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</div><h2>SL Paper 1</h2><div class="specification">
<p><span style="font-size: medium; font-family: times new roman,times;">A function is represented by the equation</span></p>
<p style="margin-left: 30px;"><span style="font-size: medium; font-family: times new roman,times;">\[f(x) = a{x^2} + \frac{4}{x} - 3\]</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>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>The function \(f (x)\) has a local maximum at the point where \(x = −1\).</span></p>
<p><span>Find the value of <em>a</em>.</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>\(f(x) = a{x^2} + 4{x^{ - 1}} - 3\)</span></p>
<p><span>\(f'(x) = 2ax - 4{x^{ - 2}}\) <em><strong>(A3)</strong></em></span></p>
<p><em><span><strong>(A1)</strong> for 2ax, <strong>(A1)</strong> for </span><span>–4x <sup>–2</sup></span><span> and <strong>(A1)</strong> for derivative of –3 being zero. <strong>(C3)</strong></span></em></p>
<p><em><span><strong>[3 marks]</strong></span></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span>\(2ax - 4x^{-2} = 0\) <em><strong>(M1)</strong></em></span></p>
<p><span>\(2a( - 1) - 4{( - 1)^{ - 2}} = 0\)</span><span> <em><strong>(M1)</strong></em></span></p>
<p><span>\( -2a - 4 = 0\)</span></p>
<p><span>\(a = -2\) <em><strong>(A1)</strong></em><strong>(ft)</strong></span></p>
<p><em><span><strong>(M1)</strong> for setting derivative function equal to 0</span>. <span><strong>(M1)</strong> for inserting</span></em><span> \(x = -1\)</span><em><span> but do not award <strong>(M0)(M1) (C3)</strong></span></em></p>
<p><em><span><strong>[3 marks]</strong></span></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-size: medium; font-family: times new roman,times;">(a) Many candidates gave up at this point. Those who attempted the derivative did so with varying success. Many could not differentiate a term with a negative index.</span></p>
<p> </p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-size: medium; font-family: times new roman,times;">(b) In part (b) most substituted the -1 into the original function rather than the differentiated one. They did not realize they had to put the differentiated function equal to zero.</span></p>
<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 function \(f(x)\) is such that \(f'(x) < 0\) for \(1 < x < 4\). At the point \({\text{P}}(4{\text{, }}2)\) on the graph of \(f(x)\) the gradient is zero.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Write down the equation of the tangent to the graph of \(f(x)\) at \({\text{P}}\).</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>State whether \(f(4)\) is greater than, equal to or less than \(f(2)\).</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>Given that \(f(x)\) is increasing for \(4 \leqslant x < 7\), what can you say about the point </span><span><span><span>\({\text{P}}\)</span></span>?</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>\(y = 2\). <em><strong>(A1)(A1) (C2)</strong></em> </span></p>
<p><span><strong>Note:</strong> Award <em><strong>(A1)</strong></em> for \(y = \ldots \), <em><strong>(A1)</strong></em> for \(2\).</span><br><span>Accept \(f(x) = 2\) and \(y = 0x + 2\) <br></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span>Less (than). <strong><em>(A2) (C2)</em></strong></span></p>
<p><span><strong><em>[2 marks]<br></em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span>Local minimum (<em>accept minimum, smallest or equivalent</em>) <em><strong>(A2) (C2)</strong></em></span></p>
<p><span><strong>Note: </strong>Award <em><strong>(A1)</strong></em> for stationary or turning point mentioned.<br>No mark is awarded for \({\text{gradient}} = 0\) as this is given in the question.</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;">This question was poorly answered by many of the candidates. They could not write down the equation of the tangent, they could not say whether one value was greater or less than another and they could not answer that </span><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: times new roman,times; font-size: medium;">\({\text{P}}\)</span> was a minimum point. Most attempted the question so it was not a case that the paper was too long. This was a very good discriminator for the paper.</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 poorly answered by many of the candidates. They could not write down the equation of the tangent, they could not say whether one value was greater or less than another and they could not answer that </span><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: times new roman,times; font-size: medium;">\({\text{P}}\)</span> was a minimum point. Most attempted the question so it was not a case that the paper was too long. This was a very good discriminator for the paper.</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;">This question was poorly answered by many of the candidates. They could not write down the equation of the tangent, they could not say whether one value was greater or less than another and they could not answer that </span><span style="font-family: times new roman,times; font-size: medium;"><span style="font-family: times new roman,times; font-size: medium;">\({\text{P}}\)</span> was a minimum point. Most attempted the question so it was not a case that the paper was too long. This was a very good discriminator for the paper.</span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the function \(f\left( x \right) = \frac{{{x^4}}}{4}\).</p>
</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">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the gradient of the graph of <em>f</em> at \(x = - \frac{1}{2}\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the <em>x</em>-coordinate of the point at which the <strong>normal</strong> to the graph of <em>f</em> has gradient \({ - \frac{1}{8}}\).</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><em>x</em><sup>3</sup> <em><strong>(A1) (C1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(A0)</strong></em> for \(\frac{{4{x^3}}}{4}\) and not simplified to <em>x</em><sup>3</sup>.</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>\({\left( { - \frac{1}{2}} \right)^3}\) <em><strong>(M1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for correct substitution of \({ - \frac{1}{2}}\) into their derivative.</p>
<p>\({ - \frac{1}{8}}\) (−0.125) <em><strong> (A1)</strong></em><strong>(ft)</strong><em><strong> (C2)</strong></em></p>
<p><strong>Note:</strong> Follow through from their part (a).</p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><em>x</em><sup>3</sup> = 8 <em><strong>(A1)(M1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(A1)</strong></em> for 8 seen maybe seen as part of an equation <em>y</em> = 8<em>x</em> + <em>c</em>, <em><strong>(M1)</strong></em> for equating their derivative to 8.</p>
<p>(<em>x</em> =) 2 <em><strong>(A1) (C3)</strong></em></p>
<p><strong>Note:</strong> Do not accept (2, 4).</p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Consider the curve \(y = {x^2}\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Write down \(\frac{{{\text{d}}y}}{{{\text{d}}x}}\).</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>The point \({\text{P}}(3{\text{, }}9)\) lies on the curve \(y = {x^2}\) . Find the gradient of the tangent to the curve at P .</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>The point \({\text{P}}(3{\text{, }}9)\) lies on the curve \(y = {x^2}\) . Find the equation of the normal to the curve at P . Give your answer in the form \(y = mx + c\) .</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><span>\(2x\) </span><span> <em><strong>(A1)</strong></em> <em><strong>(C1)</strong></em></span></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span>\(2 \times 3\) <em><strong>(M1)</strong></em></span><br><span>\( = 6\) <em><strong>(A1)</strong></em> <em><strong>(C2)</strong></em></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span>\(m({\text{perp}}) = - \frac{1}{6}\) <strong><em>(A1)</em>(ft)</strong></span></p>
<p> </p>
<p><span><strong>Note:</strong> Follow through from their answer to part (b).</span></p>
<p> </p>
<p><span>Equation \((y - 9) = - \frac{1}{6}(x - 3)\) <em><strong>(M1)</strong></em></span></p>
<p> </p>
<p><span><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for correct substitution in any formula for equation of a line.</span></p>
<p> </p>
<p><span>\(y = - \frac{1}{6}x + 9\frac{1}{2}\) <strong><em>(A1)</em>(ft)</strong> <em><strong>(C3)</strong></em></span></p>
<p> </p>
<p><span><strong>Note:</strong> Follow through from correct substitution of their gradient of the normal.</span><br><span><strong>Note:</strong> There are no extra marks awarded for rearranging the equation to the form \(y = mx + c\) .</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><span style="font-size: medium; font-family: times new roman,times;">The equation of a curve is given as \(y = 2x^{2} - 5x + 4\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Find \(\frac{{{\text{d}}y}}{{{\text{d}}x}}\)</span><span>.</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>The equation of the line <em>L</em> is \(6x + 2y = -1\).</span></p>
<p><span>Find the <em>x</em>-coordinate of the point on the curve \(y = 2x^2 - 5x + 4\) where the tangent is parallel to <em>L</em>.</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>\(\frac{{{\text{d}}y}}{{{\text{d}}x}} = 4x - 5\)</span><span> <em><strong>(A1)(A1) </strong> <strong>(C2)</strong></em></span></p>
<p><span><strong>Notes:</strong> Award <em><strong>(A1)</strong></em> for each correct term. Award <em><strong>(A1)(A0)</strong></em> if any other terms are given.<br></span></p>
<p><span><em><strong>[2 marks]</strong></em></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span>\(y = - 3x - \frac{1}{2}\) <em><strong>(M1)</strong></em></span></p>
<p><span><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for rearrangement of equation</span></p>
<p><br><span>gradient of line is –3 <em><strong>(A1)</strong></em></span></p>
<p><span>\(4x - 5 = -3\) <em><strong>(M1)</strong></em></span></p>
<p><span><strong>Notes:</strong> Award <em><strong>(M1)</strong></em> for equating their gradient to their derivative from part (a).</span> <span>If \(4x - 5 = -3\) is seen with no working award <em><strong>(M1)(A1)(M1)</strong></em>.</span></p>
<p><span> </span></p>
<p><span>\(x = \frac{1}{2}\) </span><span> <em><strong>(A1)</strong></em><strong>(ft)</strong> <em><strong>(C4)</strong></em></span></p>
<p><span><strong>Note:</strong> Follow through from their part (a). If answer is given as (0.5, 2) with no</span> <span>working award the final <em><strong>(A1)</strong></em> only.</span></p>
<p><span><em><strong>[4 marks]</strong></em></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;">
<p><span style="font-size: medium; font-family: times new roman,times;">The derivative of the function was correctly found by most candidates. Rearranging the equation of the line to find the gradient was also successfully performed. Most candidates could not find the <em>x</em>-coordinate of the point on the curve whose tangent was parallel to a given line. To most candidates, part (b) appeared to be disconnected to part (a).</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-size: medium; font-family: times new roman,times;">The derivative of the function was correctly found by most candidates. Rearranging the equation of the line to find the gradient was also successfully performed. Most candidates could not find the <em>x</em>-coordinate of the point on the curve whose tangent was parallel to a given line. To most candidates, part (b) appeared to be disconnected to part (a).</span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-size: medium; font-family: times new roman,times;">Consider the function \(f (x) = ax^3 − 3x + 5\), where \(a \ne 0\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>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>Write down the value of \(f ′(0)\).</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>The function has a local maximum at <em>x</em> = −2.</span></p>
<p><span>Calculate the value of <em>a</em>.</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>\( f '(x) = 3ax^2 - 3\) <em><strong>(A1)(A1)</strong></em> <em><strong>(C2)</strong></em></span></p>
<p><br><span><strong>Note:</strong> Award a maximum of <em><strong>(A1)(A0)</strong></em> if any extra terms are seen.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span>−3 <em><strong>(A1)</strong></em><strong>(ft)</strong> <em><strong>(C1)</strong></em></span></p>
<p><br><span><strong>Note:</strong> Follow through from their part (a).</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span>\(f '(x) = 0\) <em><strong>(M1)</strong></em></span></p>
<p><br><span><strong>Note:</strong> This may be implied from line below.</span></p>
<p><br><span>\(3a(-2)^2 - 3 = 0\) <em><strong>(M1)</strong></em></span></p>
<p><span>\((a =) \frac{1}{4}\) <em><strong>(A1)</strong></em><strong>(ft)</strong> <em><strong>(C3)</strong></em></span></p>
<p><br><span><strong>Note:</strong> Follow through from their part (a).</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;">Many candidates could find the derivative of the cubic function and find the value of the derivative at \(x = 0\). For part (c) many candidates calculated the value of the function rather than the derivative at \(x = - 2\).</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 could find the derivative of the cubic function and find the value of the derivative at \(x = 0\).<br></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 could find the derivative of the cubic function and find the value of the derivative at \(x = 0\). For part (c) many candidates calculated the value of the function rather than the derivative at \(x = - 2\). However only the best realized that the derivative is zero at the maximum and so calculated the value of \(a\).</span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-size: medium; font-family: times new roman,times;">Let \(f (x) = 2x^2 + x - 6\)</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>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>Find the value of \(f'( - 3)\).</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>Find the value of \(x\) for which \(f'(x) = 0\).</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>\(f'(x) = 4x + 1\) <em><strong>(A1)(A1)(A1) </strong> <strong>(C3)</strong></em></span></p>
