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<h2>HL Paper 2</h2><div class="specification">
<p>An ice-skater is skating such that her position vector when viewed from above at time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> seconds can be modelled by</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mi>x</mi></mtd></mtr><mtr><mtd><mi>y</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mtable><mtr><mtd><mi>a</mi><mo> </mo><msup><mtext>e</mtext><mrow><mi>b</mi><mi>t</mi></mrow></msup><mtext> </mtext><mi>cos</mi><mo> </mo><mi>t</mi></mtd></mtr><mtr><mtd><mi>a</mi><mo> </mo><msup><mtext>e</mtext><mrow><mi>b</mi><mi>t</mi></mrow></msup><mtext> sin</mtext><mo> </mo><mi>t</mi></mtd></mtr></mtable></mfenced></math></p>
<p>with respect to a rectangular coordinate system from a point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math>, where the non-zero constants <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math> can be determined. All distances are in metres.</p>
</div>
<div class="specification">
<p>At time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn></math>, the displacement of the ice-skater is given by <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>5</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr></mtable></mfenced></math> and the velocity of the ice‑skater is given by <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mo>-</mo><mn>3</mn><mo>.</mo><mn>5</mn></mtd></mtr><mtr><mtd><mn>5</mn></mtd></mtr></mtable></mfenced></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the velocity vector at time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>.</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 the magnitude of the velocity of the ice-skater at time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> is given by</p>
<p style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo> </mo><msup><mtext>e</mtext><mrow><mi>b</mi><mi>t</mi></mrow></msup><msqrt><mfenced><mrow><mn>1</mn><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow></mfenced></msqrt></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> and the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the magnitude of the velocity of the ice-skater when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>2</mn></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>At a point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math>, the ice-skater is skating parallel to the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-axis for the first time.</p>
<p>Find <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>OP</mtext></math>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the following system of coupled differential equations.</p>
<p style="padding-left: 210px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mo>-</mo><mn>4</mn><mi>x</mi></math></p>
<p style="padding-left: 210px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mn>3</mn><mi>x</mi><mo>-</mo><mn>2</mn><mi>y</mi></math></p>
</div>
<div class="specification">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac></math></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the eigenvalues and corresponding eigenvectors of the matrix <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mo>-</mo><mn>4</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>3</mn></mtd><mtd><mo>-</mo><mn>2</mn></mtd></mtr></mtable></mfenced></math>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, write down the general solution of the system.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine, with justification, whether the equilibrium point <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>0</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math> is stable or unstable.</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>(i) at <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>4</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math>.</p>
<p>(ii) at <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mo>-</mo><mn>4</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch a phase portrait for the general solution to the system of coupled differential equations for <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>−</mo><mn>6</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mn>6</mn></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>−</mo><mn>6</mn><mo>≤</mo><mi>y</mi><mo>≤</mo><mn>6</mn></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p>A ball is attached to the end of a string and spun horizontally. Its position relative to a given point, <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math>, at time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> seconds, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>≥</mo><mn>0</mn></math>, is given by the equation</p>
<p style="padding-left: 30px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">r</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mn>1</mn><mo>.</mo><mn>5</mn><mo> </mo><mi>cos</mi><mo> </mo><mo>(</mo><mn>0</mn><mo>.</mo><mn>1</mn><msup><mi>t</mi><mn>2</mn></msup><mo>)</mo></mtd></mtr><mtr><mtd><mn>1</mn><mo>.</mo><mn>5</mn><mo> </mo><mi>sin</mi><mo> </mo><mo>(</mo><mn>0</mn><mo>.</mo><mn>1</mn><msup><mi>t</mi><mn>2</mn></msup><mo>)</mo></mtd></mtr></mtable></mfenced></math> where all displacements are in metres.</p>
</div>
<div class="specification">
<p>The string breaks when the magnitude of the ball’s acceleration exceeds <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>20</mn><mo> </mo><msup><mtext>ms</mtext><mrow><mo>-</mo><mn>2</mn></mrow></msup></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the ball is moving in a circle with its centre at <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math> and state the radius of the circle.</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>Find an expression for the velocity of the ball at time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence show that the velocity of the ball is always perpendicular to the position vector of the ball.</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>Find an expression for the acceleration of the ball at time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> at the instant the string breaks.</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>How many complete revolutions has the ball completed from <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn></math> to the instant at which the string breaks?</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.iii.</div>
</div>
<br><hr><br><div class="specification">
<p>A change in grazing habits has resulted in two species of herbivore, <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>X</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Y</mtext></math>, competing for food on the same grasslands. At time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn></math> environmentalists begin to record the sizes of both populations. Let the size of the population of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>X</mtext></math> be <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>, and the size of the population <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Y</mtext></math> be <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>. The following model is proposed for predicting the change in the sizes of the two populations:</p>
<p style="padding-left: 60px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>x</mi><mo>˙</mo></mover><mo>=</mo><mn>0</mn><mo>.</mo><mn>3</mn><mi>x</mi><mo>-</mo><mn>0</mn><mo>.</mo><mn>1</mn><mi>y</mi></math></p>
<p style="padding-left: 60px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>y</mi><mo>˙</mo></mover><mo>=</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>0</mn><mo>.</mo><mn>4</mn><mi>y</mi></math></p>
<p style="padding-left: 60px;">for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>,</mo><mo> </mo><mi>y</mi><mo>></mo><mn>0</mn></math></p>
</div>
<div class="specification">
<p>For this system of coupled differential equations find</p>
</div>
<div class="specification">
<p>When <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>X</mtext></math> has a population of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2000</mn></math>.</p>
</div>
<div class="specification">
<p>It is known that <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Y</mtext></math> has an initial population of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2900</mn></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>the eigenvalues.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>the eigenvectors.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence write down the general solution of the system of equations.</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>Sketch the phase portrait for this system, for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>,</mo><mo> </mo><mi>y</mi><mo>></mo><mn>0</mn></math>.</p>
<p>On your sketch show</p>
<ul>
<li>the equation of the line defined by the eigenvector in the first quadrant</li>
<li>at least two trajectories either side of this line using arrows on those trajectories to represent the change in populations as <em>t</em> increases</li>
</ul>
<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 a condition on the size of the initial population of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Y</mtext></math> if it is to avoid its population reducing to zero.</p>
<div class="marks">[1]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> at which <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn></math>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">e.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the population of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Y</mtext></math> at this value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>. Give your answer to the nearest <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn></math> herbivores.</p>
<div class="marks">[2]</div>
<div class="question_part_label">e.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>A particle <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math> moves along the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis. The velocity of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><msup><mtext> m s</mtext><mrow><mo>−</mo><mn>1</mn></mrow></msup></math> at time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> seconds, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mo>=</mo><mo>−</mo><mn>2</mn><msup><mi>t</mi><mn>2</mn></msup><mo>+</mo><mn>16</mn><mi>t</mi><mo>−</mo><mn>24</mn></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>≥</mo><mn>0</mn></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the times when <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math> is at instantaneous rest.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the magnitude of the particle’s acceleration at <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn></math> seconds.</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 greatest speed of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math> in the interval <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>≤</mo><mi>t</mi><mo>≤</mo><mn>6</mn></math>.</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>The particle starts from the origin <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math>. Find an expression for the displacement of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math> from <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math> at time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> seconds.</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the total distance travelled by <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math> in the interval <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>≤</mo><mi>t</mi><mo>≤</mo><mn>4</mn></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p>A shock absorber on a car contains a spring surrounded by a fluid. When the car travels over uneven ground the spring is compressed and then returns to an equilibrium position.</p>
<p style="text-align: center;"><img 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"></p>
<p>The displacement, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>, of the spring is measured, in centimetres, from the equilibrium position of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn></math>. The value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> can be modelled by the following second order differential equation, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> is the time, measured in seconds, after the initial displacement.</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>x</mi><mo>¨</mo></mover><mo>+</mo><mn>3</mn><mover><mi>x</mi><mo>˙</mo></mover><mo>+</mo><mn>1</mn><mo>.</mo><mn>25</mn><mi>x</mi><mo>=</mo><mn>0</mn></math></p>
</div>
<div class="specification">
<p>The differential equation can be expressed in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mover><mi>x</mi><mo>˙</mo></mover></mtd></mtr><mtr><mtd><mover><mi>y</mi><mo>˙</mo></mover></mtd></mtr></mtable></mfenced><mo>=</mo><mi mathvariant="bold-italic">A</mi><mfenced><mtable><mtr><mtd><mi>x</mi></mtd></mtr><mtr><mtd><mi>y</mi></mtd></mtr></mtable></mfenced></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">A</mi></math> is a <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>×</mo><mn>2</mn></math> matrix.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mover><mi>x</mi><mo>˙</mo></mover></math>, show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>y</mi><mo>˙</mo></mover><mo>=</mo><mo>−</mo><mn>1</mn><mo>.</mo><mn>25</mn><mi>x</mi><mo>−</mo><mn>3</mn><mi>y</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the matrix <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">A</mi></math>.</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 eigenvalues of matrix <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">A</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the eigenvectors of matrix <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">A</mi></math>.</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>Given that when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn></math> the shock absorber is displaced <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn><mo> </mo><mtext>cm</mtext></math> and its velocity is zero, find an expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>The cross-sectional view of a tunnel is shown on the axes below. The line <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>[</mo><mtext>AB</mtext><mo>]</mo></math> represents a vertical wall located at the left side of the tunnel. The height, in metres, of the tunnel above the horizontal ground is modelled by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>1</mn><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mo> </mo><mn>0</mn><mo>.</mo><mn>8</mn><msup><mi>x</mi><mn>2</mn></msup><mo>,</mo><mo> </mo><mn>2</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mn>8</mn></math>, relative to an origin <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math>.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<p style="text-align: left;">Point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> has coordinates <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>2</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math>, point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math> has coordinates <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>2</mn><mo>,</mo><mo> </mo><mn>2</mn><mo>.</mo><mn>4</mn><mo>)</mo></math>, and point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>C</mtext></math> has coordinates <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>8</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math>.</p>
</div>
<div class="specification">
<p>Find the height of the tunnel when</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find the maximum height of the tunnel.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>4</mn></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>6</mn></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use the trapezoidal rule, with three intervals, to estimate the cross-sectional area of the tunnel.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the integral which can be used to find the cross-sectional area of the tunnel.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find the cross-sectional area of the tunnel.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>A particle moves such that its displacement, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> metres, from a point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math> at time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> seconds is given by the differential equation</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>x</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>+</mo><mn>5</mn><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>+</mo><mn>6</mn><mi>x</mi><mo>=</mo><mn>0</mn></math></p>
</div>
<div class="specification">
<p>The equation for the motion of the particle is amended to</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>x</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>+</mo><mn>5</mn><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>+</mo><mn>6</mn><mi>x</mi><mo>=</mo><mn>3</mn><mi>t</mi><mo>+</mo><mn>4</mn></math>.</p>
</div>
<div class="specification">
<p>When <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn></math> the particle is stationary at <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use the substitution <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></math> to show that this equation can be written as</p>
<p style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></mtd></mtr><mtr><mtd><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mtable><mtr><mtd><mn>0</mn><mo> </mo><mo> </mo></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>6</mn><mo> </mo><mo> </mo></mtd><mtd><mo>-</mo><mn>5</mn></mtd></mtr></mtable></mfenced><mfenced><mtable><mtr><mtd><mi>x</mi></mtd></mtr><mtr><mtd><mi>y</mi></mtd></mtr></mtable></mfenced></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the eigenvalues for the matrix <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>0</mn><mo> </mo><mo> </mo></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>6</mn><mo> </mo><mo> </mo></mtd><mtd><mo>-</mo><mn>5</mn></mtd></mtr></mtable></mfenced></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence state the long-term velocity of the particle.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use the substitution <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></math> to write the differential equation as a system of coupled, first order differential equations.</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>Use Euler’s method with a step length of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>1</mn></math> to find the displacement of the particle when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>1</mn></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the long-term velocity of the particle.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.iii.</div>
</div>
<br><hr><br><div class="specification">
<p>The curve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f\left( x \right)">
<mi>y</mi>
<mo>=</mo>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span> is shown in the graph, for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0 \leqslant x \leqslant 10">
<mn>0</mn>
<mo>⩽<!-- ⩽ --></mo>
<mi>x</mi>
<mo>⩽<!-- ⩽ --></mo>
<mn>10</mn>
</math></span>.</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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"></p>
<p>The curve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f\left( x \right)">
<mi>y</mi>
<mo>=</mo>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span> passes through the following points.</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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"></p>
<p>It is required to find the area bounded by the curve, the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span><em>-</em>axis, the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span><em>-</em>axis and the line <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 10">
<mi>x</mi>
<mo>=</mo>
<mn>10</mn>
</math></span>.</p>
</div>
<div class="specification">
<p>One possible model for the curve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f\left( x \right)">
<mi>y</mi>
<mo>=</mo>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span> is a cubic function.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use the trapezoidal rule to find an estimate for the area.</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>Use all the coordinates in the table to find the equation of the least squares cubic regression curve.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the coefficient of determination.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down an expression for the area enclosed by the cubic regression curve, the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>-axis, the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span>-axis and the line <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 10">
