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</div><h2>SL Paper 2</h2><div class="specification">
<p>A function \(f\) is given by \(f(x) = (2x + 2)(5 - {x^2})\).</p>
</div>
<div class="specification">
<p>The graph of the function \(g(x) = {5^x} + 6x - 6\) intersects the graph of \(f\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the <strong>exact </strong>value of each of the zeros 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>Expand the expression for \(f(x)\).</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find \(f’(x)\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use your answer to part (b)(ii) to find the values of \(x\) for which \(f\) is increasing.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><strong>Draw </strong>the graph of \(f\) for \( - 3 \leqslant x \leqslant 3\) and \( - 40 \leqslant y \leqslant 20\). Use a scale of 2 cm to represent 1 unit on the \(x\)-axis and 1 cm to represent 5 units on the \(y\)-axis.</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the coordinates of the point of intersection.</p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The front view of the edge of a water tank is drawn on a set of axes shown below.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The edge is modelled by \(y = a{x^2} + c\).</span></p>
<p style="font: normal normal normal 21px/normal 'Times New Roman'; text-align: center; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;"><br><img src="images/Schermafbeelding_2014-09-02_om_11.23.28.png" alt><br></span></p>
<p style="font: normal normal normal 21px/normal 'Times New Roman'; text-align: left; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;">Point \({\text{P}}\) has coordinates \((-3, 1.8)\), point \({\text{O}}\) has coordinates \((0, 0)\) and point \({\text{Q}}\) has coordinates \((3, 1.8)\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Write down the value of \(c\).</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>Find the value of \(a\).</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>Hence write down the equation of the quadratic function which models the edge of the water tank.</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>The water tank is shown below. It is partially filled with water.</span></p>
<p><br><span><img src="images/Schermafbeelding_2014-09-02_om_10.48.45_1.png" alt></span></p>
<p><span>Calculate the value of <em>y </em>when \(x = 2.4{\text{ m}}\).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>The water tank is shown below. It is partially filled with water.</span></p>
<p><br><span><img src="images/Schermafbeelding_2014-09-02_om_10.48.45.png" alt></span></p>
<p><span>State what the value of \(x\) and the value of \(y\) represent for this water tank.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>The water tank is shown below. It is partially filled with water.</span></p>
<p><br><span><img src="images/Schermafbeelding_2014-09-02_om_10.48.45_3.png" alt></span></p>
<p><span>Find the value of \(x\) when the height of water in the tank is \(0.9\) m.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>The water tank is shown below. It is partially filled with water.</span></p>
<p><br><span><img src="images/Schermafbeelding_2014-09-02_om_10.48.45_2.png" alt></span></p>
<p> </p>
<p><span>The water tank has a length of 5 m.</span></p>
<p> </p>
<p><span>When the water tank is filled to a height of \(0.9\) m, the front cross-sectional area of the water is \({\text{2.55 }}{{\text{m}}^2}\).</span></p>
<p><span>(i) Calculate the volume of water in the tank.</span></p>
<p><span>The total volume of the tank is \({\text{36 }}{{\text{m}}^3}\).</span></p>
<p><span>(ii) Calculate the percentage of water in the tank.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">g.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-size: medium; font-family: times new roman,times;">A function is defined by \(f(x) = \frac{5}{{{x^2}}} + 3x + c,{\text{ }}x \ne 0,{\text{ }}c \in \mathbb{Z}\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Write down an expression for \(f ′(x)\).</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Consider the graph of <em>f</em>. The graph of <em>f</em> passes through the point P(1, 4).</span></p>
<p><span>Find the value of <em>c</em>.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>There is a local minimum at the point Q.</span></p>
<p><span>Find the coordinates of Q.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">c, i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>There is a local minimum at the point Q.</span></p>
<p><span>Find the set of values of <em>x</em> for which the function is decreasing.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c, ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Let<em> T</em> be the tangent to the graph of <em>f</em> at P.</span></p>
<p><span>Show that the gradient of <em>T</em> is –7.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">d, i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Let<em> T</em> be the tangent to the graph of <em>f</em> at P.</span></p>
<p><span>Find the equation of <em>T</em>.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">d, ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span><em>T</em> intersects the graph again at R. Use your graphic display calculator to find the coordinates of R.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Sketch the graph of <em>y</em> = 2<sup><em>x</em></sup> for \( - 2 \leqslant x \leqslant 3\). Indicate clearly where the curve intersects the <em>y</em>-axis.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">A, a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Write down the equation of the asymptote of the graph of <em>y</em> = 2<sup><em>x</em></sup>.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">A, b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>On the same axes sketch the graph of <em>y</em> = 3 + 2<em>x</em> − <em>x</em><sup>2</sup>. Indicate clearly where this curve intersects the <em>x</em> and <em>y</em> axes.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">A, c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Using your graphic display calculator, solve the equation 3 + 2<em>x</em> − <em>x</em><sup>2</sup> = 2<sup><em>x</em></sup>.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">A, d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Write down the maximum value of the function <em>f</em> (<em>x</em>) = 3 + 2<em>x</em> − <em>x</em><sup>2</sup>.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">A, e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Use Differential Calculus to verify that your answer to (e) is correct.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">A, f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>The curve <em>y</em> = <em>px</em><sup>2</sup> + <em>qx</em> − 4 passes through the point (2, –10).</span></p>
<p><span>Use the above information to write down an equation in <em>p</em> and <em>q</em>.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">B, a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>The gradient of the curve \(y = p{x^2} + qx - 4\) at the point (2, –10) is 1.</span></p>
<p><span>Find \(\frac{{{\text{d}}y}}{{{\text{d}}x}}\).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">B, b, i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>The gradient of the curve \(y = p{x^2} + qx - 4\) at the point (2, –10) is 1.</span></p>
<p><span>Hence, find a second equation in <em>p</em> and <em>q</em>.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">B, b, ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>The gradient of the curve \(y = p{x^2} + qx - 4\) at the point (2, –10) is 1.</span></p>
<p><span>Solve the equations to find the value of <em>p</em> and of <em>q</em>.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">B, c.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Consider the function \(f(x) = {x^3} + \frac{{48}}{x}{\text{, }}x \ne 0\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Calculate \(f(2)\) .</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>Sketch the graph of the function \(y = f(x)\) for \( - 5 \leqslant x \leqslant 5\) and \( - 200 \leqslant y \leqslant 200\) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Find \(f'(x)\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Find \(f'(2)\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Write down the coordinates of the local maximum point on the graph of \(f\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<address><span>Find the range of \(f\) .</span></address>
<div class="marks">[3]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Find the gradient of the tangent to the graph of \(f\) at \(x = 1\).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">g.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>There is a second point on the graph of \(f\) at which the tangent is parallel to the tangent at \(x = 1\). </span></p>
<p><span>Find the \(x\)-coordinate of this point.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">h.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The diagram shows a sketch of the function <em>f</em> (<em>x</em>) = 4<em>x</em><sup>3</sup> − 9<em>x</em><sup>2</sup> − 12<em>x</em> + 3.</span></p>
<p style="text-align: center;"><span style="font-family: times new roman,times; font-size: medium;"><img 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" alt></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Write down the values of <em>x</em> where the graph of <em>f</em> (<em>x</em>) intersects the <em>x</em>-axis.</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 <em>f </em>′(<em>x</em>).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Find the value of the local maximum of <em>y</em> = <em>f</em> (<em>x</em>).</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Let P be the point where the graph of <em>f</em> (<em>x</em>) intersects the <em>y</em> axis.<br></span></p>
<p><span>Write down the coordinates of P.</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><span>Let P be the point where the graph of <em>f</em> (<em>x</em>) intersects the <em>y</em> axis.</span></span></p>
<p><span>Find the gradient of the curve at P.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>The line, <em>L</em>, is the tangent to the graph of <em>f</em> (<em>x</em>) at P.</span></p>
<p><span>Find the equation of <em>L</em> in the form <em>y</em> = <em>mx</em> +<em> c</em>.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>There is a second point, Q, on the curve at which the tangent to <em>f</em> (<em>x</em>) is parallel to <em>L</em>.</span></p>
<p><span>Write down the gradient of the tangent at Q.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">g.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>There is a second point, Q, on the curve at which the tangent to <em>f</em> (<em>x</em>) is parallel to <em>L</em>.</span></p>
<p><span>Calculate the <em>x</em>-coordinate of Q.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">h.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-size: medium; font-family: times new roman,times;">When Geraldine travels to work she can travel either by car (<em>C</em>), bus (<em>B</em>) or train (<em>T</em>). She travels by car on one day in five. She uses the bus 50 % of the time. The probabilities of her being late (<em>L</em>) when travelling by car, bus or train are 0.05, 0.12 and 0.08 respectively.</span></p>
</div>
<div class="specification">
<p><em><span style="font-size: medium; font-family: times new roman,times;">It is <strong>not</strong> necessary to use graph paper for this question.</span></em></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Copy the tree diagram below and fill in all the probabilities, where <em>NL</em> represents not late, to represent this information.</span></p>
<p><img src="data:image/png;base64,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" alt></p>
<div class="marks">[5]</div>
<div class="question_part_label">i.a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Find the probability that Geraldine travels by bus and is late.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">i.b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Find the probability that Geraldine is late.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">i.c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Find the probability that Geraldine travelled by train, given that she is late.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">i.d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Sketch the curve of the function \(f (x) = x^3 − 2x^2 + x − 3\) for values of \(x\) from −2 to 4, giving the intercepts with both axes.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">ii.a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>On the same diagram, sketch the line \(y = 7 − 2x\) and find the coordinates of the point of intersection of the line with the curve.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">ii.b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Find the value of the gradient of the curve where \(x = 1.7\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">ii.c.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-size: medium; font-family: times new roman,times;">The following graph shows the temperature in degrees Celsius of Robert’s cup of coffee, \(t\) minutes after pouring it out. The equation of the cooling graph is \(f (t) = 16 + 74 \times 2.8^{−0.2t}\) where \(f (t)\) is the temperature and \(t\) is the time in minutes after pouring the coffee out.</span></p>
<p><span style="font-size: medium; font-family: times new roman,times;"><img 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" alt></span></p>
</div>
<div class="specification">
<p><span style="font-size: medium; font-family: times new roman,times;">Robert, who lives in the UK, travels to Belgium. The exchange rate is 1.37 euros to one British Pound (GBP) with a commission of 3 GBP, which is subtracted before the exchange takes place. Robert gives the bank 120 GBP.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Find the initial temperature of the coffee.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">i.a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Write down the equation of the horizontal asymptote.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">i.b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Find the room temperature.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">i.c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Find the temperature of the coffee after 10 minutes.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">i.d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Find the temperature of Robert’s coffee after being heated in the microwave for 30 <strong>seconds</strong> after it has reached the temperature in part (d).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">i.e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Calculate the length of time it would take a similar cup of coffee, initially at 20°C, to be heated in the microwave to reach 100°C.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">i.f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Calculate <strong>correct to 2 decimal places</strong> the amount of euros he receives.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">ii.a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>He buys 1 kilogram of Belgian chocolates at 1.35 euros per 100 g.</span></p>