<p><br><span><strong>Note:</strong> Award <em><strong>(A1)</strong></em> for each term differentiated correctly.</span></p>
<p><span>Award at most <em><strong>(A1)(A1)(A0)</strong></em> if any extra terms seen.</span></p>
<p><span> </span></p>
<p><em><strong><span>[3 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span>\(f'( - 3) = - 11\) <em><strong>(A1)</strong></em><strong>(ft)</strong> <em><strong>(C1)</strong></em></span></p>
<p><span><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>\(4x + 1 = 0\) <em><strong>(M1)</strong></em></span></p>
<p><span>\(x = - \frac{{1}}{{4}}\)</span><span> <em><strong>(A1)</strong></em><strong>(ft)</strong> <em><strong>(C2)</strong></em></span></p>
<p><span><em><strong>[2 marks]</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 style="margin: 0.0px 0.0px 10.0px 0.0px; font: 16.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">This was a fairly standard question. However, some candidates found <em>f</em> (−3) instead of \(f'\)(−3). Quite a few candidates were unable to answer part (c) as they tried to find \(f'\)(0) instead of finding <em>x</em> when \(f'\)(<em>x</em>) = 0.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 10.0px 0.0px; font: 16.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">This was a fairly standard question. However, some candidates found <em>f</em> (−3) instead of \(f'\)(−3). Quite a few candidates were unable to answer part (c) as they tried to find \(f'\)(0) instead of finding <em>x</em> when \(f'\)(<em>x</em>) = 0.</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin: 0.0px 0.0px 10.0px 0.0px; font: 16.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">This was a fairly standard question. However, some candidates found <em>f</em> (−3) instead of \(f'\)(−3). Quite a few candidates were unable to answer part (c) as they tried to find \(f'\)(0) instead of finding <em>x</em> when \(f'\)(<em>x</em>) = 0.</span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>The coordinates of point A are \((6,{\text{ }} - 7)\) and the coordinates of point B are \(( - 6,{\text{ }}2)\). Point M is the midpoint of AB.</p>
</div>
<div class="specification">
<p>\({L_1}\) is the line through A and B.</p>
</div>
<div class="specification">
<p>The line \({L_2}\) is perpendicular to \({L_1}\) and passes through M.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the coordinates of M.</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 gradient of \({L_1}\).</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>Write down the gradient of \({L_2}\).</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down, in the form \(y = mx + c\), the equation of \({L_2}\).</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>\((0,{\text{ }}2.5)\)\(\,\,\,\)<strong>OR</strong>\(\,\,\,\)\(\left( {0,{\text{ }} - \frac{5}{2}} \right)\) <strong><em>(A1)(A1) (C2)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>(A1) </em></strong>for 0 and <strong><em>(A1) </em></strong>for –2.5 written as a coordinate pair. Award at most <strong><em>(A1)(A0) </em></strong>if brackets are missing. Accept “\(x = 0\) and \(y = - 2.5\)”.</p>
<p> </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>\(\frac{{2 - ( - 7)}}{{ - 6 - 6}}\) <strong><em>(M1)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>(M1) </em></strong>for correct substitution into gradient formula.</p>
<p> </p>
<p>\( = - \frac{3}{4}{\text{ }}( - 0.75)\) <strong><em>(A1) (C2)</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\(\frac{4}{3}{\text{ }}(1.33333 \ldots )\) <strong><em>(A1)</em>(ft) <em>(C1)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>(A0) </em></strong>for \(\frac{1}{{0.75}}\). Follow through from part (b).</p>
<p> </p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\(y = \frac{4}{3}x - \frac{5}{2}{\text{ }}(y = 1.33 \ldots x - 2.5)\) <strong><em>(A1)</em>(ft) <em>(C1)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Follow through from parts (c)(i) and (a). Award <strong><em>(A0) </em></strong>if final answer is not written in the form \(y = mx + c\).</p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">c.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>The diagram shows part of the graph of a function \(y = f(x)\). The graph passes through point \({\text{A}}(1,{\text{ }}3)\).</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-16_om_06.22.37.png" alt="M17/5/MATSD/SP1/ENG/TZ2/13"></p>
</div>
<div class="specification">
<p>The tangent to the graph of \(y = f(x)\) at A has equation \(y = - 2x + 5\). Let \(N\) be the normal to the graph of \(y = f(x)\) at A.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the value of \(f(1)\).</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the equation of \(N\). Give your answer in the form \(ax + by + d = 0\) where \(a\), \(b\), \(d \in \mathbb{Z}\).</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>Draw the line \(N\) on the diagram above.</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>3 <strong><em>(A1)</em></strong> <strong><em>(C1)</em></strong></p>
<p> </p>
<p><strong>Notes:</strong> Accept \(y = 3\)</p>
<p> </p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\(3 = 0.5(1) + c\)\(\,\,\,\)<strong>OR</strong>\(\,\,\,\)\(y - 3 = 0.5(x - 1)\) <strong><em>(A1)(A1)</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>(A1) </em></strong>for correct gradient, <strong><em>(A1) </em></strong>for correct substitution of \({\text{A}}(1,{\text{ }}3)\) in the equation of line.</p>
<p> </p>
<p>\(x - 2y + 5 = 0\) or any integer multiple <strong><em>(A1)</em>(ft)</strong> <strong><em>(C3)</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>(A1)</em>(ft) </strong>for their equation correctly rearranged in the indicated form.</p>
<p>The candidate’s answer <strong>must </strong>be an equation for this mark.</p>
<p> </p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img src="images/Schermafbeelding_2017-08-16_om_08.26.38.png" alt="M17/5/MATSD/SP1/ENG/TZ2/13.c/M"> <strong><em>(M1)(A1)</em>(ft)</strong> <strong><em>(C2)</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>M1) </em></strong>for a straight line, with positive gradient, passing through \((1,{\text{ }}3)\), <strong><em>(A1)</em>(ft) </strong>for line (or extension of their line) passing approximately through 2.5 or their intercept with the \(y\)-axis.</p>
<p> </p>
<p><strong><em>[2 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><span style="font-family: times new roman,times; font-size: medium;">Consider the graph of the function \(y = f(x)\) defined below.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><img src="data:image/png;base64,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" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Write down <strong>all</strong> the labelled points on the curve</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>that are local maximum points;</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>where the function attains its least value;</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>where the function attains its greatest value;</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>where the gradient of the tangent to the curve is positive;</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>where \(f(x) > 0\) and \(f'(x) < 0\) .</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>B, F <em><strong>(C1)</strong></em></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span><span>H </span><span> <em><strong>(C1)</strong></em></span></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span><span>F </span><span> <em><strong>(C1)</strong></em></span></span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span><span>A, E </span><span> <em><strong>(C1)</strong></em></span></span></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span><span>C </span><span> <em><strong>(C2)</strong></em></span></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;">Consider the function \(f(x) = \frac{1}{2}{x^3} - 2{x^2} + 3\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>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>Find \(f''(x)\).</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>Find the equation of the tangent to the curve of \(f\) at the point \((1{\text{, }}1.5)\).</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>\(\frac{{3{x^2}}}{2} - 4x\) <em><strong>(A1)(A1) (C2)</strong></em></span></p>
<p><span><strong>Note:</strong> Award <em><strong>(A1)</strong></em> for each correct term and no extra terms; award <em><strong>(A1)(A0)</strong></em> for both terms correct and extra terms; <em><strong>(A0)</strong></em> otherwise.</span></p>
<p><span><em><strong>[2 marks]</strong></em><br></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span>\(3x - 4\) <em><strong>(A1)</strong></em><strong>(ft)<em>(A1)</em>(ft) <em>(C2)</em></strong></span></p>
<p><span><strong>Note: </strong>accept \(3{x^1} - {4^0}\)</span></p>
<p><span><em><strong>[2 marks]</strong></em><br></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span>\(y = - 2.5x + 4\) or equivalent <em><strong>(A1)</strong></em><strong>(ft)<em>(A1) (C2)<br><br></em></strong></span></p>
<p><span><strong>Note: </strong>Award <strong><em>(A1)</em>(ft)</strong> on their (a) for \( - 2.5x\) (must have \(x\)), <em><strong>(A1)</strong></em> for \(4\) or equivalent correct answer only.</span><br><span>Accept \(y - 1.5 = - 2.5(x - 1)\)</span></p>
<p><span><em><strong>[2 marks]</strong></em><br></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;">The final part of this question was not well answered. Most candidates could gain 4 marks in this question as most knew how to differentiate and they were required to do it twice. However, few realized that they could find the gradient of the tangent from their answer to part (a). This part was badly answered by most candidates.</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 final part of this question was not well answered. Most candidates could gain 4 marks in this question as most knew how to differentiate and they were required to do it twice. However, few realized that they could find the gradient of the tangent from their answer to part (a). This part was badly answered by most candidates.</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;">The final part of this question was not well answered. Most candidates could gain 4 marks in this question as most knew how to differentiate and they were required to do it twice. However, few realized that they could find the gradient of the tangent from their answer to part (a). This part was badly answered by most candidates.</span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">A cuboid has a rectangular base of width \(x\)<span class="s1"><em> </em>cm </span>and length <span class="s1">2\(x\) cm </span>. The height of the cuboid is \(h\) <span class="s1">cm </span>. The total length of the edges of the cuboid is \(72\)<span class="s1"> cm</span>.</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2015-12-20_om_08.27.58.png" alt></p>
<p class="p1">The volume, \(V\), of the cuboid can be expressed as \(V = a{x^2} - 6{x^3}\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the value of \(a\).</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 \(x\) that makes the volume a maximum.</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">\(72 = 12x + 4h\;\;\;\)(or equivalent) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><strong>Note: </strong>Award <strong><em>(M1) </em></strong>for a correct equation obtained from the total length of the edges.</p>
<p class="p2"> </p>
<p class="p1">\(V = 2{x^2}(18 - 3x)\) <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1">\((a = ){\text{ }}36\) <span class="Apple-converted-space"> </span><strong><em>(A1) <span class="Apple-converted-space"> </span>(C3)</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">\(\frac{{{\text{d}}V}}{{{\text{d}}x}} = 72x - 18{x^2}\) <span class="Apple-converted-space"> </span><strong><em>(A1)</em></strong></p>
<p class="p1">\(72x - 18{x^2} = 0\;\;\;\)<strong>OR</strong>\(\;\;\;\frac{{{\text{d}}V}}{{{\text{d}}x}} = 0\) <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1"><strong>Notes:<span class="Apple-converted-space"> </span></strong>Award <strong><em>(A1) </em></strong>for<span class="Apple-converted-space"> </span>\( - 18{x^2}\)<span class="Apple-converted-space"> </span>seen. Award <strong><em>(M1) </em></strong>for equating derivative to zero.</p>
<p class="p2"> </p>
<p class="p1">\((x = ){\text{ 4}}\) <span class="Apple-converted-space"> </span><strong><em>(A1)</em>(ft) <span class="Apple-converted-space"> </span><em>(C3)</em></strong></p>
<p class="p1"><strong>Note: </strong>Follow through from part (a).</p>
<p class="p2"> </p>
<p class="p1"><strong>OR</strong></p>
<p class="p1">Sketch of \(V\) with visible maximum <span class="Apple-converted-space"> </span><strong><em>(M1)</em></strong></p>
<p class="p1">Sketch with \(x \geqslant 0,{\text{ }}V \geqslant 0\) and indication of maximum (e.g. coordinates) <span class="Apple-converted-space"> </span><strong><em>(A1)</em>(ft)</strong></p>
<p class="p1">\((x = ){\text{ 4}}\) <span class="Apple-converted-space"> </span><strong><em>(A1)</em>(ft) <span class="Apple-converted-space"> </span><em>(C3)</em></strong></p>
<p class="p2"> </p>
<p class="p1"><strong>Notes: </strong>Follow through from part (a).</p>
<p class="p1">Award <strong><em>(M1)(A1)(A0) </em></strong>for \((4,{\text{ }}192)\).</p>