<mi>x</mi>
<mo>=</mo>
<mn>10</mn>
</math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of this area.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>At an archery tournament, a particular competition sees a ball launched into the air while an archer attempts to hit it with an arrow.</p>
<p>The path of the ball is modelled by the equation</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mi>x</mi></mtd></mtr><mtr><mtd><mi>y</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mtable><mtr><mtd><mn>5</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mi>t</mi><mfenced><mtable><mtr><mtd><msub><mi>u</mi><mi>x</mi></msub></mtd></mtr><mtr><mtd><msub><mi>u</mi><mi>y</mi></msub><mo>-</mo><mn>5</mn><mi>t</mi></mtd></mtr></mtable></mfenced></math></p>
<p>where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> is the horizontal displacement from the archer and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> is the vertical displacement from the ground, both measured in metres, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> is the time, in seconds, since the ball was launched.</p>
<ul>
<li><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mi>x</mi></msub></math> is the horizontal component of the initial velocity</li>
<li><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mi>y</mi></msub></math> is the vertical component of the initial velocity.</li>
</ul>
<p>In this question both the ball and the arrow are modelled as single points. The ball is launched with an initial velocity such that <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mi>x</mi></msub><mo>=</mo><mn>8</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mi>y</mi></msub><mo>=</mo><mn>10</mn></math>.</p>
</div>
<div class="specification">
<p>An archer releases an arrow from the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>0</mn><mo>,</mo><mo> </mo><mn>2</mn><mo>)</mo></math>. The arrow is modelled as travelling in a straight line, in the same plane as the ball, with speed <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>60</mn><mo> </mo><msup><mtext>m s</mtext><mrow><mo>-</mo><mn>1</mn></mrow></msup></math> and an angle of elevation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn><mo>°</mo></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the initial speed of the ball.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the angle of elevation of the ball as it is launched.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the maximum height reached by the ball.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Assuming that the ground is horizontal and the ball is not hit by the arrow, find the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> coordinate of the point where the ball lands.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>For the path of the ball, find an expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the two positions where the path of the arrow intersects the path of the ball.</p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the time when the arrow should be released to hit the ball before the ball reaches its maximum height.</p>
<div class="marks">[4]</div>
<div class="question_part_label">f.</div>
</div>
<br><hr><br><div class="specification">
<p>A biologist introduces <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>100</mn></math> rabbits to an island and records the size of their population <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>x</mi><mo>)</mo></math> over a period of time. The population growth of the rabbits can be approximately modelled by the following differential equation, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> is time measured in years.</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mn>2</mn><mi>x</mi></math></p>
</div>
<div class="specification">
<p>A population of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>100</mn></math> foxes is introduced to the island when the population of rabbits has reached <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1000</mn></math>. The subsequent population growth of rabbits and foxes, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> is the population of foxes at time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>, can be approximately modelled by the coupled equations:</p>
<p style="padding-left: 240px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mi>x</mi><mfenced><mrow><mn>2</mn><mo>-</mo><mn>0</mn><mo>.</mo><mn>01</mn><mi>y</mi></mrow></mfenced></math></p>
<p style="padding-left: 240px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mi>y</mi><mfenced><mrow><mn>0</mn><mo>.</mo><mn>0002</mn><mi>x</mi><mo>-</mo><mn>0</mn><mo>.</mo><mn>8</mn></mrow></mfenced></math></p>
</div>
<div class="specification">
<p>Use Euler’s method with a step size of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>25</mn></math>, to find</p>
</div>
<div class="specification">
<p>The graph of the population sizes, according to this model, for the first <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math> years after the foxes were introduced is shown below.</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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"></p>
<p>Describe the changes in the populations of rabbits and foxes for these <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math> years at</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the population of rabbits <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math> year after they were introduced.</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>(i) the population of rabbits 1 year after the foxes were introduced.</p>
<p>(ii) the population of foxes 1 year after the foxes were introduced.</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>point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math>.</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>Find the non-zero equilibrium point for the populations of rabbits and foxes.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>Jorge is carefully observing the rise in sales of a new app he has created.</p>
<p>The number of sales in the first four months is shown in the table below.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<p style="text-align: left;">Jorge believes that the increase is exponential and proposes to model the number of sales <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math> in month <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> with the equation</p>
<p style="text-align: left; padding-left: 30px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mo>=</mo><mi>A</mi><msup><mtext>e</mtext><mrow><mi>r</mi><mi>t</mi></mrow></msup><mo>,</mo><mo> </mo><mi>A</mi><mo>,</mo><mo> </mo><mi>r</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math></p>
</div>
<div class="specification">
<p>Jorge plans to adapt Euler’s method to find an approximate value for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math>.</p>
<p>With a step length of one month the solution to the differential equation can be approximated using Euler’s method where</p>
<p style="padding-left: 30px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>≈</mo><mi>N</mi><mfenced><mi>n</mi></mfenced><mo>+</mo><mn>1</mn><mo>×</mo><mi>N</mi><mo>'</mo><mfenced><mi>n</mi></mfenced><mo>,</mo><mo> </mo><mi>n</mi><mo>∈</mo><mi mathvariant="normal">ℕ</mi></math></p>
</div>
<div class="specification">
<p>Jorge decides to take the mean of these values as the approximation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math> for his model. He also decides the graph of the model should pass through the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>2</mn><mo>,</mo><mo> </mo><mn>52</mn><mo>)</mo></math>.</p>
</div>
<div class="specification">
<p>The sum of the square residuals for these points for the least squares regression model is approximately <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mo>.</mo><mn>555</mn></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that Jorge’s model satisfies the differential equation</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>N</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mi>r</mi><mi>N</mi></math></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>≈</mo><mfrac><mrow><mi>N</mi><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>-</mo><mi>N</mi><mfenced><mi>n</mi></mfenced></mrow><mrow><mi>N</mi><mfenced><mi>n</mi></mfenced></mrow></mfrac></math></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find three approximations for the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the equation for Jorge’s model.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the sum of the square residuals for Jorge’s model using the values <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo> </mo><mn>2</mn><mo>,</mo><mo> </mo><mn>3</mn><mo>,</mo><mo> </mo><mn>4</mn></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Comment how well Jorge’s model fits the data.</p>
<div class="marks">[1]</div>
<div class="question_part_label">f.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Give two possible sources of error in the construction of his model.</p>
<div class="marks">[2]</div>
<div class="question_part_label">f.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>A student investigating the relationship between chemical reactions and temperature finds the Arrhenius equation on the internet.</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mi>A</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mfrac><mi>c</mi><mi>T</mi></mfrac></mrow></msup></math></p>
<p>This equation links a variable <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> with the temperature <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math> are positive constants and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mo>></mo><mn>0</mn></math>.</p>
</div>
<div class="specification">
<p>The Arrhenius equation predicts that the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mi>k</mi></math> against <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mi>T</mi></mfrac></math> is a straight line.</p>
</div>
<div class="specification">
<p>Write down</p>
</div>
<div class="specification">
<p>The following data are found for a particular reaction, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math> is measured in Kelvin and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> is measured in <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>cm</mtext><mn>3</mn></msup><mo> </mo><msup><mtext>mol</mtext><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo> </mo><msup><mtext>s</mtext><mrow><mo>−</mo><mn>1</mn></mrow></msup></math>:</p>
