2gflk/teh6rmbE2nlQ3t11ntx78+Vyoht/KgetXh1rk2eeELnn3uvk81oFlh5MpxzZUS+PbVgqf9jaLHUNrR7ebqau2vxENlTdLwv+6WtS1a7+IPSWND2+SOoO6HiraLoLr6MskPbrRJTTJLfoCByWHatekA2LH5EFy+ukQddxAV3gndLRtFKekFvkrhF6/aLeFWK3F0ektWG53L+gXea8+G0Z0TURW7dC+q18ZpR8qVzrX4OCNVNXn2auSN46Zr30iEwZUapx5ztYmtCPpvW52inHDv1Zlt//I3lrzlOad0g/kvbGN0XumSwjbH9wCL2N0wPobJVfPfKctEu7vPfH38qk8WPlhjl10lrUFYYOaXr0KkkkbpAn5HZ5dOIX5PT+l8g1Y9tk8Q+ek/f+frpMtHVs00PiNQJ+C9C58FuY+j0V6Gz9tTyyuElENsjiOybJ+PKb9B542NF44i9rd1Xo/wSPIw0yp/85Uj7+Dlm88g+y/k/vePgXNk9Pg7TKOuVI42bZdddkqeLLNM2Fl/oJ6H6uHpSGOZWy6Onn5Y7Fv5Y/rX9F3irqF2A/W+DEH21ekUvkLt2vyPUaLFPXtyXHst35ta/K7LEimxfPkH96prGIz9c+MuJ7L4llrZdHJg4+8d3Sp1K+91KbWC8tlIknbxv1swWoG4GeBOhc9KTDe9oJqPut1ycH8jZK/bLZMlZ1MiZ9T55p+ki7WNUgyaZnG+SChXeZcQWg7zh5pE0Nkn5FnpstsnjSTfKduvf0ffqKavGORvnZinNloQmdNw3PUEIKUED7c/VcGfdIo9R8/15pfO5W9QEgY79TLwd0HCDc0SjPrvmCjP/vA/T/o03qFOtVKqXll8sjm3bKpvsvlc2P1cuOrocupArxE4GICFguy4l5dFzeZDMCmggcb9tg3T+21Cqdvcn6OJCYjloty25S01n3/F/VT63GV/7Zun/Nu9bxrrg+tDbNHmFJ1TKr5dTGrne9fnG8ZZlVlS1Oucla1nI089DH91jLqoJyzdXU7vb/WY2PPWQt2xNMy1snTXpu+1KratmeU20eqGNmM+a7xZzP/ZPnTOn91qaPA/jHlC9kRvmAz9WM4+e7QWdfZflDa83+v55K6uNN1uxS27+9U+/q9yoZ7whr9qYP9YuNiBDIQ8DtO4MrFxHpJJqcRmfrcrm269nfqWeA23/eLDfOmJeRZq/S8XLf928T6fbozYxiHm5QTwFadeqJH12DylODy0/8rBn1n/LxmqXyw0mD5LSu3D4n49UtXRvukPLT+vv+TPbUVZ4Tg967x3dq2yp574UfZvq4zJeQWdCLLbmZpp6ucmIiqmNyYO2/ypvXz5SpgwMaa5G6vcGlzZ0eLvDgQ096ARRIHb5M8OVj5E/U/4ePtXtZ9TG59+v/GOy5WkT4J86DE480riqiHt92PfKqrFpyv0waeOapOSPOGS+L29tlwx1D5DTP5o/wPoOuf2PJyS7LpXxgH+8PQo0IaCBA50KDRoh7CLn+Ejy83+kOVL2kz8CL5NJLL5KBWg0+/kzyFoNTv8SrX+4/lE2zR4hULZOW47o/4apD9rd8LNd/6UINbzs4Lu0Ny2XV2TfIrV0diyOyd8XzspnbDBz+jbApPIFjyXP1P84oN+xcPTE3ytvXV0i5Vp+rInLy9k312VpTU3PiDz0fb5LZpaVStWyPHLdWydQyzedz6WiTt0bcIf+ge5zh/cPhyIYL0LkwvAEJX30J/m8ZMefGSDzDP5T2tD5Ozm57VdfsweoXoqflkQ9ukVlV52n2lKEjcmjvFrll/P+Qe8cPTLsqdI4MefH/Sn/dfhEKpUE5qB4CR6S17qHkubrh+cc1PlePyYG6GdL/qjmyoqk9eb9ncm6URw7L3FlXd3tEtR6uZkXR2d4ka9c2JeecUZF3tm+Xxxe8KlffPUbvx/yaxUy0ugm43Vrldh+VW3m2I+C/wF+ttsZ/s+