<p class="p1">Award <strong><em>(C3) </em></strong>for \(x = 4,{\text{ }}y = 192\).</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>The model in this question seemed to be too difficult for the vast majority of the candidates, and therefore was a strong discriminator between grade 6 and grade 7 candidates. An attempt to find an equation for the volume of the cube often started with <em>V</em> = <em>x</em> x 2<em>x</em> x <em>h</em> . Many struggled to translate the total length of the edges into a correct equation, and consequently were unable to substitute <em>h</em>. Some tried to write <em>x</em> in terms of <em>h</em> and got lost, others tried to work backwards from the expression given in the question.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>As very few found a value for <em>a</em>, often part (b) was not attempted. When a derivative was calculated this was usually done correctly.</p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The point A has coordinates (4 , −8) and the point B has coordinates (−2 , 4).</p>
</div>
<div class="specification">
<p>The point D has coordinates (−3 , 1).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the coordinates of C, the midpoint of line segment AB.</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 gradient of the line DC.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the equation of the line DC. Write your answer in the form <em>ax</em> + <em>by</em> + <em>d</em> = 0 where <em>a</em> , <em>b</em> and <em>d</em> are integers.</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>(1, −2) <em><strong>(A1)(A1) (C2)</strong></em><br><strong>Note:</strong> Award <em><strong>(A1)</strong></em> for 1 and <em><strong>(A1)</strong></em> for −2, seen as a coordinate pair.</p>
<p>Accept <em>x</em> = 1, <em>y</em> = −2. Award <em><strong>(A1)(A0)</strong></em> if <em>x</em> and <em>y</em> coordinates are reversed.</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>\(\frac{{1 - \left( { - 2} \right)}}{{ - 3 - 1}}\) <em><strong>(M1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for correct substitution, of their part (a), into gradient formula.</p>
<p>\( = - \frac{3}{4}\,\,\,\left( { - 0.75} \right)\) <em><strong>(A1)</strong></em><strong>(ft)</strong><em><strong> (C2)</strong></em></p>
<p><strong>Note:</strong> Follow through from part (a).</p>
<p><em><strong>[2 marks]</strong></em></p>
<p> </p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\(y - 1 = - \frac{3}{4}\left( {x + 3} \right)\) <em><strong>OR </strong></em> \(y + 2 = - \frac{3}{4}\left( {x - 1} \right)\) <em><strong>OR</strong></em> \(y = - \frac{3}{4}x - \frac{5}{4}\) <em><strong>(M1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for correct substitution of their part (b) and a given point.</p>
<p><em><strong>OR</strong></em></p>
<p>\(1 = - \frac{3}{4} \times - 3 + c\) <em><strong>OR</strong></em> \( - 2 = - \frac{3}{4} \times 1 + c\) <em><strong>(M1) </strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for correct substitution of their part (b) and a given point.</p>
<p>\(3x + 4y + 5 = 0\) (accept any integer multiple, including negative multiples) <em><strong>(A1)</strong></em><strong>(ft)</strong><em><strong> (C2)</strong></em></p>
<p><strong>Note:</strong> Follow through from parts (a) and (b). Where the gradient in part (b) is found to be \(\frac{5}{0}\), award at most <em><strong>(M1)(A0)</strong></em> for either \(x = - 3\) or \(x + 3 = 0\).</p>
<p><em><strong>[2 marks]</strong></em></p>
<p> </p>
<p> </p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">A sketch of the function \(f(x) = 5{x^3} - 3{x^5} + 1\) is shown for \( - 1.5 \leqslant x \leqslant 1.5\) and \( - 6 \leqslant y \leqslant 6\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><img src="data:image/png;base64,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" alt></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Write down \(f'(x)\) .</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>Find the equation of the tangent to the graph of \(y = f(x)\) at \((1{\text{, }}3)\) .</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>Write down the coordinates of the second point where this tangent intersects the graph of \(y = f(x)\) .</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>\(f'(x) = 15{x^2} - 15{x^4}\) <em><strong>(A1)(A1) (C2)</strong></em></span></p>
<p> </p>
<p><span><strong>Note:</strong> Award a maximum of <em><strong>(A1)(A0)</strong></em> if extra terms seen.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span>\(f'(1) = 0\) <em><strong>(M1)</strong></em></span></p>
<p> </p>
<p><span><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for \(f'(x) = 0\) .</span></p>
<p> </p>
<p><span>\(y = 3\) <em><strong>(A1)</strong></em><strong>(ft)</strong><em><strong> (C2)</strong></em></span></p>
<p> </p>
<p><span><strong>Note:</strong> Follow through from their answer to part (a).</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span>\(( - 1.38{\text{, }}3)\) \(( - 1.38481 \ldots {\text{, }}3)\) <em><strong>(A1)</strong></em><strong>(ft)</strong><em><strong>(A1)</strong></em><strong>(ft)</strong><em><strong> (C2)</strong></em></span></p>
<p> </p>
<p><span><strong>Note:</strong> Follow through from their answer to parts (a) and (b).</span></p>
<p><span><strong>Note:</strong> Accept \(x = - 1.38\), \(y = 3\) (\(x = - 1.38481 \ldots\) , \(y = 3\)) .</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>A quadratic function \(f\) is given by \(f(x) = a{x^2} + bx + c\). The points \((0,{\text{ }}5)\) and \(( - 4,{\text{ }}5)\) lie on the graph of \(y = f(x)\).</p>
</div>
<div class="specification">
<p>The \(y\)-coordinate of the minimum of the graph is 3.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the equation of the axis of symmetry of the graph of \(y = f(x)\).</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 value of \(c\).</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of \(a\) and of \(b\).</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>\(x = - 2\) <strong><em>(A1)(A1) (C2)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>(A1) </em></strong>for \(x = \) (a constant) and <strong><em>(A1) </em></strong>for \( - 2\).</p>
<p> </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>\((c = ){\text{ }}5\) <strong><em>(A1) (C1)</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>\( - \frac{b}{{2a}} = - 2\)</p>
<p>\(a{( - 2)^2} - 2b + 5 = 3\) or equivalent</p>
<p>\(a{( - 4)^2} - 4b + 5 = 5\) or equivalent</p>
<p>\(2a( - 2) + b = 0\) or equivalent <strong><em>(M1)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>(M1) </em></strong>for two of the above equations.</p>
<p> </p>
<p>\(a = 0.5\) <strong><em>(A1)</em>(ft)</strong></p>
<p>\(b = 2\) <strong><em>(A1)</em>(ft) <em>(C3)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award at most <strong><em>(M1)(A1)</em>(ft)<em>(A0) </em></strong>if the answers are reversed.</p>
<p>Follow through from parts (a) and (b).</p>
<p> </p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</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">Consider the curve \(y = {x^2} + \frac{a}{x} - 1,{\text{ }}x \ne 0\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find \(\frac{{{\text{d}}y}}{{{\text{d}}x}}\).</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">The gradient of the tangent to the curve is \( - 14\) when \(x = 1\).</p>
<p class="p1">Find the value of \(a\).</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">\(2x - \frac{a}{{{x^2}}}\) <span class="Apple-converted-space"> </span><strong><em>(A1)(A1)(A1) (C3)</em></strong></p>
<p class="p1"><strong>Notes: </strong>Award <strong><em>(A1) </em></strong>for \(2x\), <strong><em>(A1) </em></strong>for \( - a\) and <strong><em>(A1) </em></strong>for \({x^{ - 2}}\).</p>
<p class="p1">Award at most <strong><em>(A1)(A1)(A0) </em></strong>if extra terms are present.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="p1">\(2(1) - \frac{a}{{{1^2}}} = - 14\) <strong><em>(M1)(M1)</em></strong></p>
<p class="p1"><strong>Note: </strong>Award <strong><em>(M1) </em></strong>for substituting \(1\) into their gradient function, <strong><em>(M1) </em></strong>for equating their <strong>gradient </strong>function to \( - 14\).</p>
<p class="p1">Award <strong><em>(M0)(M0)(A0) </em></strong>if the original function is used instead of the gradient function.</p>
<p class="p2"> </p>
<p class="p1">\(a = 16\) <span class="Apple-converted-space"> </span><strong><em>(A1)</em>(ft) <span class="Apple-converted-space"> </span><em>(C3)</em></strong></p>
<p class="p1"><strong>Note: </strong>Follow through from their gradient function from part (a).</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" style="padding-left: 20px; padding-right: 20px;">
<p>The equation of line \({L_1}\) is \(y = 2.5x + k\). Point \({\text{A}}\) \(\,(3,\, - 2)\) lies on \({L_1}\).</p>
<p>Find the value of \(k\).</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 line \({L_2}\) is perpendicular to \({L_1}\) and intersects \({L_1}\) at point \({\text{A}}\).</p>
<p>Write down the gradient of \({L_2}\).</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>Find the equation of \({L_2}\). Give your answer in the form \(y = mx + c\) .</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>Write your answer to part (c) in the form \(ax + by + d = 0\) where \(a\), \(b\) and \(d \in \mathbb{Z}\).</p>
<div class="marks">[1]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>\( - 2 = 2.5\, \times 3 + k\) <em><strong>(M1)</strong></em></p>
<p><strong>Note:</strong> Award <strong><em>(M1)</em></strong> for correct substitution of \((3,\, - 2)\) into equation of \({L_1}\).</p>
<p>\((k = ) - 9.5\) <em><strong>(A1) (C2)</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\( - 0.4\,\left( { - \frac{2}{5}} \right)\) <em><strong>(A1) (C1)</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\(y - ( - 2) = - 0.4\,(x - 3)\) <em><strong>(M1)</strong></em></p>
<p><strong>OR</strong></p>
<p>\( - 2 = - 0.4\,(3) + c\) <em><strong>(M1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for their gradient and given point substituted into equation of a straight line. Follow through from part (b).</p>
<p>\(y = - 0.4x - 0.8\) \(\left( {y = - \frac{2}{5}x - \frac{4}{5}} \right)\) <em><strong>(A1)</strong></em><strong>(ft)</strong><em><strong> (C2)</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\(2x + 5y + 4 = 0\) (or any integer multiple) <strong><em>(A1)</em>(ft) <em>(C1)</em></strong></p>
<p><strong>Note:</strong> Follow through from part (c).</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>Question 7: Perpendicular Line<br>The response to this question was mixed.<br>Part (a) was well attempted by the majority.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>In part (b), the gradient was not fully calculated (being left as a reciprocal) by a large number of candidates.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>In part (c), the common error was the use of c from part (a) in the line.</p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>In part (d), the notation for integer was not understood by a large number of candidates.</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 small manufacturing company makes and sells \(x\) machines each month. The monthly cost \(C\) , in dollars, of making \(x\) machines is given by</span><br><span style="font-family: times new roman,times; font-size: medium;">\[C(x) = 2600 + 0.4{x^2}{\text{.}}\]</span><span style="font-family: times new roman,times; font-size: medium;">The monthly income \(I\) , in dollars, obtained by selling \(x\) machines is given by</span><br><span style="font-family: times new roman,times; font-size: medium;">\[I(x) = 150x - 0.6{x^2}{\text{.}}\]</span><span style="font-family: times new roman,times; font-size: medium;">\(P(x)\) is the monthly profit obtained by selling \(x\) machines.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Find \(P(x)\) .</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>Find the number of machines that should be made and sold each month to maximize \(P(x)\) .</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><span>Use your answer to part (b) to find the selling price of</span> <span><strong>each machine</strong> in order to maximize \(P(x)\) .</span></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>\(P(x) = I(x) - C(x)\) <em><strong>(M1)</strong></em></span><br><span>\( = - {x^2} + 150x - 2600\) <em><strong>(A1)</strong></em> <em><strong>(C2)</strong></em></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span>\( - 2x + 150 = 0\) <em><strong>(M1)</strong></em></span></p>
<p> </p>
<p><span><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for setting \(P'(x) = 0\) .</span></p>
<p> </p>
<p><span><strong>OR</strong></span></p>
<p> </p>
<p><span>Award <em><strong>(M1)</strong></em> for sketch of \(P(x)\) and maximum point identified. <em><strong>(M1)</strong></em></span><br><span>\(x = 75\) <em><strong>(A1)</strong></em><strong>(ft)</strong><em><strong> (C2)</strong></em></span></p>
<p> </p>
<p><span><strong>Note:</strong> Follow through from their answer to part (a).</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span>\(\frac{{7875}}{{75}}\) <em><strong>(M1)</strong></em></span></p>
<p> </p>
<p><span><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for \(7875\) seen.</span></p>
<p> </p>
<p><span>\( = 105\) <em><strong>(A1)</strong></em><strong>(ft)</strong><em><strong> (C2)</strong></em></span></p>
<p> </p>
<p><span><strong>Note:</strong> Follow through from their answer to part (b).</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>The equation of line \({L_1}\) is \(y = - \frac{2}{3}x - 2\).</p>
</div>
<div class="specification">
<p>Point P lies on \({L_1}\) and has \(x\)-coordinate \( - 6\).</p>
</div>
<div class="specification">
<p>The line \({L_2}\) is perpendicular to \({L_1}\) and intersects \({L_1}\) when \(x = - 6\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the gradient of \({L_1}\).</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the \(y\)-coordinate 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>Determine the equation of \({L_2}\). Give your answer in the form \(ax + by + d = 0\), where \(a\), \(b\) and \(d\) are integers.</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>\( - \frac{2}{3}\) <strong><em>(A1)</em></strong> <strong><em>(C1)</em></strong></p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\(y = - \frac{2}{3}( - 6) - 2\) <strong><em>(M1)</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>(M1) </em></strong>for correctly substituting \( - 6\) into the formula for \({L_1}\).</p>