<p style="text-align: center;"><img 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"></p>
</div>
<div class="specification">
<p>Find an estimate of</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>k</mi></mrow><mrow><mo>d</mo><mi>T</mi></mrow></mfrac></math> is always positive.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi>lim</mi><mrow><mi>T</mi><mo>→</mo><mo>∞</mo></mrow></munder><mi>k</mi><mo>=</mo><mi>A</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi>lim</mi><mrow><mi>T</mi><mo>→</mo><mn>0</mn></mrow></munder><mi>k</mi><mo>=</mo><mn>0</mn></math>, sketch the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> against <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>(i) the gradient of this line in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math>;</p>
<p>(ii) the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-intercept of this line in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the equation of the regression line for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mi>k</mi></math> on <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mi>T</mi></mfrac></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math>.</p>
<p>It is not required to state units for this value.</p>
<div class="marks">[1]</div>
<div class="question_part_label">e.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math>.</p>
<p>It is not required to state units for this value.</p>
<div class="marks">[2]</div>
<div class="question_part_label">e.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>The voltage <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v">
<mi>v</mi>
</math></span> in a circuit is given by the equation</p>
<p style="text-align: center;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v\left( t \right) = 3\,{\text{sin}}\left( {100\pi t} \right)">
<mi>v</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>3</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>100</mn>
<mi>π<!-- π --></mi>
<mi>t</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t \geqslant 0">
<mi>t</mi>
<mo>⩾<!-- ⩾ --></mo>
<mn>0</mn>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> is measured in seconds.</p>
</div>
<div class="specification">
<p>The current <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="i">
<mi>i</mi>
</math></span> in this circuit is given by the equation</p>
<p style="text-align: center;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="i\left( t \right) = 2\,{\text{sin}}\left( {100\pi \left( {t + 0.003} \right)} \right)">
<mi>i</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>100</mn>
<mi>π<!-- π --></mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>t</mi>
<mo>+</mo>
<mn>0.003</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
</div>
<div class="specification">
<p>The power <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
<mi>p</mi>
</math></span> in this circuit is given by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p\left( t \right) = v\left( t \right) \times i\left( t \right)">
<mi>p</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mi>v</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>×<!-- × --></mo>
<mi>i</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</math></span>.</p>
</div>
<div class="specification">
<p>The average power <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{p_{av}}">
<mrow>
<msub>
<mi>p</mi>
<mrow>
<mi>a</mi>
<mi>v</mi>
</mrow>
</msub>
</mrow>
</math></span> in this circuit from <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t = 0">
<mi>t</mi>
<mo>=</mo>
<mn>0</mn>
</math></span> to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t = T">
<mi>t</mi>
<mo>=</mo>
<mi>T</mi>
</math></span> is given by the equation</p>
<p style="text-align: center;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{p_{av}}\left( T \right) = \frac{1}{T}\int_0^T {p\left( t \right){\text{d}}t} ">
<mrow>
<msub>
<mi>p</mi>
<mrow>
<mi>a</mi>
<mi>v</mi>
</mrow>
</msub>
</mrow>
<mrow>
<mo>(</mo>
<mi>T</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mi>T</mi>
</mfrac>
<msubsup>
<mo>∫<!-- ∫ --></mo>
<mn>0</mn>
<mi>T</mi>
</msubsup>
<mrow>
<mi>p</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="T > 0">
<mi>T</mi>
<mo>></mo>
<mn>0</mn>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the maximum and minimum value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v">
<mi>v</mi>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down two transformations that will transform the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = v\left( t \right)">
<mi>y</mi>
<mo>=</mo>
<mi>v</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</math></span> onto the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = i\left( t \right)">
<mi>y</mi>
<mo>=</mo>
<mi>i</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = p\left( t \right)">
<mi>y</mi>
<mo>=</mo>
<mi>p</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</math></span> for 0 ≤ <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> ≤ 0.02 , showing clearly the coordinates of the first maximum and the first minimum.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the total time in the interval 0 ≤ <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> ≤ 0.02 for which <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p\left( t \right)">
<mi>p</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</math></span> ≥ 3.</p>
<p> </p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{p_{av}}">
<mrow>
<msub>
<mi>p</mi>
<mrow>
<mi>a</mi>
<mi>v</mi>
</mrow>
</msub>
</mrow>
</math></span>(0.007).</p>
<p> </p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>With reference to your graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = p\left( t \right)">
<mi>y</mi>
<mo>=</mo>
<mi>p</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</math></span> explain why <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{p_{av}}\left( T \right)">
<mrow>
<msub>
<mi>p</mi>
<mrow>
<mi>a</mi>
<mi>v</mi>
</mrow>
</msub>
</mrow>
<mrow>
<mo>(</mo>
<mi>T</mi>
<mo>)</mo>
</mrow>
</math></span> > 0 for all <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="T">
<mi>T</mi>
</math></span> > 0.</p>
<p> </p>
<div class="marks">[2]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p\left( t \right)">
<mi>p</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</math></span> can be written as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p\left( t \right) = a\,{\text{sin}}\left( {b\left( {t - c} \right)} \right) + d">
<mi>p</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mi>a</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>b</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>t</mi>
<mo>−</mo>
<mi>c</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mi>d</mi>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b">
<mi>b</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c">
<mi>c</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d">
<mi>d</mi>
</math></span> > 0, use your graph to find the values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b">
<mi>b</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c">
<mi>c</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d">
<mi>d</mi>
</math></span>.</p>
<p> </p>
<div class="marks">[6]</div>
<div class="question_part_label">g.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the system of paired differential equations</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\dot x = 3x + 2y">
<mrow>
<mover>
<mi>x</mi>
<mo>˙<!-- ˙ --></mo>
</mover>
</mrow>
<mo>=</mo>
<mn>3</mn>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
<mi>y</mi>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\dot y = 2x + 3y">
<mrow>
<mover>
<mi>y</mi>
<mo>˙<!-- ˙ --></mo>
</mover>
</mrow>
<mo>=</mo>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mn>3</mn>
<mi>y</mi>
</math></span>.</p>
<p>This represents the populations of two species of symbiotic toadstools in a large wood.</p>
<p>Time <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> is measured in decades.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use the eigenvalue method to find the general solution to this system of equations.</p>
<div class="marks">[10]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given the initial conditions that when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t = 0">
<mi>t</mi>
<mo>=</mo>
<mn>0</mn>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 150">
<mi>x</mi>
<mo>=</mo>
<mn>150</mn>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 50">
<mi>y</mi>
<mo>=</mo>
<mn>50</mn>
</math></span>, find the particular solution.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find the solution when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t = 1">
<mi>t</mi>
<mo>=</mo>
<mn>1</mn>
</math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>As <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t \to \infty ">
<mi>t</mi>
<mo stretchy="false">→</mo>
<mi mathvariant="normal">∞</mi>
</math></span>, find an asymptote to the trajectory of the particular solution found in (b)(i) and state if this trajectory will be moving towards or away from the origin.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the curve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><msqrt><mi>x</mi></msqrt></math>.</p>
</div>
<div class="specification">
<p>The shape of a piece of metal can be modelled by the region bounded by the functions <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math>, the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis and the line segment <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>[AB]</mtext></math>, as shown in the following diagram. The units on the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> axes are measured in metres.</p>
<p style="text-align: center;"><img 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"></p>
<p>The piecewise function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> is defined by</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mo>{</mo><mtable><mtr><mtd><msqrt><mi>x</mi></msqrt><mo> </mo><mo> </mo></mtd><mtd><mn>0</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mn>0</mn><mo>.</mo><mn>16</mn></mtd></mtr><mtr><mtd><mn>1</mn><mo>.</mo><mn>25</mn><mi>x</mi><mo>+</mo><mn>0</mn><mo>.</mo><mn>2</mn><mo> </mo><mo> </mo></mtd><mtd><mn>0</mn><mo>.</mo><mn>16</mn><mo><</mo><mi>x</mi><mo>≤</mo><mn>0</mn><mo>.</mo><mn>5</mn></mtd></mtr></mtable></math></p>
<p>The graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math> is obtained from the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> by:</p>
<ul>
<li>a stretch scale factor of <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac></math> in the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> direction,</li>