<p><span>Calculate the cost of his chocolates in GBP <strong>correct to 2 decimal places</strong>.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">ii.b.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Consider the function \(f:x \mapsto \frac{{kx}}{{{2^x}}}\).</span></p>
</div>
<div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The cost per person, in euros, when \(x\) people are invited to a party can be determined by the function </span></p>
<p style="text-align: center;"><span style="font-family: times new roman,times; font-size: medium;">\(C(x) = x + \frac{{100}}{x}\)</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Given that \(f(1) = 2\), show that \(k = 4\).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">i.a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Write down the values of \(q\) and \(r\) for the following table.</span></p>
<p><span><img src="data:image/png;base64,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" alt></span></p>
<div class="marks">[2]</div>
<div class="question_part_label">i.b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>As \(x\) increases from \( - 1\), the graph of \(y = f(x)\) reaches a maximum value and then decreases, behaving asymptotically.</span></p>
<p><span><span>Draw the graph of \(y = f(x)\) for \( - 1 \leqslant x \leqslant 8\). Use a scale of \({\text{1 cm}}\) to represent 1 unit on both axes. The position of the maximum, </span></span><span><span><span><span>\({\text{</span></span>M}}\), the</span> <span>\(y\)-intercept and the asymptotic behaviour should be clearly shown.</span></span></p>
<div class="marks">[4]</div>
<div class="question_part_label">i.c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Using your graphic display calculator, find the coordinates of \({\text{M}}\), the maximum point on the graph of \(y = f(x)\).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">i.d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Write down the equation of the horizontal asymptote to the graph of \(y = f(x)\).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">i.e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>(i) Draw and label the line \( y = 1\) on your graph.</span></p>
<p><span>(ii) The equation \(f(x) = 1\) has two solutions. One of the solutions is \(x = 4\). Use your <strong>graph</strong> to find the other solution.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">i.f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Find \(C'(x)\).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">ii.a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Show that the cost per person is a minimum when \(10\) people are invited to the party.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">ii.b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Calculate the minimum cost per person.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">ii.c.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the curve <em>y</em> = 2<em>x</em><sup>3</sup> − 9<em>x</em><sup>2</sup> + 12<em>x</em> + 2, for −1 < <em>x</em> < 3</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the curve for −1 < <em>x</em> < 3 and −2 < <em>y</em> < 12.</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>A teacher asks her students to make some observations about the curve.</p>
<p>Three students responded.<br><strong>Nadia</strong> said <em>“The x-intercept of the curve is between −1 and zero”.</em><br><strong>Rick</strong> said <em>“The curve is decreasing when x < 1 ”.</em><br><strong>Paula</strong> said <em>“The gradient of the curve is less than zero between x = 1 and x = 2 ”.</em></p>
<p>State the name of the student who made an <strong>incorrect</strong> observation.</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 <em>y</em> when <em>x</em> = 1 .</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find \(\frac{{{\text{dy}}}}{{{\text{dx}}}}\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the stationary points of the curve are at <em>x</em> = 1 and <em>x</em> = 2.</p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that <em>y</em> = 2<em>x</em><sup>3</sup> − 9<em>x</em><sup>2</sup> + 12<em>x</em> + 2 = <em>k</em> has <strong>three</strong> solutions, find the possible values of <em>k</em>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">f.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-size: medium; font-family: times new roman,times;">On Monday Paco goes to a running track to train. He runs the first lap of the track</span> <span style="font-size: medium; font-family: times new roman,times;">in 120 seconds. Each lap Paco runs takes him 10 seconds longer than his previous lap.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Find the time, in seconds, Paco takes to run his fifth lap.</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>Paco runs his last lap in 260 seconds.</span></p>
<p><span>Find how many laps he has run on Monday.</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>Find the <strong>total</strong> time, in <strong>minutes,</strong> run by Paco on Monday.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>On Wednesday Paco takes Lola to train. They both run the first lap of the track in 120 seconds. Each lap Lola runs takes 1.06 times as long as her previous lap.</span></p>
<p><span>Find the time, in seconds, Lola takes to run her third lap.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Find the <strong>total</strong> time, in seconds, Lola takes to run her first four laps.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Each lap Paco runs again takes him 10 seconds longer than his previous lap. After a certain number of laps Paco takes less time per lap than Lola.</span></p>
<p><span>Find the number of the lap when this happens.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">f.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the function \(f(x) = - {x^4} + a{x^2} + 5\), where \(a\) is a constant. Part of the graph of \(y = f(x)\) is shown below.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-16_om_17.47.40.png" alt="M17/5/MATSD/SP2/ENG/TZ2/06"></p>
</div>
<div class="specification">
<p>It is known that at the point where \(x = 2\) the tangent to the graph of \(y = f(x)\) is horizontal.</p>
</div>
<div class="specification">
<p>There are two other points on the graph of \(y = f(x)\) at which the tangent is horizontal.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the \(y\)-intercept of the graph.</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 \(f'(x)\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that \(a = 8\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find \(f(2)\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the \(x\)-coordinates of these two points;</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the intervals where the gradient of the graph of \(y = f(x)\) is positive.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the range of \(f(x)\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the number of possible solutions to the equation \(f(x) = 5\).</p>
<div class="marks">[1]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The equation \(f(x) = m\), where \(m \in \mathbb{R}\), has four solutions. Find the possible values of \(m\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">g.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">A water container is made in the shape of a cylinder with internal height \(h\) cm and internal base radius \(r\) cm.</p>
<p class="p2" style="text-align: center;"><img src="images/Schermafbeelding_2017-03-07_om_08.31.01.png" alt="N16/5/MATSD/SP2/ENG/TZ0/06"></p>
<p class="p1">The water container has no top. The inner surfaces of the container are to be coated with a water-resistant material.</p>
</div>
<div class="specification">
<p class="p1">The volume of the water container is \(0.5{\text{ }}{{\text{m}}^3}\).</p>
</div>
<div class="specification">
<p class="p1">The water container is designed so that the area to be coated is minimized.</p>
</div>
<div class="specification">
<p class="p1">One can of water-resistant material coats a surface area of \(2000{\text{ c}}{{\text{m}}^2}\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Write down a formula for \(A\), <span class="s1">the surface area to be coated.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Express this volume in \({\text{c}}{{\text{m}}^3}\).</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1"><span class="s1">Write down, in terms of \(r\) </span>and \(h\), an equation for the volume of this water container.</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 class="p1">Show that \(A = \pi {r^2}\frac{{1\,000\,000}}{r}\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Show that \(A = \pi {r^2} + \frac{{1\,000\,000}}{r}\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find \(\frac{{{\text{d}}A}}{{{\text{d}}r}}\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Using your answer to part (e), find the value of \(r\) <span class="s1">which minimizes \(A\).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the value of this minimum area.</p>
<div class="marks">[2]</div>
<div class="question_part_label">g.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the least number of cans of water-resistant material that will coat the area in <span class="s1">part (g).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">h.</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 sequence \({u_1},{\text{ }}{u_2},{\text{ }}{u_3},{\text{ }} \ldots ,{\text{ }}{{\text{u}}_n},{\text{ }} \ldots \) where</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">\[{u_1} = 600,{\text{ }}{u_2} = 617,{\text{ }}{u_3} = 634,{\text{ }}{u_4} = 651.\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The sequence continues in the same manner.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Find the value of \({u_{20}}\).</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 sum of the first 10 terms of the sequence.</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>Now consider the sequence \({v_1},{\text{ }}{v_2},{\text{ }}{v_3},{\text{ }} \ldots ,{\text{ }}{v_n},{\text{ }} \ldots \) where</span></p>
<p><span>\[{v_1} = 3,{\text{ }}{v_2} = 6,{\text{ }}{v_3} = 12,{\text{ }}{v_4} = 24\]</span></p>
<p><span>This sequence continues in the same manner.</span></p>
<p><span>Find the exact value of \({v_{10}}\).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Now consider the sequence \({v_1},{\text{ }}{v_2},{\text{ }}{v_3},{\text{ }} \ldots ,{\text{ }}{v_n},{\text{ }} \ldots \) where</span></p>
<p><span>\[{v_1} = 3,{\text{ }}{v_2} = 6,{\text{ }}{v_3} = 12,{\text{ }}{v_4} = 24\]</span></p>
<p><span>This sequence continues in the same manner.</span></p>
<p><span>Find the sum of the first 8 terms of this sequence.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>\(k\) is the smallest value of \(n\) for which \({v_n}\) is greater than \({u_n}\).</span></p>
<p><span>Calculate the value of \(k\).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">In the month before their IB Diploma examinations, eight male students recorded the number of hours they spent on social media.</p>
<p class="p2">For each student, the number of hours spent on social media <span class="s1">(\(x\)) </span>and the number of IB Diploma points obtained <span class="s1">(\(y\)) </span>are shown in the following table.</p>
<p class="p2" style="text-align: center;"><img src="images/Schermafbeelding_2017-03-07_om_07.43.52.png" alt="N16/5/MATSD/SP2/ENG/TZ0/01"></p>
</div>
<div class="specification">
<p class="p1">Use your graphic display calculator to find</p>
</div>
<div class="specification">
<p class="p1">Ten female students also recorded the number of hours they spent on social media in the month before their IB Diploma examinations. Each of these female students spent between <span class="s1">3 </span>and <span class="s1">30 </span>hours on social media.</p>
<p class="p1">The equation of the regression line <span class="s1"><em>y </em></span>on <span class="s1"><em>x </em></span>for these ten female students is</p>
<p class="p1">\[y = - \frac{2}{3}x + \frac{{125}}{3}.\]</p>
<p class="p1">An eleventh girl spent <span class="s1">34 </span>hours on social media in the month before her IB Diploma examinations.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1"><span class="s1">On graph paper, draw a scatter diagram for these data. Use a scale of </span><span class="s2">2 cm </span>to represent <span class="s2">5 </span>hours on the \(x\)-axis and <span class="s2">2 cm </span>to represent <span class="s2">10 </span>points on the \(y\)-axis.</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 class="p1"><span class="s1">(i) <span class="Apple-converted-space"> \({\bar x}\)</span>, </span>the mean number of hours spent on social media;</p>
<p class="p2">(ii) <span class="Apple-converted-space"> \({\bar y}\)</span>, the mean number of IB Diploma points.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1"><span class="s1">Plot the point \((\bar x,{\text{ }}\bar y)\) </span>on your scatter diagram and label this point <span class="s2">M</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 class="p1"><span class="s1">Write down the value of \(r\), </span>the Pearson’s product–moment correlation coefficient, for these data.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Write down the equation of the regression line \(y\) <span class="s1">on \(x\) for these eight male students.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Draw the regression line, from part (e), on your scatter diagram.</p>
<div class="marks">[2]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Use the given equation of the regression line to estimate the number of IB Diploma <span class="s1">points that this girl obtained.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">g.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Write down a reason why this estimate is not reliable.</p>
<div class="marks">[1]</div>
<div class="question_part_label">h.</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 function \(f(x) = \frac{3}{4}{x^4} - {x^3} - 9{x^2} + 20\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Find \(f( - 2)\).</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">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>The graph of the function \(f(x)\) has a local minimum at the point where \(x = - 2\).</span></p>