<p> </p>
<p>\((y = ){\text{ }}2\) <strong><em>(A1)</em></strong> <strong><em>(C2)</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>(A0)(A1) </em></strong>for \(( - 6,{\text{ }}2)\) with or without working.</p>
<p> </p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>gradient of \({L_2}\) is \(\frac{3}{2}\) <strong><em>(A1)</em>(ft)</strong></p>
<p> </p>
<p><strong>Note: </strong>Follow through from part (a).</p>
<p> </p>
<p>\(2 = \frac{3}{2}( - 6) + c\)\(\,\,\,\)<strong>OR</strong>\(\,\,\,\)\(y - 2 = \frac{3}{2}\left( {x - ( - 6)} \right)\) <strong><em>(M1)</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>(M1) </em></strong>for substituting their part (b), their gradient and \( - 6\) into equation of a straight line.</p>
<p> </p>
<p>\(3x - 2y + 22 = 0\) <strong><em>(A1)</em>(ft)</strong> <strong><em>(C3)</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Follow through from parts (a) and (b). Accept any integer multiple.</p>
<p>Award <strong><em>(A1)(M1)(A0)</em></strong> for \(y = \frac{3}{2}x + 11\).</p>
<p> </p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</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" style="padding-left: 20px; padding-right: 20px;">
<p><span>Expand the expression \(x(2{x^3} - 1)\).</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><span>Differentiate \(f(x) = x(2{x^3} - 1)\).</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>Find the \(x\)-coordinate of the local minimum of the curve \(y = f(x)\).</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>\(2{x^4} - x\) <strong><em>(A1)(A1) (C2)</em></strong></span></p>
<p> </p>
<p><span><strong>Note: </strong>Award <strong><em>(A1) </em></strong>for \(2{x^4}\), <em><strong>(A1)</strong></em> for \( - x\).</span></p>
<p> </p>
<p><span><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span>\(8{x^3} - 1\) <strong><em>(A1)</em>(ft)<em>(A1)</em>(ft) <em>(C2)</em></strong></span></p>
<p> </p>
<p><span><strong>Note: </strong>Award <strong><em>(A1)</em>(ft) </strong>for \(8{x^3}\), <strong><em>(A1)</em>(ft) </strong>for \(–1\). Follow through from their part (a).</span></p>
<p><span> Award at most <strong><em>(A1)(A0) </em></strong>if extra terms are seen.</span></p>
<p> </p>
<p><span><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span>\(8{x^3} - 1 = 0\) <strong><em>(M1)</em></strong></span></p>
<p> </p>
<p><span><strong>Note: </strong>Award <strong><em>(M1) </em></strong>for equating their part (b) to zero.</span></p>
<p> </p>
<p><span>\((x = )\frac{1}{2}{\text{ (0.5)}}\) <strong><em>(A1)</em>(ft) <em>(C2)</em></strong></span></p>
<p> </p>
<p><span><strong>Notes: </strong>Follow through from part (b).</span></p>
<p><span> \(0.499\) is the answer from the use of trace on the GDC; award <strong><em>(A0)(A0)</em></strong>.</span></p>
<p><span> For an answer of \((0.5, –0.375)\), award <strong><em>(M1)(A0)</em></strong>.</span></p>
<p> </p>
<p><span><strong><em>[2 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;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">A surprising number of candidates were unable to correctly expand the expression given in part (a). Most candidates were able to differentiate their function but a considerable number were unable to find the x-coordinate of the minimum point. Candidates must read the questions correctly as answers giving ordered pairs were not awarded the final mark. A number of candidates did not use calculus to determine the local minimum but graphed the function, often achieving full marks for part (c), even when parts (b) or (a) were incorrect or left blank.</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: 19.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">A surprising number of candidates were unable to correctly expand the expression given in part (a). Most candidates were able to differentiate their function but a considerable number were unable to find the x-coordinate of the minimum point. Candidates must read the questions correctly as answers giving ordered pairs were not awarded the final mark. A number of candidates did not use calculus to determine the local minimum but graphed the function, often achieving full marks for part (c), even when parts (b) or (a) were incorrect or left blank.</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: 19.0px Arial;"><span style="font-family: 'times new roman', times; font-size: medium;">A surprising number of candidates were unable to correctly expand the expression given in part (a). Most candidates were able to differentiate their function but a considerable number were unable to find the x-coordinate of the minimum point. Candidates must read the questions correctly as answers giving ordered pairs were not awarded the final mark. A number of candidates did not use calculus to determine the local minimum but graphed the function, often achieving full marks for part (c), even when parts (b) or (a) were incorrect or left blank.</span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Consider the function \(f(x) = a{x^2} + c\).</p>
<p>Find \(f'(x)\)</p>
<p> </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>Point \({\text{A}}( - 2,\,5)\) lies on the graph of \(y = f(x)\) . The gradient of the tangent to this graph at \({\text{A}}\) is \( - 6\) .</p>
<p>Find the value of \(a\) .</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of \(c\) .</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>\(2ax\) <em><strong>(A1) (C1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(A1)</strong></em> for \(2ax\). Award <em><strong>(A0)</strong></em> if other terms are seen.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\(2a( - 2) = - 6\) <em><strong>(M1)(M1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for correct substitution of \(x = - 2\) in their gradient function, <em><strong>(M1)</strong> </em>for equating their gradient function to \( - 6\) . Follow through from part (a).</p>
<p>\((a = )1.5\,\,\,\left( {\frac{3}{2}} \right)\) <em><strong>(A1)</strong></em><strong>(ft)</strong><em><strong> (C3)</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\({\text{their }}1.5 \times {( - 2)^2} + c = 5\) <em><strong>(M1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for correct substitution of their \(a\) and point \({\text{A}}\). Follow through from part (b).</p>
<p>\((c = ) - 1\) <em><strong>(A1)</strong></em><strong>(ft)</strong><em><strong> (C2)</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>Question 11: Equation of tangent<br>Part (a) was generally well answered.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>In part (b), many candidates substituted the value of the function, rather than its gradient; this was usually correctly followed through into part (c).</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>In part (b), many candidates substituted the value of the function, rather than its gradient; this was usually correctly followed through into part (c).</p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Maria owns a cheese factory. The amount of cheese, in kilograms, Maria sells in one week, \(Q\), is given by</p>
<p style="text-align: center;">\(Q = 882 - 45p\),</p>
<p>where \(p\) is the price of a kilogram of cheese in euros (EUR).</p>
</div>
<div class="specification">
<p>Maria earns \((p - 6.80){\text{ EUR}}\) for each kilogram of cheese sold.</p>
</div>
<div class="specification">
<p>To calculate her weekly profit \(W\), in EUR, Maria multiplies the amount of cheese she sells by the amount she earns per kilogram.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down how many kilograms of cheese Maria sells in one week if the price of a kilogram of cheese is 8 EUR.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find how much Maria earns in one week, from selling cheese, if the price of a kilogram of cheese is 8 EUR.</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>Write down an expression for \(W\) in terms of \(p\).</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the price, \(p\), that will give Maria the highest weekly profit.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>522 (kg) <strong><em>(A1) (C1)</em></strong></p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\(522(8 - 6.80)\) or equivalent <strong><em>(M1)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>(M1) </em></strong>for multiplying their answer to part (a) by \((8 - 6.80)\).</p>
<p> </p>
<p>626 (EUR) (626.40) <strong><em>(A1)</em>(ft) <em>(C2)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Follow through from part (a).</p>
<p> </p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\((W = ){\text{ }}(882 - 45p)(p - 6.80)\) <strong><em>(A1)</em></strong></p>
<p><strong>OR</strong></p>
<p>\((W = ) - 45{p^2} + 1188p - 5997.6\) <strong><em>(A1) (C1)</em></strong></p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>sketch of \(W\) with some indication of the maximum <strong><em>(M1)</em></strong></p>
<p><strong>OR</strong></p>
<p>\( - 90p + 1188 = 0\) <strong><em>(M1)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>(M1) </em></strong>for equating the correct derivative of their part (c) to zero.</p>
<p> </p>
<p><strong>OR</strong></p>
<p>\((p = ){\text{ }}\frac{{ - 1188}}{{2 \times ( - 45)}}\) <strong><em>(M1)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>(M1) </em></strong>for correct substitution into the formula for axis of symmetry.</p>
<p> </p>
<p>\((p = ){\text{ }}13.2{\text{ (EUR)}}\) <strong><em>(A1)</em>(ft) <em>(C2)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Follow through from their part (c), if the value of \(p\) is such that \(6.80 < p < 19.6\).</p>
<p> </p>
<p><strong><em>[2 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 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 curve \(y = {x^3} + kx\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Write down \(\frac{{{\text{d}}y}}{{{\text{d}}x}}\).</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>The curve has a local minimum at the point where \(x = 2\).</span></p>
<p><span>Find the value of \(k\).</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>The curve has a local minimum at the point where \(x = 2\).</span></p>
<p><span>Find the value of \(y\) at this local minimum.</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>\(3{x^2} + k\) <strong><em>(A1) (C1)</em></strong></span></p>
<p><span><strong><em>[1 mark]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span>\(3{(2)^2} + k = 0\) <strong><em>(A1)</em>(ft)<em>(M1)</em></strong></span></p>
<p> </p>
<p><span><strong>Note:</strong> Award <strong><em>(A1)</em>(ft) </strong>for substituting 2 in their \(\frac{{{\text{d}}y}}{{{\text{d}}x}}\), <strong><em>(M1) </em></strong>for setting their \(\frac{{{\text{d}}y}}{{{\text{d}}x}} = 0\).</span></p>
<p> </p>
<p><span>\(k = - 12\) <strong><em>(A1)</em>(ft) <em>(C3)</em></strong></span></p>
<p> </p>
<p><span><strong>Note: </strong>Follow through from their derivative in part (a).</span></p>
<p> </p>
<p><span><strong><em>[3 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span>\({2^3} - 12 \times 2\) <strong><em>(M1)</em></strong></span></p>
<p> </p>
<p><span><strong>Note: </strong>Award <strong><em>(M1) </em></strong>for substituting 2 and their –12 into equation of the curve.</span></p>
<p> </p>
<p><span>\( = - 16\) <strong><em>(A1)</em>(ft) <em>(C12)</em></strong></span></p>
<p> </p>
<p><span><strong>Note: </strong>Follow through from their value of \(k\) found in part (b).</span></p>
<p> </p>
<p><span><strong><em>[2 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><span style="font-family: times new roman,times; font-size: medium;"><em>f</em> (<em>x</em>) = 5<em>x</em><sup>3</sup> − 4<em>x</em><sup>2</sup> + <em>x</em></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Find <em>f</em>'(<em>x</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>Find using your answer to part (a) the <em>x</em>-coordinate of</span></p>
<p><span>(i) the local maximum point;</span></p>
<p><span>(ii) the local minimum point.</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>15<em>x</em><sup>2</sup> – 8<em>x</em> + 1 <em><strong>(A1)(A1)(A1)</strong></em> <em><strong>(C3)</strong></em></span></p>
<p><span><strong>Note:</strong> Award <em><strong>(A1)</strong></em> for each correct term.</span></p>
<p><span><em><strong>[3 marks]</strong></em><br></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span>15<em>x</em><sup>2</sup> – 8<em>x</em> +1 = 0 <em><strong>(A1)</strong></em><strong>(ft)</strong></span></p>
<p><span><strong>Note:</strong> Award <em><strong>(A1)</strong></em><strong>(ft)</strong> for setting their derivative to zero.</span></p>
<p> </p>
<p><span>(i) \((x =)\frac{1}{5}(0.2)\) <em><strong>(A1)</strong></em><strong>(ft)</strong></span></p>
<p><span><strong> </strong></span></p>
<p><span>(ii) \((x =)\frac{1}{3}(0.333)\) <em><strong>(A1)</strong></em><strong>(ft)</strong> <em><strong>(C3)</strong></em></span></p>
<p><span><strong>Notes:</strong> Follow through from their answer to part (a).</span></p>
<p><br><span><em><strong>[3 marks]</strong></em></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;">
<p><span style="font-size: medium; font-family: times new roman,times;">Many candidates lost 1 mark in part (a) through not realizing that the derivative of<em> x</em> is 1. As a consequence, 15<em>x</em><sup>2</sup> – 8<em>x</em> proved to be a popular answer. <br></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-size: medium; font-family: times new roman,times;">Very few candidates gained the marks in part (b) to find the maximum and minimum point. Although the question indicated to use their answer to part (a), very few candidates set the derivative to zero which would have given them 1 mark. It seemed as if many candidates were trying to use their calculators to find the coordinates but could not find which was the maximum and which was the minimum.</span></p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The equation of the straight line \({L_1}\) is \(y = 2x - 3.\)</p>