ob207e0/5Xq+2Vf7Zmf3+D1WbCbddWsGMucm+Pj609y6ZapanxGaWTrSVrGjU0/au1f823T8WZijf5U7f7rY9bH2+63xary/iW3BvK95JGfO4n71dPH/OkW9urZjLrXP1kz3PW5NKU6SXW5CVrrMa2tDENvp95RR5A4zEXqXGB6t+WqM/W1S9ZLZ8Y8YVVZKOwexwE3L4z1CVFx8VtB8fCbEQgAIGamnlW26aHrLEnf6ksnbzEWr2pxfokgGPne4iampp8dwmtvEmxKiST4jUtVpM+902zDe0feJ4HxjVPsDyKm2SbR1oUjbGA23eG003sul1cIR4ETgqcJqXjHpSXrAcRQQABBBBAAAEEENBQgDEXGjYKISGAAAIIIIAAAgggYKIAnQsTW42YEUAAAQQQQAABBBDQUIDOhYaNQkgIIIAAAggggAACCJgoQOfCxFYjZgQQQAABBBBAAAEENBSgc6FhoxASAggggAACCCCAAAImCtC5MLHViBkBBBBAAAEEEEAAAQ0F6Fxo2CiEhAACCCCAAALREqitrY1WQmSDgIsAnQsXGDYjgAACCCCAAAJeCTz4IHM0eWVJPXoL0LnQu32IDgEEEEAAAQQQQAABYwToXBjTVASKAAIIIIAAAggggIDeAnQu9G4fokMAAQSyCByR1oa18qtHp8hFcxrkSJbSvI0AAuEIMOYiHHeOGrwAnYvgzTkiAggg4JHAX6R1+UypXfGc3DNrpXzqUa1UgwAC3gsw5sJ7U2rUU4DOhZ7tQlQIIIBADgJnSdnUZfLCz38oD1eV5lCeIggggAACCPgrQOfCX19qRwABBBBAAAEEEEAgNgIJy7Isp2wTiYS4vOVUnG0IIIAAAmEJdO6V5dePkweG/YvsfWSc9C0wDj73C4RjNwRyEFC3RTHuIgcoihgj4PadwZULY5qQQE26X5VY/TtfsfXH1iRXJWBSvMTKOeuPALUioKcAnQs924WoEEAAAQQQQAABBBAwToDbooxrsvgGnPrrn7qsnHqtNOzrTkL2MvZ19nESyLTFLdPESc7uZF/vcR/roDS/+Lz85u1PnIqd3Ha2XHjDrdL666elVs36e3Kfxkn1MvmsLXKmw57z58932Jq5qaamJmOjPX77esYODv8u2SeAc+dkQ3hh7VSHfVu2dRVOtjL293Xfx36u2+PPZd1eB+sImCrgdluUGlfhuIi4vuVYno0I+C1QU1Pj9yE8q59YPaPMqAjbDBLLOr7HWlZVapXO3mR97PB2LpuUq0mf+5wHubRq/mVwzd8s1z1Mss01J8rFW8DtO4MrF6Z2F4kbAQQQSAkwoDslwU8EEEAAgYAE3K5cMOYioAbgMMULpN8KVXxt/tZArP75YuuPrUmuSsCkeImVc9a0c9afFqPWuAjQuYhLS5MnAghEUKBTjjTMlf6nDZE7NrRL++Lxck7iZlne+pcI5kpKCCCAAAImCJxuQpDEiAACCCDgJNBL+o5bKG3WQqc32YYAAggggEDgAoy5CJycAyKAAAJ6CrjdP6tntESFAAIIIBCmgNt3BrdFhdkqHDsvAe5bzosr58ImuaqkTIqXWHM+DfMuiG3eZDntgGtOTAUVMsm2oATZCYGTAnQuOBUQQAABBBBAAAEEEEDAEwE6F54wUgkCCCCAAAIIIIAAAgjQueAcQAABBBBAAAEEEEAAAU8E6Fx4wkglCCCAAAIIIIAAAgggQOeCcwABBBBAAAEEEEAAAQQ8EaBz4QkjlSCAAAIIIIAAAggggACdC84BBBBAAAEEEEAAAQQQ8ESAzoUnjFSCAAIIIIAAAgi4C9TW1rq/yTsIREiAzkWEGpNUEEAAAQQQQEBPASbR07NdiMp7AToX3ptSIwIIIIAAAggggAACsRSgcxHLZidpBBBAAAEEEEAAAQS8F6Bz4b0pNSKAAAIIIIAAAt0EGHPRjYOVCAvQuYhw45IaAggggAACCOghwJgLPdqBKPwXoHPhvzFHQAABBBBAAAEEEEAgFgJ0LmLRzCSJAAIIIIAAAggggID/AgnLsiynwyQSCXF5y6k42xBAAAEEDBfgc9/wBiR8rQXUbVGMu9C6iQguTwG37wyuXOQJSfHwBEy6X5VY/TtPsPXH1iRXJWBSvMTKOeuPALUioKcAnQs924WoEEAAAQQQQAABBBAwToDOhXFNRsAIIIAAAggggAACCOgpQOdCz3YhKgQQQAABBBBAAAEEjBNgQLdxTUbACCCAgD8CboPz/DkatSKAAAIImCzg9p3BlQuTWzVmsTMo0p8GN8lVCZgUL7H6c85yHuBq2jlgYrz+nWXUHHUBOhdRb2HyQwABBBBAAAEEEEAgIAE6FwFBcxgEEEAAAQQQQAABBKIuwJiLqLcw+SGAAAI5CrjdP5vj7hRDAAEEEIiRgNt3BlcuYnQSmJ4q96/704ImuSoBk+IlVn/OWc4DXE07B0yM17+zjJqjLkDnIuotTH4IIIAAAggggAACCAQkQOciIGgOgwACCCCAAAIIIIBA1AXoXES9hckPAQQiLHBEWv/9cZnSPyHq3tfEVXNkRVO7dEY4Y1JDAAEEENBbgM6F3u1DdAgggICLwDE5sPZ5+Tf5e3m6zRLreJu8cuMHcn/FnfJY00cu+7AZAQQQQAABfwXoXPjrS+0IIICAPwKdB+Q/z7lRvnNNmfRRR+hVKpXfuV8ernpVHlv1qhzx56jUigACCCCAQI8Cp/f4Lm8igAACCOgp0OuLMnasLbRe58qgYf1tG1lFAAEEEEAgOAGuXARnzZEQQACBAAT6yfVfuvDE1YwAjsYhEEAAAQQQSBegc5GuwWsEEEDAZIEjr8v6XdfJ9KrzhA93kxuS2KMoUFtbG8W0yAmBDAG+fzJI2IAAAgiYKPCRNP3s1/K5hbfLiD58tJvYgsQcbQGTJtWMdkuQnd8CCcuyLKeDuE3p7VSWbQgEIZD6YFZ//Um9Vse1rzvFYi9jX2cfJ4FMW9wyTZzk7E729R73sQ5K84vPy2/e/sSp2MltZ8uFN9wqrb9+WmoffFBELDnW3iSdox6RXu+tljMc9pw/f77D1sxNNTU1GRvt8dvXM3Zw+HfJPgGcOycbwgtrpzrs27Ktq3CylbG/r/s+9nPdHn8u6/Y6WEfAVAHXvoLqXDgt6tuKBQGdBGpqanQKp8dYiLVHnqLexDaT7/j+f7Mee+4N65PMt3LeolxN+tznPMi5afMqiGteXHkVNsk2r8QoHFsBt+8Mrp2b2l0kbgQQQEBEOtsb5MerzpKv33rJyUHcndKx91eyfPNBfBBAQCMBdVWDBYE4CNC5iEMrkyMCCERQoFM6Wuvk/lu+KffeO176n3Zylu7EaXL2kFUi/ZOzX0Qwb1JCwEyB9Nt5zcyAqBHITYB5LnJzohQCCCCglUDngXr5zthJsrzdIayq62TMRWc5vMEmBBBAAAEE/BWgc+GvL7UjgAACvgj0GjBRlrVZssyX2qkUAQQQQACBwgS4LaowN/ZCAAEEEEAAAQQQQAABmwCdCxsIqwgggAACCCCAAAIIIFCYAJ2LwtzYCwEEEEAAAQQQQAABBGwCdC5sIKwigAACCCCAAAIIIIBAYQJ0LgpzYy8EEEAAAQQQQAABBBCwCSTUtIK2bclV1ym9nQqzDQEEEEDAeAE+941vQhJAAAEEAhNw+87gykVgTcCBihUwaQIiYi22td33x9bdpph3THJVeZoUL7EWc2a672uSq2nnrLs67yCQXYDORXYjSiCAAAIIIIAAAggggEAOAnQuckCiCAIIIIAAAggggAACCGQXYMxFdiNKIIAAArEQcLt/NhbJkyQCCCCAQF4Cbt8ZXLnIi5HCYQqYdH8tsfp3pmDrj61JrkrApHiJlXPWtHPWnxaj1rgI0LmIS0uTJwIIIIAAAggggAACPgvQufAZmOoRQAABBBBAAAEEEIiLAJ2LuLQ0eSKAAAIIIIAAAggg4LMAnQufgakeAQQQQAABBBBAAIG4CNC5iEtLkycCCCCAAAIIIIAAAj4L0LnwGZjqEUAAAQQQQAABBBCIiwCdi7i0NHkigAACCCCAAAIIIOCzAJ0Ln4GpHgEEEEAAAQQQqK2tBQGBWAjQuYhFM5MkAggggAACCIQpYNJkimE6cWzzBehcmN+GZIAAAggggAACCCCAgBYCdC60aAaCQAABBBBAAAEEEEDAfAE6F+a3IRkggAACCCCAgOYCjLnQvIEIzzMBOheeUVIRAggggAACCCDgLMCYC2cXtkZPgM5F9NqUjBBAAAEEEEAAAQQQCEWAzkUo7BwUAQQQQAABBBBAAIHoCSQsy7Kc0kokEuLyllNxtiGAAAIIGC7A577hDUj4Wguo26IYd6F1ExFcngJu3xlcucgTkuLhCZh0vyqx+neeYOuPrUmuSsCkeImVc9YfAWpFQE8BOhd6tgtRIYAAAggggAACCCBgnACdC+OajIARQAABBBBAAAEEENBTgM6Fnu1CVAgggAACCCCAAAIIGCfAgG7jmoyAEUAAARHpbJcdP75PJty7UtqlSmav+bF8f+Jg6VMEjtvgvCKqZFcEEEAAgYgKuH1ncOUiog0exbQYFOlPq5rkqgRMite/WD+S5vrtIl9/Vtqsv0rbKzfKBzO+KT9oOFjwSeJfrAWH1OOOJsVLrD02ZcFvmuSqkjQt3oIbhh1jL0DnIvanAAAIIGCaQOfb/yEff+lGqSw9Q0TOkNLKr8mUW0WeX/+6HDEtGeJFAAEEEIiUAJ2LSDUnySCAQBwEel04WsYOUB2Lk0vnQXl313ly782XS9/UNn4igAACCCAQggBjLkJA55AIIICANwKd0tG6WVYtWyVvXft9eXjcACnmL0Zu9896Eyu1IIAAAghEScDtO6OY76Eo+ZCLAQIm3a9KrP6dUNimbA9Kw5xKObt8vNyx+Nfyp/WvyFsdnak38/5pkqtKzqR4iTXv0zGnHUxyNe2czakBKISAiwBXLlxg2KyfQOqLpLa2ttsvFvZ1p8jtZezr7OMkIGJ3sq877WUvY19nHyeBIqxr/qe0v/6ybFj7Rzk2dY18fcBrcnYi8xjz58/P3OiwpaamJmOrvQ3t6xk7SBH5PPhgV3UcJ9OxCyfthd3Jvp5WtOulvUy2dbVjtjL296O4TxfgyRf2nHNZt9fBOgKmCrhduRDLZRFxfctlDzYj4K9ATU2NvwfwsHZi9RDTVlV0bY9aLctustRnb4//VS2zWo7bUKyT+5beb236OONNe2HHdeVq0ud+dM8Dx+YJbCOu/lGbZOufAjVHScDtO+N0U3tLxI0AAghES+AsKZu6SqyphWR1llw05jqpkrcK2Zl9EEAAAQQQ8EyAMReeUVIRAgggEJZAp3Tsf0vevr5CyvvwsR5WK3BcBBBAAAEp6sEi+CGAAAIIBC5wTA7UzZD+V82RFU3tooZwd7ZvkkWPHJa5s66WAfQtAm8RDogAAgggcEqAr6FTFrxCAAEEDBA4Xc65eKRc07JYbqvoL6clLpXbX/hEJj33hEwdzCwXBjQgISKAAAKRFmDMRaSbl+QQQCB6Ar2kz+ApsqJtiqyIXnJkhAACCCBguABXLgxvQMJHAAEEEEAAAf0F1GNqWRCIgwCdizi0MjkigAACCCCAQKgCqbmaQg2CgyMQgACdiwCQOQQCCCCAAAIIIIAAAnEQoHMRh1YmRwQQQAABBBBAAAEEAhCgcxEAModAAAEEEEAAgXgLMOYi3u0fp+zpXMSptckVAQQQQAABBEIRYMxFKOwcNAQBOhchoHNIBBBAAAEEEEAAAQSiKEDnIoqtSk4IIIAAAggggAACCIQgkLAsy3I6biKREJe3nIqzDQEEEEDAcAE+9w1vQMLXWkDdFsW4C62biODyFHD7zuDKRZ6QFA9PwKT7VYnVv/MEW39sTXJVAibFS6ycs/4