<p><span>Using your answer to part (b), show that there is a second local minimum at \(x = 3\).</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>The graph of the function \(f(x)\) has a local minimum at the point where \(x = - 2\).</span></p>
<p><span>Sketch the graph of the function \(f(x)\) for \( - 5 \leqslant x \leqslant 5\) and \( - 40 \leqslant y \leqslant 50\). Indicate on your</span></p>
<p><span>sketch the coordinates of the \(y\)-intercept.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>The graph of the function \(f(x)\) has a local minimum at the point where \(x = - 2\).</span></p>
<p><span>Write down the coordinates of the local maximum.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Let \(T\) be the tangent to the graph of the function \(f(x)\) at the point \((2, –12)\).</span></p>
<p><span>Find the gradient of \(T\).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>The line \(L\) passes through the point \((2, −12)\) and is perpendicular to \(T\).</span></p>
<p><span>\(L\) has equation \(x + by + c = 0\), where \(b\) and \(c \in \mathbb{Z}\).</span></p>
<p><span>Find</span></p>
<p><span>(i) the gradient of \(L\);</span></p>
<p><span>(ii) the value of \(b\) and the value of \(c\).</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">g.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Consider the function \(f(x) = \frac{{96}}{{{x^2}}} + kx\), where \(k\) is a constant and \(x \ne 0\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Write down \(f'(x)\).</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 class="p1">The graph of \(y = f(x)\) has a local minimum point at \(x = 4\).</p>
<p class="p1">Show that \(k = 3\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">The graph of \(y = f(x)\) has a local minimum point at \(x = 4\).</p>
<p class="p1">Find \(f(2)\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">The graph of \(y = f(x)\) has a local minimum point at \(x = 4\).</p>
<p class="p1">Find \(f'(2)\)</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">The graph of \(y = f(x)\) has a local minimum point at \(x = 4\).</p>
<p class="p1">Find the equation of the normal to the graph of \(y = f(x)\) at the point where \(x = 2\).</p>
<p class="p1">Give your answer in the form \(ax + by + d = 0\) where \(a,{\text{ }}b,{\text{ }}d \in \mathbb{Z}\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">The graph of \(y = f(x)\) has a local minimum point at \(x = 4\).</p>
<p class="p1"><span class="s1">Sketch the graph of \(y = f(x)\)</span>, for \( - 5 \leqslant x \leqslant 10\) and \( - 10 \leqslant y \leqslant 100\)<span class="s1">.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">The graph of \(y = f(x)\) has a local minimum point at \(x = 4\).</p>
<p class="p1">Write down the coordinates of the point where the graph of \(y = f(x)\) intersects the \(x\)-axis.</p>
<div class="marks">[2]</div>
<div class="question_part_label">g.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">The graph of \(y = f(x)\) has a local minimum point at \(x = 4\).</p>
<p class="p1">State the values of \(x\) for which \(f(x)\) is decreasing.</p>
<div class="marks">[2]</div>
<div class="question_part_label">h.</div>
</div>
<br><hr><br><div class="specification">
<p>Violeta plans to grow flowers in a rectangular plot. She places a fence to mark out the perimeter of the plot and uses 200 metres of fence. The length of the plot is \(x\) metres.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-16_om_17.40.47.png" alt="M17/5/MATSD/SP2/ENG/TZ2/05"></p>
</div>
<div class="specification">
<p>Violeta places the fence so that the area of the plot is maximized.</p>
</div>
<div class="specification">
<p>By selling her flowers, Violeta earns 2 Bulgarian Levs (BGN) per square metre of the plot.</p>
</div>
<div class="specification">
<p>Violeta wants to invest her 5000 BGN.</p>
<p>A bank offers a nominal annual interest rate of 4%, compounded <strong>half-yearly</strong>.</p>
</div>
<div class="specification">
<p>Another bank offers an interest rate of \(r\)% compounded <strong>annually</strong>, that would allow her to double her money in 12 years.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the width of the plot, in metres, is given by \(100 - x\).</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 area of the plot in terms of \(x\).</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 \(x\) that maximizes the area of the plot.</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>Show that Violeta earns 5000 BGN from selling the flowers grown on the plot.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the amount of money that Violeta would have after 6 years. Give your answer correct to two decimal places.</p>
<div class="marks">[3]</div>
<div class="question_part_label">e.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find how long it would take for the interest earned to be 2000 BGN.</p>
<div class="marks">[3]</div>
<div class="question_part_label">e.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the lowest possible value for \(r\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">f.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the function \(f(x) = 0.3{x^3} + \frac{{10}}{x} + {2^{ - x}}\).</p>
</div>
<div class="specification">
<p>Consider a second function, \(g(x) = 2x - 3\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate \(f(1)\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of \(y = f(x)\) for \( - 7 \leqslant x \leqslant 4\) and \( - 30 \leqslant y \leqslant 30\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the equation of the vertical asymptote.</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 down the coordinates of the \(x\)-intercept.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the possible values of \(x\) for which \(x < 0\) and \(f’(x) > 0\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the solution of \(f(x) = g(x)\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">f.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Consider the functions \(f(x) = \frac{{2x + 3}}{{x + 4}}\) and \(g(x) = x + 0.5\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Sketch the graph of the function \(f(x)\), for \( - 10 \leqslant x \leqslant 10\) . Indicating clearly the axis intercepts and any asymptotes.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Write down the equation of the vertical asymptote.</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>On the same diagram as part (a) sketch the graph of \(g(x) = x + 0.5\) .</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>Using your graphical display calculator write down the coordinates of <strong>one</strong> of the points of intersection on the graphs of \(f\) and \(g\), <strong>giving your answer correct to five decimal places</strong>.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Write down the gradient of the line \(g(x) = x + 0.5\) .</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>The line \(L\) passes through the point with coordinates \(( - 2{\text{, }} - 3)\) and is perpendicular to the line \(g(x)\) . Find the equation of \(L\).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">f.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the function \(g(x) = {x^3} + k{x^2} - 15x + 5\).</p>
</div>
<div class="specification">
<p>The tangent to the graph of \(y = g(x)\) at \(x = 2\) is parallel to the line \(y = 21x + 7\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find \(g'(x)\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that \(k = 6\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the equation of the tangent to the graph of \(y = g(x)\) at \(x = 2\). Give your answer in the form \(y = mx + c\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use your answer to part (a) and the value of \(k\), to find the \(x\)-coordinates of the stationary points of the graph of \(y = g(x)\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find \(g’( - 1)\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence justify that \(g\) is decreasing at \(x = - 1\).</p>
<div class="marks">[1]</div>
<div class="question_part_label">d.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the \(y\)-coordinate of the local minimum.</p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The line \({L_1}\) has equation \(2y - x - 7 = 0\) and is shown on the diagram.</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2017-03-07_om_07.59.34.png" alt="N16/5/MATSD/SP2/ENG/TZ0/03"></p>
<p class="p1" style="text-align: left;">The point <span class="s1">A </span>has coordinates \((1,{\text{ }}4)\).</p>
</div>
<div class="specification">
<p class="p1">The point <span class="s1">C </span>has coordinates \((5,{\text{ }}12)\). <span class="s1">M </span>is the midpoint of <span class="s1">AC</span>.</p>
</div>
<div class="specification">
<p class="p1">The straight line, \({L_2}\), is perpendicular to <span class="s1">AC </span>and passes through <span class="s1">M</span>.</p>
</div>
<div class="specification">
<p class="p1">The point <span class="s1">D </span>is the intersection of \({L_1}\) and \({L_2}\).</p>
</div>
<div class="specification">
<p class="p1">The length of <span class="s1">MD </span>is \(\frac{{\sqrt {45} }}{2}\).</p>
</div>
<div class="specification">
<p class="p1">The point <span class="s1">B </span>is such that <span class="s1">ABCD </span>is a rhombus.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1"><span class="s1">Show that </span><span class="s2">A </span>lies on \({L_1}\).</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">M</span><span class="s2">.</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 class="p1">Find the length of <span class="s1">AC</span><span class="s2">.</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 class="p1">Show that the equation of \({L_2}\) <span class="s1">is \(2y + x - 19 = 0\).</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the coordinates of <span class="s1">D</span><span class="s2">.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1"><span class="s1">Write down the length of </span><span class="s2">MD </span>correct to five significant figures.</p>
<div class="marks">[1]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the area of <span class="s1">ABCD</span><span class="s2">.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">g.</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 defined by \(f(x) = 1.5x + 4 + \frac{6}{x}{\text{, }}x \ne 0\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Write down the equation of the vertical asymptote.</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">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Find the gradient of the graph of the function at \(x = - 1\).</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>Using your answer to part (c), decide whether the function \(f(x)\) is increasing or decreasing at \(x = - 1\). Justify your answer.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Sketch the graph of \(f(x)\) for \( - 10 \leqslant x \leqslant 10\) and \( - 20 \leqslant y \leqslant 20\) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>\({{\text{P}}_1}\) is the local maximum point and \({{\text{P}}_2}\) is the local minimum point on the graph of \(f(x)\) .</span></p>
<p><span>Using your graphic display calculator, write down the coordinates of</span></p>
<p><span>(i) \({{\text{P}}_1}\) ;</span></p>
<p><span>(ii) \({{\text{P}}_2}\) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Using your sketch from (e), determine the range of the function \(f(x)\) for \( - 10 \leqslant x \leqslant 10\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">g.</div>
</div>
<br><hr><br><div class="specification">
<p>A pan, in which to cook a pizza, is in the shape of a cylinder. The pan has a diameter of 35 cm and a height of 0.5 cm.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-16_om_11.14.51.png" alt="M17/5/MATSD/SP2/ENG/TZ1/04"></p>
</div>
<div class="specification">
<p>A chef had enough pizza dough to exactly fill the pan. The dough was in the shape of a sphere.</p>
</div>
<div class="specification">
<p>The pizza was cooked in a hot oven. Once taken out of the oven, the pizza was placed in a dining room.</p>
<p>The temperature, \(P\), of the pizza, in degrees Celsius, °C, can be modelled by</p>
<p>\[P(t) = a{(2.06)^{ - t}} + 19,{\text{ }}t \geqslant 0\]</p>
<p>where \(a\) is a constant and \(t\) is the time, in minutes, since the pizza was taken out of the oven.</p>
<p>When the pizza was taken out of the oven its temperature was 230 °C.</p>
</div>
<div class="specification">
<p>The pizza can be eaten once its temperature drops to 45 °C.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate the volume of this pan.</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 radius of the sphere in cm, correct to one decimal place.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of \(a\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the temperature that the pizza will be 5 minutes after it is taken out of the oven.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate, to the nearest second, the time since the pizza was taken out of the oven until it can be eaten.</p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>In the context of this model, state what the value of 19 represents.</p>
<div class="marks">[1]</div>
<div class="question_part_label">f.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The following table shows the number of bicycles, \(x\), produced daily by a factory and their total production cost, \(y\)<span class="s1">, in US dollars (USD)</span>. The table shows data recorded over seven days.</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2015-12-22_om_10.06.31.png" alt></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) Write down the Pearson’s product–moment correlation coefficient, \(r\), for these data.</p>
<p class="p1">(ii) Hence comment on the result.</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 class="p1">Write down the equation of the regression line \(y\) on \(x\) for these data, in the form \(y = ax + b\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Estimate the total cost, <strong>to the nearest </strong><span class="s1"><strong>USD</strong></span>, of producing \(13\)<span class="s1"> </span>bicycles on a particular day.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">All the bicycles that are produced are sold. The bicycles are sold for <span class="s1">304 USD </span><strong>each</strong>.</p>
<p class="p1">Explain why the factory does <strong>not </strong>make a profit when producing \(13\)<span class="s1"> </span>bicycles on a particular day.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">All the bicycles that are produced are sold. The bicycles are sold for <span class="s1">304 USD </span><strong>each</strong>.</p>
<p class="p1">(i) Write down an expression for the total selling price of \(x\) bicycles.</p>
<p class="p1">(ii) Write down an expression for the <strong>profit </strong>the factory makes when producing \(x\) bicycles on a particular day.</p>