<p>Write down the \(y\)-intercept of \({L_1}\) .</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>Write down the gradient of \({L_1}\) .</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>The line \({L_2}\) is parallel to \({L_1}\) and passes through the point \((0,\,\,3)\) .</p>
<p>Write down the equation of \({L_2}\) .</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The line \({L_3}\) is perpendicular to \({L_1}\) and passes through the point \(( - 2,\,\,6).\)</p>
<p>Write down the gradient of \({L_3}.\)</p>
<div class="marks">[1]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the equation of \({L_3}\) . Give your answer in the form \(ax + by + d = 0\) , where \(a\) , \(b\) and \(d\) are integers.</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>\((0,\,\, - 3)\) <em><strong>(A1) (C1)</strong></em></p>
<p>Note: Accept \( - 3\) or \(y = - 3.\)</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\(2\) <em><strong>(A1) (C1)</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\(y = 2x + 3\) <em><strong>(A1)</strong></em><strong>(ft)</strong><em><strong> (C1)</strong></em></p>
<p><strong>Note:</strong> Award <strong><em>(A1)</em>(ft)</strong> for correct equation. Follow through from part (b)<br>Award <em><strong>(A0)</strong></em> for \({L_2} = 2x + 3\).</p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\( - \frac{1}{2}\) <em><strong>(A1)</strong></em><strong>(ft)</strong><em><strong> (C1)</strong></em></p>
<p><strong>Note:</strong> Follow through from part (b).</p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\(6 = - \frac{1}{2}( - 2) + c\) <em><strong>(M1)</strong></em></p>
<p>\(c = 5\) (may be implied)</p>
<p><strong>OR</strong></p>
<p>\(y - 6 = - \frac{1}{2}(x + 2)\) <em><strong>(M1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for correct substitution of their gradient in part (d) and the point \(( - 2,\,\,6)\). Follow through from part (d).</p>
<p>\(x + 2y - 10 = 0\) (or any integer multiple) <em><strong>(A1)</strong></em><strong>(ft)</strong><em><strong> (C2)</strong></em></p>
<p><strong>Note:</strong> Follow through from (d). The answer must be in the form \(ax + by + d = 0\) for the <strong><em>(A1)</em>(ft)</strong> to be awarded. Accept any integer multiple.</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>Question 12: Linear function.</p>
<p>Many candidates demonstrated a good understanding of linear functions so successfully found the \(y\)-intercepts, gradient and equation in the form \(y = mx + c\). However only the very best were able to rewrite this in the form \(ax + by + d = 0\) where \(a\), \(b\) and \(d\) are integers.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Question 12: Linear function.</p>
<p>Many candidates demonstrated a good understanding of linear functions so successfully found the \(y\)-intercepts, gradient and equation in the form \(y = mx + c\). However only the very best were able to rewrite this in the form \(ax + by + d = 0\) where \(a\), \(b\) and \(d\) are integers.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin-top: 0cm;"><span style="font-size: 11.0pt; font-family: 'Arial','sans-serif'; color: #3f3f3f;">Question 12: Linear function.</span></p>
<p style="margin-top: 0cm; font-variant-ligatures: normal; font-variant-caps: normal; orphans: 2; text-align: start; widows: 2; -webkit-text-stroke-width: 0px; word-spacing: 0px;"><span style="font-size: 11.0pt; font-family: 'Arial','sans-serif'; color: #3f3f3f;">Many candidates demonstrated a good understanding of linear functions so successfully found the \(y\)-intercepts, gradient and equation in the form \(y = mx + c\). However only the very best were able to rewrite this in the form \(ax + by + d = 0\) where \(a\), \(b\) and \(d\) are integers.</span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin-top: 0cm;"><span style="font-size: 11.0pt; font-family: 'Arial','sans-serif'; color: #3f3f3f;">Question 12: Linear function.</span></p>
<p style="margin-top: 0cm;">Many candidates demonstrated a good understanding of linear functions so successfully found the \(y\)-intercepts, gradient and equation in the form \(y = mx + c\). However only the very best were able to rewrite this in the form \(ax + by + d = 0\) where \(a\), \(b\) and \(d\) are integers.</p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="margin-top: 0cm;"><span style="font-size: 11.0pt; font-family: 'Arial','sans-serif'; color: #3f3f3f;">Question 12: Linear function.</span></p>
<p style="margin-top: 0cm; font-variant-ligatures: normal; font-variant-caps: normal; orphans: 2; text-align: start; widows: 2; -webkit-text-stroke-width: 0px; word-spacing: 0px;"><span style="font-size: 11.0pt; font-family: 'Arial','sans-serif'; color: #3f3f3f;">Many candidates demonstrated a good understanding of linear functions so successfully found the \(y\)-intercepts, gradient and equation in the form \(y = mx + c\). However only the very best were able to rewrite this in the form \(ax + by + d = 0\) where \(a\), \(b\) and \(d\) are integers.</span></p>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>A company sells fruit juices in cylindrical cans, each of which has a volume of \(340\,{\text{c}}{{\text{m}}^3}\). The surface area of a can is \(A\,{\text{c}}{{\text{m}}^2}\) and is given by the formula</p>
<p>\(A = 2\pi {r^2} + \frac{{680}}{r}\) ,</p>
<p>where \(r\) is the radius of the can, in \({\text{cm}}\).</p>
<p>To reduce the cost of a can, its surface area must be minimized.</p>
<p>Find \(\frac{{{\text{d}}A}}{{{\text{d}}r}}\)</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate the value of \(r\) that minimizes the surface area of a can.</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>\(\left( {\frac{{{\text{d}}A}}{{{\text{d}}r}}} \right) = 4\pi r - \frac{{680}}{{{r^2}}}\) <em><strong>(A1)(A1)(A1) (C3)</strong></em></p>
<p><strong>Note: </strong>Award <em><strong>(A1)</strong></em> for \(4\pi r\) (accept \(12.6r\)), <em><strong>(A1)</strong></em> for \( - 680\), <em><strong>(A1)</strong></em> for \(\frac{1}{{{r^2}}}\) or \({r^{ - 2}}\)</p>
<p>Award at most <em><strong>(A1)(A1)(A0)</strong></em> if additional terms are seen.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\(4\pi r - \frac{{680}}{{{r^2}}} = 0\) <em><strong>(M1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for equating their \(\frac{{{\text{d}}A}}{{{\text{d}}r}}\) to zero.</p>
<p>\(4\pi {r^3} - 680 = 0\) <em><strong>(M1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for initial correct rearrangement of the equation. This may be assumed if \({r^3} = \frac{{680}}{{4\pi }}\) or \(r = \sqrt[3]{{\frac{{680}}{{4\pi }}}}\) seen.</p>
<p><strong>OR</strong></p>
<p>sketch of \(A\) with some indication of minimum point <em><strong>(M1)(M1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for sketch of \(A\), <em><strong>(M1)</strong></em> for indication of minimum point.</p>
<p><strong>OR</strong></p>
<p>sketch of \(\frac{{{\text{d}}A}}{{{\text{d}}r}}\) with some indication of zero <em><strong>(M1)(M1)</strong></em></p>
<p><strong>Note: </strong>Award <em><strong>(M1)</strong></em> for sketch of \(\frac{{{\text{d}}A}}{{{\text{d}}r}}\), <em><strong>(M1)</strong></em> for indication of zero.</p>
<p>\((r = )\,\,3.78\,({\text{cm}})\,\,\,\,\,(3.78239...)\) <em><strong>(A1)</strong></em><strong>(ft)</strong><em><strong> (C3)</strong> </em> </p>
<p><strong>Note:</strong> Follow through from part (a).</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>Question 15: Optimization<br>Many candidates were able to differentiate in part (a), but then were unable to relate this to part (b). However, it seemed that many more had not studied the calculus at all.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Question 15: Optimization<br>Many candidates were able to differentiate in part (a), but then were unable to relate this to part (b). However, it seemed that many more had not studied the calculus at all.</p>
<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;">Consider the function \(f(x) = 2{x^3} - 5{x^2} + 3x + 1\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>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>Write down the value of \(f'(2)\).</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>Find the equation of the tangent to the curve of \(y = f(x)\) at the point \((2{\text{, }}3)\).</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>\(f'(x) = 6{x^2} - 10x + 3\) <em><strong>(A1)(A1)(A1) (C3)<br></strong></em></span></p>
<p><span><strong>Notes: </strong>Award <em><strong>(A1)</strong></em> for each correct term and no extra terms.</span><br><span>Award <em><strong>(A1)(A1)(A0)</strong></em> if each term correct and extra term seen.</span><br><span>Award<em><strong>(A1)(A0)(A0)</strong></em> if two terms correct and extra term seen.</span><br><span>Award <em><strong>(A0)</strong></em> otherwise.</span></p>
<p><span><em><strong>[3 marks]</strong></em><br></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span>\(f'(2) = 7\) <em><strong>(A1)</strong></em><strong>(ft) <em>(C1)</em></strong></span></p>
<p><span><strong><em>[1 mark]<br></em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span>\(y = 7x - 11\) or equivalent <em><strong>(A1)</strong></em><strong>(ft)<em>(A1)</em>(ft) <em>(C2)</em></strong></span></p>
<p><span><strong>Note: </strong>Award <strong><em>(A1)</em>(ft)</strong> on their (b) for \(7x\) (must have \(x\)), <strong><em>(A1)</em>(ft)</strong> for \( - 11\). Accept \(y - 3 = 7(x - 2)\) .</span></p>
<p><span><em><strong>[2 marks]</strong></em><br></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;">Most candidates were able to score full marks for parts (a) and (b). When mistakes were made in part (a) follow-through marks could be awarded for part (b) provided working was shown. Part (c) was disappointing with many candidates not realizing that the answer in (b) was the gradient of the tangent line.</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 score full marks for parts (a) and (b). When mistakes were made in part (a) follow-through marks could be awarded for part (b) provided working was shown. Part (c) was disappointing with many candidates not realizing that the answer in (b) was the gradient of the tangent line.</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 score full marks for parts (a) and (b). When mistakes were made in part (a) follow-through marks could be awarded for part (b) provided working was shown. Part (c) was disappointing with many candidates not realizing that the answer in (b) was the gradient of the tangent line.</span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The equation of a curve is \(y = \frac{1}{2}{x^4} - \frac{3}{2}{x^2} + 7\).</p>
</div>
<div class="specification">
<p class="p1">The gradient of the tangent to the curve at a point <span class="s1">P </span>is \( - 10\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find \(\frac{{{\text{d}}y}}{{{\text{d}}x}}\).</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 coordinates of <span class="s1">P</span><span class="s2">.</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 class="p1"><span class="s1">\(2{x^3} - 3x\) <span class="Apple-converted-space"> </span></span><strong><em>(A1)(A1) <span class="Apple-converted-space"> </span>(C2)</em></strong></p>
<p class="p2"> </p>
<p class="p1"><strong>Note: <span class="Apple-converted-space"> </span></strong>Award <strong><em>(A1) </em></strong>for \(2{x^3}\), award <strong><em>(A1) </em></strong>for \( - 3x\).</p>
<p class="p1">Award at most <strong><em>(A1)(A0) </em></strong>if there are any extra terms.</p>
<p class="p2"> </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"><span class="Apple-converted-space">\(2{x^3} - 3x = - 10\) </span><strong><em>(M1)</em></strong></p>
<p class="p2"> </p>
<p class="p1"><strong>Note: <span class="Apple-converted-space"> </span></strong>Award <strong><em>(M1) </em></strong>for equating their answer to part (a) to \( - 10\).</p>
<p class="p2"> </p>
<p class="p1"><span class="Apple-converted-space">\(x = - 2\) </span><strong><em>(A1)</em>(ft)</strong></p>
<p class="p2"> </p>
<p class="p1"><strong>Note: <span class="Apple-converted-space"> </span></strong>Follow through from part (a). Award <strong><em>(M0)(A0) </em></strong>for \( - 2\) seen without working.</p>
<p class="p2"> </p>
<p class="p1"><span class="Apple-converted-space">\(y = \frac{1}{2}{( - 2)^4} - \frac{3}{2}{( - 2)^2} + 7\) </span><strong><em>(M1)</em></strong></p>
<p class="p2"> </p>
<p class="p1"><strong>Note: <span class="Apple-converted-space"> </span></strong>Award <strong><em>(M1) </em></strong>substituting their \( - 2\) into the original function.</p>
<p class="p2"> </p>
<p class="p1"><span class="Apple-converted-space">\(y = 9\) </span><strong><em>(A1)</em>(ft) <span class="Apple-converted-space"> </span><em>(C4)</em></strong></p>
<p class="p2"> </p>
<p class="p1"><strong>Note: <span class="Apple-converted-space"> </span></strong>Accept \(( - 2,{\text{ }}9)\).</p>
<p class="p2"> </p>
<p class="p3"><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The table given below describes the behaviour of <em>f</em> ′(<em>x</em>), the derivative function of <em>f</em> (<em>x</em>), in the domain −4 < <em>x</em> < 2.</span></p>
<p style="text-align: center;"><img src="data:image/png;base64,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" alt></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>State whether<em> f</em> (0) is greater than, less than or equal to <em>f</em> (−2). Give a reason for your answer.</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>The point P(−2, 3) lies on the graph of <em>f</em> (<em>x</em>).</span></p>
<p><span>Write down the equation of the tangent to the graph of <em>f</em> (<em>x</em>) at the point P.</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><span>The point P(−2, 3) lies on the graph of <em>f</em> (<em>x</em>).</span></span></p>
<p><span>From the information given about <em>f</em> ′(<em>x</em>), state whether the point (−2, 3) is a maximum, a minimum or neither. Give a reason for your answer.</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>greater than <em><strong> (A1)</strong></em></span></p>