IUCsCegrQudCzXYgKAQQQQAABBBBAAAHjBOhcGNdkBIwAAggggAACCCCAgJ4CdC70bBeiQgABBBBAAAEEEEDAOAEGdBvXZASMAAII+CPgNjjPn6NRKwIIIICAyQJu3xlcuTC5VWMWO4Mi/Wlwk1yVgEnxEqs/5yznAa6mnQMmxuvfWUbNURegcxH1FiY/BBBAAAEEEEAAAQQCEqBNOmDjAAAR2klEQVRzERA0h0EAAQQQQAABBBBAIOoCjLmIeguTHwIIIJCjgNv9sznuTjEEEEAAgRgJuH1ncOUiRieB6aly/7o/LWiSqxIwKV5i9eec5TzA1bRzwMR4/TvLqDnqAnQuot7C5IcAAggggAACCCCAQEACdC4CguYwCCCAAAIIIIAAAghEXYDORdRbmPwQQAABBBBAAAEEEAhIgM5FQNAcBgEEEEAAAQQQQACBqAvQuYh6C5MfAggggAACCCCAAAIBCdC5CAiawyCAAAIIIIAAAgggEHUBOhdRb2HyQwABBBBAAAEEEEAgIAE6FwFBcxgEEEAAAQQQiK9AbW1tfJMn81gJ0LmIVXOTLAIIIIAAAgiEIWDSpJph+HDM6AjQuYhOW5IJAggggAACCCCAAAKhCtC5CJWfgyOAAAIIIIAAAgggEB2BhGVZllM6iURCXN5yKs42BBBAAIHQBI7Jgbp7ZeRPL5fN66ZKWYF/NuJzP7QG5MAIIICAcQJu3xkFfgUZlz8BR0DApPtVidW/Ew7bTNvOA7+VmhlPS3vmWzlvMclVJWVSvMSa82mYV0GTXE07Z/NqCAojYBOgc2EDYRUBBBAwSqBjhzw29w9y7jWXGBU2wSKAAAIIRFOAzkU025WsEEAgFgIfSdMzL4rcM12u/fyZsciYJBFAAAEE9BZgzIXe7UN0CCCAgItAp3Q0PSXffmmUPP29C2THnOtk/K7p0sKYCxcvNiMQroC6jYu5LsJtA47urQBjLrz1pLYQBEy6v5ZY/TtBsD1p29Eozzwhcs9dFdLHA26TXFW6JsVLrB6coA5VmOTqED6bEIisALdFRbZpSQwBBKIr8JE0PdsgFyy8S0b04WM8uu1MZggggIB5AtwWZV6bxTbi1F+p1GXl1GuFYV93ArKXsa+zj5NApi1umSZOcnYn+7rzPvfLjBtHyNO/2e309qltF35V7hzbKXs++W+y5Zc/ktoHHxSRT+XPG1+Q5/s8LN/+0nvSL3GqeOrV/PnzUy97/FlTU5Pxvj1++3rGDg7/LtnHz3PH+89Ep/ayb8u2rs6LbGXs7+u+j/1ct8efy7q9DtYRMFXA7bYoNZeF4yLi+pZjeTYi4LdATU2N34fwrH5i9YwyoyJsP7Q2zR6h5idy+a/Uqlq2xzqeIdfzBuVq0uc+50HP7Vnou7gWKpd9P5Nss2dDCQSS8+Q5MnDlwtTuInEjgAACXQIHpYEB3V0avEAAAQQQ8F/A7coFN+v6b88RPBJIvxXKoyp9q4ZYfaPtdkucf0fxpmbOA28cnWrB1kml+G24Fm/oVoNJtm45sB2BXAToXOSiRBkEEEAAAQQQQAABBBDIKnB61hIUQAABBBDQXOBcGfdIY3IQhuaBEh4CCCCAQMQFGHMR8QYmPQQQQCBXAbf7Z3Pdn3IIIIAAAvERcPvO4Lao+JwDxmdq0v2qxOrf6YatP7YmuSoBk+IlVs5Z085Zf1qMWuMiQOciLi1NnggggAACCCCAAAII+CxA58JnYKpHAAEEEEAAAQQQQCAuAnQu4tLS5IkAAggggAACCCCAgM8CdC58BqZ6BBBAAAEEEEAAAQTiIkDnIi4tTZ4IIIAAAggggAACCPgsQOfCZ2CqRwABBBBAAAEEEEAgLgJ0LuLS0uSJAAIIIIAAAggggIDPAnQufAamegQQQAABBBBAoLa2FgQEYiFA5yIWzUySCCCAAAIIIBCmgEmTKYbpxLHNF6BzYX4bkgECCCCAAAIIIIAAAloI0LnQohkIAgEEEEAAAQQQQAAB8wUSlmVZTmkkEglxecupONsQQAABBAwX4HPf8AYkfAQQQCBAAbfvDK5cBNgIHKo4AZPuVyXW4tq6p72x7Umn8PdMclVZmhQvsRZ+Xva0p0mupp2zPbnzHgLZBOhcZBPifQQQQAABBBBAAAEEEMhJgM5FTkwU+v/bu98Qy8oyAODPNSmRTSSEcg3Kihk/tBWs+MUl1GS3JRTJYklbFrcPQsQGi/8yGZlBCmTZkDIWYi01iTaLTMgMXFZRItm1yLB2ESow54N90G0gE7w3rjbbdOaeuXfu3vfe+573t7DsnHvOed/n+T3v3sMz95wZAgQIECBAgAABAgT6CXjmop+Q/QQIEChEoO7+2ULSlyaBpALd27j8roukxAYfs0DdNcMnF2MuhOmGF8jp/lqxDl/nfmey7Sc03P6cXLsZ5hSvWIdbk/3Oysm1Xy72E2iSgOaiSdWUCwECBAgQIECAAIEJCmguJohvagIECBAgQIAAAQJNEtBcNKmaciFAgAABAgQIECAwQQEPdE8Q39QECBCYJoG6h/OmKUaxECBAgMB0CNRdM3xyMR31EcUAAjk9vCfWAQo65CFsh4Trc1pOrt1UcopXrH0W35C7c3LNbc0OWRKnEXhLQHNhIRAgQIAAAQIECBAgMBIBzcVIGA1CgAABAgQIECBAgIBnLqwBAgQIEHhLoO7+WTwECBAgQKAqUHfN8MlFVcr21ArkdH+tWNMtI7ZpbHNy7QrkFK9Yrdnc1myaihm1FAHNRSmVlicBAgQIECBAgACBxAKai8TAhidAgAABAgQIECBQioDmopRKy5MAAQIECBAgQIBAYgHNRWJgwxMgQCCdwD/i8K0XR/ehurf/boxt9/052ukmNDIBAgQIEFhTQHOxJo+dBAgQmF6B9t+fiocePLYiwC2xY8sHwxv7ChJfEiBAgMBYBc4c62wmI0CAAIERCbwav/vRI3HeD1+JzhXnjWhMwxAgQIAAgdMT0Fycnp+zCRAgMBmBk8/Fof0PxN2z74vZV7fHlq2fjJkNPrOYTDHMSoAAAQLLAq5EyxL+JUCAQDYCr8eJhw/E3YsRceTu+NK1l8fsVXfEz06czCYDgRIoTWB+fr60lOVbqIDmotDCS5sAgZwFzoqZ3Yei0/l3vHz00Th4y9aII9+Ma288GMeWPM6dc2XF3lyBnH6ZYnOrILNxCLQ6nU6n10R1v9K717FeIzAOgeU35u53f5a/7s5b3e4VS/WY6rZzegmstuW22qSXXNWpuj36c96Mpb88Hc89uBRv7Lw8rrzw7FVTLCwsrHqt1wtzc3OrXq7GX91edUKP/5fOmda183b1qvWpbnePqr7Wb7uJ51TXej+DXvurY9gmkKtAba/QbS56/YmIXi97jcDEBObm5iY293onFut6xQY/vrm2/+ocP/j57jd71v679WDn+Ju9vF7pPHHL5s75tzzRea3X7j6vdV1zet9v7jroU6jEu7mmA87JNp2CkZskUHfN8EB3ru2iuAkQaJjAf2912j1sWhvi/bOzsSk2xoZhh3AeAQLJBLqfYvhDoAQBz1yUUGU5EiBQgMBSvPTiprj1czN+z0UB1ZZifgIrb+fNL3oRExhcQHMxuJUjCRAgMB0C7cU49sgv49jiG2/H016MZ7+1P45cuTMuO8fb+nQUSRQECBAoU8BVqMy6y5oAgawF2vHab++Jize+K1qtTbFr/1Ox9JnbYuGKC3xqkXVdBU+AAIH8Bfy0qPxrKAMCBAiMRKD2J3+MZHSDECBAgECTBOquGT65aFKVG55LTverijXdYmSbxjYn165ATvGK1ZrNbc2mqZhRSxHQXJRSaXkSIECAAAECBAgQSCyguUgMbHgCBAgQIECAAAECpQhoLkqptDwJECBAgAABAgQIJBbwQHdiYMMTIEAgF4G6h/NyiV+cBAgQIDA+gbprhk8uxlcDM52mgIciTxOw5vScXLsp5BSvWGsW3QheZjsCxB5DcO2BMqKXcrIdUcqGKVRAc1Fo4aVNgAABAgQIECBAYNQCmotRixqPAAECBAgQIECAQKECnrkotPDSJkCAQFWg7v7Z6nG2CRAgQIBA3TXDJxfWRjYCOd2vKtZ0y4ptGtucXLsCOcUrVms2tzWbpmJGLUVAc1FKpeVJgAABAgQIECBAILGA5iIxsOEJECBAgAABAgQIlCKguSil0vIkQIAAAQIEJiYwPz8/sblNTGCcApqLcWqbiwABAgQIEChSIKdnb4oskKRHJqC5GBmlgQgQIECAAAECBAiULaC5KLv+sidAgAABAgQIECAwMgHNxcgoDUSAAAECBAgQIECgbAHNRdn1lz0BAgQIECAwBgEPdI8B2RRTIaC5mIoyCIIAAQIECBBosoAHuptcXbmtFNBcrNTwNQECBAgQIECAAAECQwtoLoamcyIBAgQIECBAgAABAisFWp1Op7PyheWvW61W1OxaPsS/BAgQINAgAe/7DSqmVAgQIJBYoO6a4ZOLxPCGH51ATverinV0da+OxLYqMprtnFy7GecUr1hHs0aro+TkmtuarVrbJrAeAc3FerQcS4AAAQIECBAgQIBArYDmopbGDgIECBAgQIAAAQIE1iPgmYv1aDmWAAECDRaou3+2wSlLjQABAgSGFKi7ZvjkYkhQp41fIKf7a8Wabn2wTWObk2tXIKd4xWrN5rZm01TMqKUIaC5KqbQ8CRAgQIAAAQIECCQW0FwkBjY8AQIECBAgQIAAgVIENBelVFqeBAg0VqC9eCweeXhf7NrYitbG2+PwyXZjc5UYAQIECEy3gAe6p7s+oiNAgMAaAm/E4rPfi9uuORCx947YdfX2uGLmnDWOX3tX3cN5a59lLwECBAiUKFB3zfDJRYmrIdOcPRSZpnA5uXYFcoo3baztWDp2b1x3zbPxiZ//Or5/047TaizSxjr6tZtTvGIdff1zey/IMd40VTNqCQJnlpCkHAkQINA4gaWjceCmh+JD3/lpfPWS88N3ihpXYQkRIEAgSwHXoyzLJmgCBMoWeD1OHNoXN8en4tL2T+KG7rMWrU2xa9/jcWLJ8xZlrw3ZEyBAYLICmovJ+pudAAEC6xdo/zWe/vFzcdklH42PX7on7n/5zfjnn26K2H9D3HjgaCytf0RnECCQWGB+fj7xDIYnMB0CmovpqIMoCBAgMLjA0stx/Plz45Jt22Pz+e+MiDNiw0U74ut3bYkjN++LQydeH3wsRxIgMBaBnJ69GQuISRor4KdFNba0zUts+Y25+92f5a+7WVa3e2VePaa67ZxeAqttua026SVXdapu9z7na/GVqzfHvY++0Gv3/1778FXx5W0Rj3/3aFz7xB/irCe/fWrfndd/IGZm74j37rw+rrzw7FOvd79YWFj4v+26jbm5uVW7qvFXt1ed0OP/pXNSrp3Rvyf2qlf1tX7b3XXR75jq/mk/p7rWq/EPsl0dwzaBXAXqflqU5iLXioqbAIFyBU4ejlsv+mL8/q7D8djui049zN0+cV9sn/1V7Dj+QOyeOatcH5kTIECAQHKBuubCbVHJ6U1AgACBEQuc87HYtnNjPP/MC7F46vntdiy99GI8v/XTseUjGosRixuOAAECBAYU0FwMCOUwAgQITI/AeXHZnttj+2N3xu0/+ONbD3C3F38TB+//W+z9xmdjxjv79JRKJAQIEChMwCWosIJLlwCBZgicccE1cc+RfbHpyS/Eu1uteMd1v4j37NkXezef24wEZUGAAAECWQp45iLLsgmaAAECBAgQIECAwOQEPHMxOXszEyBAgAABAgQIEChCwG1RRZRZkgQIECBAgAABAgTSC2gu0hubgQABAgQIECBAgEARApqLIsosSQIECBAgQIAAAQLpBTQX6Y3NQIAAAQIECBAgQKAIAc1FEWWWJAECBAgQIECAAIH0