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</div><h2>SL Paper 2</h2><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>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 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_4.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_1.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.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_5.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_3.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_2.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>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">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><span style="font-size: medium; font-family: times new roman,times;">A shipping container is to be made with six rectangular faces, as shown in the diagram.</span></p>
<p> </p>
<p> </p>
<p><span style="font-size: medium; font-family: times new roman,times;"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAiUAAAC8CAIAAAAo+UkeAAATS0lEQVR4nO3d8W8TZ57H8f4nYzFH1eSIDkqBtCsRdY/SNFqlWVohJHYLNOQK1XHAdq9UcRq0DSuVIjkUriCoWltw2aMsMC5d2pQkOJReDthaBEqq7NwSHY2DVVAA15VDmtjP/TAzjp+QBBLyzNjJ+yX/ksZ2xtMvz8fPfBx4TAAAoN5jXh8AAGBWIG8AFJCEGTl5IrC+pNqICyHEYJ+xpURb0Ri94/WBzWbppHk2fDxQM78+kkhPcD/yBkChGIobm3TNp2vFVcHutBBC9IarF+iab2EgOuT1wc1aaTNYpfl0zadXBs2J4iYnb1paWpb98p+3vfmmv7aWGzfVt/379qdSKeV/FDDDpK+HN5Tq2uqQOSCEmGX7m0R3cNdh+4XfJ3k1VH9k4uVenaFoYOHIm4BxjeRN2DCe/Kf5ixc+de7cORNQ7L2dO1/fuDEWi6n+k4AZJW7UaD59/uzbziS7w/7KcTYQ6aRp1JUveOByr8yAGVytay80RpMT30/Km4aGP+7aFdA137lz5xQfHyDChvFsWVlHR4fXB4I8NxiPNtWVF+ta5Vv+tSWar8QfSQgh0vHoyeMhf2XuxbR0/OJhf6WulW78aP9bRTlXeNLxqBGoKfLpmk8vrw+biZGnT5oRu3u40x3cUFLklBDpePSQv0LzlWz4j4+3Lc3ZVCXMiBHyr62L3LJjwP5WwjTqK7TiisDFsddd+4DX1kVuOU8+6s6D8eipkL9S13y6VllndCeFECLRHdxQYl2w0ny6XVxlD/5qqLrU/pa2oMboHf95Rh9N0jTqyot1zVdSHThxPBi+lL3XYDxqNFpPW7R+z8V4Onv8Y5+QWxH/Ur1ILm/S8ejJj5znP9ydTIv786a1rf3QoabSRYvf3717zDMGTKPOzs5ny8r+8/Bhrw8EeWswHtlRoRVX1LfGh7pDlcW6trQucste4zSf86UQQqTjrdvLi/XyPdHkD9Z37ShK90XqK52FbygRqbeW7xJ/JOE0QCX+1r7ongrNZy+j9kNWNEb77fvbm6psh7Qp3HctvMFa619ojPb3GVtKxowEWzIaeEG3nr/7mnU88p2tXCndaFxP26/OWdDTuS98NLs+GVnux38e6WHdocpiK4/T8dbt5VvC8aGch1fWGd1J6yRYRzjuCREiEakrct4ECJHdb1X4DTN5PVy9QNc2WU8+Rt60trUb4c9eWvFy9avVXGGHarFY7PWNG/21tQwb7pfuMzYWZRdTKxuyq2dvuHrByDprVzvWomwt7vYC7RTa1gPtwBhZH+NGjVZc9dHRjyuLdc1XssHoS1u1kH2foWhgYe79h6KN83165buhwI7Qd+dDlcV6kT/cur8ueLkvUl8yfo1hH0bl/rbj774T6b0r3dn5iRuMvrTz0pz9yn2Jksu6lpU9vImeR2K9iqINoe6EEOnEpUvX0tmH27uutBmssnNrghOSTsivOif10/b/PmeXOXbetLa1f/7Fl1u3vvGLp5/hCjtUS6VS7+3cuWrlSoYNMuvtuXPRKRGpy71EZn1p7w/SSWt3Ur4nmswmk/222l6v7ffj9sZIXi5X7Qr4Nwav2leUkhcby4t1+2MIVj7lLNlxo0bzlZRvaIz0pa0+qeiFmkAknpZC7j52MOjLVm0/md15OJ2H9VqyUSp9aR1h7gbi/lPk/NCJnmeUQWdDVrk90mfnmHX/+4NtohMyaguVPZ4f7Ot15Tsi8UHracbNG+u2d+8+6hy4o6WlhToHkrGWXaetGfW2+lbuBbRRb6vtrU/5nmhyyI6lkY+0DZjB1XrRsir/SWfRTUvXi0Z/Ii539XfuaYWcdYVq7F3IyBHaOw9pZXeexz7gUV/aoTv2tmmsUzTO84zxYLsZkg/jvmCb8IRYByBfW7MvFZb7Qyej8Zwf/YC8aW1rP/LJMeocuMM0TeocZFnXbezV0H6LvbQu0t0ZHa/hsN5034kGVuiab2Gg4/rZDusyUTx6pNG/vsKqbaoD4ewyaIfElnDfoPNjnY1ItRHPbpvmB/56/fzZawPy23lpb2HtosbZhUy8ZbEv8Tltk/VanEScaI8yKlEmfB7pceZnJ60Mss6Vtc2SHy6EEIPxS1fj6YlOyKhXbf8vc3aWQqST5vnoQ+5vstfWqHPgjv7+fuocWEby5k5ftDVYN9+na6Xr/Ec7k+mR/UFf99loPD3SqP+QNE9bn60qWfvOiUt37CW13B8yjHDEHPVJrbFCIqcRSZptgfUlmk8vWl13/PLoa3r2D7XWVutRxVXB891nL8Xv20845Y0VDNaVtwU1xlXz7KV4Ojd+cur63AfOD0TvmGejo57YPtSFgYt3zfPR+MAEzyNJROrmWx88G+wztjgfyXOyxC51EmbkdDQ+OOEJsb61oMboTcejn0XMhF2VWSGXTppnP8s55ofKG+tGnQN3pFKp/fv2U+fAuXRTWXfoYjxxsbG8eOSjzHberG9stSLEKbr9wYh5LeJfOu7nibXiCr9hJq01cGS5lH6sdTmu3B+KdFsfAXAeIu1L5AjJHkD2yXONavWtL0trAqftO6fjf31/fYl9eMFIzse1nU9MZF9pLqfGr97TZj1k/OeRXuC1jrPdV63kGHmsEM5Hun26VloTMLL7kvFPiFMCjRxe9hnkfaQQYlJ5Q50DN1HnYDoMxqNGo//dcPRixDhi/07JQ/wmPFSYXN5Q58BN1Dl4VIlIXVFuutiXjPj71jwx6bzJ1jmvrl3HFXaoRp2DR2JVLCN/oYD126Oz5O9byztTyZuROqf0aa6wQzXqHDyKdDwatiruB7UaUG3qeUOdAzdR5wCF7pHyxqpzllDnwBXUOUBBe9S8oc6Bm6hzgMI1DXlDnQM3UecABWra8iZb5zQ3N3v4ejBLtLS06JqPOgcoINOZN7l1Dpc7oJpV5+zft9/rAwHwUKY5b3LrnP7+fg9fGGYDq855feNG3t8A+W/68ya3zjFN08PXhtnAqnOeLSujzgHynKq8oc6Bm6hzgPynMG+sOuf55c9T58AFsViMOgfIZ2rzxqpz1ryyljoHLkilUtQ5QN5SnjfWzV9bR50Dd1DnAPnJpbyx6pynFy+hzoELOjo6qHOAfONe3lDnwE3UOUC+cTVvqHPgJuocIK+4nTfUOXAZdQ6QJ7zJm9a29r1795UuWkydAxdQ5wD5wLO8aW1rP/LJseXLllPnwAXUOYDnvMyb1rb2z7/48re/eYU6By6gzgG85XHeWDd/bd0z1DlwBXUO4JW8yJtW6hy4yKpzWlpavD4QYHbJl7xppc6Bi7J1DsMGuCaP8qY1p87hcgdUy9Y5dIeAO/Irb6ybVed0dnZ6eF4wS1h1Dt0h4IJ8zJtWp845fvy4h6cGs0RHR8ezZWXUOYBqeZo3rU6ds6OhgSvsUC0Wi61auZI6B1Aqf/OmlToHLkqlUv7aWuocQJ28zhvr1tDwx5Lif/TX1nLjpvqma75ny8q8+/MIzGQFkDetbe265jMBV+iaz8M/kMAMVjB54+E5wqzCsAGKkDeAhGEDFCFvAAnDBihC3gAShg1QhLwBJAwboAh5A0gYNkAR8gaQMGyAIuQNIGHYAEXIG0DCsAGKkDeAhGEDFCFvAAnDBihC3gAShg1QhLwBJAwboAh5A0gYNkAR8gaQMGyAIuQNIGHYAEXIG0DCsAGKkDeAhGEDFCFvAAnDBihC3gAShg1QhLwBJAwboAh5A0gYNkAR8gaQMGyAIuQNIGHYAEXIG0DCsAGKkDeAhGEDFCFvAAnDBihC3gAShg1QhLwBJAwboAh5A0gYNkAR8gaQMGyAIuQNIGHYAEXIG0DCsAGKkDeAhGEDFCFvAAnDBihC3gAShg1QhLwBJAwboAh5A0gYNkAR8gaQMGyAIuQNIGHYAEXIG0DCsAGKkDeAhGEDFCFvAAnDBihC3gAShg1QhLwBJAwboAh5A0gYNkAR8gaQMGyAIuQNIGHYAEXIG0DCsAGKkDeAhGEDFCFvAAnDBihC3gAShg1QhLwBJAwboAh5A0gYNkAR8gaQMGyAIuQNIGHYAEUKN29+/O5oQ035Ql2raowmhBBC3Iz4f6k/tePrH9OunT7MPOQNoEjh5o0QQmRuntr8D3OfP9g1LIQQ926c27Pmqd9/fnPYnXOHGYm8ARQp7LwRw10Hn5u74N3z94QQQmRun9m+9dMbGTdOHGYq8gZQpMDzRvSGqxfo1UZcCCESlz/Y/mFX0o3ThpmLvAEUKfS8uRXxL9VfOdqbGU5eDtaHugbcOW2YucgbQJFCz5tkNPCCPj8QvX1x3/YTPT9zKQ2PirwBFCn0vBkwg6v1eev+vb7x9I177pwyzGzkDaBIoefNUNzYpM9Zve/yHbY2mBbkDaBIoefNQE/Ttm2nrg+5c7YwC5A3gCIFnTfDya5ju//r2yRbG0wf8gZQpBDzJjNgfrrjncORtoNvv9d2Y4i0wXQibwBFCjFv7vU0VT8+Z/m/HeyIEzaYbuQNoEgh5g2gEMMGKELeABKGDVCEvAEkDBugCHkDSBg2QBHyBpAwbIAi5A0gYdgARcgbQMKwAYqQN4CEYQMUIW8ACcMGKELeABKGDVCEvAEkDBugCHkDSBg2QJGRvOns7CxdtPjQoSbP04W8gYcYNkCRx3K/OHfu3DOlT+/aFfA8YMgbeIVhAxR5bNTXsVisqvLFrVvf+PyLLz2PGfIG7mPYAEVG540QIpVK/f6NN15a8fKRT455njTkDVzGsAGKjJE3lmAwWLpo8cEPP/I8bMgbuIlhAxQZN2+EU+fkw4fWWALgGoYNUGSivBFOnbPhtY3e1jksAXANwwYo8oC8EUKkUql3/vAHb+sclgC4hmEDFHlw3liO/vnPHtY5LAFwDcMGKPKweSOEiEajXtU5LAFwDcMGKDKJvBFCxGKxtWvWul/nsATANQwboMjk8kZ4VOewBMA1DBugyKTzxuJyncMSANcwbIAiU8wb4W6dwxIA1zBsgCJTzxvhYp3DEgDXMGyAIo+UN0KIVCq1o6FBdZ3DEgDXMGyAIo+aN5bjx48rrXNYAuAahg1QZHryRgjR2dmprs5hCYBrGDZAkWnLGyFELBZ7de06FXUOSwBcw7ABikxn3ghldQ5LAFzDsAGKTHPeWKa9zmEJgGsYNkCRSedNJn529+9Wlc3Rl7z55c3McLLrT5vLivUnt0duD+fezapz/LV15A0KC8MGKDK1/c3A90c3PTHntaaOE9u3Bb/65uuWdjOZGX0nq8757W9eefQ6hyUArmHYAEWmeD0t09O0Spu7sGLX1/K2ZhSrzlm+bPkj1jksAXANwwYoMtX+Zrjr4HNzS/yRxEPc16pz9u7dR94g/zFsgCJT3d/cPvP2U7r+3MHvJtrejDBN81HqHJYAuIZhAxSZUt5k+i/s3vnxgc1PaOuaeu6JoR/v/vTg2Onv759yncMSANcwbIAik8qb4WRP9ELP7dvnD+z4tOeeGazSSjf/pavr0AfHrqUe5vGpVOr93bunUOewBMA1DBugyKTy5lbEv1TX5v2qof1mRojh7tCL8/Q5v244ExuazLM0NzdPts5hCYBrGDZAkcldT8ske69c6XU++pwZuvX3b3vv/yD0g022zmEJgGsYNkARJX+/wMOYVJ3DEgDXMGyAIp7ljXDqnOeXP//AOoclAK5h2ABFvMwbS3Nz89OLl0xc57AEwDUMG6CI93kjhDBN8xcT1jksAXANwwYokhd5I5w6Z80ra8esc1gC4BqGDVAkX/JGTFjnsATANQwboEge5Y1lzDqHJQCuYdgARfIub8RYdQ5LAFzDsAGK5GPeCKfOeWnFy1adwxIA1zBsgCJ5mjfCqXOWLFp85JNjLAFwDcMGKJK/eWNpbm7WNZ+u+UzAFeQNoEi+540QwjTNqher/LW13Li5cHtv506vRx6YmQogbwAAMwB5AwBwA3kDAHADeQMAcAN5AwBwA3lTwDLJ6xdOnwx/evq//3ZrUv+kNwC4j7wpUMPJrsOvPalbv5yka4tW7jxzY2gK/7Q3ALiEvClMqW8aK6vfMy58n0zdvHz07Yp5urZog3GdwAGQt8ibQpS5Fz1QdyrmpEvm5+8+rNR8j//rqZueHhYATIC8mRESkboiX4k/kvD6QABgPOTNTJDpPbpGK918qi8jRCZ+dvfvVpXN0Ze8+eXNzHCy60+by4r1J7dHbg97fZgAZjXyZgb4+YaxuWRlyPw5W98MfH900xNzXmvqOLF9W/Crb75uaTeTdDsAPEXeFLzM7a8alr0W+ltS+o89Tau0uQsrdn3NtgZAfiBvClwmHql/veFMbPTv3wx3HXxuLo0OgPxB3hSyzN2u0Ftbj/7vz/d/5/aZt5/S9ecOfsf2BkB+IG8KVuZuV+itLaEr2WImk+y58n8/CSFEpv/C7p0fH9j8hLauqeeeGPrx7k/EDgCPkTeFKXO3K/T6kgp/0DDChhE2DKNpz5sr/uXAmf+50HP79vkDOz7tuWcGq7TSzX/p6jr0wbFrKa+PGMBsR94UoqTZtGmR/TfZjNweX/PR0dqlujbvVw3tNzNCDHeHXpynz/n1GO0OALiOvJlRMsneK1d6nStsmaFbf/+2lw9CA8gL/w9Ru3XJfg9qvwAAAABJRU5ErkJggg==" alt></span></p>
<p><span style="font-size: medium; font-family: times new roman,times;">The dimensions of the container are</span></p>