<li>followed by a stretch scale factor <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac></math> in the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> direction,</li>
<li>followed by a translation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>2</mn></math> units to the right.</li>
</ul>
<p>Point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> lies on the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> and has coordinates <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>0</mn><mo>.</mo><mn>5</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>.</mo><mn>825</mn><mo>)</mo></math>. Point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math> is the image of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> under the given transformations and has coordinates <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>p</mi><mo>,</mo><mo> </mo><mi>q</mi><mo>)</mo></math>.</p>
</div>
<div class="specification">
<p>The piecewise function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math> is given by</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mo>{</mo><mtable><mtr><mtd><mi>h</mi><mfenced><mi>x</mi></mfenced><mo> </mo><mo> </mo></mtd><mtd><mn>0</mn><mo>.</mo><mn>2</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mi>a</mi></mtd></mtr><mtr><mtd><mn>1</mn><mo>.</mo><mn>25</mn><mi>x</mi><mo>+</mo><mi>b</mi><mo> </mo><mo> </mo></mtd><mtd><mi>a</mi><mo><</mo><mi>x</mi><mo>≤</mo><mi>p</mi></mtd></mtr></mtable></math></p>
</div>
<div class="specification">
<p>The area enclosed by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo></math>, the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis and the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mi>p</mi></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>0627292</mn><mo> </mo><msup><mtext>m</mtext><mn>2</mn></msup></math> correct to six significant figures.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence show that the equation of the tangent to the curve at the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>0</mn><mo>.</mo><mn>16</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>.</mo><mn>4</mn></mrow></mfenced></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>1</mn><mo>.</mo><mn>25</mn><mi>x</mi><mo>+</mo><mn>0</mn><mo>.</mo><mn>2</mn></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math> and the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an expression for<math xmlns="http://www.w3.org/1998/Math/MathML"><mo> </mo><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the area enclosed by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math>, the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis and the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>5</mn></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the area of the shaded region on the diagram.</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>A sector of a circle, centre <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math> and radius <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>.</mo><mn>5</mn><mo> </mo><mtext>m</mtext></math>, is shown in the following diagram.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<div class="specification">
<p>A square field with side <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn><mo> </mo><mtext>m</mtext></math> has a goat tied to a post in the centre by a rope such that the goat can reach all parts of the field up to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>.</mo><mn>5</mn><mo> </mo><mtext>m</mtext></math> from the post.</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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"></p>
<p style="text-align: center;"><sup>[Source: mynamepong, n.d. Goat [image online] Available at: <a href="https://thenounproject.com/term/goat/1761571/">https://thenounproject.com/term/goat/1761571/</a></sup><br><sup>This file is licensed under the Creative Commons Attribution-ShareAlike 3.0 Unported (CC BY-SA 3.0)</sup><br><sup><a href="https://creativecommons.org/licenses/by-sa/3.0/deed.en">https://creativecommons.org/licenses/by-sa/3.0/deed.en</a> [Accessed 22 April 2010] Source adapted.]</sup></p>
</div>
<div class="specification">
<p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi></math> be the volume of grass eaten by the goat, in cubic metres, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> be the length of time, in hours, that the goat has been in the field.</p>
<p>The goat eats grass at the rate of <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>V</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mn>0</mn><mo>.</mo><mn>3</mn><mo> </mo><mi>t</mi><msup><mtext>e</mtext><mrow><mo>-</mo><mi>t</mi></mrow></msup></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the angle <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>AÔB</mtext></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the area of the shaded segment.</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the area of the field that can be reached by the goat.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> at which the goat is eating grass at the greatest rate.</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>The goat is tied in the field for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn></math> hours.</p>
<p>Find the total volume of grass eaten by the goat during this time.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = \frac{{\sqrt x }}{{\sin x}},{\text{ }}0 < x < \pi ">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mrow>
<msqrt>
<mi>x</mi>
</msqrt>
</mrow>
<mrow>
<mi>sin</mi>
<mo><!-- --></mo>
<mi>x</mi>
</mrow>
</mfrac>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>0</mn>
<mo><</mo>
<mi>x</mi>
<mo><</mo>
<mi>π<!-- π --></mi>
</math></span>.</p>
</div>
<div class="specification">
<p>Consider the region bounded by the curve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f(x)">
<mi>y</mi>
<mo>=</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span>, the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>-axis and the lines <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \frac{\pi }{6},{\text{ }}x = \frac{\pi }{3}">
<mi>x</mi>
<mo>=</mo>
<mfrac>
<mi>π<!-- π --></mi>
<mn>6</mn>
</mfrac>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>x</mi>
<mo>=</mo>
<mfrac>
<mi>π<!-- π --></mi>
<mn>3</mn>
</mfrac>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>-coordinate of the minimum point on the curve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f(x)">
<mi>y</mi>
<mo>=</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span> satisfies the equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\tan x = 2x">
<mi>tan</mi>
<mo></mo>
<mi>x</mi>
<mo>=</mo>
<mn>2</mn>
<mi>x</mi>
</math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> for which <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x)">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span> is a decreasing function.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f(x)">
<mi>y</mi>
<mo>=</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span> showing clearly the minimum point and any asymptotic behaviour.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the coordinates of the point on the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> where the normal to the graph is parallel to the line <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = - x">
<mi>y</mi>
<mo>=</mo>
<mo>−</mo>
<mi>x</mi>
</math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>This region is now rotated through <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2\pi ">
<mn>2</mn>
<mi>π</mi>
</math></span> radians about the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>-axis. Find the volume of revolution.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>An environmental scientist is asked by a river authority to model the effect of a leak from a power plant on the mercury levels in a local river. The variable <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> measures the concentration of mercury in micrograms per litre.</p>
<p>The situation is modelled using the second order differential equation</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mtext>d</mtext><mn>2</mn></msup><mi>x</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>+</mo><mn>3</mn><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>+</mo><mn>2</mn><mi>x</mi><mo>=</mo><mn>0</mn></math></p>
<p>where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>≥</mo><mn>0</mn></math> is the time measured in days since the leak started. It is known that when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn><mo>,</mo><mo> </mo><mi>x</mi><mo>=</mo><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mn>1</mn></math>.</p>
</div>
<div class="specification">
<p>If the mercury levels are greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>1</mn></math> micrograms per litre, fishing in the river is considered unsafe and is stopped.</p>
</div>
<div class="specification">
<p>The river authority decides to stop people from fishing in the river for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn><mo>%</mo></math> longer than the time found from the model.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the system of coupled first order equations:</p>
<p style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mi>y</mi></math></p>
<p style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mo>-</mo><mn>2</mn><mi>x</mi><mo>-</mo><mn>3</mn><mi>y</mi></math></p>
<p style="text-align:left;">can be written as the given second order differential equation.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the eigenvalues of the system of coupled first order equations given in part (a).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find the exact solution of the second order differential equation.</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> against <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>, labelling the maximum point of the graph with its coordinates.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use the model to calculate the total amount of time when fishing should be stopped.</p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down one reason, with reference to the context, to support this decision.</p>
<div class="marks">[1]</div>
<div class="question_part_label">f.</div>
</div>
<br><hr><br><div class="specification">