<p class="p1">(iii) Find the least number of bicycles that the factory should produce, on a particular day, in order to make a profit.</p>
<div class="marks">[5]</div>
<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;">The diagram below shows the graph of a line \(L\) passing through (1, 1) and (2 , 3) and the graph \(P\) of the function \(f (x) = x^2 − 3x − 4\)</span></p>
<p><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>Find the gradient of the line <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>Differentiate \(f (x)\) .</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 coordinates of the point where the tangent to <em>P</em> is parallel to the line <em>L</em>.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Find the coordinates of the point where the tangent to <em>P</em> is perpendicular to the line<em> L</em>.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Find</span></p>
<p><span>(i) the gradient of the tangent to <em>P</em> at the point with coordinates (2, − 6).</span></p>
<p><span>(ii) the equation of the tangent to <em>P</em> at this point.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>State the equation of the axis of symmetry of <em>P</em>.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Find the coordinates of the vertex of <em>P</em> and state the gradient of the curve at this point.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">g.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-size: medium; font-family: times new roman,times;">The diagram shows an <strong>aerial</strong> view of a bicycle track. The track can be modelled by the quadratic function</span></p>
<p><span style="font-size: medium; font-family: times new roman,times;">\(y = \frac{{ - {x^2}}}{{10}} + \frac{{27}}{2}x\), where \(x \geqslant 0,{\text{ }}y \geqslant 0\)</span></p>
<p><span style="font-size: medium; font-family: times new roman,times;">(<em>x</em> , <em>y</em>) are the coordinates of a point <em>x</em> metres east and <em>y</em> metres north of O , where O is the origin (0, 0) . B is a point on the bicycle track with coordinates (100, 350) .<br></span></p>
<p><span style="font-size: medium; font-family: times new roman,times;"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAY0AAAFfCAIAAAA9MweiAAAgAElEQVR4nO3db2wb530H8HuXvvAGuO2bFA2Oy82FJZRLUsOrgnVWDLiDqg6MajSOURWpM1mI5aGrl+ZPIarSkrktZKNSrGZW4C6lYtpDMVSlYKN/Zjq4OnHZTDYBBnEw5rK4cEyaqwXGJS6hrMqPnr2gdHx4PJIixbt77rnvB3ph6WTeI4r68nme+z3PSdQuhbd/OvrYQ1sk6aHDv7tFKaX0D68+86C09V9eKxDbTgoAApJsffSVP5w9sHnzF6bfukMppcs3E8ce3XJg9sayrScFAMHYm1P0zlvTX9h8z3df+5BSSulKIfGDb8eyK/aeEwAEY3NO0fdnH1Okx2ZvUEqpfuWlsZeu6DafEQBEY3dO5V/77oPSwyevrqwsXnn5n196c9Hm8wGAeOzOKf3y0Yck5eil/H9PDf/svT9hyAcATbM7p25fPdkv3f3ot54eP5u9bfO5AMAbVvT337xc8tb7+h32yPKCdrlMW1heofbn1PKN2SekTQ8fvZRHVwoASlb0a5fPnz786H2SdM/OMfVmOR1Wlhe0yxdeObT97q0H/v03TuXUn7Kz3/nH2auoRAAAk5U/vvrsPZIkPbDv5NuVM9f65aNffmz2feNzW3Pqjn7lP74fSenoSgFAteXLRxVJkiRp05e/l1hgcsKJnFpZfCc2NvrK+fj0s9+P31hGSgGAleXLR5XHf/yz8S9ukqQt35q9ZnSqnMip2++d/PqmTV94YvoiQgoAalq+fFR5YvZG4b2fDm2RpE1fnLxUKM2pOzruAwCobTWnlunKQuJ7X94kbdp66Gx2eQU5BQDcMHKKUrr49sl9D0hSIDR1WV9BTgEAJ9iconTlpjq28x5JeujZX7+dsM6pgjrSEVDk0kf/jFa+Skhy89FwSJEDwYHpS7kl9ix1DgEANFCZU5Te0a/M7NuySdrc88juB6tzaikbOxSU13IqFNGMHaL0+cme3WNqltClnHq4t2cqqZPGhwAAGjLnFKX09o1fj2yXJElSqnJKn58MPa9abF+3qEX6g2G1sPrpghre1RdJkwaHAADW4fbvDgeeMG9It/LHt176+uaqnCIFdTQod/aGX55TtYpdV0h6JvQAkz6koI4GS72tOocAABr48OprsVcOf22LdN+jhyOzr12tqEdfvnb20Bcrc4qkZ0KdazNTnb3hmLY2fCNapE/eMaIurH1zKdH6Z7TFOodWH9eKrT83AAhjRc/+fqG8c8FadpBc8szLIz2ditzZOzGvU1odPcxXPqp9qOYGU8gpAGhNZXaQrDoaUjpG1QJBTgEAJ6qyo6COdKwO6DYy7rM4E3IKAGq4fv16aXbo+vXr1UersoOkZ0L7VuOGpGdCu5gwWtQi/Up5Hr3GoRqQUwBg6fr169u3by/l1Pbt26ujyqI/NTZqVBu0sy4BOQUA1Uoh9aMf/aiUUxcvXqyOKonkkmfPJHOEUkpJLjEVnlDZyvL21XkipwDAxAgpulYkQCmtjiqJ5M6P9XQqckDp2D85e0GryhqSS0wNdClyZ2/4dNK8bqbmoWrIKQBg5fN5I6Qok1OU0osXLz799NPGdzqXHcgpAGAVi8V33nnH+NRUZZnP58uHHGsTcgoA6qhTDY6cAgAuIKcAgHfIKQDgHXIKAHiHnAIA3iGnAIB3yCkA4B1yCgB4h5wCAN4hpwCAd8gpAOBOPp9/mlHKKePT2dlZ4zuRUwDgmncYpZwyPsU6ZADgDsZ9AMA75BQA8A45BQC8Q04BFzKZjKZpc7HYXCw2fXw6PDxc56P0bfF4XNM0dkoVRIWcAnfk8/lEIjF9fPrg0JAiBzbysbO7uxReqVSqWCy6/ZNB+yGnwDmlbDoyPr53z54NZlOdj4NDQ6eiUWSWSJBTYLt8Ph+Px9fZbzo4NBQeHj4yPl4a3Jk+TkWjpaHfepJuZ3f3kfHxRCKBwPI65BTYKJFI1I+ng0ND08enW55pMma16vfRdnZ3Tx+f1jTNjp8RHICcgvbL5/OnotGd3d0Oj8uMOa9ap967Z088Hkf3ynOQU9BOmqaFh4ctMyI8PByPxx27PJfJZE5Fo5adrFL3CmnlIcgpaI9UKmU5xCt1YVysHtA0bfr4tGV0Th+fRlmDJyCnYKNq9aGOjI/zMyVULBYTiYRl9wppxT/kFLQun88fGR+v/sufi8W4/cu3TNWd3d1zsRhGgtxCTkErisXiqWjUu3/tmUymejC4s7s7kUi43TSwgJyCpiUSieoLal5JKJZlf/Dg0FAmk3G7aVABOQVNyOfz1YMmr8/vaJpWfQXAi7ErMOQUrFc8Hq8uNRCm61E9y753zx5hfjqBIadgVSaTMXWjdnZ3x+Nxt9vVZpaTbqeiUbfbBfUgp4BSq9moI+PjAo+JqoeBmLHiGXLK74rFommaeWd3dyqVcrtdTqjuWInXfxQDcsrXMpmMqVvht7UmmqaZZqz89gx4AnLKv0xT5r4tLKqescIYkDfIKZ8yFUDiLzOVSrEzdP4Z/HoCcsp3isWi6boeRjol1aPguVjM7UYBpcgpv6n+U/TnWK+WYrFo6mmKfd3TK5BTPqJpmmlo4/OxXi2mmbuDQ0OIKnchp/wikUjgb2/9TJmOp8tdyClfMHUQMCG1HplMhi1ZQPfTRcgp8ZkmXKaPT7vdIs8oFouYzuMBckpwppBCvXWzEFU8QE6JjA0pFARthCnuEVUOQ06JyXR9HXMrG2eqWUdUOQk5JSDTUAUh1S6ma6aIKsdUZge5NndgR18kTdiv5eaj4ZAiB4ID05dyS3R9hyzOhJxyBELKVogqV7DZsZSNHQrKnRU5pc9P9uweU7OELuXUw709U0mdND5keSbklP0QUg5AVDnPyA6iJ18cfPbZwQ42pxa1SH8wrBZWP11Qw7vWjtY5VONMyCn7YU7KGYgqh61lhz4/OfBiMnt+hM0pkp4JPcCkDymoo8FQRCN1D9U6E3LKZggpJyGqnFTKjlvJiacmk7doQWVzimiRPnnHiLqw9s2koI4G5f4ZbbHOoZpnQk7ZCSHlPESVY6TVEd/EvE5pZU5VR4/xlY9qH1r9imTF4Z/NPxBSbmGjCs+8fSSqz0+NnsmWelDtyymLMyGn7MGu3cOfivMQVQ6QkseOzmXXSgow7vMa0x8JKs5dYerPevqGrHySRjoq7rex9tE/oy1Skp4J7WLCaFGL9CvlefQah2qdCTnVbpqmsb81hJSL2KjCJjBtV5kdlf0p1CXwLJPJsBskYRLXdWzlGqKqvernFOo8OVUsFtmtkXA7Xx6Yimyxf04bNcopSkkuMTXQpcidveHTSfO6mZqHLM6EnGoT/D1wy/T+gdtAtAvWIXsPe/vig0NDbjcHKmA8bgfklMdgvpZ/pusbmqa53SLPQ055iakKAde/uWX6TeHtZIOQU56RyWRQT+gh7L566PluEHLKG/L5PGY9PIe93HFkfNzt5ngYcsobcIHPi0xXZnETjZYhpzyAHUGEh4fdbg40wXT5D3PqrUFO8Q5z516XSqUwp75ByCmumd6NMXfuUXOxGObUNwI5xS/T7Abmzj0tPDyMdU4tQ07xi607x9y515nedbC5RVOQU5xid7/DSEEMbJ1681ONtxe0VNLs3YXlFbuayxPkFI9Q0imqDbz93F7Q5v/rxSe2ywHl3t5/+sEPJyd+OB7e33v/lm27R15548ayja12H3KKOxggiI0dzjc9UZWLDcoB5f6J5GosrSzf/O0Lj9yn3Lv3394s1P+vnoac4g670hgTruIxvQ81V1FlzilK6fLNXzy1VQ4wm1YKCDnFF7bWBtNSojJNVDXxW7bIqTsfnPvOVjmw9alzH9jSWC4gpzjCLuLDtJTYTBVV6/1vVTm1or/548e2KXLvv168KfCMOnKKI1gL5itsRdV6f92lnLp3x1f/Yf/gwP7Bga/13q8o937l+V+8qwucUsgpfrBvsFhb7wfFYtG02EDTtAad6FJOffY7v/y/23R5QUu+/ssTT/feG1A+9/gLr2cEvuSHnOICO2Gxd88eTEv5BDsdaXw8+NefPxWNWr8GLOen1Oc+LweUz4XP3RQ2qZBT7jNt/o8l9b4S+clPqqNKkQP7vvENi6iyyClKb78x/lcBRf7SZFJ3sOGOQk65j922BXco8ZtvP/mkZU4pcmDq2DHzd1vl1Er+3DMda/cGFhRyymXsiA83j/GbfD5fK6RKH+YulcX1vnd/+dxXPiN/9u/Hf/uBuFPpyCk3mUZ8KETwG8v5KfaDmQT48PcXz5x6bu9WOaDIyraerw0O7B/s/9I2OfCZv/3GczNv5IRe6IecchNGfD7HLvdrlFO+hpxyDUZ8YLrTX/UHtm8tQU65AyM+KOkLPWyZUH8Z+ItvP/mk263jBXLKHexiY4z4/KzOFNXrr7/udut4gZxyAUZ8wEqlUn/T9WB1TqHi14Cccho74sNiYzCkUqm5WGwuFjv2wgtGVGFjnxLklNNwjQ8aYlek452MIqcchhEfrAe78fTePXvcbo77kFOOYrfywPsk1IF+Nws55Ry2qA+vPGiIrVzxeSEVcsohpr06cR0HGmJnCcLDw243x03IKYewIz7cQgbWia2z8/PLBjnlBPaNEXt1wvqxe376uRuOnLKdqWDK5xMN0Cx2WtO35VTIKdvhwg1sEC4TI6fsxRbCoGAKWsO+ivw5oY6cshf7Toi9hKBlbK/ch/dMQ07ZKJFIGK+t6ePTbjcHPMw0oe63WU7klF1M0+e+vVID7cLuAOO3tz3klF3Y+4b6sKMOdvDthDpyyhbsfUQwfQ7twr6ufDWhjpyyBabPwSb+nFBHTrUfW33ut3kEsJs/K9SRU+2H6nOwFXsd2SeVw8ipNmOnz33yGgLnsRMLfngvlCillOQuHdsflAOKHBqJpfXK7yC5+Wg4pMiB4MD0pdzSOg9ZnMkHOcVu3oJtGME+ftvyRaL0Vio2eym3ROlSbn56sGPHiLpQPq7PT/bsHlOzhC7l1MO9PVNJnTQ+ZHkmH+QUduEAx7AvNuGv1UjkvcSFrNEVWlDDO4JhtbD66aIW6Wc+XVDDu/oiadLgUI0ziZ5T7CIsbN4CdvNV570yO0h6JrR3MnmL+fQBJn1IQR0NhiIaqXuo1plEzym/TRmA69jJ0EQi4XZzbGRkB9E1dSb8zTE1a0QN0SJ9MjsMJAV1NCj3z2iLdQ7VPJPQOYVaBHCef2oUStmxoIZ3KHJAkTt7wzFtdZqpOnqMr3xU+9DqVyQrTv5gDsNSPnCFT2oUmOwgueTJcK8cCB6IZQndYE5ZnEncnPLJawX4xN6UVNQJB1N2LGqRfqVjVC0QinHf+rD7Igg/nQkc8sOcgzk7iBbpW8spStIzoV1MGC1qkX6lPI9e41CtMwmaU/6ZywRuHRkfF7tLZcoOUlBHt62O+yjqEhry1bVh4JbwGxNL2dihYE84mswRSknu/FjomzPpQvk46jzr8lWtHfBM7JeipKejgx2lH69rcCKWrFr+QnKJqYEuRe7sDZ9OmtfN1DxkcSbhcsq3mwEBh9iuvXivRqxDbh0KO4Er7FSpYF0q5FSL/HCRBbyFLfsUbLYUOdUidKaAQ6JefUZOtYLtTPn2VtrAIVM1nzBLI5BTrTAqgLFKBngj5OoI5FTT2NuoCfM6AJGIt9oUOdU08V4EIBh2XkKMt1LkVHOE7FSDeIzrPGK8myKnmoPOFHiCYF0q5FQT4vG4kBd9QUgilc4gp5ogahEdCEmk6hnk1HqxM1PoTIEnCDNLhZxaF4FXJIDAhFndhZxaF3SmwKPEmKVCTjWGzhR4lxhdKuRUY7jMB54mQJcKOdUYbtMAnsbuSuzRLhVyqgFRN8oAX2F3JfZilwo51QA6UyAAdo9sL3apkFP1oDMFwvB0lwo5VQ8u84EwPH3hDzlVE2qmQDDevR0pcqomzEyBYNgLf95a8YecsobOFAjJ6FJ5a8UfcsoaOlMgJI/uS4WcsoDOFAjMi5soIKcsoDMFAvNilwo5ZcbeTgadKRCS57pUyCkz9lfodlsAbMF2qeLxuNvNaQw5VcGLXWKAFnhrcgM5VcFz/WGA1njrYhFyqoytgkNnCoTnoS4VcqqMXaiJzhQIz0NdKuTUKq9vfAHQAq90qZBTqzy96wVAa9idizRNc7s5NSGnKKW0WCyiMwU+xN6jJDw87HZzakJOUVr5rpLJZNxuDoBzTkWj/L/4kVOeeUsBsIMnZmaRU54ZogPYhP/JWeRU+ZIHOlPgT2yXis/98/yeU2wJSSqVcrs5AO7gfCWG33PKK/UjALbifGWrr3OK3cKFw98NgJN43inE1znFeV8XwEk8L6Pxb06xc4foTAFQjqdB/JtT/F+LBXAYt10qn+YUFsoAWOKz5tmnOYXaTgBL7J8GP+MMP+YUFsoA1MLnMhqJUqJr5yYHuhQ5oMihkZPzOVLxHSQ3Hw2HFDkQHJi+lFta5yGLM3GTU6jtBKiDw6lbiWR/NXXsnKYTSpdy89ODHZ29E/O6cVyfn+zZPaZmCV3KqYd7e6aSOml8yPJM3OQUtxc1AHjA4Qbc0nsX3siW42VRi/QrHaNqgRifBsNqYfXoghre1RdJkwaHapyJj5zi9ooGAD+M0kJO3stN2UEK6mjQyCmSngk9wKQPKaijwVBEI3UP1ToTHzl1ZHycq18AAId4u9uuRU5tOxAr9bCIFumTd4yoC+zRoNw/oy3WOVTzTBzkFP/rwgE4YUyPHBwacrst5pxaUMP7J5O3KKVW0WN85aPah7jOKQ4nCAH4xFXtDpsdRE++OFieRN9QTklWnPiBamPLEfi54ArAJ/bv5cj4uLuNYbJDn58Kn04z1+wEG/fF43F+3h8A+Mdune7u+GMtO8i1s8f+M20qLCDpmdAuJowWtUi/Up5Hr3Go1pnczins2wnQFH5qPiVKKSVZ9dgJ1SjU1K9EI4kCpSLVJbDbgPFw/QLAE4wpXXf3PpKonp4Lh4y/YUUOKHJnOXFEqfPEVlMALWDf4OPxuFvNkOYOdFWGVECpnGYiucTUQJcid/aGTyfN62ZqHrI4k3s5xWF9LYBXHBwacr3k0BfrkNlyBHSmAJrCLuFw6wKU+DmFraYANoKHAgXxc4otR+D2ttQAPHO9QEH8nOKq/B/Ai1xfcCZ4TqEcAaAtjAX8rtw1S/CcYp9czKADtIydTXd+/kTknMKdrwDa6ODQ0KloFPNTbeb65B8AtIXIOeX6xVQAaAthc4qH4jQAaAthc4q3DZ4BoGVi5hQW9AGIRMycwoI+AJEImFPYXxhAMALmFPYXBhCMgDllbJeDBX0AYhAtpzjZfhAA2ki0nMIMOoB4hMopbIkHICShcgpb4gEISaicwpZ4AEISJ6dSqRS2xAMQkjg55e5+gwBgH0FyyvX9mwHAPoLkFDuDji3xAAQjSE5hBh1AYCLkFG4qAyA2EXLKqEHHTWUAhOT5nEINOoDwPJ9T7t5WDAAc4Pmcwi4uAMLzdk6x+6BjFxcAUXk7p7CLC4AfeDinsA86gE94OKdwJ1EAn/BwThkz6LiTKIDYvJpTmEEH8A+v5tSpaBQz6AA+4dWcwgw6gH94MqfYrTtTqVS7HhYA+OTJnMLWnQC+4r2cYhceY+tOAD/wXk5h604Av/FeTmHhMYDfeCyn2Ps1YOtOAJ/wWE6hbArAhzyWU0bZ1JHx8Y0/GgB4gpdyCmVTAP7kpZwyyqaw8BjAV0rZQXTtwtzsxOD9o2qBmL6D5Oaj4ZAiB4ID05dyS+s8ZHGmjeUU7ngM4FsSpZRoka/u2/94R0DpqMopfX6yZ/eYmiV0Kace7u2ZSuqk8SHLM20sp1A2BeBbRnYsapH+qpxa1CL9wbBaWP10QQ3v6oukSYNDNc60sZzCHY8BfKtuTpH0TOgBJn1IQR0NhiIaqXuo1pk2kFPsblMomwLwm3o5RbRIn7xjRF0wvlBQR4Ny/4y2WOdQzTNtIKdQNgXgZ3Vyqjp6jK98VPuQLTmFsikAP7MrpyQrrTURZVMAPueBcZ9xkz7sNgXgTw3n0XcxYbSoRfqV8jx6jUO1ztRSTmG3KQDgvS6BLZvKZDItPAIAeF39nHK/ztNYK4OyKQDfkiiltKCOdATWui2dpm4RySWmBroUubM3fDppXjdT85DFmZrPKXatDG7SB+BbXK9DxloZAKCc55SxViY8PGxHkwDAE/jNKayVAYASfnMKa2UAoITfnMKd2QGghNOcwloZADBwmlNYKwMABk5zCoM+ADDwmFOJRMIY9GmaZmurAIB/POYU7isDACzucopdKzMXi9ndKgDgH3c5hbUyAGDCXU4dHBrCBgkAwOIrp7BBAgBU4yun2EEf1soAQAlfOWUM+rBBAgAYOMopdtCHDRIAwMBRTmGDBACwxFFOGbvi4WaiAMDiJafYXfGwQQIAsHjJKXbQ51iTAMATeMkpDPoAoBYuckrTNAz6AKAWLnIKu+IBQB1c5BR2xQOAOtzPKWyFDgD1uZ9TxqAPu+IBgCWXc6pYLBqdqVPRqGONAQAPcTmn2K3QM5mMY40BAA9xOaewFToANORmTmHQBwDr4WZOYdAHAOvhZk5h0AcA6+FaTmHQBwDr5FpOYdAHAOvkWk4Zgz7c/woA6nMnp9hBH256DAD1uZNTGPQBwPq5k1O40gcA6+dCTuFKHwA0xYWcwqAPAJriQk5h0AcATXE6p9hBXzwed+zsAOBdTucUO+jL5/OOnR0AvMvpnDo4NITyTgBoiqM5lc/nMegDgGY5mlMY9AFACxzNKazpA4AWOJdTH7vrLpR3AkALNpRTJDcfDYcUORAcmL6UW6r/zZ/8+CdQ3gkALdhATunzkz27x9QsoUs59XBvz1RSJ3W+/dN3fwrlnQDQgpZzalGL9AfDamH10wU1vKsvkq4VVFjTBwAtazWnSHom9AATTKSgjgZDEa1GUGFNH4AvLBdufXhn9d8rH936Y4PpoHVqMaeIFumTd4yoC8YXCupoUO6f0RYtvx83ZwcQ28qN898fDG3btGnL0JkbK3f0K6cObL9b2vTNszfvNP7PjbSWU9Wp1CCndnZ3Y9AHILo/ZWeHNkuP/Pg3s8NPRhKp+dfeuKqvtOFx7copqdLH7rrrkx//xKfv/tSfb/ozCQA8q340rNyY3bdps7LzB6990IZulMGhcR+tcd/2jfDbA3LePDwghw/oQvPI/7z8d3ff/cyrhQbf1+R5W/x/JD0T2sXk1KIW6Vdqz6NT7n+j/D8g583DA3L4gM43byV//tmtm6T7jl6+3Y7xnnHeVv9jc3UJlPvfKP8PyHnz8IAcPqDTzVvJX5oaP/7C/s3SIy+/s1hx7W+D5239vzZZ58n5b5T/B+S8eXhADh/Qqebd0a9efuPqB/nE8bHY1aWrJx+WPnvg7P9ei029fEVvz3k38p9JLjE10KXInb3h08lG62Y4/43y/4CcNw8PyOEDOtW8hVef2SZJ9+wcU2+uUEq0k30BadOXRs9fX27Xedv0OADgXyv6+2+++f5aCcLK8sK75c/aATkFALxDTgEA75BTAMA75BQA8A45BQC8cyanlnLJ0yM9nYrcNXgskatXZeWMgnZ+arAjoMgBpSccTZpaxFVrl7KxQ6YNc5raRtVeJJc889PJgS6lYhEVJ08g0bVzkwNdihxQ5K7BiXNaRX2fW40saOqZn0/s31aukTbam0ueDPfKAaVj/9S8W69JomsX5mYnBu8fVQuVp9G1Vyf2B+WAZR2S3a9JB3KK6Mmp3p7Dam6Jkqw6urt3Yr49tV8tWsqemZ46r+mUUpK7dGx/UN49mby1dpSv1pJsbKgjULEgqcnyWhvblktMDXQFByZ+rmrMU8TLE0iysaGOrsHIFZ1SSrLqaIhZPuFWIxe1yIHHB/YG5UDQnFO3khN7e0fP5wglufNjPXtdeU0SLfLVffsf7wgoHZU5Ra6dPXbiVa1AjapJ9oVn/2vS/pwyrQQsqCMd9ZYr29+eqxcuXC8/iyQ9E+osv2i4aq0+Pznw1MhAF5NTTS9XsrFtPTss3tt5eQIXtUg/+8dGtEif8am7jTS95KqbZ9p10unWmp86Sil5L3EhW+4lVW5D4MRr0vacIlqkr2IfBff+tKwtqOEdxrPMU2tvJSeemkxqaniHUvGSbWIbVTvbtjt4IJatOi83T2BpAw+jp0wK6ug2Tn7LFjlVtYyfCSPHW2uRU2YFdaRjLacceU3anVPVP/NCxR+e+xbUcGgodo1QylNriZ58cXBiXq9sQAvb6djSOC3SJ4dGIqdLsz/BganSiICnJ7DU4+tUOg7MpAskd/758Im1eRO3G1mdU9VfKagjHZ19kTRxobXry6n7D81ll6hTr0m7c6r6OeUspwrqSMgYTnPTWn1+cuDFpE4qG9D0Nqr2WNQi/eXrD/qVmfJsBTdPIKW0vP400KhJbudUOZWqv+J8axvmFCmoz/etzpE59Jr0eU6VxlbGhCUnrb2VPHZ0bnU6gMOcqhgp09V3VLf+qOrS03MTE5PhkCJ3lqaoazQJOcVqlFPlN1EqTE653ceuh+jJEyMRduMJHlpL9OSJsdVxqLkBXIz76g1beHgC1+hXZg6E57JLtHQRSu5cu0zmdiO9Pe67lTx2eCbNtF2IcZ/pQkb1rJtrSPZXUyfNu+Nw0NoFNbzDuIcY89HZF0mT5rdRtauF7EmZZ4mDJ7DE9MwQPTnVKzMz0y42ssY8elUX1a3W1smppeyZE9F0ZUGFI69J/9UlUEopJTl16piaM17E6dmZC8bFC65aawoFHuoSSEEdDXYcmjMuVLPPEi9PYFWY8tNID9YlUEopXcqpJ6ZU4xpvIX3y9IUCEaQugZ/Cv3J7tNhIT2dlb8X4xfPW2qq/Nx7qPMm1uQNdwYFoWieU5C4dO9BXfpY4eQJLHai1Ok9aSEcObCsXUrjaSKuc4qTOk1JaI6cKWmy019THN16WItR5UkrpUm5+erAjoLpi0VcAAABgSURBVMihkZPmFQEOW63wrvWMU8pVay0nI5raRtUu5VUU1c8SJ0+gsdakxroZFxpZmrupfncsHSytjqixlsuZ1hbUkY7KqQZKVxdvWU5ElNtu72sS65ABgHfIKQDg3f8D7NGj6l4VawkAAAAASUVORK5CYII=" alt></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>The coordinates of point A are (75, 450). Determine whether point A is on the bicycle track. Give a reason for your answer.</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 derivative of \(y = \frac{{ - {x^2}}}{{10}} + \frac{{27}}{2}x\).</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>Use the answer in part (b) to determine if A (75, 450) is the point furthest north on the track between O and B. Give a reason for your answer.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>(i) Write down the midpoint of the line segment OB.</span></p>
<p><span>(ii) Find the gradient of the line segment OB.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Scott starts from a point C(0,150) . He hikes along a straight road towards the bicycle track, parallel to the line segment OB.</span></p>
<p><span>Find the equation of Scott’s road. Express your answer in the form \(ax + by = c\), where \(a, b {\text{ and }} c \in \mathbb{R}\).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Use your graphic display calculator to find the coordinates of the point where Scott first crosses the bicycle track.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">f.</div>
</div>
<br><hr><br><div class="specification">
<p><strong><span style="font-size: medium; font-family: times new roman,times;">Give all answers in this question to the nearest whole currency unit.</span></strong></p>
<p><span style="font-size: medium; font-family: times new roman,times;">Ying and Ruby each have 5000 USD to invest.</span></p>