<p><span style="font-size: medium; font-family: times new roman,times;">length 2<em>x</em></span><br><span style="font-size: medium; font-family: times new roman,times;">width <em>x</em></span><br><span style="font-size: medium; font-family: times new roman,times;">height <em>y</em>.</span></p>
<p><span style="font-size: medium; font-family: times new roman,times;">All of the measurements are in metres. The total length of all twelve edges is 48 metres.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Show that <em>y</em> =12 − 3<em>x </em>.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Show that the volume <em>V</em> m<sup>3</sup> of the container is given by</span></p>
<p><span><em>V</em> = 24<em>x</em><sup>2</sup> − 6<em>x</em><sup>3</sup></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 \( \frac{{\text{d}V}}{{\text{d}x}}\).</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 value of <em>x</em> for which <em>V</em> is a maximum.</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 maximum volume of the container.</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>Find the length and height of the container for which the volume is a maximum.</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>The shipping container is to be painted. One litre of paint covers an area of 15 m<sup>2</sup> .</span> <span>Paint comes in tins containing four litres.</span></p>
<p><span>Calculate the number of tins required to paint the shipping container.</span></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;">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 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">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><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="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;">A dog food manufacturer has to cut production costs. She wishes to use as little aluminium as possible in the construction of cylindrical cans. In the following diagram, <em>h</em> represents the height of the can in cm and <em>x</em>, the radius of the base of the can in cm.</span></p>
<p style="text-align: center;"><span style="font-family: times new roman,times; font-size: medium;"><img src="data:image/png;base64,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" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">The volume of the dog food cans is 600 cm<sup>3</sup>.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Show that \(h = \frac{{600}}{{\pi {x^2}}}\).</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 an expression for the curved surface area of the can, in terms of <em>x</em>. Simplify your answer.<br></span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Hence write down an expression for <em>A</em>, the total surface area of the can, in terms of <em>x</em>.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Differentiate <em>A</em> in terms of <em>x</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 value of <em>x</em> that makes <em>A</em> a minimum.</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>Calculate the minimum total surface area of the dog food can.</span></p>
<div class="marks">[2]</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;">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="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><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 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" 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><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 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 lobster trap is made in the shape of half a cylinder. It is constructed from a steel frame with netting pulled tightly around it. The steel frame consists of a rectangular base, two semicircular ends and two further support rods, as shown in the following diagram.</span></p>