<p><span>Gradient between <em>x</em> = </span><span><span>−</span>2 and <em>x</em> = 0 is positive. <em><strong> (R1)</strong></em></span></p>
<p><strong><span>OR</span></strong></p>
<p><span>The function is increased between these points or equivalent. <em><strong> (R1) (C2)</strong></em></span></p>
<p><span> </span></p>
<p><span><strong>Note:</strong> Accept a sketch. Do not award<em><strong> (A1)(R0)</strong></em>.</span></p>
<p><span> </span></p>
<p><em><strong><span>[2 marks]</span></strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span><em>y</em> = 3 <em><strong> (A1)(A1) (C2)</strong></em></span></p>
<p><span> </span></p>
<p><span><strong>Note:</strong> Award<em><strong> (A1)</strong> </em>for <em>y</em> = a constant,<em><strong> (A1)</strong></em> for 3.</span></p>
<p><span> </span></p>
<p><em><strong><span>[2 marks]</span></strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span>minimum <em><strong> (A1)</strong></em></span></p>
<p><span>Gradient is negative to the left and positive to the right or equivalent. <em><strong> (R1) (C2)</strong></em></span></p>
<p><span><strong> </strong></span></p>
<p><span><strong>Note:</strong> Accept a sketch. Do not award <em><strong>(A1)(R0)</strong></em>.</span></p>
<p><span> </span></p>
<p><em><strong><span>[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;">Very few candidates received full marks for this question and many omitted the question completely. A sketch showing the information provided in the table would have been very useful but few candidates chose this approach.</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;">Very few candidates received full marks for this question and many omitted the question completely. A sketch showing the information provided in the table would have been very useful but few candidates chose this approach.</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;">Very few candidates received full marks for this question and many omitted the question completely. A sketch showing the information provided in the table would have been very useful but few candidates chose this approach.</span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-size: medium; font-family: times new roman,times;">The straight line, <em>L</em>, has equation \(2y - 27x - 9 = 0\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Find the gradient of <em>L</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><span>Sarah wishes to draw the tangent to \(f (x) = x^4\) parallel to <em>L</em>.</span></p>
<p><span>Write down \(f ′(x)\).</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>Find the <em>x</em> coordinate of the point at which the tangent must be drawn.<br></span></p>
<div class="marks">[2]</div>
<div class="question_part_label">c, i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Write down the value of \(f (x)\) at this point.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">c, ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span><em>y</em> = 13.5<em>x</em> + 4.5 <em><strong>(M1)</strong></em></span></p>
<p><br><span><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for 13.5<em>x</em> seen.</span></p>
<p><br><span>gradient = 13.5 <em><strong>(A1)</strong></em> <em><strong>(C2)</strong></em></span></p>
<p><span><em><strong>[2 marks]</strong></em></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span>4<em>x</em><sup>3</sup> <em><strong>(A1)</strong></em> <em><strong>(C1)</strong></em></span></p>
<p><span><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>4<em>x</em><sup>3</sup> = 13.5 <em><strong>(M1)</strong></em></span></p>
<p><span><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for equating their answers to (a) and (b).</span></p>
<p><br><span><em>x</em> = 1.5 <em><strong>(A1)</strong></em><strong>(ft)</strong></span></p>
<p><em><span><strong>[2 marks]</strong></span></em></p>
<div class="question_part_label">c, i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span>\(\frac{{81}}{{16}}\) (5.0625, 5.06) <strong><em>(A1)</em>(ft)</strong> <strong><em>(C3)</em></strong></span></p>
<p><span><strong>Note:</strong> Award <strong><em>(A1)</em>(ft)</strong> for substitution of their (c)(i) into <em>x</em><span>4</span> with working seen.</span></p>
<p><span><strong><em>[1 mark]</em></strong></span></p>
<div class="question_part_label">c, ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="font-size: medium; font-family: times new roman,times;">The structure of this question was not well understood by the majority; the links between parts not being made. Again, this question was included to discriminate at the grade 6/7 level.</span></p>
<p><span style="font-size: medium; font-family: times new roman,times;">Most were successful in this part.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-size: medium; font-family: times new roman,times;">The structure of this question was not well understood by the majority; the links between parts not being made. Again, this question was included to discriminate at the grade 6/7 level.</span></p>
<p><span style="font-size: medium; font-family: times new roman,times;">This part was usually well attempted.</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-size: medium; font-family: times new roman,times;">The structure of this question was not well understood by the majority; the links between parts not being made. Again, this question was included to discriminate at the grade 6/7 level.</span></p>
<p><span style="font-size: medium; font-family: times new roman,times;">Only the best candidates succeeded in this part.</span></p>
<div class="question_part_label">c, i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-family: 'times new roman', times; font-size: medium;">The structure of this question was not well understood by the majority; the links between parts not being made. Again, this question was included to discriminate at the grade 6/7 level.</span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">Only the best candidates succeeded in this part.</span></p>
<div class="question_part_label">c, ii.</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 function \(f(x) = {x^3} + 2{x^2} - 5\).</span></p>
<div style="text-align: center;"><br><img src="images/Schermafbeelding_2014-09-02_om_15.08.15.png" alt></div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Label the local maximum as \({\text{A}}\) on the graph.</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>Label the local minimum as B on the graph.</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>Write down the interval where \(f'(x) < 0\).</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Draw the tangent to the curve at \(x = 1\) on the graph.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Write down the equation of the tangent at \(x = 1\).</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> <br><img src="images/up_and_up_1.jpg" alt></p>
<p><span>correct label on graph <strong><em>(A1) (C1)</em></strong></span></p>
<p><span><strong><em>[1 mark]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p> </p>
<p><span>correct label on graph <strong><em>(A1) (C1)</em></strong></span></p>
<p><span><strong><em>[1 mark]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span>\( - 1.33 < x < 0\) \(\left( { - \frac{4}{3} < x < 0} \right)\) <strong><em>(A1) (C1)</em></strong></span></p>
<p><span><strong><em>[1 mark]</em></strong></span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p> <br><img src="images/up_and_up.jpg" alt></p>
<p><span>tangent drawn at \(x = 1\) on graph <strong><em>(A1) (C1)</em></strong></span></p>
<p><span><strong><em>[1 mark]</em></strong></span></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span>\(y = 7x - 9\) <strong><em>(A1)(A1) (C2)</em></strong></span></p>
<p> </p>
<p><span><strong>Notes: </strong>Award <strong><em>(A1) </em></strong>for \(7\), <strong><em>(A1) </em></strong>for \(-9\).</span></p>
<p><span>If answer not given as an equation award at most <strong><em>(A1)(A0)</em></strong><em>.</em></span></p>
<p> </p>
<p><span><strong><em>[2 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-size: medium; font-family: times new roman,times;">A curve is described by the function \(f (x) = 3x - \frac{2}{{x^2}}\), \(x \ne 0\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>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>The gradient of the curve at point A is 35.</span></p>
<p><span>Find the <em>x</em>-coordinate of point A.</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>\(f'(x) = 3 + \frac{4}{{{x^3}}}\) <em><strong>(A1)(A1)(A1)</strong></em> <em><strong>(C3)</strong></em></span></p>
<p><br><span><strong>Notes:</strong> Award <em><strong>(A1)</strong></em> for 3, <em><strong>(A1)</strong></em> for + 4 and <em><strong>(A1)</strong></em> for \(\frac{1}{{{x^3}}}\)</span><span> </span><span>or \(x^{-3}\). Award at most</span> <span><em><strong>(A1)(A1)(A0)</strong></em> if additional terms are seen.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span>\(3 + \frac{4}{{{x^3}}} = 35\) <em><strong>(M1)</strong></em></span></p>
<p><span><br><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for equating their derivative to 35 only if the derivative is <strong>not</strong> a constant.<br></span></p>
<p><span><br>\({x^3} = \frac{1}{8}\) <em><strong>(A1)</strong></em><strong>(ft)</strong><br></span></p>
<p><span>\(\frac{1}{2}(0.5)\) <em><strong>(A1)</strong></em><strong>(ft)</strong> <em><strong>(C3)</strong></em><br></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-size: medium; font-family: times new roman,times;">\[f(x) = \frac{1}{3}{x^3} + 2{x^2} - 12x + 3\]</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>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>Find the interval of \(x\) for which \(f(x)\) is decreasing.</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>\(f'(x) = {x^2} + 4x - 12\) <em><strong>(A1)(A1)(A1) (C3)</strong></em></span></p>
<p><span><strong>Notes:</strong> Award <em><strong>(A1)</strong></em> for each term. Award at most <em><strong>(A1)(A1)(A0)</strong></em> if other terms are seen.</span></p>
<p><span><em><strong>[3 marks]</strong></em><br></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span>\( - 6 \leqslant x \leqslant 2\) <strong>OR</strong> \( - 6 < x < 2\) <strong><em>(A1)</em>(ft)<em>(A1)</em>(ft)<em>(A1)</em> <em>(C3)</em></strong></span></p>
<p><span><strong>Notes:</strong> Award <strong><em>(A1)</em>(ft)</strong> for \( - 6\), <strong><em>(A1)</em>(ft)</strong> for \(2\), <em><strong>(A1)</strong></em> for consistent use of strict (\(<\)) or weak (\(\leqslant\)) inequalities. Final <em><strong>(A1)</strong></em> for correct interval notation (accept alternative forms). This can only be awarded when the left hand side of the inequality is less than the right hand side of the inequality. Follow through from their solutions to their \(f'(x) = 0\) only if working seen.</span></p>
<p><span><em><strong>[3 marks]</strong></em><br></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;">
<p><span style="font-size: medium; font-family: times new roman,times;">This question was quite a good differentiator with many able to score at least one mark in part (a). Part (b) proved however to be quite a challenge as many candidates did not seem to understand what was required and were unable to use their answer to part (a) to help them to meet the demands of this question part. The top quartile scored well with virtually everyone scoring at least three marks. The picture was somewhat reversed with the lower quartile with the majority of candidates scoring 2 or fewer marks.</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-size: medium; font-family: times new roman,times;">This question was quite a good differentiator with many able to score at least one mark in part (a). Part (b) proved however to be quite a challenge as many candidates did not seem to understand what was required and were unable to use their answer to part (a) to help them to meet the demands of this question part. The top quartile scored well with virtually everyone scoring at least three marks. The picture was somewhat reversed with the lower quartile with the majority of candidates scoring 2 or fewer marks.</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 \(f(x) = {x^4}\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Write down \(f'(x)\).</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>Point \({\text{P}}(2,6)\) lies on the graph of \(f\).</span></p>
<p><span>Find the gradient of the tangent to the graph of \(y = f(x)\) at \({\text{P}}\).</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>Point \({\text{P}}(2,16)\) lies on the graph of \(f\).</span></p>
<p><span>Find the equation of the normal to the graph at \({\text{P}}\). Give your answer in the form \(ax + by + d = 0\), where \(a\), \(b\) and \(d\) are integers.</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>\(\left( {f'(x) = } \right)\) \(4{x^3}\) <strong><em>(A1) (C1)</em></strong></span></p>
<p> </p>
<p><span><strong><em>[1 mark]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span>\(4 \times {2^3}\) <strong><em>(M1)</em></strong></span></p>
<p> </p>
<p><span><strong>Note: </strong>Award <strong><em>(M1) </em></strong>for substituting 2 into their derivative.</span></p>
<p> </p>
<p><span>\( = 32\) <em><strong>(A1)</strong></em><strong>(ft) <em>(C2)</em></strong></span></p>
<p> </p>
<p><span><strong>Note: </strong>Follow through from their part (a).</span></p>
<p> </p>
<p><span><strong><em>[2 marks]</em></strong></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span>\(y - 16 = - \frac{1}{{32}}(x - 2)\) <strong>or</strong> \(y = - \frac{1}{{32}}x + \frac{{257}}{{16}}\) <strong><em>(M1)(M1)</em></strong></span></p>
<p> </p>
<p><span><strong>Note: </strong>Award <strong><em>(M1) </em></strong>for their gradient of the normal seen, <strong><em>(M1) </em></strong>for point substituted into equation of a straight line in only \(x\) and \(y\) (with any constant ‘\(c\)’ eliminated).</span></p>
<p> </p>
<p><span>\(x + 32y - 514 = 0\) <strong>or </strong>any integer multiple <strong><em>(A1)</em>(ft) <em>(C3)</em></strong></span></p>