<p><span style="font-size: medium; font-family: times new roman,times;">Ying invests his 5000 USD in a bank account that pays a nominal annual interest rate of 4.2 % <strong>compounded yearly</strong>. Ruby invests her 5000 USD in an account that offers a fixed interest of 230 USD each year.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Find the amount of money that Ruby will have in the bank after 3 years.</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>Show that Ying will have 7545 USD in the bank at the end of 10 years.</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>Find the number of complete years it will take for Ying’s investment to first exceed 6500 USD.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Find the number of complete years it will take for Ying’s investment to exceed Ruby’s investment.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Ruby moves from the USA to Italy. She transfers 6610 USD into an Italian bank which has an exchange rate of 1 USD = 0.735 Euros. The bank charges 1.8 % commission.</span></p>
<p><span>Calculate the amount of money Ruby will invest in the Italian bank after commission.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Ruby returns to the USA for a short holiday. She converts 800 Euros at a bank in Chicago and receives 1006.20 USD. The bank advertises an exchange rate of 1 Euro = 1.29 USD.</span></p>
<p><span>Calculate the percentage commission Ruby is charged by the bank.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">f.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Daniel wants to invest \(\$ 25\,000\) for a total of three years. There are two investment options.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Option One</strong> pays compound interest at a nominal annual rate of interest of 5 %, compounded <strong>annually</strong>.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Option Two</strong> pays compound interest at a nominal annual rate of interest of 4.8 %, compounded <strong>monthly</strong>.</span></p>
</div>
<div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">An arithmetic sequence is defined as</span></p>
<p style="text-align: center;"><span style="font-family: times new roman,times; font-size: medium;"><em>u<sub>n</sub></em> = 135 + 7<em>n</em>, <em>n</em> = 1, 2, 3, …</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Calculate the value of his investment at the end of the third year for each investment option, <strong>correct to two decimal places</strong>.</span></p>
<div class="marks">[8]</div>
<div class="question_part_label">A.a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Determine Daniel’s best investment option.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">A.b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Calculate <em>u</em><sub>1</sub>, the first term in the sequence.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">B.a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Show that the common difference is 7.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">B.b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span><em>S<sub>n</sub></em> is the sum of the first <em>n</em> terms of the sequence.</span></p>
<p><span><span>Find an expression for <em>S<sub>n</sub></em>. Give your answer in the form <em>S<sub>n</sub></em></span><span> = <em>An</em><sup>2</sup> + <em>Bn</em>, where <em>A</em> and<em> B</em> are constants.</span></span></p>
<div class="marks">[3]</div>
<div class="question_part_label">B.c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>The first term, <em>v</em><sub>1</sub>, of a geometric sequence is 20 and its fourth term <em>v</em><sub>4</sub> is 67.5.</span></p>
<p><span>Show that the common ratio, <em>r</em>, of the geometric sequence is 1.5.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">B.d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span><em>T<sub>n</sub></em> is the sum of the first <em>n</em> terms of the geometric sequence.</span></p>
<p><span>Calculate <em>T</em><sub>7</sub>, the sum of the first seven terms of the geometric sequence.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">B.e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span><span><em>T</em><sub><span><em>n</em></span></sub> is the sum of the first <em>n</em> terms of the geometric sequence.</span></span></p>
<p><span>Use your graphic display calculator to find the smallest value of <em>n</em> for which <em>T<sub>n</sub></em> > <em>S<sub>n</sub></em>.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">B.f.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1"><span class="s1">A cup of boiling water is placed in a room to cool. The temperature of the room is 20°C </span>. This situation can be modelled by the exponential function \(T = a + b({k^{ - m}})\), where \(T\) is the temperature of the water, in °<span class="s1">C </span>, and \(m\) is the number of minutes for which the cup has been placed in the room. A sketch of the situation is given as follows.</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2015-12-21_om_09.45.14.png" alt></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Explain why \(a = 20\).</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"><span class="s1">Initially, at \(m = 0\)</span>, the temperature of the water is 100°C<span class="s1">.</span></p>
<p class="p2">Find the value of \(b\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">After being placed in the room for one minute, the temperature of the water is 84°C<span class="s1">.</span></p>
<p class="p2">Show that \(k = 1.25\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">After being placed in the room for one minute, the temperature of the water is 84°C<span class="s1">.</span></p>
<p class="p1">Find the temperature of the water three minutes after it has been placed in the room.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">After being placed in the room for one minute, the temperature of the water is 84°C<span class="s1">.</span></p>
<p class="p1">Find the total time needed for the water to reach a temperature of <span class="s1">35°</span><span class="s1">C</span>. Give your answer in minutes and seconds, correct to the nearest second.</p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Antonio and Barbara start work at the same company on the same day. They each earn an annual salary of \(8000\) euros during the first year of employment. The company gives them a salary increase following the completion of each year of employment. Antonio is paid using plan A and Barbara is paid using plan B.</p>
<p>Plan A: The annual salary increases by \(450\) euros each year.</p>
<p>Plan B: The annual salary increases by \(5\,\% \) each year.</p>
<p>Calculate</p>
<p>i) Antonio’s annual salary during his second year of employment;</p>
<p>ii) Barbara’s annual salary during her second year of employment.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down an expression for</p>
<p>i) Antonio’s annual salary during his \(n\) th year of employment;</p>
<p>ii) Barbara’s annual salary during her \(n\) th year of employment.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the number of years for which Antonio’s annual salary is greater than or equal to Barbara’s annual salary.</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>Both Antonio and Barbara plan to work at the company for a total of \(15\) years.</p>
<p>i) Calculate the <strong>total amount</strong> that <strong>Barbara</strong> will be paid during these \(15\) years.</p>
<p>ii) Determine whether Antonio earns more than Barbara during these \(15\) years.</p>
<div class="marks">[7]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-size: medium; font-family: times new roman,times;">Given \(f (x) = x^2 − 3x^{−1}, x \in {\mathbb{R}}, - 5 \leqslant x \leqslant 5, x \ne 0\),</span></p>
</div>
<div class="specification">
<p><span style="font-size: medium; font-family: times new roman,times;">A football is kicked from a point A (a, 0), 0 < a < 10 on the ground towards a goal to the right of A.</span></p>
<p><span style="font-size: medium; font-family: times new roman,times;">The ball follows a path that can be modelled by <strong>part</strong> of the graph</span></p>
<p style="margin-left: 30px;"><span style="font-size: medium; font-family: times new roman,times;">\(y = − 0.021x^2 + 1.245x − 6.01, x \in {\mathbb{R}}, y \geqslant 0\).</span></p>
<p><span style="font-size: medium; font-family: times new roman,times;"><em>x</em> is the horizontal distance of the ball from the origin</span></p>
<p><span style="font-size: medium; font-family: times new roman,times;"><em>y</em> is the height above the ground</span></p>
<p><span style="font-size: medium; font-family: times new roman,times;">Both <em>x</em> and <em>y</em> are measured in metres.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Write down the equation of the vertical asymptote.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">i.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">i.b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Using your graphic display calculator or otherwise, write down the coordinates of any point where the graph of \(y = f (x)\) has zero gradient.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">i.c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Write down all intervals in the given domain for which \(f (x)\) is increasing.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">i.d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Using your graphic display calculator or otherwise, find the value of <em>a</em>.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">ii.a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Find \(\frac{{dy}}{{dx}}\).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">ii.b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>(i) Use your answer to part (b) to calculate the horizontal distance the ball has travelled from A when its height is a maximum.</span></p>
<p><span>(ii) Find the maximum vertical height reached by the football.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">ii.c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Draw a graph showing the path of the football from the point where it is kicked to the point where it hits the ground again. Use 1 cm to represent 5 m on the horizontal axis and 1 cm to represent 2 m on the vertical scale.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">ii.d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>The goal posts are 35 m from <strong>the point where the ball is kicked</strong>.</span></p>
<p><span>At what height does the ball pass over the goal posts?</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">ii.e.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-size: medium; font-family: times new roman,times;">The graph of the function \(f(x) = \frac{{14}}{x} + x - 6\), for 1 ≤ <em>x</em> ≤ 7 is given below.</span></p>
<p><span style="font-size: medium; font-family: times new roman,times;"><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>Calculate \(f (1)\). </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>Find \(f ′(x)\).</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><strong>Use your answer to part (b)</strong> to show that the <em>x</em>-coordinate of the local minimum point of the graph of \(f\) is 3.7 correct to 2 significant figures.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Find the range of \(f\).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Points A and B lie on the graph of \(f\). The <em>x</em>-coordinates of A and B are 1 and 7 respectively.</span></p>
<p><span>Write down the <em>y</em>-coordinate of B.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Points A and B lie on the graph of f . The <em>x</em>-coordinates of A and B are 1 and 7 respectively.<br></span></p>
<p><span>Find the gradient of the straight line passing through A and B.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>M is the midpoint of the line segment AB.</span></p>
<p><span>Write down the coordinates of M.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">g.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span><em>L</em> is the tangent to the graph of the function \(y = f (x)\), at the point on the graph with the same <em>x</em>-coordinate as M.</span></p>
<p><span>Find the gradient of <em>L</em>.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">h.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Find the equation of <em>L</em>. Give your answer in the form \(y = mx + c\).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">i.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Consider the function \(f(x) = 0.5{x^2} - \frac{8}{x},{\text{ }}x \ne 0\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find \(f( - 2)\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find \(f'(x)\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the gradient of the graph of \(f\) at \(x = - 2\).</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>Let \(T\) be the tangent to the graph of \(f\) at \(x = - 2\).</p>
<p>Write down the equation of \(T\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Let \(T\) be the tangent to the graph of \(f\) at \(x = - 2\).</p>
<p>Sketch the graph of \(f\) for \( - 5 \leqslant x \leqslant 5\) and \( - 20 \leqslant y \leqslant 20\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Let \(T\) be the tangent to the graph of \(f\) at \(x = - 2\).</p>
<p>Draw \(T\) on your sketch.</p>
<div class="marks">[2]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">The tangent, \(T\), intersects the graph of \(f\) <span class="s1">at a second point, P.</span></p>
<p class="p2">Use your graphic display calculator to find the coordinates of P.</p>
<div class="marks">[2]</div>
<div class="question_part_label">g.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">A closed rectangular box has a height \(y{\text{ cm}}\) and width \(x{\text{ cm}}\). Its length is twice its width. It has a fixed outer surface area of \(300{\text{ c}}{{\text{m}}^2}\) .</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><img style="display: block; margin-left: auto; margin-right: auto;" 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>Factorise \(3{x^2} + 13x - 10\).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">i.a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Solve the equation \(3{x^2} + 13x - 10 = 0\).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">i.b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Consider a function \(f(x) = 3{x^2} + 13x - 10\) .</span></p>
<p><span>Find the equation of the axis of symmetry on the graph of this function.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">i.c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Consider a function \(f(x) = 3{x^2} + 13x - 10\) .</span></p>