<p style="font: normal normal normal 21px/normal 'Times New Roman'; min-height: 25px; text-align: center; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;"><span style="font-family: 'times new roman', times; font-size: medium;"><br><img src="images/Schermafbeelding_2014-09-20_om_14.54.16.png" alt><br></span></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;"> </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 semicircular ends each have radius \(r\) and the support rods each have length \(l\).</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;">Let \(T\) be the total length of steel used in the frame of the lobster trap.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Write down an expression for \(T\) in terms of \(r\), \(l\) and \(\pi \).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>The volume of the lobster trap is \(0.75{\text{ }}{{\text{m}}^{\text{3}}}\).</span></p>
<p><span>Write down an equation for the volume of the lobster trap in terms of \(r\), \(l\) and \(\pi \).</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 volume of the lobster trap is \(0.75{\text{ }}{{\text{m}}^{\text{3}}}\).</span></p>
<p><span>Show that \(T = (2\pi + 4)r + \frac{6}{{\pi {r^2}}}\).</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>The volume of the lobster trap is \(0.75{\text{ }}{{\text{m}}^{\text{3}}}\).</span></p>
<p><span>Find \(\frac{{{\text{d}}T}}{{{\text{d}}r}}\).</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 lobster trap is designed so that the length of steel used in its frame is a minimum.</span></p>
<p><span>Show that the value of \(r\) for which \(T\) is a minimum is \(0.719 {\text{ m}}\), correct to three significant figures.</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 lobster trap is designed so that the length of steel used in its frame is a minimum.</span></p>
<p><span>Calculate the value of \(l\) for which \(T\) 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 lobster trap is designed so that the length of steel used in its frame is a minimum.</span></p>
<p><span>Calculate the minimum value of \(T\).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">g.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Tepees were traditionally used by nomadic tribes who lived on the Great Plains of North America. They are cone-shaped dwellings and can be modelled as a cone, with vertex O, shown below. The cone has radius, \(r\), height, \(h\), and slant height, \(l\).</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2015-12-04_om_10.28.13.png" alt></p>
<p class="p1">A model tepee is displayed at a Great Plains exhibition. The curved surface area of this tepee is covered by a piece of canvas that is \(39.27{\text{ }}{{\text{m}}^2}\), and has the shape of a semicircle, as shown in the following diagram.</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2015-12-04_om_10.29.53.png" alt></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Show that the slant height, \(l\), is \(5\) m, correct to the nearest metre.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Find the circumference of the base of the cone.</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Find the radius, \(r\), of the base.</p>
<p class="p1">(iii) <span class="Apple-converted-space"> </span>Find the height, \(h\).</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">A company designs cone-shaped tents to resemble the traditional tepees.</p>
<p class="p1">These cone-shaped tents come in a range of sizes such that the sum of the diameter and the height is equal to <strong>9.33 m</strong>.</p>
<p class="p1">Write down an expression for the height, \(h\), in terms of the radius, \(r\), of these cone-shaped tents.</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">A company designs cone-shaped tents to resemble the traditional tepees.</p>
<p class="p1">These cone-shaped tents come in a range of sizes such that the sum of the diameter and the height is equal to <strong>9.33 m</strong>.</p>