<p> </p>
<p><span><strong>Note: </strong>Follow through from their part (b).</span></p>
<p> </p>
<p><span><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><span style="font-size: medium; font-family: times new roman,times;">Consider \(f:x \mapsto {x^2} - 4\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Find \(f ′(x)\).</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>Let <em>L</em> be the line with equation <em>y</em> = 3<em>x</em> + 2.</span></p>
<p><span>Write down the gradient of a line parallel to <em>L</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><span>Let <em>L</em> be the line with equation <em>y</em> = 3<em>x</em> + 2.</span></p>
<p><span>Let P be a point on the curve of <em>f</em>. At P, the tangent to the curve is parallel to <em>L</em>. Find the coordinates of P.</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>\(2x\) <em><strong>(A1)</strong></em> <em><strong>(C1)</strong></em></span></p>
<p><span><em><strong>[1 mark]</strong></em></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span>3 <em><strong>(A1)</strong></em> <em><strong>(C1)</strong></em></span></p>
<p><span><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>\(2x = 3\) <em><strong>(M1)</strong></em></span></p>
<p><br><span><strong>Note: <em>(M1)</em></strong> for equating their (a) to their (b).</span></p>
<p><br><span>\(x =1.5\) <em><strong>(A1)</strong></em><strong>(ft)</strong></span></p>
<p><span>\(y = (1.5)^2 - 4\) <em><strong>(M1)</strong></em></span></p>
<p><br><span><strong>Note: <em>(M1)</em></strong> for substituting their <em>x</em> in <em>f</em> (<em>x</em>). </span></p>
<p><br><span>(1.5, −1.75) (accept <em>x</em> = 1.5, <em>y</em> = −1.75) <em><strong>(A1)</strong></em><strong>(ft)</strong> <em><strong>(C4)</strong></em></span></p>
<p><br><span><strong>Note:</strong> Missing coordinate brackets receive <em><strong>(A0)</strong></em> if this is the first time it occurs.</span></p>
<p><span> </span></p>
<p><em><strong><span>[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-size: medium; font-family: times new roman,times;">This question was generally answered well in parts (a) and (b).</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-size: medium; font-family: times new roman,times;">This question was generally answered well in parts (a) and (b).</span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-size: medium; font-family: times new roman,times;">This part proved to be difficult as candidates did not realise that to find the value of the <em>x</em> coordinate they needed to equate their answers to the first two parts. They did not understand that the first derivative is the gradient of the function. Some found the value of <em>x</em>, but did not substitute it back into the function to find the value of <em>y</em>.</span></p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>A function \(f\) is given by \(f(x) = 4{x^3} + \frac{3}{{{x^2}}} - 3,{\text{ }}x \ne 0\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the derivative of \(f\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the point on the graph of \(f\) at which the gradient of the tangent is equal to 6.</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>\(12{x^2} - \frac{6}{{{x^3}}}\) or equivalent <strong><em>(A1)(A1)(A1) (C3)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>(A1) </em></strong>for \(12{x^2}\), <strong><em>(A1) </em></strong>for \( - 6\) and <strong><em>(A1) </em></strong>for \(\frac{1}{{{x^3}}}\) or \({x^{ - 3}}\). Award at most <strong><em>(A1)(A1)(A0) </em></strong>if additional terms seen.</p>
<p> </p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\(12{x^2} - \frac{6}{{{x^3}}} = 6\) <strong><em>(M1)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>(M1) </em></strong>for equating their derivative to 6.</p>
<p> </p>
<p>\((1,{\text{ }}4)\)\(\,\,\,\)<strong>OR</strong>\(\,\,\,\)\(x = 1,{\text{ }}y = 4\) <strong><em>(A1)</em>(ft)<em>(A1)</em>(ft) <em>(C3)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>A frequent wrong answer seen in scripts is \((1,{\text{ }}6)\) for this answer with correct working award <strong><em>(M1)(A0)(A1) </em></strong>and if there is no working award <strong><em>(C1)</em></strong>.</p>
<p> </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><span style="font-size: medium; font-family: times new roman,times;">The figure below shows the graphs of functions \(f_1 (x) = x\) and \(f_2 (x) = 5 - x^2\).</span></p>
<p><span style="font-size: medium; font-family: times new roman,times;"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZ0AAAFvCAIAAAAe54HnAAAfKklEQVR4nO2dv2vj6LqA/W+kmoG0+xfcxpMmzamPSZViO+cUt1tY5y9wBgYuXOIqsFynHZzmVmK4zUEwkEKwxTCqTiGmOYjDENBhWYJu8e5oPbIjS7L0fr+eBxe7mUzms2M/er/v/aFJCaDF6avXppcAQTAxvQAICLwGOuA10AOvgQ54DfTAa6ADXgM98BrogNdAD7wGOuA10AOvgQ54DfQ4ffVaHqYXAp6D10APjAY64DXQA6+BDngN9MBroANeAz3wGuiA10APvAY64DXQA6+BDngN9MBroANeAz3wGuiA10APvAY64DXQA6+BDngN9KCJCnTAa6AHUgMd8BrogddAB7wGeuA10AGvgR54DXTAa6AHXgMd8BrogddAB7wGeuA10AGvgR54DXTAa6AHXgMd8BrogddAB7wGeuA10AGvgR54DXTAa6AH9w8FHfAa6IHRQAe8BnrgNdABr4EeeA10wGugB14DHfAa6IHXQAe8BnrgNdABr4EeeA10wGugB14DHfAa6IHXQAe8BnrgNdABr4Ee9FGBDngN9MBooANeAz3wGuiA10APvAY64DXQA6+BDngN9GjvtTzP8zwfdTHgMXgN9DjotaIoVreri9nsZrlM01RnVeAfeA30aPZaHMdn0+nqdlUUhdqSwEvwGujR4LU0TU9fvV7drjTXA76C10CPl7xWFMXZdHoxmxGpwSDgNdDjJa+tblenr15HUZTneRzH2A2OBK+BHi95TZqrrubzq/n89NXrs+n0YbNRXhv4BF4DPfZ6TU7WruZzCdPSND2bTk9fvU6SRH2B4Al4DfSo+t63BSdeu14sqq88bDanr17fr9cm1gg+gNdAj4Z4bdtr8pWb5VJxaeAVeA302Ou1oijkTK36SpIkp69ec8QGvcFroEdzPrQ6UJP/pY8KeoPXQI+G+rWr+fxiNkvTVLoO4jhWXhv4BF4DPZr7qJIkuVku6QyF48FroAdzikAHvAZ64DXQAa+BHngNdMBroAdeAx3wGuiB10AHvAZ6cP9Q0AGvgR4YDXTAa6AHXgMd8BrogddAB7wGeuA10AGvgR54DXTAa6AHXgMd8BrogddAB7wGeuA10AGvgR54DXTAa6AHXgMd8BrogddAB7wGejjttecvH3/56XwymUzevHt8eja9HGgCr4Ee7va9P2fvfzz54fLu16+f784nf737/G/TK4Im8Bro4aLRyrIsn//x/scfTn768NX0QqAleA30cNNrz0+P795M/vL28V+mVwJtwWugh5Nee/r49s3J5PzuM0dq7oDXQA/XvPb89cPPJ5NvnL59/N30iqAdeA30cM1rZVmW5Zf3l5OT87tPL4drBHLWgddADwe9JiFbcwKU3Kh14DUYnvQbD5vN9uP01Wv5jyRJ5BvyPDe92GaeHt++mZz8/OFrQ1CG16wDr8FR5HmeJMnDZnOzXF4vFlWFWqfH2XR6vVisblcPm41dsnv+dHd+Mrl8/+WP///ty8f/vjyZTCY/XN79+vTHF/GadeA16EyWZVEU3SyXZ9Np5aaL2ex6sZBwrIrXan+x2odW3xBF0cNms7pdXS8W2z9NTCc/Sv35bVE7XHv+9MvP//Pp6fenx3dv/tyccr5mHXgNWlEURRzHq9tVZZ+L2Wx1u4qiqL16Dp6vFUUhslvdrq7m80pz4rgsy45+Hp14+XDt+dPd+X/89OGfuuuBtuA1aEJ0drNcVmHU6nYVx3G/rWLXvEFRFEmS3K/XleNkAUmS9PjXu/P0+PbN/sq15093f/nP99lvKsuAzuA12E+aplV0djGb3a/Xx4dLx+RD8zzfNey4Edzzp7vz030VHs9Pj/91+fbj076/BDaA1+A7iqKIouhiNqvcMeAJ1yB977UQ8mI2i6KoKIqhFlnx/PnufN/ojucvH969+/CFUzWLwWvwB3meVwHa1Xwex/Hgshi2fk0UXG1RB1Lwb9n7v52c333K/v7u8q8/f8jq+nr69Zd3/5s9l+Xzl8f/+0zIZid4Df4w2qB22M9IdblZllXrv14s4jg+4odJl/tkcnL57mM9Jnv+8vd3lz9866tqbkIAk+C1oKkZbezCsVH7DYqieNhsZAd9MZsdZzdwG7wWKNtGe9hsdEphdfqo4jjGboGD14JD4ho5R1OI0bbR7A/FbiGD18IijmMx2s1yqd+upN/3XtnterEw3LoAiuC1UMiyTPo3r+ZzU59wU/M8qvjUiM1BH7zmP7LxlHq0KIoMrsTgnKKiKOQ88Ww6fdhsTC0DdMBrnpOmqWzEbAhVjM9fq4LWi9mMbanH4DVvKYrifr2WCEWrofIAxr0mVIeM9+v1GI0KYBy85idZllVhmj0fXUu8VpZlURTSiUXg5iV4zUOq0zRLwrQKe7wmJElC4OYleM0r8jyX86PrxcLCD6qF93vfDtzU57vBWOA1f6iiD7NJzwasMto21UtHqtQP8JonyN7T8qDDWq+V1oe60Am85jxFUcgH0qoUwV5s9ppQHU2STHAavOY2Vd7T2r3nNvZ7rSzLNE0t387DQfCaw0gdlkPBRRuvSeypsJgGiqKQcZX2h8CwF7zmKrJjuprPjXcRtOegsKIosidhKn1Xbr3CIOA196haHZ2LJpqFJXtq2VarLamZOI7luM3mbAzsgtcco9oirW5XptfSmWZhXc3nURTZsA/dJssyOW5jiJtD4DWXyLJMpOboZ6xBWA+bzfViUdpxvlYjz3N52e/Xa9NrgVbgNWeQwOFsOnVUauXLXpMUpOypLfRauVVM42KYHCB4zQ3kk+/6QU/VR7VtrqIoLmazqpXVTq8JVSbBrWPNAMFrDuDN6fVeYa1uVzKkW5AdX5qmdmYhqzQ0arMZvGY7IjU/Pkh7vbYdxG0/rG3VrH4jrl9mPAavWY1PUitf8Fr6PZbHa4I3EbSv4DV78UxqpTv9Bm3w48TTV/CapfgntdIvr5VbGWrUZht4zUa8lFrZzmtxHFt7srYLarMTvGYdvkqtdGSeR1dQm4XgNbvwWGqlp14rUZt94DWL8Ftqpb9eK1GbZeA1W/BeaqXXXitRm03gNSsIQWql714rUZs14DXzBCK1MgCvlVtqs7mu2HvwmmHkY3Axm3kvtdLK+4eOQZqmgVyorAWvmSS0bYv3RqsIJwa3E7xmjDzPg5JaGZLXyi21mV5IiOA1M1TjvMORWhmY18pvQ40YRakPXjNAJTV3J9/2IzSvld9GUaK2kXj+8vGXn84nk8nkzbvHp+fq63jNAPJeD01qZZBeKwP+dY/Nc/b+x5MfLu9+/fr57nzy17vP/67+CK9pE/IFPEyvBRuej8vzP97/+MPJTx++7vtDvKaKnCXfLJemF2KGML1WfruHQ1A5opF5fnp892byl7eP/9r7x3hND8qagvVauVXTE+xvf0iePr59czI5v/v8vPePn/GaEryty7C9VnJhG4bnrx9+Ppl84/Tt4++73/NPvKYB2xAhcK+VwR9EDMaX95eTk/O7T/vDta8f8JoGHBsLgfRRNSOJI4dmAtuHhGzfJUC/++PPd3htdHgfVwRutAq5hwPXub48Pb59Mzn5+cPXF07X8NrYyL4jzKqOXfCaIJUfnEv05PnT3fnJ5PL9l5f+HK+NSpZlnBNvg9cqgprjMjC1w7Uv7y9rCQTO18ajKArmcNXAa9skSXL66vX1YmF6IW6x73Dty/vL7xKj5ENHo7pvuemFWAReqyGN8Zy9duHp8e2beuVa3WvUr42D5AqiKDK9ELvAa7tIDiFJEtMLcYTnT3fnp/UKj7rX6DcYAWqUXgKv7VLVNnJe0Ybnz3fn34/uKEu8Nj5yHkyuYC94bS9Vfsn0Qqzlt+z9307O7z5lf393+defP2T1+g68Nir0FTSD114iiqLTV6/v12vTC7ET6XKfTE4u3338sqdoDa+Nys1ySb1lA3itATmT5aCtM78/vj2t94ritcHgknsQvNYAB20DgteGgSOSNuC1ZngXxXE8SNULXhsArrQtoe/9IGFG/UVRPGw2F7PZ6avXF7PZ8T8Qrw2AHKtxMnIQjNaGoN5OeZ7fr9dn06kEqkOdTeO1YwnzAtsPvNaGKvz3u1Qoz3NJlUgz2bCdOXjtKOhs7wRea4lM1vW1dTRNU+mykFE3YxRF4bX+MG2mK3itPdI66lkrXhzHYrSz6fR+vR7vPBqv9ed+vfbvnTcqeK0TMjrBg6tmURRxHEta4Gw6fdhsxt7f4LWeyJAZmkA7gdc6kee59OSZXkh/JNEpaYGL2UytZB2v9UFmqzEUsCt4rSvuZqUkLSBGu14slJtw8Fof5IyA2WpdwWs9kLIPh95saZpWic6b5dLIyvFaZ9y9hBoHr/XAoc1BLdFpsEwdr3WjGkNkeiFOgtf6IYe5Nl9Ka2kB4403eK0b3qSojEAfVW/s3I0WRRFFUdX/pJDobAle64CXJUWaYLTe2LYbzfPcSKKzJXitLdJa4GsJuA547Rgs2Y2O2v80FHitFVVrgfGDA6fBa0didjeapqksQNICdhpNwGutoLVgEPDakZjajSZJYkmisyV47TB+NyFrgteOR3k3qtz/NBR47QDMjByQZq9Jck32+/frNUnnl1DYje72P7liNAGvHYAd6IA0e012OteLhXycvB9A1ptRd6MjDXpUBq81wQ50WBq8JgXP8kEtikIcN8ioey+R3eiwr48Tic6W4LUm2IEOS4PXkiTZ/iDJ55YrSgOi/kF26/b0Pw0FXnsRqnAHp33eoCgK4rVmBpliFMextNCMPehRGby2H6pwx6C91yReI3XQjIxg6HHp1R/0qAxe2w9VuGPQ3mvXi4Xxwnon6PpGNTXoURm8tgd2oCNR9b03d79LtYdnEcRIyMaizdzmWlrA7/v44bU6HgxftpY28VqapkTKnZBSpIb0pQ2DHpXBa3UGTDNBjYNey7LsYjbjxe+ElI7vLWfzL9HZErz2HXEck4Ybj2av1aRWFMX9eu3rAdCw7Jaz2TboURm89ie2jbjyj4N1ubUDuNNXr/ldtKTaZzxsNtWgxyiKwnwB8dqfyBlECKcPpmjwWhzHD5tN7UGw1h7pjZGHx4nOluC1P5C3BfcDHRXmeYzBdqKTUxQBr/2BtEyFGbSrgdeGpTboMUmSlxIIoYHXypKCNS3w2lDEcSwHamfT6Xaic4x+eBfBa38UrNEypQBeO56D/U/iu9ASoDXwGgVreuC13rQf9JjnOa3NoXvNknv8BAJe60GPQY9yrhJyZj9or1Uzvjln1YH7Inciy7J+gx6rSsxRl2czQXtNLmuBV/pogtFacnz/k3TOBJsKC9drHEPog9cOMuCgRxlhFOZeJFyvkS7QB6+9xPagx4vZbJBBj1JqvrpdDbJCtwjUa6QLjIDXdhl10KNU7QZY8xGi10gXmAKvbaMw6DHYw5YQvUa6wBR4TdAc9BhmzUdwXqO7wCB4LUkS5UGPYdZ8BOc1OXEI7fJlCSF7zeD9n6TmI6gNSlheCzlDZAMBek3SAsYHPYZ2oByW10Ku6LGBoLyW57k9d7STK3o4cz4C8hr3LjBOIF6rJTotOfS4XizCudFXKF4L8/TUNrzvD631P1lV9R3UIUwoXpNst9/3grUfj4320qBHq5Ao0s61DUsQXgu2OtE2vPSawURnV8L5IAThNantsGpTECY+ea39oEerOHhzeD/w32tBHStYjh9ek7SAGO16sXCrLkwOmr0P2fz3mqSBnLiWeo/rXus96NEqQuis8txr1HZYhbteO37Qoz2EUBvgude4naJVuOi17UGPD5uN00ariKLI784qn73G3A7bcMhrRVFEUTTsoEerkEu+6VWMhbdek2D7aj43vRD4Eye8NuqgR3vwuxneW68Fks92C8u9pjDo0So8PqXx02vh1B+6hbVeS9NUihwlLRDI5dDjrJqfXpOrLoW4tmFhf6j+oEer8LUKykOvUYhrLVYZzaH+p/HwdX6Rh16Ty29oF14nsMFrlgx6tAcvQzbfvObr9ccPzHotz/P79VoSnVfzua+pwK54+ZHxzWtMxLUZU16zc9CjPfgXsnnlNY/zO36g7zWbBz3ag38hm1deC+3mFM6h6TUnBj3ag2chmz9e87t+2g8UvFYUBYnOHngWsnnitRBGFHjAqF4LpP9pPHwK2TzxGi3uTjCS15we9GgPPoVsPniNYM0VBveaH4Me7cGbkM0Hr4Uw/9MPBvSaT4Me7cGbkM15rwUyr90PBukPraUFMNqw+BGyOe81gjWHOMZo3g96tAQ/Qja3vcY8Irfo5zUSncp4ELK57TU5MyZYc4WuXqP/yQgehGwOe41gzTnaey3MQY/24HrI5rDXCNaco43XAh/0aAnSvRNFkemF9MRVr0mwdrNcml4IdKDZa/Q/WYXTN6xy1WsSrHExd4u9XttNC2A0G3C64dpJr0mwxqRv56h5jUGPluNuyOak1wjWHKXyGolOJ3A3ZHPPawRr7iJ5HgY9OoSjIZt7XiNYcxS5+Eta4H695jfoBI7287zotaqVjwcPHjyUH2N5rexSRdnmOwf5advBWssfOOy3tfzOYX+aqW9r+Z0vfc9uotPml67ldxr5aWafwsGQzdSbswGXvFY7WXPxo977O916si8NerT5pWv5nQF6TUbmNNSKeus1HfqdrNn2LPrhyrNI07RKdN4sl7WLfMOzcCWB4MovooEeT8G5Q21nvNY7DWrVs+iN/c+izaDHvc8ijmOZoHezXNpvN/t/EQfp8RScK0JwxmvOXTGGxarfRY32gx53n0WWZWfTaZIk5beEqfy3tdj8ixgVtz6AbtR50A0qt4+zqkKyNugxiqKD/U+7Uriaz7fLo+yvlrI/omxGDgqiKOpqqCzLTt0ZXuSG144Z3ZEkyep2ZZURulJlFS15V+V53m/QY81rMudr+3Ils4nsrJaqNtqmF9KToiiu5vPtWoqu4zp0hhfleX6zXF7N5zKCrCgK+Xc7ibjJa1mW3a/Xba7Do9J7zprscarf4tl0aucH5iBpmiZJYoPXjux/qklBCgi2n9TuVyxBro6DlFaZIoqiq/k8TVOpv5FPRKefINchheFFRVFUoczVfH6/Xt8sl8N47X69tsQIvSueb5ZLOYrOskwCAXd3ssZHmA4y6NFdrwmy6Ta9ip5cLxbbm2iJ3bqeZqodFEg0Iye2Pf76fq+laSqnuUVRiOBMGeGY201dzGZVpFkUhXwmHZ2BY9BrcRzL/utsOj1y0KPrXnN6H1qjX7ig2Ql/jEP3ey3P85oRTI3bHrA9zek3pRGvDT7oEa/Zg0yI6vEXL2azq/l86OXs/4d6ByKH8wYSEBrJvg97b9Cr+Zx9aBvGG/RYk4Jc/O/X6+orcqpibamHN16TT1a/t5NOJ7ycBvZ+MxzwmuRQTF0/B3wFxQuO5g1KLa+NPejx9Puu5t2twNV8bvPtQrzx2v163TvmOthWdSRFUWRZdr1YSER1v14nSdI1WdHkteqirbaj3kZevqEOKSWrMsiPGo/Tl6cajO21LMsUBj3uSkH+UTnPVku39cYPr8VxfDWfH3PxGK9GVy6rF7OZvCUuZrN+Dj0Qr8kF3MiJe6cTygYjlGW5ul05sQM14rU2/U9DsSsF2RCcTafyYbP81+SB17Isq6zRm/HaqrIs225ZybKsX53ZpNz5OO1+k+hZOWTrlA1pMMLxVycbGMNr8sqcKg563Pvukr6Fh83G2mO1Cte9tiu13r/3m+XS5hODVl6TD5Wm14ZKJ9ekJh+ho1dnAPkVDHKFlJYsI3e0c1oKpeNekzL1i9nserGQx9V83vuUzXhBZTOt+qiiKBppO/0Sg5T/iRylIUMeF7OZ/adsu9yv16KhIwtudhOdAy6yDe5KoSgKafmQDJ0r7d8Vtd6b6nHMZb7W22sV+70mXSNyeJymqbIOBukZqt6FtYfrfcv9qPU/mdrxues1aYmtHu4m1gfE5rtV7fdalSs4m05717n0Rqe9NhCaBz0q467XYC9qNbpdadqHpmmqH91Yvm93CM1EZ0vwmmdYe7cq6+YUyUeRYO0Y2g96VAaveYZUVls4R9curzk3btgqJC3QadCjMnjNP+yco2uX1+x8jeyn96BHZRpqicBR7Jyja5HXCNZ6cOSgR2UwmpdIos/0Kr7DIq9JEjbMOowe1AY9OvG64TUvsbDgwxavDTuSyG8GHPSoDF7zFdtuuGOL16xNGFuFqf6nocBrvmLb59cWr9nme6sYb9CjMnjNV2wr+LDCaxbuzy1B0gJitOvFwvWXCK95jFXFDFZ4jWBtF51Bj8rgNY+xquDDvNdonKphYf/TUOA1v7Gn4MO81+hyr9Af9KgMXvMbew6UDHuNWtzy27TLqv/JxURnS/Ca95xNpzZM+DDsNavOGvUxPuhRGbzmPVLwYbxK3KTXbMsNa2LJoEdl8Jr3WLIDM+k122r5dLBq0KMy9L2HgLy9zZ6lmPRaaI1TSZL4muhsCUYLARvuA2vMa/akThRwvf9pKPBaIBivSDXmNePPXAH7Bz0qg9cCQe5gZ/CMxYzXvK/F3R70eDWfBxKWHgSvBYKM57lZLk0twIzXLL9Z9DG4NehRGbwWDmZLuAx4zZJM8ODU+p+Ml/BYCF4LB7Ptoga8JnNxfcoGujvoURm8FhQGbwiv7TWf5uIWRUGisxN4LSgM1jxoe814omQQQut/Ggq8FhQGgxhtr7le3uHZoEdl8FpomDp0UvVakiTu1uJ6OehRGfqoQkOShPfrtfK/q+o1R0eteTzoURmMFiBGhk3qec2UuY+hlhbAaEeC1wLESPZAz2sOjVoLZ9CjMngtTPSzB0peM95X0RISnaOC18JEP3ug5DWJRW2enhjmoEdl8FqY6J9BKXnN5vKONE1vlssqLUCiczw6eS3Pc3rRvEE5e6DhNRvmzO2FQY/KtPRalmXVlUbON8deGIyNcvZAw2sWTu+g/8kILb12MZtdzecPm01lNwI3D9DMHozuNaumdzDo0SxtvJam6faN2uTI2ZL3DxyDZvZgdK9Zct+tPM/v12sGPZqljddqx2pyXfRjUELgaGYPRvea8ekdDHq0hx75ULkZI0dsfqCWPRjXa2ZvzsKgR9vo4TV5C5HS8QM1IYzrNSOtYWVZxnF8NZ9LWuB+veZTYQlV33tLwRVFQT7UM3Q2cCN6TbbTmm9KBj1azq7Otk2367vV7YqMgWfoZA9G9JpmQyj9T07QyWsSdHNl8gyd7MFYXlNrCGXQo0O0P1+rSa0oCg4TvEHheGosryk0hKZpWiU6b5ZLEp3209JrcRyfTadJkqRpmqZpkiQ3yyVXLG9QkMNYXhv1VjQMenSUNl6TN33tYVu/ChyDwmZuFK/JrQPHaAhl0KPTtPFalmXpDtToeMbYh++jeE0WPeAFlkGPfsCcIhDGvmvy8F6TAvGh0vN5npPo9Aa8BhWjHlUN77Wh7hBK/5N/4DWoGDV7MLzXjh8hyaBHX8FrUDFq9mBgrx05QpJBj37D/UNhm/GyBwN7rXfGgP6nEMBosM142YMhvdYjY7Db/4TRPAavQY2RsgdDeq1TxoBBjwGC16DGSNmDIb3WUr0kOoMFr0GNkbIHg3mtTY8B/U+Bg9dglzEmFw3mteaMAYMeocRrsI8xJhcN4zUJJnczBgx6hG3wGuxFJhcNKIdhvLZ7+MegR9gFr8FekiQZ9r4Hw3htO2NQSwuMOmUJ3AKvwUsc36e0zQBeq4rrGPQIzeA1eIlhb1U1gNcknSGHaCQ6oQG8Bg2cTadX8/kgP+pYr8nIZgY9QhvwGjQghf2DzKPt6TVJC1Qx2v16TaITDkLfOzQzVOtRZ6/VBj1ezGZG7nwMLoLRQIcOXtvtf9K5FSB4A14DHVp57aVBjw+bzemr19xTA1qC10CHA16L41g6OqWdoJYWuJjNhspfQAjgNdDhRa8VRdHc/ySjcWkkgPbgNdChKV5b3a4aBj0OfjM98B68Bjr0r/PY2+gO0ABeAx16em3Ue2SBr+A10KGn164Xi/HuaQq+gtdAhz5eo2wN+oHXQIc+XpOyNVpBoSt4DXTo4zXK1qAf9IeCDp29Rtka9AajgQ6dvUbZGvQGr4EOnb02xs3+IBDwGujQzWuUrcEx4DXQoZvXbpZLpq1Bb/Aa6NDBa0VRULYGx4DXQIcOXpPp40xbg97gNdChg9e2bxIK0AO8Bjq09Zr0Tg1yqxgIFrwGOrT1mtwklN4pOAa8Bjq09Rq9U3A8eA10aOW1LMvonYLjoT8UdGjlNXqnYBAwGujQymv0TsEg4DXQ4bDXkiRhEwqDgNdAh8NeW92uzqZTNqFwPHgNdDjsNe47BUOB10CHA16TTSgDPGAQengtyzKqJqErB7zGAA8YkK5ekwKjh81mpPWArzR5TQZ4sAmFoejktaIoLmYzvAY9aPIaUyRhWDp57X69xmvQjyavsQmFYWnvtSRJLmYzOd7Fa9CVJq9dzedsQmFAWnqtKIqz6TRNU7n5GV6DrvS5fyhAP6r+0GbB3SyX4jK8Bv3Aa6BHm3gtiqJqcgxeg37gNdBj12vbEZwM+DubTqMokk2ojJ5f3a7SNKXjBdqD10CPg16TAG3vI01TI2sGF8FroMfBfWhRFOkWxGvQD7wGenTtN+B8DfqB18Be8Br0A6+BveR5/rDZcLIGXcFrAOAb/w8o6FsChLNrXgAAAABJRU5ErkJggg==" alt></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>(i) Differentiate \(f_1 (x) \) with respect to <em>x</em>.</span></p>