<p>A water trough which is 10 metres long has a uniform cross-section in the shape of a semicircle with radius 0.5 metres. It is partly filled with water as shown in the following diagram of the cross-section. The centre of the circle is O and the angle KOL is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\theta ">
<mi>θ<!-- θ --></mi>
</math></span> radians.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-09_om_11.09.30.png" alt="M17/5/MATHL/HP2/ENG/TZ1/08"></p>
</div>
<div class="specification">
<p>The volume of water is increasing at a constant rate of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0.0008{\text{ }}{{\text{m}}^3}{{\text{s}}^{ - 1}}">
<mn>0.0008</mn>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>m</mtext>
</mrow>
<mn>3</mn>
</msup>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>s</mtext>
</mrow>
<mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an expression for the volume of water <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="V{\text{ }}({{\text{m}}^3})">
<mi>V</mi>
<mrow>
<mtext> </mtext>
</mrow>
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mrow>
<mtext>m</mtext>
</mrow>
<mn>3</mn>
</msup>
</mrow>
<mo stretchy="false">)</mo>
</math></span> in the trough in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\theta ">
<mi>θ</mi>
</math></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>Calculate <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}\theta }}{{{\text{d}}t}}">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>θ</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
</math></span> when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\theta = \frac{\pi }{3}">
<mi>θ</mi>
<mo>=</mo>
<mfrac>
<mi>π</mi>
<mn>3</mn>
</mfrac>
</math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Charlotte decides to model the shape of a cupcake to calculate its volume.</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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"></p>
<p>From rotating a photograph of her cupcake she estimates that its cross-section passes through the points <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>0</mn><mo>,</mo><mo> </mo><mn>3</mn><mo>.</mo><mn>5</mn><mo>)</mo><mo>,</mo><mo> </mo><mo>(</mo><mn>4</mn><mo>,</mo><mo> </mo><mn>6</mn><mo>)</mo><mo>,</mo><mo> </mo><mo>(</mo><mn>6</mn><mo>.</mo><mn>5</mn><mo>,</mo><mo> </mo><mn>4</mn><mo>)</mo><mo>,</mo><mo> </mo><mo>(</mo><mn>7</mn><mo>,</mo><mo> </mo><mn>3</mn><mo>)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>7</mn><mo>.</mo><mn>5</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math>, where all units are in centimetres. The cross-section is symmetrical in the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis, as shown below:</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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"></p>
<p>She models the section from <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>0</mn><mo>,</mo><mo> </mo><mn>3</mn><mo>.</mo><mn>5</mn><mo>)</mo></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>4</mn><mo>,</mo><mo> </mo><mn>6</mn><mo>)</mo></math> as a straight line.</p>
</div>
<div class="specification">
<p>Charlotte models the section of the cupcake that passes through the points <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>4</mn><mo>,</mo><mo> </mo><mn>6</mn><mo>)</mo><mo>,</mo><mo> </mo><mo>(</mo><mn>6</mn><mo>.</mo><mn>5</mn><mo>,</mo><mo> </mo><mn>4</mn><mo>)</mo><mo>,</mo><mo> </mo><mo>(</mo><mn>7</mn><mo>,</mo><mo> </mo><mn>3</mn><mo>)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>7</mn><mo>.</mo><mn>5</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math> with a quadratic curve.</p>
</div>
<div class="specification">
<p>Charlotte thinks that a quadratic with a maximum point at <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>4</mn><mo>,</mo><mo> </mo><mn>6</mn><mo>)</mo></math> and that passes through the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>7</mn><mo>.</mo><mn>5</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math> would be a better fit.</p>
</div>
<div class="specification">
<p>Believing this to be a better model for her cupcake, Charlotte finds the volume of revolution about the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis to estimate the volume of the cupcake.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the equation of the line passing through these two points.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the equation of the least squares regression quadratic curve for these four points.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By considering the gradient of this curve when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>4</mn></math>, explain why it may not be a good model.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the equation of the new model.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down an expression for her estimate of the volume as a sum of two integrals.</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of Charlotte’s estimate.</p>
<div class="marks">[1]</div>
<div class="question_part_label">d.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = - 1 + \ln \left( {\sqrt {{x^2} - 1} } \right)">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo>+</mo>
<mi>ln</mi>
<mo><!-- --></mo>
<mrow>
<mo>(</mo>
<mrow>
<msqrt>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</msqrt>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
</div>
<div class="specification">
<p>The function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> is defined by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = - 1 + \ln \left( {\sqrt {{x^2} - 1} } \right),{\text{ }}x \in D">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo>+</mo>
<mi>ln</mi>
<mo><!-- --></mo>
<mrow>
<mo>(</mo>
<mrow>
<msqrt>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</msqrt>
</mrow>
<mo>)</mo>
</mrow>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>x</mi>
<mo>∈<!-- ∈ --></mo>
<mi>D</mi>
</math></span></p>
</div>
<div class="specification">
<p>The function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g">
<mi>g</mi>
</math></span> is defined by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g(x) = - 1 + \ln \left( {\sqrt {{x^2} - 1} } \right),{\text{ }}x \in \left] {1,{\text{ }}\infty } \right[">
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo>+</mo>
<mi>ln</mi>
<mo><!-- --></mo>
<mrow>
<mo>(</mo>
<mrow>
<msqrt>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</msqrt>
</mrow>
<mo>)</mo>
</mrow>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>x</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mo>]</mo>
<mrow>
<mn>1</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi mathvariant="normal">∞<!-- ∞ --></mi>
</mrow>
<mo>[</mo>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the largest possible domain <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="D">
<mi>D</mi>
</math></span> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> to be a function.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f(x)">
<mi>y</mi>
<mo>=</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span> showing clearly the equations of asymptotes and the coordinates of any intercepts with the axes.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Explain why <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> is an even function.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Explain why the inverse function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f^{ - 1}}">
<mrow>
<msup>
<mi>f</mi>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</math></span> does not exist.</p>
<div class="marks">[1]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the inverse function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{g^{ - 1}}">
<mrow>
<msup>
<mi>g</mi>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</math></span> and state its domain.</p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g'(x)">
<msup>
<mi>g</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, show that there are no solutions to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g'(x) = 0">
<msup>
<mi>g</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0</mn>
</math></span>;</p>
<div class="marks">[2]</div>
<div class="question_part_label">g.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, show that there are no solutions to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="({g^{ - 1}})'(x) = 0">
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mi>g</mi>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<msup>
<mo stretchy="false">)</mo>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0</mn>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">g.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = 2{\sin ^2}x + 7\sin 2x + \tan x - 9,{\text{ }}0 \leqslant x < \frac{\pi }{2}">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>2</mn>
<mrow>
<msup>
<mi>sin</mi>
<mn>2</mn>
</msup>
</mrow>
<mi>x</mi>
<mo>+</mo>
<mn>7</mn>
<mi>sin</mi>
<mo><!-- --></mo>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mi>tan</mi>
<mo><!-- --></mo>
<mi>x</mi>
<mo>−<!-- − --></mo>
<mn>9</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>0</mn>
<mo>⩽<!-- ⩽ --></mo>
<mi>x</mi>
<mo><</mo>
<mfrac>
<mi>π<!-- π --></mi>
<mn>2</mn>
</mfrac>
</math></span>.</p>
</div>
<div class="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u = \tan x">
<mi>u</mi>
<mo>=</mo>
<mi>tan</mi>
<mo><!-- --></mo>
<mi>x</mi>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine an expression for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f’(x)">
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span> in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch a graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f’(x)">
<mi>y</mi>
<mo>=</mo>
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="0 \leqslant x < \frac{\pi }{2}">
<mn>0</mn>
<mo>⩽</mo>
<mi>x</mi>
<mo><</mo>
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
</math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>-coordinate(s) of the point(s) of inflexion of the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f(x)">
<mi>y</mi>
<mo>=</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span>, labelling these clearly on the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f’(x)">
<mi>y</mi>
<mo>=</mo>
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Express <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sin x">
<mi>sin</mi>
<mo></mo>
<mi>x</mi>
</math></span> in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mu ">
<mi>μ</mi>
</math></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>Express <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sin 2x">
<mi>sin</mi>
<mo></mo>
<mn>2</mn>
<mi>x</mi>
</math></span> in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="u">