<p class="p1">Show that the volume of the tent, \(V\), can be written as</p>
<p class="p1">\[V = 3.11\pi {r^2} - \frac{2}{3}\pi {r^3}.\]</p>
<div class="marks">[1]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">A company designs cone-shaped tents to resemble the traditional tepees.</p>
<p class="p1">These cone-shaped tents come in a range of sizes such that the sum of the diameter and the height is equal to <strong>9.33 m</strong>.</p>
<p class="p1">Find \(\frac{{{\text{d}}V}}{{{\text{d}}r}}\).</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">A company designs cone-shaped tents to resemble the traditional tepees.</p>
<p class="p1">These cone-shaped tents come in a range of sizes such that the sum of the diameter and the height is equal to <strong>9.33 m</strong>.</p>
<p class="p1">(i) <span class="Apple-converted-space"> </span>Determine the exact value of \(r\) for which the volume is a maximum.</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Find the maximum volume.</p>
<div class="marks">[4]</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 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><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><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-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-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 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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-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;">Nadia designs a wastepaper bin made in the shape of an <strong>open</strong> cylinder with a volume of \(8000{\text{ c}}{{\text{m}}^3}\).</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><img src="data:image/png;base64,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" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Nadia decides to make the radius, \(r\) , of the bin \(5{\text{ cm}}\).</span></p>
</div>
<div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">Merryn also designs a cylindrical wastepaper bin with a volume of \(8000{\text{ c}}{{\text{m}}^3}\). She decides to fix the radius of its base so that the <strong>total external surface area</strong> of the bin is minimized.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"><img src="data:image/png;base64,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" alt></span></p>
<p><span style="font-family: times new roman,times; font-size: medium;">Let the radius of the base of Merryn’s wastepaper bin be \(r\) , and let its height be \(h\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Calculate</span><br><span>(i) the area of the base of the wastepaper bin;</span><br><span>(ii) the height, \(h\) , of Nadia’s wastepaper bin;</span><br><span>(iii) the total <strong>external</strong> surface area of the wastepaper bin.</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>State whether Nadia’s design is practical. Give a reason.</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 an equation in \(h\) and \(r\) , using the given volume of the bin.</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>Show that the total external surface area, \(A\) , of the bin is \(A = \pi {r^2} + \frac{{16000}}{r}\) .</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 \(\frac{{{\text{d}}A}}{{{\text{d}}r}}\).</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>(i) Find the value of \(r\) that minimizes the total external surface area of the wastepaper bin.</span><br><span>(ii) Calculate the value of \(h\) corresponding to this value of \(r\) .</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span>Determine whether Merryn’s design is an improvement upon Nadia’s. Give a reason.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">g.</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;">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>A manufacturer makes trash cans in the form of a cylinder with a hemispherical top. The trash can has a height of 70 cm. The base radius of both the cylinder and the hemispherical top is 20 cm.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<div class="specification">