<p><span>Calculate the minimum value of this function.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">i.d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Show that \(4{x^2} + 6xy = 300\).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">ii.a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Find an expression for \(y\) in terms of \(x\).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">ii.b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Hence show that the volume \(V\) of the box is given by \(V = 100x - \frac{4}{3}{x^3}\).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">ii.c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Find \(\frac{{{\text{d}}V}}{{{\text{d}}x}}\).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">ii.d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>(i) Hence find the value of \(x\) and of \(y\) required to make the volume of the box a maximum.</span></p>
<p><span>(ii) Calculate the maximum volume.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">ii.e.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;"><strong>Throughout this question <em>all</em> the numerical answers must be given correct to the nearest whole number.</strong></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Park School started in January 2000 with \(100\) students. Every full year, there is an increase of \(6\% \) in the number of students.</span></p>
<p><span>Find the number of students attending Park School in</span></p>
<p><span>(i) January 2001;</span></p>
<p><span>(ii) January 2003.</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>Park School started in January 2000 with \(100\) students. Every full year, there is an increase of \(6\% \) in the number of students.</span></p>
<p><span>Show that the number of students attending Park School in January 2007 is \(150\).</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>Grove School had \(110\) students in January 2000. Every full year, the number of students is \(10\) more than in the previous year.</span></p>
<p><span>Find the number of students attending Grove School in January 2003.</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>Grove School had \(110\) students in January 2000. Every full year, the number of students is \(10\) more than in the previous year.</span></p>
<p><span>Find the year in which the number of students attending Grove School will be first \(60\% \) <strong>more than</strong> in January 2000.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Each January, one of these two schools, the one that has more students, is given extra money to spend on sports equipment.</span></p>
<p><span>(i) Decide which school gets the money in 2007. Justify your answer.</span></p>
<p><span>(ii) Find the first year in which Park School will be given this extra money.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">A parcel is in the shape of a rectangular prism, as shown in the diagram. It has a length \(l\) cm, width \(w\) cm and height of \(20\) cm.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">The total volume of the parcel is \(3000{\text{ c}}{{\text{m}}^3}\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Express the volume of the parcel in terms of \(l\) and \(w\).</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>Show that \(l = \frac{{150}}{w}\).</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 parcel is tied up using a length of string that fits <strong>exactly </strong>around the parcel, as shown in the following diagram.</span></p>
<p><span><br><img src="images/Schermafbeelding_2014-09-02_om_11.55.29_5.png" alt><br></span></p>
<p><span>Show that the length of string, \(S\) cm, required to tie up the parcel can be written as</span></p>
<p><span>\[S = 40 + 4w + \frac{{300}}{w},{\text{ }}0 < w \leqslant 20.\]</span></p>
<p><span> </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>The parcel is tied up using a length of string that fits <strong>exactly </strong>around the parcel, as shown in the following diagram.</span></p>
<p><span><br><img src="images/Schermafbeelding_2014-09-02_om_11.55.29_4.png" alt><br></span></p>
<p><span>Draw the graph of \(S\) for \(0 < w \leqslant 20\) and \(0 < S \leqslant 500\), clearly showing the local minimum point. Use a scale of \(2\) cm to represent \(5\) units on the horizontal axis \(w\)<em> </em>(cm), and a scale of \(2\) cm to represent \(100\) units on the vertical axis \(S\) (cm).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>The parcel is tied up using a length of string that fits <strong>exactly </strong>around the parcel, as shown in the following diagram.</span></p>
<p><span><br><img src="images/Schermafbeelding_2014-09-02_om_11.55.29_3.png" alt><br></span></p>
<p><span>Find \(\frac{{{\text{d}}S}}{{{\text{d}}w}}\).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>The parcel is tied up using a length of string that fits <strong>exactly </strong>around the parcel, as shown in the following diagram.</span></p>
<p><span><br><img src="images/Schermafbeelding_2014-09-02_om_11.55.29_2.png" alt><br></span></p>
<p><span>Find the value of \(w\) for which \(S\) is a minimum.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>The parcel is tied up using a length of string that fits <strong>exactly </strong>around the parcel, as shown in the following diagram.</span></p>
<p><span><br><img src="images/Schermafbeelding_2014-09-02_om_11.55.29.png" alt><br></span></p>
<p><span>Write down the value, \(l\), of the parcel for which the length of string is a minimum.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">g.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>The parcel is tied up using a length of string that fits <strong>exactly </strong>around the parcel, as shown in the following diagram.</span></p>
<p><span><br><img src="images/Schermafbeelding_2014-09-02_om_11.55.29_1.png" alt><br></span></p>
<p><span>Find the minimum length of string required to tie up the parcel.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">h.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-size: medium; font-family: times new roman,times;">Consider the function \(g(x) = bx - 3 + \frac{1}{{{x^2}}},{\text{ }}x \ne 0\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Write down the equation of the vertical asymptote of the graph of <em>y</em> = <em>g</em>(<em>x</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>Write down <em>g</em>′(<em>x</em>) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>The line <em>T</em> is the tangent to the graph of <em>y</em> = <em>g</em>(<em>x</em>) at the point where <em>x</em> = 1. The gradient of <em>T</em> is 3.</span></p>
<p><span>Show that <em>b</em> = 5.</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><span>The line <em>T</em> is the tangent to the graph of <em>y</em> = <em>g</em>(<em>x</em>) at the point where <em>x</em> = 1. The gradient of <em>T</em> is 3.</span></span></p>
<p><span>Find the equation of <em>T</em>.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Using your graphic display calculator find the coordinates of the point where the graph of <em>y</em> = <em>g</em>(<em>x</em>) intersects the <em>x</em>-axis.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>(i) Sketch the graph of <em>y</em> = <em>g</em>(<em>x</em>) for −2 ≤ <em>x</em> ≤ 5 and −15 ≤ <em>y</em> ≤ 25, indicating clearly your answer to part (e).</span></p>
<p><span>(ii) Draw the line <em>T</em> on your sketch.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Using your graphic display calculator find the coordinates of the local minimum point of <em>y</em> = <em>g</em>(<em>x</em>) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">g.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Write down the interval for which <em>g</em>(<em>x</em>) is increasing in the domain 0 < <em>x</em> < 5 .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">h.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Consider the function <em>f </em>(<em>x</em>) = <em>x</em><sup>3 </sup><em>–</em> 3x– 24<em>x</em> + 30.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Write down <em>f</em> (0).</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Find \(f'(x)\).</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>Find the gradient of the graph of <em>f</em> (<em>x</em>) at the point where <em>x</em> = 1.</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>(i) Use </span><em>f '</em><span>(</span><em>x</em><span>) to find the </span><em>x</em><span>-coordinate of M and of N.</span></p>
<p><span>(ii) Hence or otherwise write down the coordinates of M and of N.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Sketch the graph of <em>f</em> (<em>x</em>) for \( - 5 \leqslant x \leqslant 7\) and \( - 60 \leqslant y \leqslant 60\). Mark clearly M and N on your graph.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Lines <em>L</em><sub>1</sub> and <em>L</em><sub>2</sub> are parallel, and they are tangents to the graph of <em>f</em> (<em>x</em>) at points A and B respectively. <em>L<sub>1</sub></em> has equation <em>y</em> = 21<em>x</em> + 111.</span></p>
<p><span>(i) Find the <em>x</em>-coordinate of A and of B.</span></p>
<p><span>(ii) Find the <em>y</em>-coordinate of B.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">f.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The diagram shows part of the graph of \(f(x) = {x^2} - 2x + \frac{9}{x}\) , where \(x \ne 0\) .</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</span></p>
<p><span>(i) the equation of the vertical asymptote to the graph of \(y = f (x)\) ;</span></p>
<p><span>(ii) the solution to the equation \(f (x) = 0\) ;</span></p>
<p><span>(iii) the coordinates of the local minimum point.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Find \(f'(x)\) . </span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Show that \(f'(x)\) can be written as \(f'(x) = \frac{{2{x^3} - 2{x^2} - 9}}{{{x^2}}}\) .</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>Find the gradient of the tangent to \(y = f (x)\) at the point \({\text{A}}(1{\text{, }}8)\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>The line, \(L\), passes through the point A and is perpendicular to the tangent at A. </span></p>
<p><span>Write down the gradient of \(L\) .</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>The line, \(L\) , passes through the point A and is perpendicular to the tangent at A. </span></p>
<p><span>Find the equation of \(L\) . Give your answer in the form \(y = mx + c\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span><span>The line, \(L\) , passes through the point A and is perpendicular to the tangent at A. </span></span></p>
<p><span>\(L\) also intersects the graph of \(y = f (x)\) at points B and C . Write down the <strong><em>x</em>-coordinate</strong> of B and of C .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">g.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Mal is shopping for a school trip. He buys \(50\) tins of beans and \(20\) packets of cereal. The total cost is \(260\) Australian dollars (\({\text{AUD}}\)).</span></p>
</div>
<div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The triangular faces of a square based pyramid, \({\text{ABCDE}}\), are all inclined at \({70^ \circ }\) to the base. The edges of the base \({\text{ABCD}}\) are all \(10{\text{ cm}}\) and \({\text{M}}\) is the centre. \({\text{G}}\) is the mid-point of \({\text{CD}}\).</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 an equation showing this information, taking \(b\) to be the cost of one tin of beans and \(c\) to be the cost of one packet of cereal in \({\text{AUD}}\).</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">i.a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Stephen thinks that Mal has not bought enough so he buys \(12\) more tins of beans and \(6\) more packets of cereal. He pays \(66{\text{ AUD}}\).</span></p>
<p><span>Write down another equation to represent this information.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">i.b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span><span>Stephen thinks that Mal has not bought enough so he buys \(12\) more tins of beans and \(6\) more packets of cereal. He pays \(66{\text{ AUD}}\).</span></span></p>
<p><span>Find the cost of one tin of beans.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">i.c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>(i) Sketch the graphs of the two equations from parts (a) and (b).</span></p>
<p><span>(ii) Write down the coordinates of the point of intersection of the two graphs.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">i.d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Using the letters on the diagram draw a triangle showing the position of a \({70^ \circ }\) angle.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">ii.a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Show that the height of the pyramid is \(13.7{\text{ cm}}\), to 3 significant figures.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">ii.b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Calculate</span></p>
<p><span>(i) the length of \({\text{EG}}\);</span></p>
<p><span>(ii) the size of angle \({\text{DEC}}\).</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">ii.c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Find the total surface area of the pyramid.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">ii.d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Find the volume of the pyramid.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">ii.e.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Consider the curve \(y = {x^3} + \frac{3}{2}{x^2} - 6x - 2\)</span> .</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>(i) Write down the value of \(y\) when \(x\) is \(2\).</span></p>
<p><span>(ii) Write down the coordinates of the point where the curve intercepts the \(y\)-axis.</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>Sketch the curve for \( - 4 \leqslant x \leqslant 3\) and \( - 10 \leqslant y \leqslant 10\). Indicate clearly the information found in (a).</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Find \(\frac{{{\text{d}}y}}{{{\text{d}}x}}\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Let \({L_1}\) be the tangent to the curve at \(x = 2\).</span></p>
<p><span>Let \({L_2}\) be a tangent to the curve, parallel to \({L_1}\).</span></p>
<p><span>(i) Show that the gradient of \({L_1}\) is \(12\).</span></p>
<p><span>(ii) Find the \(x\)-coordinate of the point at which \({L_2}\) and the curve meet.</span></p>
<p><span>(iii) Sketch and label \({L_1}\) and \({L_2}\) on the diagram drawn in (b).</span></p>
<div class="marks">[8]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>It is known that \(\frac{{{\text{d}}y}}{{{\text{d}}x}} > 0\) for \(x < - 2\) and \(x > b\) where \(b\) is positive.</span></p>
<p><span>(i) Using your graphic display calculator, or otherwise, find the value of \(b\).</span></p>
<p><span>(ii) Describe the behaviour of the curve in the interval \( - 2 < x < b\) .</span></p>