<p><span>(ii) Differentiate \(f_2 (x) \) with respect to <em>x</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>Calculate the value of <em>x</em> for which the gradient of the two graphs is the same.</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>Draw the tangent to the <strong>curved</strong> graph for this value of <em>x</em> on the figure, showing clearly the property in part (b).</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><span>(i) \(f_1 ' (x) = 1\) <em><strong>(A1)</strong></em></span></p>
<p> </p>
<p><span>(ii) \(f_2 ' (x) = - 2x\) <em><strong>(A1)(A1)</strong></em></span></p>
<p><em><span><strong>(A1)</strong> for correct differentiation of each term. <strong>(C3)</strong></span></em></p>
<p> </p>
<p><em><span><strong>[3 marks]</strong></span></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span>\(1 = - 2x\) <em><strong>(M1)</strong></em></span></p>
<p><span>\(x = - \frac{1}{2}\) <em><strong>(A1)</strong></em><strong>(ft)</strong> <em><strong>(C2)</strong></em></span></p>
<p><span><em><strong>[2 marks]</strong></em></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span><em><strong>(A1)</strong> is for the tangent drawn at</em> \(x = \frac{1}{2}\) <em>and reasonably parallel to the line</em> \(f_1\) <em>as shown.</em></span></p>
<p><strong><span><img src="data:image/png;base64,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" alt><em> (A1) (C1)</em></span></strong></p>
<p><strong><span><em>[1 mark]</em></span></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-size: medium; font-family: times new roman,times;">Most candidates were able to differentiate correctly, but only a third were able to calculate the value of <em>x</em> for which the gradients of the graphs were the same and a similar number did not attempt to. Some found the <em>x</em>-coordinate of the point of intersection.</span></p>
<p> </p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-size: medium; font-family: times new roman,times;">Most candidates were able to differentiate correctly, but only a third were able to calculate the value of <em>x</em> for which the gradients of the graphs were the same and a similar number did not attempt to. Some found the <em>x</em>-coordinate of the point of intersection.</span></p>
<p> </p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="font-size: medium; font-family: times new roman,times;">c) Very few candidates were able to draw the tangent correctly. Some tangents were drawn horizontally and some at the point of intersection. The line could have been drawn without any knowledge of calculus so the indication here was that many of the candidates misunderstood the question.</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 function is given as \(f(x) = 2{x^3} - 5x + \frac{4}{x} + 3,{\text{ }} - 5 \leqslant x \leqslant 10,{\text{ }}x \ne 0\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Write down the derivative of the function.</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>Use your graphic display calculator to find the coordinates of the local minimum point of \(f(x)\) in the given domain.</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><span>\(6{x^2} - 5 - \frac{4}{{{x^2}}}\) <strong><em>(A1)(A1)(A1)(A1) (C4)</em></strong></span></p>
<p> </p>
<p><span><strong>Note: </strong>Award <strong><em>(A1) </em></strong>for \(6{x^2}\), <strong><em>(A1) </em></strong>for \(–5\), <strong><em>(A1) </em></strong>for \(–4\), <strong><em>(A1) </em></strong>for \({x^{ - 2}}\) or \(\frac{1}{{{x^2}}}\).</span></p>
<p><span> Award at most <strong><em>(A1)(A1)(A1)(A0) </em></strong>if additional terms are seen.</span></p>
<p> </p>
<p><span><strong><em>[4 marks]</em></strong></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span>\((1.15,{\text{ }} 3.77)\) \(\left( {{\text{(1.15469..., 3.76980...)}}} \right)\) <strong><em>(A1)(A1) (C2)</em></strong></span></p>
<p> </p>
<p><span><strong>Notes: </strong>Award <strong><em>(A1)(A1) </em></strong>for “\(x = 1.15\) and \(y = 3.77\)”.</span></p>
<p><span> Award at most <strong><em>(A0)(A1)</em>(ft) </strong>if parentheses are omitted.</span></p>
<p> </p>
<p><span><strong><em>[2 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="question" style="padding-left: 20px; padding-right: 20px;">
<p>Consider the function \(f(x) = {x^3} - 3{x^2} + 2x + 2\) . Part of the graph of \(f\) is shown below.</p>
<p><img src="data:image/png;base64,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" alt></p>
<p>Find \(f'(x)\) .</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>There are two points at which the gradient of the graph of \(f\) is \(11\). Find the \(x\)-coordinates of these points.</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>\((f'(x) = )\,\,3{x^2} - 6x + 2\) <em><strong>(A1)(A1)(A1) (C3)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(A1)</strong></em> for \(3{x^2}\), <em><strong>(A1)</strong></em> for \( - 6x\) and <em><strong>(A1)</strong></em> for \( + 2\).<br>Award at most <em><strong>(A1)(A1)(A0)</strong></em> if there are extra terms present.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>\(11 = 3{x^2} - 6x + 2\) <em><strong>(M1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for equating their answer from part (a) to \(11\), this may be implied from \(0 = 3{x^2} - 6x - 9\) .</p>
<p>\((x = )\,\, - 1\,\,,\,\,\,\,(x = )\,\,3\) <em><strong>(A1)</strong></em><strong>(ft)<em>(A1)</em><strong>(ft) <em>(C3)</em></strong></strong></p>
<p><strong>Note: </strong>Follow through from part (a).<br>If final answer is given as coordinates, award at most <strong><em>(M1)(A0)(A1</em>)(ft)</strong> for \(( - 1,\,\, - 4)\) and \((3,\,\,8)\) .</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>Question 15: Differential calculus.</p>
<p>Many candidates correctly differentiated the cubic equation. Most candidates were unable to use differential calculus to find the point where a cubic function had a specified gradient.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Question 15: Differential calculus.</p>
<p>Many candidates correctly differentiated the cubic equation. Most candidates were unable to use differential calculus to find the point where a cubic function had a specified gradient.</p>
<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 figure shows the graphs of the functions \(f(x) = \frac{1}{4}{x^2} - 2\) and \(g(x) = x\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><img src="data:image/png;base64,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" alt></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Differentiate \(f(x)\) with respect to \(x\) .</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>Differentiate \(g(x)\) with respect to \(x\) .</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>Calculate the value of \(x\) for which the gradients of the two graphs are the same.</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><span>Draw the tangent to the parabola at the point with the value of \(x\) found in part (c).</span></p>
<div class="marks">[2]</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>\(\frac{1}{2}x{\text{ }}\left( {\frac{2}{4}x} \right)\) <em><strong>(A1) (C1)</strong></em></span></p>
<p><span><strong>Note:</strong> Accept an equivalent, unsimplified expression (i.e. \(2 \times \frac{1}{4}x\)).</span></p>
<p><span><em><strong>[1 mark]</strong></em><br></span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span>\(1\) <em><strong>(A1)</strong></em> <em><strong>(C1)</strong></em></span></p>
<p><span><em><strong>[1 mark]<br></strong></em></span></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span>\(\frac{1}{2}x = 1\) <em><strong>(M1)</strong></em></span></p>
<p><span>\(x = 2\) <strong><em>(A1)</em>(ft)</strong> <em><strong>(C2)</strong></em></span></p>
<p><span><strong>Notes:</strong> Award <em><strong>(M1)(A0)</strong></em> for coordinate pair \((2{\text{, }} - 1)\) seen with or without working. Follow through from their answers to parts (a) and (b).</span></p>
<p><span><em><strong>[2 marks]</strong></em><br></span></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span><img 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" alt></span></p>
<p><span>tangent drawn to the parabola at the \(x\)-coordinate found in part (c) <strong><em>(A1)</em>(ft)</strong></span></p>
<p><span>candidate’s attempted tangent drawn parallel to the graph of \(g(x)\) <strong><em>(A1)</em>(ft)</strong> <em><strong>(C2)</strong></em></span></p>
<p><span><em><strong>[2 marks]<br></strong></em></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;">
<p><span style="font-family: times new roman,times; font-size: medium;">Parts (a) and (b) were reasonably well attempted indicating that candidates are well drilled in the process of differentiation. Correct answers however in part (c) proved elusive to many as frequent attempts to equate the two given functions rather than the gradients of the given functions resulted in a popular, but incorrect, answer of \(x = - 1.46\) . Part (d) was poorly attempted with many candidates simply either not attempting to draw a tangent or drawing it in the wrong place.</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 reasonably well attempted indicating that candidates are well drilled in the process of differentiation. Correct answers however in part (c) proved elusive to many as frequent attempts to equate the two given functions rather than the gradients of the given functions resulted in a popular, but incorrect, answer of \(x = - 1.46\) . Part (d) was poorly attempted with many candidates simply either not attempting to draw a tangent or drawing it in the wrong place.</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 (a) and (b) were reasonably well attempted indicating that candidates are well drilled in the process of differentiation. Correct answers however in part (c) proved elusive to many as frequent attempts to equate the two given functions rather than the gradients of the given functions resulted in a popular, but incorrect, answer of \(x = - 1.46\) . Part (d) was poorly attempted with many candidates simply either not attempting to draw a tangent or drawing it in the wrong place.</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;">Parts (a) and (b) were reasonably well attempted indicating that candidates are well drilled in the process of differentiation. Correct answers however in part (c) proved elusive to many as frequent attempts to equate the two given functions rather than the gradients of the given functions resulted in a popular, but incorrect, answer of \(x = - 1.46\) . Part (d) was poorly attempted with many candidates simply either not attempting to draw a tangent or drawing it in the wrong place.</span></p>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>A factory produces shirts. The cost, <em>C</em>, in Fijian dollars (FJD), of producing<em> x</em> shirts can be modelled by</p>
<p style="text-align: center;"><em>C</em>(<em>x</em>) = (<em>x</em> − 75)<sup>2</sup> + 100.</p>
</div>
<div class="specification">
<p>The cost of production should not exceed 500 FJD. To do this the factory needs to produce at least 55 shirts and at most <em>s</em> shirts.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the cost of producing 70 shirts.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <em>s</em>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the number of shirts produced when the cost of production is lowest.</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>(70 − 75)<sup>2</sup> + 100 <em><strong>(M1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for substituting in <em>x</em> = 70.</p>
<p>125 <em><strong>(A1) (C2)</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>s</em> − 75)<sup>2</sup> + 100 = 500 <em><strong>(M1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for equating <em>C</em>(<em>x</em>) to 500. Accept an inequality instead of =.</p>
<p><strong>OR</strong></p>
<p><img src="data:image/png;base64,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"> <em><strong>(M1)</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for sketching correct graph(s).</p>
<p>(<em>s</em> =) 95 <em><strong>(A1) (C2)</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img 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"> <em><strong>(M1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for an attempt at finding the minimum point using graph.</p>
<p><strong>OR</strong></p>
<p>\(\frac{{95 + 55}}{2}\) <em><strong>(M1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for attempting to find the mid-point between their part (b) and 55.</p>
<p><strong>OR</strong></p>
<p>(<em>C'</em>(<em>x</em>) =) 2<em>x</em> − 150 = 0 <em><strong>(M1)</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>(M1)</strong></em> for an attempt at differentiation that is correctly equated to zero.</p>
<p>75 <em><strong>(A1) (C2)</strong></em></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>