<mi>u</mi>
</math></span>.</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>Hence show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = 0">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0</mn>
</math></span> can be expressed as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u^3} - 7{u^2} + 15u - 9 = 0">
<mrow>
<msup>
<mi>u</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>7</mn>
<mrow>
<msup>
<mi>u</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>15</mn>
<mi>u</mi>
<mo>−</mo>
<mn>9</mn>
<mo>=</mo>
<mn>0</mn>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Solve the equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = 0">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0</mn>
</math></span>, giving your answers in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\arctan k">
<mi>arctan</mi>
<mo></mo>
<mi>k</mi>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k \in \mathbb{Z}">
<mi>k</mi>
<mo>∈</mo>
<mrow>
<mi mathvariant="double-struck">Z</mi>
</mrow>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>A point P moves in a straight line with velocity <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v">
<mi>v</mi>
</math></span> ms<sup>−1</sup> given by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v\left( t \right) = {{\text{e}}^{ - t}} - 8{t^2}{{\text{e}}^{ - 2t}}">
<mi>v</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−<!-- − --></mo>
<mi>t</mi>
</mrow>
</msup>
</mrow>
<mo>−<!-- − --></mo>
<mn>8</mn>
<mrow>
<msup>
<mi>t</mi>
<mn>2</mn>
</msup>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−<!-- − --></mo>
<mn>2</mn>
<mi>t</mi>
</mrow>
</msup>
</mrow>
</math></span> at time <em>t</em> seconds, where <em>t</em> ≥ 0.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the first time <em>t</em><sub>1</sub> at which P has zero velocity.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an expression for the acceleration of P at time <em>t</em>.</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 value of the acceleration of P at time <em>t</em><sub>1</sub>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>A curve <em>C</em> is given by the implicit equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x + y - {\text{cos}}\left( {xy} \right) = 0">
<mi>x</mi>
<mo>+</mo>
<mi>y</mi>
<mo>−<!-- − --></mo>
<mrow>
<mtext>cos</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mi>y</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span>.</p>
</div>
<div class="specification">
<p>The curve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="xy = - \frac{\pi }{2}">
<mi>x</mi>
<mi>y</mi>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mfrac>
<mi>π<!-- π --></mi>
<mn>2</mn>
</mfrac>
</math></span> intersects <em>C</em> at P and Q.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}} = - \left( {\frac{{1 + y\,{\text{sin}}\left( {xy} \right)}}{{1 + x\,{\text{sin}}\left( {xy} \right)}}} \right)">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mo>−</mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>y</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mi>y</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>x</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mi>y</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></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>Find the coordinates of P and Q.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that the gradients of the tangents to <em>C</em> at P and Q are <em>m</em><sub>1</sub> and <em>m</em><sub>2</sub> respectively, show that <em>m</em><sub>1</sub> × <em>m</em><sub>2</sub> = 1.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the coordinates of the three points on <em>C</em>, nearest the origin, where the tangent is parallel to the line <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = - x">
<mi>y</mi>
<mo>=</mo>
<mo>−</mo>
<mi>x</mi>
</math></span>.</p>
<div class="marks">[7]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>The following graph shows the two parts of the curve defined by the equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^2}y = 5 - {y^4}">
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mi>y</mi>
<mo>=</mo>
<mn>5</mn>
<mo>−<!-- − --></mo>
<mrow>
<msup>
<mi>y</mi>
<mn>4</mn>
</msup>
</mrow>
</math></span>, and the normal to the curve at the point P(2 , 1).</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<p> </p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that there are exactly two points on the curve where the gradient is zero.</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>Find the equation of the normal to the curve at the point P.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The normal at P cuts the curve again at the point Q. Find the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>-coordinate of Q.</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>The shaded region is rotated by 2<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\pi ">
<mi>π</mi>
</math></span> about the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span>-axis. Find the volume of the solid formed.</p>
<div class="marks">[7]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="question">
<p>A function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> satisfies the conditions <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( 0 \right) = - 4">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mn>0</mn>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mo>−</mo>
<mn>4</mn>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( 1 \right) = 0">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mn>1</mn>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span> and its second derivative is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f''\left( x \right) = 15\sqrt x + \frac{1}{{{{\left( {x + 1} \right)}^2}}}">
<msup>
<mi>f</mi>
<mo>″</mo>
</msup>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>15</mn>
<msqrt>
<mi>x</mi>
</msqrt>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> ≥ 0.</p>
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right)">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span>.</p>
</div>
<br><hr><br><div class="specification">
<p>Xavier, the parachutist, jumps out of a plane at a height of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="h">
<mi>h</mi>
</math></span> metres above the ground. After free falling for 10 seconds his parachute opens. His velocity, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v\,{\text{m}}{{\text{s}}^{ - 1}}">
<mi>v</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>m</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>s</mtext>
</mrow>
<mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> seconds after jumping from the plane, can be modelled by the function</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v(t) = \left\{ {\begin{array}{*{20}{l}} {9.8t{\text{,}}}&{0 \leqslant t \leqslant 10} \\ {\frac{{98}}{{\sqrt {1 + {{(t - 10)}^2}} }},}&{t > 10} \end{array}} \right.">
<mi>v</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow>
<mo>{</mo>
<mrow>
<mtable columnalign="left" rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>9.8</mn>
<mi>t</mi>
<mrow>
<mtext>,</mtext>
</mrow>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>0</mn>
<mo>⩽<!-- ⩽ --></mo>
<mi>t</mi>
<mo>⩽<!-- ⩽ --></mo>
<mn>10</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mfrac>
<mrow>
<mn>98</mn>
</mrow>
<mrow>
<msqrt>
<mn>1</mn>
<mo>+</mo>
<mrow>
<msup>
<mrow>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo>−<!-- − --></mo>
<mn>10</mn>
<mo stretchy="false">)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
</mrow>
</mfrac>
<mo>,</mo>
</mrow>
</mtd>
<mtd>
<mrow>
<mi>t</mi>
<mo>></mo>
<mn>10</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo fence="true" stretchy="true" symmetric="true"></mo>
</mrow>
</math></span></p>
</div>
<div class="specification">
<p>His velocity when he reaches the ground is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2.8{\text{ m}}{{\text{s}}^{ - 1}}">
<mn>2.8</mn>
<mrow>
<mtext> m</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>s</mtext>
</mrow>
<mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find his velocity when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t = 15">
<mi>t</mi>
<mo>=</mo>
<mn>15</mn>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate the vertical distance Xavier travelled in the first 10 seconds.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="h">
<mi>h</mi>
</math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>The region <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A">
<mi>A</mi>
</math></span> is enclosed by the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 2\arcsin (x - 1) - \frac{\pi }{4}">
<mi>y</mi>
<mo>=</mo>
<mn>2</mn>
<mi>arcsin</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>−<!-- − --></mo>
<mfrac>
<mi>π<!-- π --></mi>
<mn>4</mn>
</mfrac>
</math></span>, the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span>-axis and the line <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = \frac{\pi }{4}">
<mi>y</mi>
<mo>=</mo>
<mfrac>
<mi>π<!-- π --></mi>
<mn>4</mn>
</mfrac>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down a definite integral to represent the area of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A">
<mi>A</mi>
</math></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>Calculate the area of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A">
<mi>A</mi>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>The function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> is defined by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = {\left( {x - 1} \right)^2}">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> ≥ 1 and the function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g">
<mi>g</mi>
</math></span> is defined by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g\left( x \right) = {x^2} + 1">
<mi>g</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>1</mn>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> ≥ 0.</p>
<p>The region <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="R">
<mi>R</mi>
</math></span> is bounded by the curves <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f\left( x \right)">
<mi>y</mi>
<mo>=</mo>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = g\left( x \right)">
<mi>y</mi>
<mo>=</mo>
<mi>g</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span> and the lines <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 0">
<mi>y</mi>
<mo>=</mo>
<mn>0</mn>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 0">
<mi>x</mi>
<mo>=</mo>
<mn>0</mn>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 9">