<p><span>(iii) Write down the equation of the tangent to the curve at \(x = - 2\).</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The diagram shows a Ferris wheel that moves with constant speed and completes a rotation every 40 seconds. The wheel has a radius of \(12\) m and its lowest point is \(2\) m above the ground.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><img 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" alt></span></p>
<p> </p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Initially, a seat C is vertically below the centre of the wheel, O. It then rotates in an anticlockwise (counterclockwise) direction.</span></p>
<p><span>Write down</span></p>
<p><span>(i) the height of O above the ground;</span></p>
<p><span>(ii) the maximum height above the ground reached by C .</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>In a revolution, C reaches points A and B , which are at the same height above the ground as the centre of the wheel. Write down the number of seconds taken for C to first reach A and then B .</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 sketch below shows the graph of the function, \(h(t)\) , for the height above ground of C, where \(h\) is measured in metres and \(t\) is the time in seconds, \(0 \leqslant t \leqslant 40\) .</span></p>
<p><span><img src="data:image/png;base64,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" alt></span></p>
<p><span><strong>Copy</strong> the sketch and show the results of part (a) and part (b) on your diagram. Label the points clearly with their coordinates.</span></p>
<div class="marks">[4]</div>
<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;">A manufacturer claims that fertilizer has an effect on the height of rice plants. He measures the height of fertilized and unfertilized plants. The results are given in the following table.</span></p>
<p><span style="font-size: medium; font-family: times new roman,times;"><img src="data:image/png;base64,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" alt></span></p>
<p><span style="font-size: medium; font-family: times new roman,times;">A chi-squared test is performed to decide if the manufacturer’s claim is justified</span> <span style="font-size: medium; font-family: times new roman,times;">at the<strong> 1 %</strong> level of significance.</span></p>
</div>
<div class="specification">
<p><span style="font-size: medium; font-family: times new roman,times;">The population of fleas on a dog after <em>t</em> days, is modelled by</span></p>
<p><span style="font-size: medium; font-family: times new roman,times;">\[N = 4 \times {(2)^{\frac{t}{4}}},{\text{ }}t \geqslant 0\]</span></p>
<p><span style="font-size: medium; font-family: times new roman,times;">Some values of<em> N</em> are shown in the table below.</span></p>
<p><span style="font-size: medium; font-family: times new roman,times;"><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 the null and alternative hypotheses for this test.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">i, a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>For the number of fertilized plants with height greater than 75 cm,</span> <span>show that the expected value is 97.5.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">i, b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Write down the value of \(\chi_{calc}^2\).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">i, c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Write down the number of degrees of freedom.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">i, d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Is the manufacturer’s claim justified? Give a reason for your answer.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">i, f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Write down the value of </span><span><em>p</em>.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">ii, a, i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Write down the value of <em>q</em>.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">ii, a, ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Using the values in the table above, draw the graph of <em>N</em> for 0 ≤ <em>t</em> ≤ 20. Use 1 cm to represent 2 days on the horizontal axis and 1 cm to represent 10 fleas on the vertical axis.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">ii, b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span><strong>Use your graph</strong> to estimate the number of days for the population of fleas to reach 55.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">ii, c.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hugo is given a rectangular piece of thin cardboard, \(16\,{\text{cm}}\) by \(10\,{\text{cm}}\). He decides to design a tray with it.</p>
<p>He removes from each corner the shaded squares of side \(x\,{\text{cm}}\), as shown in the following diagram.</p>
<p><img src="data:image/png;base64,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" alt></p>
<p>The remainder of the cardboard is folded up to form the tray as shown in the following diagram.</p>
<p><img src="data:image/png;base64,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" alt></p>
<p>Write down, <strong>in terms of</strong> \(x\) , the length and the width of the tray.</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>(i) State whether \(x\) can have a value of \(5\). Give a reason for your answer.</p>
<p>(ii) Write down the interval for the possible values of \(x\) .</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the volume, \(V\,{\text{c}}{{\text{m}}^3}\), of this tray is given by</p>
<p>\[V = 4{x^3} - 52{x^2} + 160x.\]</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find \(\frac{{dV}}{{dx}}.\)</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><strong>Using your answer from part (d)</strong>, find the value of \(x\) that maximizes the volume of the tray.</p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate the maximum volume of the tray.</p>
<div class="marks">[2]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of \(V = 4{x^3} - 51{x^2} + 160x\) , for the possible values of \(x\) found in part (b)(ii), and \(0 \leqslant V \leqslant 200\) . Clearly label the maximum point.</p>
<div class="marks">[4]</div>
<div class="question_part_label">g.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The temperature in \(^ \circ {\text{C}}\) of a pot of water removed from the cooker is given by \(T(m) = 20 + 70 \times {2.72^{ - 0.4m}}\), where \(m\) is the number of minutes after the pot is removed from the cooker.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Show that the temperature of the water when it is removed from the cooker is \({90^ \circ }{\text{C}}\).</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 following table shows values for \(m\) and \(T(m)\).</span></p>
<p><span><img src="data:image/png;base64,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" alt></span></p>
<p><span>(i) Write down the value of \(s\).</span></p>
<p><span>(ii) Draw the graph of \(T(m)\) for \(0 \leqslant m \leqslant 10\) . Use a scale of \(1{\text{ cm}}\) to represent \(1\) minute on the horizontal axis and a scale of \(1{\text{ cm}}\) to represent \({10^ \circ }{\text{C}}\) on the vertical axis.</span></p>
<p><span>(iii) <strong>Use your graph</strong> to find how long it takes for the temperature to reach \({56^ \circ }{\text{C}}\). Show your method clearly.</span></p>
<p><span>(iv) Write down the temperature approached by the water after a long time. Justify your answer.<strong><br></strong></span></p>
<div class="marks">[9]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Consider the function \(S(m) = 20m - 40\) for \(2 \leqslant m \leqslant 6\) .</span></p>
<p><span>The function \(S(m)\) represents the temperature of soup in a pot placed on the cooker two minutes after the water has been removed. The soup is then heated.</span></p>
<p><span>Draw the graph of \(S(m)\) on the same set of axes used for part (b).</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>Consider the function \(S(m) = 20m - 40\) for \(2 \leqslant m \leqslant 6\) .</span></p>
<p><span>The function \(S(m)\) represents the temperature of soup in a pot placed on the cooker two minutes after the water has been removed. The soup is then heated.</span></p>
<p><span>Comment on the meaning of the constant \(20\) in the formula for \(S(m)\) in relation to the temperature of the soup.</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>Consider the function \(S(m) = 20m - 40\) for \(2 \leqslant m \leqslant 6\) .</span></p>
<p><span>The function \(S(m)\) represents the temperature of soup in a pot placed on the cooker two minutes after the water has been removed. The soup is then heated.</span></p>
<p><span>(i) <strong>Use your graph</strong> to solve the equation \(S(m) = T(m)\) . Show your method clearly.</span></p>
<p><span>(ii) Hence describe by using inequalities the set of values of \(m\) for which \(S(m) > T(m)\).</span></p>
<div class="marks">[4]</div>
<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;">A deep sea diver notices that the intensity of light, \(I\) , below the surface of the ocean decreases with depth, \(d\) , according to the formula</span><br><span style="font-family: times new roman,times; font-size: medium;">\[I = k{(1.05)^{ - d}}{\text{,}}\]where \(I\) is expressed as a percentage, \(d\) is the depth in metres below the surface and \(k\) is a constant.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The intensity of light at the surface is \(100\% \).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Calculate the value of \(k\) .</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 intensity of light at a depth \(25{\text{ m}}\) below the surface.</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>To be able to see clearly, a diver needs the intensity of light to be at least \(65\% \).</span></p>
<p><span>Using your graphic display calculator, find the greatest depth below the surface at which she can see clearly.</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>The table below gives the intensity of light (correct to the nearest integer) at different depths.</span></p>
<p><span><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAlcAAABOCAIAAACCH6SpAAARa0lEQVR4nO2dz2vbWL+H5z+RQSsbDC3dZOVVF369MCbMYiCbXhyDs+ili4G5vDIOU7IYbsGmZqBQKDINhZeBjk2Hl4Eh5WAYyuAiZhblxQjmQihGDGEIQotgyuHchSxZkuVvksbWOa0/D9q0SWPpydH5nN/9QgAAAADbyheybwAAAACQBlIQAADA9oIUBAAAsL0gBQEAAGwvSEEAAADbSywFdS2HCxcuXLhwbc+VTMFsM3jrgGECyCGAHALIoYEfAqRg1sAwAeQQQA4B5NDADwFSMGtgmAByCCCHAHJo4IcAKZg1MEwAOQSQQwA5NPBDgBTMGhgmgBwCyCGAHBr4IUAKZg0ME0AOAeQQQA4N/BAgBbMGhgkghwByCCCHBn4IkIJZA8MEkEMAOQSQQwM/BEjBrIFhAsghgBwCyKGBHwKkYNbAMAHkEEAOAeTQwA/BJ5SCrs1+6Nabffviyv9k5lgvDjsjd+kLH6zOHS2na4WaOeFCCHfUbQ9sj6/xdlehsGH5QA4B5BBADg38ENwsBV3Wyqcdy1Y2+oPRWkOFu6xd1HK6tnflFJw57LuGsTLbuG3WtFKLnc2/efykUT1izmxdd7wKRYojd8bHRlXXcsX6k7ebf+orIk8O9+zR8GWncavN3GSBUcSVNDme/bqzX9RyulaoGC+shAHuWM+NipbT8/u9sZNFQzINqa9VWDvldC2nV017YWHmWC9a5YKu7TQev5FmR4nCkyvWe6/taJdEITk37Av6JSASTp7NTP+taPYny92wazGbjt78Gdjhtlm7agpyz+rVmoPpSrMXtrmn56NVHvcmxwfN48mGe4RKpKA37pZ3D9mUi5nDjirlnpVJP/hSZMnhtvnlvf3/yufiRUIIoZArOXL46U+Pn/qVF3fe9Oo7eszAudX5qtI+cbjgzslh+auudS7hJuW+Vvx02NwJ+gDB8JIQfi1UKR8xZyb4lLV3K52xJ+keZRWeYXOnWPcr1ZnDjiqL90stOetOQSH8RGnkc/oNqwxv3G0e2x+RgnzSr94jv/OMGaV4q00IcWGb9yKFeCMokIIXtrlXNFjQQjljxt1NP/UVkSpnuWEklHIlRQ7/881ouuj8xUdQBLfNWqRec1m7mHynMkLmKILVqy1KSPQrk371buhKuKyVv/pQ1pqRU3jipSUmRDE5m0hBIcRsOnhQjLWMrsu51dmNBtWVU/AKb+MHq3sr5d7ib/VGkJ+CfNKv3o62WCVWXgmUS0GVXMkvOcKvrcJ67cI292JNSXl1mTw5Z8wo6Vq1ZQ5YbLhvub7auiaUcFkrX1h08lzWCmYcVJOzoRT0nzMySr4YIN5pdH7xp+u4Y/1kGpXqM2syaJULkYmHc6uzG040+o3xQNxf9qBNjriuEuraJ71GPqdr1Vbnm0q0kbK46XgLZQNIr8uSDbTVv8HsUS0FlXIlveQI4VdkD4Z+75BP+tVCMdoNclkrf5OG78cjcSy9tlgSUW0NJsGw3nJxSh1/yghJfvxqfKdhvvP4lH3bDmaOlZOzqRScr53xH9Ub94yn85UF3rt+fadoMJdP+tWCruX0u83eaMqFP7VQKM7n85ItTW6bNe0fB51/vXVm87H4VGux5mp4m1PWrurl9tB2+XRwkDoDJJbb/utHdl22/PtCCvqkTBUr5Up2yRFCcJc9rEWb9onM274U9Jm35rWcru0GM6PL1foWpqAQ3Hn7eH9pYaNycjJIwQ+xNVT+Na9uEg8f/WmpKRh+UOosTvhtiRQ8tzq7ej7Whl1h/IwZpWLqKP+akF2XqVWzJ0AKEsguOUJ44279+8VMP1IwzrwRP689lKvoJXyqEEK49uD7buebipbTy+EifOXkZDAiSoz5Jh8+kmFrS0Fum7XIDCWfDg5Wvqsb/2VIf12VGuVLoFgKquVKdsk5tx4fxeYgMCKahLusXZwXIeUG/SR8qnAn5tf/PTgNx/mC9ZLKydno6pidg8Epp54wgxRMfKc/VJ02KRjcz2fdF1ye+1xa4yAP1VJQKVdS5cymr54eJ6fhEwtor7WXac3If62EEL6BoHgkl9ptfraFQN4a0egmunG3PDegmpwMdkr43xPdcsu933+1Vo2IRhtT10/BpXnB98P6bf1Wx/og5ptUtBWTgkJksFRJgddVodX/CZRLQZVcyZMzc9jTHgs337qT5y9G4Uq/bd8pEYW77NFh6sYAIX8zQOafuRwNkddHMTkb2zVfjhzF4o275UJkXjBcPnvGjFJ46gR33vTq/zgYnMbXEf39x6tfpzzxQatTMFlV+R/xYDj1nPHT1rfH/W9KevW7vvnvlNd1C9aICqHQTvAE6qWgQq5kDWrNl2RHr0XUbfmu+Zlj/fyT5a97nDnjp61vTyJnoKi1MVzCp3rjbrkQ7Jrn3uS4ES4wVkzO+k9QK9Y7P76yEifihMdQRXdKBH3BRz8+axa1nJ7f757Y3uKf+EPJR8xxbXMv+PmlFvsPM0rBH5dbEIk2adAx1aqt52OHvx/Wb6/aZbEV+wWFEOE5IKlnYslDmpxYSU7ObCniSoYcf14j8YLH/YSLAMvGcfKlzw5pKciOKmGlx+ylenzmjJ9EKh8Z9yiEkLiTZFHtp52gpowcuadpb2ZS9PKzY1LZkrNj1AVyCCCHAHJo4IfgM01B/1ij+vUOBeXTwUF14+NdKI4EkEMAOQSQQwM/BJ9rCgp/sOLKQcg9e9Cqf7c9/6eEmkAOAeQQQA4N/BBITcHoZMxGZuO4Zw9S/3/BpTsZddsZTfmgOBJADgHkEEAODfwQSO8Lbh0wTAA5BJBDADk08EOAFMwaGCaAHALIIYAcGvghQApmDQwTQA4B5BBADg38ECAFswaGCSCHAHIIIIcGfgiQglkDwwSQQwA5BJBDAz8ESMGsgWECyCGAHALIoYEfAqRg1sAwAeQQQA4B5NDAD0FKCuLChQsXLlzbc6EvmCkwTAA5BJBDADk08EOAFMwaGCaAHALIIYAcGvghQApmDQwTQA4B5BBADg38ECAFswaGCSCHAHIIIIcGfgiQglkDwwSQQwA5BJBDAz8ESMGsgWECyCGAHALIoYEfAqRg1sAwAeQQQA4B5NDADwFSMGtgmAByCCCHAHJo4IcAKZg1MEwAOQSQQwA5NPBDgBTMGhgmgBwCyCGAHBr4IfhcUpCfDpulSmfsrevnOePj9vfM5Su+fsY6j4a2+xE/+VM1nAmQQwA5BJBDAz8EN0tBl7Xy0dPY9vr2xZpv8IqsNQW5c3JYfxiG3Aercyd66Fx94AjBnTe9+t4hm67KyVVkWhw9mw3+1a1XW+ws8RXujI+Nqq7livUnb51Z4mvWc6Oi5fT8fm/sXPcBb4L0d5U71k8vO418Ts+3F22g1RqzRJ6cmTN+0sjndK1QMQa2Fy0R3LN/6dZ3dC2nazuNzi/xr2aH9JITMHOsF61yQddyxXrv9Uc1lDeBGn5c+6TX8COjbBxbmdYtBGvoC3LbrGmFmjm52iPNpqM3f27+6fn0t9GfHxXJ3rhb/Xo4TQTDpF8t6Npu1zqPfOe7fvPr/uR6BT274sgn/d29xr0dXSslq29v3C3vHrIpFzOHHVXKPWtReZ1bna8q7ROHC+6cHJa/ij3yhpH6rvp1/U6j8wOLVl6ExmyRJId7v786Hjt83vLbKRostMOng4P8TsN85wkh+JS1q9GvZokatTz3rF5t3qx0J2azmH+QrEkkoYCf2fTVk96J7QkhuPP28X4xUZ3KI/MU9Mbd5rG98RQ8tzrGR3VML2zzXsqzuKyVzy2/4dw2a1XzWo+TcXHktllLVt8XtrkXeZYzZtwNH5nbZm3RDeIuaxev+YA3Qd67yj2rV1nd903TmDVy5PD/G43eB064y9rFRfG4sM29aKc5XngyRYFa3m8r310UEn46bO7IahYkkO8nVpDm/Qp15GSZgudWZ1ffeK3KPatX+bjhWZe18in/8IPVuaPdbgzeL31UvNxfAfkpyCf96u3I7ysadRe2uRf7Ba0QsiGkvaveuFsuHQxOVxXM7U3BGBe2eS8y78Bd1o606LnL2ne2uC/IbbMWq3aSrQSJqOAnzhkzSp9rCnLPHv3YaX5p/se1B61yQdeCARM/AoPZteD5FyPFi2F0z2YvO41d03Ynw/nc1fEkHLLz/L/caXQGb0e//eHy4EP37xjMnUdgMIeX/+ePL/9ZnP9xfpPcNmuxe1g8yoquz4Vt7q2Y9Uz0qy5Hegou/Y1fl+317YuUBprLWvmrD3ffFEnv6oVt7ullo+/PCKbNbyEFhWczs91on8Q6y964Wy7o+WZ/4nLn5KHxNDnHnBUK1PKR92jl30hDAT8JzphRJdqdWbLuFJyvlylUmo+Oxw5PloNEV+PcevwwGIPyh9HbzP2LGSVdy+nl+93nvzk8URFf2GYzcDebvvph5PJwkU5YfScaZdw5OSxHez+z6cA4mGdzlBXNE39SML0Lu9R5ugzZKUi+q8uZtw0pyCf96u2K8cJyZkJwb3LcyBcSK622OwX9EuIvamgnlkb7k4W6ltv8GA+FCrX80qhYYgBZJir4ieGyVjW6HEEmGxgRjdeb8eojnhnJJaa5YAHCGTNKi2+LdlD4pF8tNToDK7msMdaJWRqaiBdHPunvPkwpmquGqp1BI6XjmPaTrwBSkEDOu+qyVj6qKKXrv90p6BOsfkys+PAmw06na1R1rVBJ9BQzRLYcnzNmlPyesQiXYUttHISo4Sfk3Op8rcjSGCE1BYmVF6tTUMymgwdFLafn97svR4thq0tS0B+6uR2OiH6ZGmnpKeiHxKoa8NqDHrJTECOiSa6gCCk4J+nBe9dvGsPpTPiLjbVkHzozVJAjhBCe/bqzH1RQz1qxISiZqOJHCCG4Zz1tpQzFSUN2CqbnB5GCQoiZY/27b1R1Lbdoll6agn58zkdc76eHVnoKnjGjpGv3h86H1Kf/1PqCyyt6or+U5DRnmskNIq8vGEv65adGCs6JFZ7EdMANVqXdGCXkxJG4YnYZdfzw6c+95wpFoFBiRDS6fdJ7x6y/L0vB4HOju5cuT8GgO/i/j/5nd9UYRdq8oH+TK4c1rr2XQH4KYqdEkjNmlIrNQXACQspTIwXnuKx1KxwRjb+nIusVxVGUkBMh+722NIr44Q7rPWZO2GyavOyPZL5TPjJSMN9m7t9/vPp1ymOrRnUtp8+3bxMpeDYa/DavvvnpsFmaf1BKCpZa7C/v91/YYg7DH03dWb0wabn6cydms6jl7nSs1J7gp7hGVAjsmk8S3f3NnTe9+r3EU29tCvLp4CBfbT33t81PWXuvsRjL8jt/4SJwd2I27ywaE5miSC0vRLDE/VbW5y7RKOCHe/NdA2ocNxZhzSeoPRt8X1vshTh5Hy4tC3cpOCeH5YJePmL+8pbwsK5wp8T8lBZ/n0ObvT9ZfETVtPnff7CfXw+etcoFXQtezuht+BnGp6xd1bVq8oQzl7VukWMUkcZs8uC0W2lJqPR+wbP5atuomYBgaV8hWBspol97+3i/KOOUI5m75sOTwJJPTWnMEjlyvHf9+QFpuWK9M0yWh8WBYThBLaiIqi2TyfKwCul++HRwkFwLKXldccga+oKfECvXxSxYcXbM6h+o+NkxnxaQQwA5BJBDAz8E25SC/HTYbF7eAffG3er9K50Oyk+HzeTQ2aV8zoZvDOQQQA4B5NDAD8E2pOBi9rF4tUkL7pwcXhqE3mRo3Ff9/5T41IAcAsghgBwa+CHYhhScL4q53qSFNxle8v8Lfvdxc2afo+G1ATkEkEMAOTTwQ7ANKagWMEwAOQSQQwA5NPBDgBTMGhgmgBwCyCGAHBr4IUAKZg0ME0AOAeQQQA4N/BAgBbMGhgkghwByCCCHBn4IkIJZA8MEkEMAOQSQQwM/BEjBrIFhAsghgBwCyKGBH4KUFMSFCxcuXLi25/piVUICAAAAnz1IQQAAANvL/wPonlXHqmSP+wAAAABJRU5ErkJggg==" alt></span></p>