ApqL9MZmIECAAAECBAgQIFCEgOaiiDJLkgABAgQIECBAgEB6Ac1FemMzECBAgAABAgQIEChCQHNRRJklSYAAAQIECBAgQCC9gOYivbEZCBAgQIAAAQIECBQhoLkoosySJECAAAECBAgQIJBeQHOR3tgMBAgQIECAAAECBIoQ0FwUUWZJEiBAgAABAgQIEEgvoLlIb2wGAgQIECBAgAABAkUIaC6KKLMkCRAgQIAAAQIECKQX0FykNzYDAQIECBAgQIAAgSIEzlwry1artdZu+wgQIECAAAECBAgQIHBKoLa56HQ6pw7yBQECBAgQIECAAAECBPoJuC2qn5D9BAgQIECAAAECBAgMJKC5GIjJQQQIECBAgAABAgQI9BPQXPQTsp8AAQIECBAgQIAAgYEE/gM/4Zz7W0NuCAAAAABJRU5ErkJggg=="></p>
<p>The graph of <em>g</em> intersects the graph of <em>f</em> at <em>x</em> = −1 and <em>x</em> = 2.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the coordinates of the vertex of the graph of <em>g</em>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>On the grid above, sketch the graph of g for −2 ≤ <em>x</em> ≤ 4.</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 area of the region enclosed by the graphs of <em>f</em> and <em>g</em>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>(1,5) (exact) <em><strong> A1 N1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img 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"> <em><strong> A1A1A1 N3</strong></em></p>
<p><strong>Note:</strong> The shape must be a concave-down parabola.<br>Only if the shape is correct, award the following for points in circles:<br><em><strong>A1</strong></em> for vertex,<br><em><strong>A1 </strong></em>for correct intersection points,<br><em><strong>A1 </strong></em>for correct endpoints.</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>integrating and subtracting functions (in any order) <em><strong>(M1)</strong></em><br><em>eg </em>\(\int {f - g} \)</p>
<p>correct substitution of limits or functions (accept missing d<em>x</em>, but do not accept any errors, including extra bits) <em><strong>(A1)</strong></em><br>eg \(\int_{ - 1}^2 {g - f,\,\,\int { - {{\left( {x - 1} \right)}^2}} } + 5 - {x^2}\)</p>
<p>area = 9 (exact) <em><strong>A1 N2</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br>