<p>A designer is asked to produce a new trash can.</p>
<p>The new trash can will also be in the form of a cylinder with a hemispherical top.</p>
<p>This trash can will have a height of <em>H</em> cm and a base radius of <em>r</em> cm.</p>
<p style="text-align: center;"><img 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"></p>
<p>There is a design constraint such that <em>H</em> + 2<em>r</em> = 110 cm.</p>
<p>The designer has to maximize the volume of the trash can.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the height of the cylinder.</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 total volume of the trash can.</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 height of the <strong>cylinder</strong>, <em>h</em> , of the new trash can, in terms of <em>r</em>.</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 the volume, <em>V</em> cm<sup>3</sup> , of the new trash can is given by</p>
<p>\(V = 110\pi {r^3}\).</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>Using your graphic display calculator, find the value of <em>r</em> which maximizes the value of <em>V</em>.</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>The designer claims that the new trash can has a capacity that is at least 40% greater than the capacity of the original trash can.</p>
<p>State whether the designer’s claim is correct. Justify your answer.</p>
<div class="marks">[4]</div>
<div class="question_part_label">f.</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;">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="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="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="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 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 \(f(x) = 3x + \frac{{12}}{{{x^2}}},{\text{ }}x \ne 0\).</span></p>
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<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>
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<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>
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<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>
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<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>
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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 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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<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>
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<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>
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<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>
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<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>
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<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>
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<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>
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<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>
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<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>
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<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>
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<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>
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<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>
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