<p><span>Using this information draw the graph of \(I\) against \(d\) for \(0 \leqslant d \leqslant 100\) . Use a scale of \(1{\text{ cm}}\) to represent 10 metres on the horizontal axis and 1 cm to represent \(10\% \) on the vertical axis.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Some sea creatures have adapted so they can see in low intensity light and cannot tolerate too much light.<br></span></p>
<p><span>Indicate clearly on your graph the range of depths sea creatures could inhabit if they can tolerate between \(5\% \) and \(35\% \) of the light intensity at the surface.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the function \(f\left( x \right) = \frac{{48}}{x} + k{x^2} - 58\), where <em>x</em> > 0 and <em>k</em> is a constant.</p>
<p>The graph of the function passes through the point with coordinates (4 , 2).</p>
</div>
<div class="specification">
<p>P is the minimum point of the graph of <em>f </em>(<em>x</em>).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <em>k</em>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Using your value of <em>k</em> , find <em>f</em> ′(<em>x</em>).</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><strong>Use your answer</strong> to part (b) to show that the minimum value of <em>f</em>(<em>x</em>) is −22 .</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the <strong>two</strong> values of<em> x</em> which satisfy<em> f </em>(<em>x</em>) = 0.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of <em>y</em> = <em>f</em> (<em>x</em>) for 0 < <em>x</em> ≤ 6 and −30 ≤ <em>y</em> ≤ 60.<br>Clearly indicate the minimum point P and the <em>x</em>-intercepts on your graph.</p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>A function, \(f\) , is given by</p>
<p>\[f(x) = 4 \times {2^{ - x}} + 1.5x - 5.\]</p>
<p>Calculate \(f(0)\)</p>
<p> </p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use your graphic display calculator to solve \(f(x) = 0.\)</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of \(y = f(x)\) for \( - 2 \leqslant x \leqslant 6\) and \( - 4 \leqslant y \leqslant 10\) , showing the \(x\) and \(y\) intercepts. Use a scale of \(2\,{\text{cm}}\) to represent \(2\) units on both the horizontal axis, \(x\) , and the vertical axis, \(y\) .</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The function \(f\) is the derivative of a function \(g\) . It is known that \(g(1) = 3.\)</p>
<p>i) Calculate \(g'(1).\)</p>
<p>ii) Find the equation of the tangent to the graph of \(y = g(x)\) at \(x = 1.\) Give your answer in the form \(y = mx + c.\)</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>A distress flare is fired into the air from a ship at sea. The height, \(h\) , in metres, of the flare above sea level is modelled by the quadratic function</p>
<p>\[h\,(t) = - 0.2{t^2} + 16t + 12\,,\,t \geqslant 0\,,\]</p>
<p>where \(t\) is the time, in seconds, and \(t = 0\,\) at the moment the flare was fired.</p>
<p>Write down the height from which the flare was fired.</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 height of the flare \(15\) seconds after it was fired.</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>The flare fell into the sea \(k\) seconds after it was fired.</p>
<p>Find the value of \(k\) .</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find \(h'\,(t)\,.\)</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>i) Show that the flare reached its maximum height \(40\) seconds after being fired.</p>
<p>ii) Calculate the maximum height reached by the flare.</p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The nearest coastguard can see the flare when its height is more than \(40\) metres above sea level.</p>
<p>Determine the total length of time the flare can be seen by the coastguard.</p>
<div class="marks">[3]</div>
<div class="question_part_label">f.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-size: medium; font-family: times new roman,times;">Consider the function \(f(x) = - \frac{1}{3}{x^3} + \frac{5}{3}{x^2} - x - 3\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Sketch the graph of <em>y</em> = <em>f</em> (<em>x</em>) for −3 ≤ <em>x</em> </span><span><span>≤</span> 6 and −10 </span><span><span>≤</span> <em>y</em> </span><span><span>≤ </span>10 showing clearly the axes intercepts and local maximum and minimum points. Use a scale of 2 cm to represent 1 unit on the <em>x</em>-axis, and a scale of 1 cm to represent 1 unit on the <em>y</em>-axis.</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>Find the value of <em>f</em> (−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>Write down the coordinates of the <em>y</em>-intercept of the graph of <em>f</em> (<em>x</em>).</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>Find <em>f '</em>(<em>x</em>).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Show that \(f'( - 1) = - \frac{{16}}{3}\).</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Explain what <em>f</em> <em>'</em>(−1) represents.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Find the equation of the tangent to the graph of <em>f</em> (<em>x</em>) at the point where <em>x</em> is –1.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">g.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Sketch the tangent to the graph of <em>f</em> (<em>x</em>) at <em>x</em> = −1 on your diagram for (a).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">h.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>P and Q are points on the curve such that the tangents to the curve at these points are horizontal. The <em>x</em>-coordinate of P is <em>a</em>, and the <em>x</em>-coordinate of Q is <em>b</em>, <em>b</em> > <em>a</em>.</span></p>
<p><span>Write down the value of</span></p>
<p><span>(i) <em>a</em> ;</span></p>
<p><span>(ii) <em>b</em> .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span><span>P and Q are points on the curve such that the tangents to the curve at these points are horizontal. The <em>x</em>-coordinate of P is <em>a</em>, and the <em>x</em>-coordinate of Q is <em>b</em>, <em>b</em> > <em>a</em>.</span></span></p>
<p><span>Describe the behaviour of <em>f</em> (<em>x</em>) for <em>a</em> < <em>x</em> < <em>b</em>.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">j.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">George leaves a cup of hot coffee to cool and measures its temperature every minute. His results are shown in the table 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>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Write down the decrease in the temperature of the coffee</span></p>
<p><span>(i) during the first minute (between <em>t</em> = 0 and <em>t</em> =1) ;</span></p>
<p><span>(ii) during the second minute;</span></p>
<p><span>(iii) during the third minute.</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>Assuming the pattern in the answers to part (a) continues, show that \(k = 19\).</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>Use the <strong>seven</strong> results in the table to draw a graph that shows how the temperature of the coffee changes during the first six minutes.</span></p>
<p><span>Use a scale of 2 cm to represent 1 minute on the horizontal axis and 1 cm to represent 10 °C on the vertical axis.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span><span>The function that models the change in temperature of the coffee is <em>y</em> = <em>p</em> (2<sup>−<em>t</em></sup> )+ <em>q</em>.</span></span></p>
<p><span>(i) Use the values <em>t</em> = 0 and <em>y</em> = 94 to form an equation in <em>p</em> and <em>q</em>.</span></p>
<p><span>(ii) Use the values <em>t</em> =1 and <em>y</em> = 54 to form a second equation in <em>p</em> and <em>q</em>.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Solve the equations found in part (d) to find the value of <em>p</em> and the value of <em>q</em>.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>The graph of this function has a horizontal asymptote.</span></p>
<p><span>Write down the equation of this asymptote.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>George decides to model the change in temperature of the coffee with a linear function using correlation and linear regression.</span></p>
<p><span>Use the <strong>seven</strong> results in the table to write down</span></p>
<p><span>(i) the correlation coefficient;</span></p>
<p><span>(ii) the equation of the regression line <em>y</em> on <em>t</em>.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">g.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Use the equation of the regression line to estimate the temperature of the coffee at <em>t</em> = 3.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">h.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Find the percentage error in this estimate of the temperature of the coffee at <em>t</em> = 3.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">i.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">A biologist is studying the relationship between the number of chirps of the Snowy Tree cricket and the air temperature. He records the chirp rate, \(x\), of a cricket, and the corresponding air temperature, \(T\), in degrees Celsius.</p>
<p class="p1">The following table gives the recorded values.</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2015-12-04_om_08.39.25.png" alt></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Draw the scatter diagram for the above data. Use a scale of 2 cm for 20 chirps on the horizontal axis and 2 cm for 4<span class="s1"><strong>°</strong></span>C on the vertical axis.</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 class="p1">Use your graphic display calculator to write down the Pearson’s product–moment correlation <span class="s1">coefficient, \(r\)</span>, between \(x\) and \(T\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Interpret the relationship between \(x\) and \(T\) using your value of \(r\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Use your graphic display calculator to write down the equation of the regression line \(T\) on \(x\). Give the equation in the form \(T = ax + b\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Calculate the air temperature when the cricket’s chirp rate is \(70\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Given that \(\bar x = 70\), draw the regression line \(T\) on \(x\) on your scatter diagram.</p>
<div class="marks">[2]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">A forest ranger uses her own formula for estimating the air temperature. She counts the number of chirps in 15 seconds, \(z\), multiplies this number by \(0.45\) and then she adds \(10\).</p>
<p class="p1">Write down the formula that the forest ranger uses for estimating the temperature, \(T\).</p>
<p class="p1">Give the equation in the form \(T = mz + n\).</p>
<div class="marks">[1]</div>
<div class="question_part_label">g.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">A cricket makes 20 chirps in <strong>15</strong> seconds.</p>
<p class="p1">For this chirp rate</p>
<p class="p1">(i) <span class="Apple-converted-space"> </span>calculate an estimate for the temperature, \(T\), <strong>using the forest ranger’s formula</strong>;</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>determine the actual temperature recorded by the biologist, <strong>using the table above</strong>;</p>
<p class="p1">(iii) <span class="Apple-converted-space"> </span>calculate the percentage error in the forest ranger’s estimate for the temperature, compared to the actual temperature recorded by the biologist.</p>
<div class="marks">[6]</div>
<div class="question_part_label">h.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-size: medium; font-family: times new roman,times;">Consider the function \(f(x) = 3x + \frac{{12}}{{{x^2}}},{\text{ }}x \ne 0\).</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">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Calculate \(f ′(x)\) when \(x = 1\).</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>Use your answer to part (b) to decide whether the function, \(f\) , is increasing or decreasing at \(x = 1\). Justify your answer.</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>Solve the equation \(f ′(x) = 0\).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>The graph of <em>f</em> has a local minimum at point P. Let <em>T</em> be the tangent to the graph of <em>f</em> at P.</span></p>
<p><span>Write down the coordinates of P.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">e, i.</div>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>The graph of <em>f</em> has a local minimum at point P. Let <em>T</em> be the tangent to the graph of <em>f</em> at P.</span></p>
<p><span>Write down the gradient of <em>T</em>.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">e, ii.</div>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>The graph of <em>f</em> has a local minimum at point P. Let <em>T</em> be the tangent to the graph of <em>f</em> at P.</span></p>
<p><span>Write down the equation of <em>T</em>.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">e, iii.</div>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Sketch the graph of the function <em>f</em>, for −3 ≤ <em>x</em> ≤ 6 and −7 ≤ <em>y</em> ≤ 15. Indicate clearly the point P and any intercepts of the curve with the axes.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">f.</div>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>On your graph draw and label the tangent <em>T</em>.<br></span></p>
<div class="marks">[2]</div>
<div class="question_part_label">g, i.</div>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span><em>T</em> intersects the graph of <em>f</em> at a second point. Write down the<em> x</em>-coordinate of this point of intersection.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">g, ii.</div>
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