<mi>y</mi>
<mo>=</mo>
<mn>9</mn>
</math></span> as shown on the following diagram.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<p style="text-align: left;">The shape of a clay vase can be modelled by rotating the region <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="R">
<mi>R</mi>
</math></span> through 360˚ about the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span>-axis.</p>
<p style="text-align: left;">Find the volume of clay used to make the vase.</p>
</div>
<br><hr><br><div class="specification">
<p>The curve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="C">
<mi>C</mi>
</math></span> is defined by equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="xy - \ln y = 1,{\text{ }}y > 0">
<mi>x</mi>
<mi>y</mi>
<mo>−<!-- − --></mo>
<mi>ln</mi>
<mo><!-- --></mo>
<mi>y</mi>
<mo>=</mo>
<mn>1</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>y</mi>
<mo>></mo>
<mn>0</mn>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}}">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
</math></span> in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></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>Determine the equation of the tangent to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="C">
<mi>C</mi>
</math></span> at the point <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\frac{2}{{\text{e}}},{\text{ e}}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>2</mn>
<mrow>
<mtext>e</mtext>
</mrow>
</mfrac>
<mo>,</mo>
<mrow>
<mtext> e</mtext>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>An object is placed into the top of a long vertical tube, filled with a thick viscous fluid, at time <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t = 0">
<mi>t</mi>
<mo>=</mo>
<mn>0</mn>
</math></span> seconds.</p>
<p>Initially it is thought that the resistance of the fluid would be proportional to the velocity of the object. The following model was proposed, where the object’s displacement, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>, from the top of the tube, measured in metres, is given by the differential equation</p>
<p style="text-align: center;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{{\text{d}}^2}x}}{{{\text{d}}{t^2}}} = 9.81 - 0.9\left( {\frac{{{\text{d}}x}}{{{\text{d}}t}}} \right)">
<mfrac>
<mrow>
<mrow>
<msup>
<mrow>
<mtext>d</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mi>x</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mrow>
<msup>
<mi>t</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
<mo>=</mo>
<mn>9.81</mn>
<mo>−<!-- − --></mo>
<mn>0.9</mn>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
</div>
<div class="specification">
<p>The maximum velocity approached by the object as it falls is known as the terminal velocity.</p>
</div>
<div class="specification">
<p>An experiment is performed in which the object is placed in the fluid on a number of occasions and its terminal velocity recorded. It is found that the terminal velocity was consistently smaller than that predicted by the model used. It was suggested that the resistance to motion is actually proportional to the velocity squared and so the following model was set up.</p>
<p><span class="mjpage mjpage__block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" alttext="\frac{{{{\text{d}}^2}x}}{{{\text{d}}{t^2}}} = 9.81 - 0.9{\left( {\frac{{{\text{d}}x}}{{{\text{d}}t}}} \right)^2}">
<mfrac>
<mrow>
<mrow>
<msup>
<mrow>
<mtext>d</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mi>x</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mrow>
<msup>
<mi>t</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
<mo>=</mo>
<mn>9.81</mn>
<mo>−<!-- − --></mo>
<mn>0.9</mn>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</math></span></p>
</div>
<div class="specification">
<p>At terminal velocity the acceleration of an object is equal to zero.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By substituting <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v = \frac{{{\text{d}}x}}{{{\text{d}}t}}"> <mi>v</mi> <mo>=</mo> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </mfrac> </math></span> into the equation, find an expression for the velocity of the particle at time <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t"> <mi>t</mi> </math></span>. Give your answer in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v = f(t)"> <mi>v</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </math></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>From your solution to part (a), or otherwise, find the terminal velocity of the object predicted by this model.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the differential equation as a system of first order differential equations.</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>Use Euler’s method, with a step length of 0.2, to find the displacement and velocity of the object when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t = 0.6"> <mi>t</mi> <mo>=</mo> <mn>0.6</mn> </math></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>By repeated application of Euler’s method, find an approximation for the terminal velocity, to five significant figures.</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>Use the differential equation to find the terminal velocity for the object.</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>Use your answers to parts (d), (e) and (f) to comment on the accuracy of the Euler approximation to this model.</p>
<div class="marks">[2]</div>
<div class="question_part_label">g.</div>
</div>
<br><hr><br><div class="question">
<p>An earth satellite moves in a path that can be described by the curve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="72.5{x^2} + 71.5{y^2} = 1"> <mn>72.5</mn> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mn>71.5</mn> <mrow> <msup> <mi>y</mi> <mn>2</mn> </msup> </mrow> <mo>=</mo> <mn>1</mn> </math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = x(t)"> <mi>x</mi> <mo>=</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = y(t)"> <mi>y</mi> <mo>=</mo> <mi>y</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </math></span> are in thousands of kilometres and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t"> <mi>t</mi> </math></span> is time in seconds.</p>
<p>Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}x}}{{{\text{d}}t}} = 7.75 \times {10^{ - 5}}"> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </mfrac> <mo>=</mo> <mn>7.75</mn> <mo>×</mo> <mrow> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>5</mn> </mrow> </msup> </mrow> </math></span> when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 3.2 \times {10^{ - 3}}"> <mi>x</mi> <mo>=</mo> <mn>3.2</mn> <mo>×</mo> <mrow> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </msup> </mrow> </math></span>, find the possible values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}t}}"> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>y</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </mfrac> </math></span>.</p>
<p>Give your answers in standard form.</p>
</div>
<br><hr><br><div class="specification">
<p>The function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> is defined by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = \frac{{2\,{\text{ln}}\,x + 1}}{{x - 3}}">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mrow>
<mi>x</mi>
<mo>−<!-- − --></mo>
<mn>3</mn>
</mrow>
</mfrac>
</math></span>, 0 < <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> < 3.</p>
</div>
<div class="specification">
<p>Draw a set of axes showing <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span> values between −3 and 3. On these axes</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f'\left( x \right)"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math></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>Hence, or otherwise, find the coordinates of the point of inflexion on the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f\left( x \right)"> <mi>y</mi> <mo>=</mo> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math></span>.</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>sketch the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f\left( x \right)"> <mi>y</mi> <mo>=</mo> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math></span>, showing clearly any axis intercepts and giving the equations of any asymptotes.</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>sketch the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = {f^{ - 1}}\left( x \right)"> <mi>y</mi> <mo>=</mo> <mrow> <msup> <mi>f</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math></span>, showing clearly any axis intercepts and giving the equations of any asymptotes.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, or otherwise, solve the inequality <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) > {f^{ - 1}}\left( x \right)"> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>></mo> <mrow> <msup> <mi>f</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="question">
<p>A particle moves along a horizontal line such that at time <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> seconds, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> ≥ 0, its acceleration <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span> is given by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span> = 2<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> − 1. When <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> = 6 , its displacement <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="s">
<mi>s</mi>
</math></span> from a fixed origin O is 18.25 m. When <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> = 15, its displacement from O is 922.75 m. Find an expression for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="s">
<mi>s</mi>
</math></span> in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span>.</p>
</div>
<br><hr><br><div class="question">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="l"> <mi>l</mi> </math></span> be the tangent to the curve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = x{{\text{e}}^{2x}}"> <mi>y</mi> <mo>=</mo> <mi>x</mi> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mn>2</mn> <mi>x</mi> </mrow> </msup> </mrow> </math></span> at the point (1, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{{\text{e}}^2}"> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mn>2</mn> </msup> </mrow> </math></span>).</p>
<p>Find the coordinates of the point where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="l"> <mi>l</mi> </math></span> meets the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>-axis.</p>
</div>
<br><hr><br>