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<h2>HL Paper 2</h2><div class="question">
<p>Suppose that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_1}">
<mrow>
<msub>
<mi>u</mi>
<mn>1</mn>
</msub>
</mrow>
</math></span> is the first term of a geometric series with common ratio <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r">
<mi>r</mi>
</math></span>.</p>
<p>Prove, by mathematical induction, that the sum of the first <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span> terms, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{s_n}">
<mrow>
<msub>
<mi>s</mi>
<mi>n</mi>
</msub>
</mrow>
</math></span> is given by</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{s_n} = \frac{{{u_1}\left( {1 - {r^n}} \right)}}{{1 - r}}">
<mrow>
<msub>
<mi>s</mi>
<mi>n</mi>
</msub>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<msub>
<mi>u</mi>
<mn>1</mn>
</msub>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mrow>
<msup>
<mi>r</mi>
<mi>n</mi>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>r</mi>
</mrow>
</mfrac>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n \in {\mathbb{Z}^ + }">
<mi>n</mi>
<mo>∈</mo>
<mrow>
<msup>
<mrow>
<mi mathvariant="double-struck">Z</mi>
</mrow>
<mo>+</mo>
</msup>
</mrow>
</math></span>.</p>
</div>
<br><hr><br><div class="specification">
<p>A geneticist uses a Markov chain model to investigate changes in a specific gene in a cell as it divides. Every time the cell divides, the gene may mutate between its normal state and other states.</p>
<p>The model is of the form</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><msub><mi>X</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mi>Z</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mtd></mtr></mtable></mfenced><mo>=</mo><mi mathvariant="bold-italic">M</mi><mfenced><mtable><mtr><mtd><msub><mi>X</mi><mi>n</mi></msub></mtd></mtr><mtr><mtd><msub><mi>Z</mi><mi>n</mi></msub></mtd></mtr></mtable></mfenced></math></p>
<p>where <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>X</mi><mi>n</mi></msub></math> is the probability of the gene being in its normal state after dividing for the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mtext>th</mtext></math> time, and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>Z</mi><mi>n</mi></msub></math> is the probability of it being in another state after dividing for the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mtext>th</mtext></math> time, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>∈</mo><mi mathvariant="normal">ℕ</mi></math>.</p>
<p>Matrix <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">M</mi></math> is found to be <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>0</mn><mo>.</mo><mn>94</mn><mo> </mo><mo> </mo></mtd><mtd><mi>b</mi></mtd></mtr><mtr><mtd><mn>0</mn><mo>.</mo><mn>06</mn><mo> </mo><mo> </mo></mtd><mtd><mn>0</mn><mo>.</mo><mn>98</mn></mtd></mtr></mtable></mfenced></math>.</p>
</div>
<div class="specification">
<p>The gene is in its normal state when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>0</mn></math>. Calculate the probability of it being in its normal state</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></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>What does <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math> represent in this context?</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the eigenvalues of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">M</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>Find the eigenvectors of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">M</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>when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>5</mn></math>.</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>in the long term.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>A transformation, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math>, of a plane is represented by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">r</mi><mo>′</mo><mo>=</mo><mi mathvariant="bold-italic">P</mi><mi mathvariant="bold-italic">r</mi><mo>+</mo><mi mathvariant="bold-italic">q</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">P</mi></math> is a <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo> </mo><mo>×</mo><mo> </mo><mn>2</mn></math> matrix, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">q</mi></math> is a <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo> </mo><mo>×</mo><mo> </mo><mn>1</mn></math> vector, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">r</mi></math> is the position vector of a point in the plane and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">r</mi><mo>′</mo></math> the position vector of its image under <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math>.</p>
<p>The triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>OAB</mtext></math> has coordinates <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>0</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>0</mn><mo>,</mo><mo> </mo><mn>1</mn><mo>)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>1</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math>. Under T, these points are transformed to <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>0</mn><mo>,</mo><mo> </mo><mn>1</mn><mo>)</mo></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mfrac><mn>1</mn><mn>4</mn></mfrac><mo>,</mo><mo> </mo><mn>1</mn><mo>+</mo><mfrac><msqrt><mn>3</mn></msqrt><mn>4</mn></mfrac></mrow></mfenced></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mfrac><msqrt><mn>3</mn></msqrt><mn>4</mn></mfrac><mo>,</mo><mo> </mo><mfrac><mn>3</mn><mn>4</mn></mfrac></mrow></mfenced></math> respectively.</p>
</div>
<div class="specification">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">P</mi></math> can be written as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">P</mi><mo>=</mo><mi mathvariant="bold-italic">R</mi><mi mathvariant="bold-italic">S</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">S</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">R</mi></math> are matrices.</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">S</mi></math> represents an enlargement with scale factor <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>5</mn></math>, centre <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>0</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math>.</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">R</mi></math> represents a rotation about <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>0</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math>.</p>
</div>
<div class="specification">
<p>The transformation <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi></math> can also be described by an enlargement scale factor <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac></math>, centre <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>a</mi><mo>,</mo><mo> </mo><mi>b</mi><mo>)</mo></math>, followed by a rotation about the same centre <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>a</mi><mo>,</mo><mo> </mo><mi>b</mi><mo>)</mo></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By considering the image of <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>0</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math>, find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">q</mi></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>By considering the image of <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>1</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>0</mn><mo>,</mo><mo> </mo><mn>1</mn><mo>)</mo></math>, show that</p>
<p style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">P</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mfrac><msqrt><mn>3</mn></msqrt><mn>4</mn></mfrac><mo> </mo></mtd><mtd><mfrac><mn>1</mn><mn>4</mn></mfrac></mtd></mtr><mtr><mtd><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mo> </mo></mtd><mtd><mfrac><msqrt><mn>3</mn></msqrt><mn>4</mn></mfrac></mtd></mtr></mtable></mfenced></math>.</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>Write down the matrix <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">S</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>Use <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">P</mi><mo>=</mo><mi mathvariant="bold-italic">R</mi><mi mathvariant="bold-italic">S</mi></math> to find the matrix <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">R</mi></math>.</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>Hence find the angle and direction of the rotation represented by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">R</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>Write down an equation satisfied by <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mi>a</mi></mtd></mtr><mtr><mtd><mi>b</mi></mtd></mtr></mtable></mfenced></math>.</p>
<div class="marks">[1]</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 <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">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>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 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 src="data:image/png;base64,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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>Phil takes out a bank loan of $150 000 to buy a house, at an annual interest rate of 3.5%. The interest is calculated at the end of each year and added to the amount outstanding.</p>
</div>
<div class="specification">
<p>To pay off the loan, Phil makes annual deposits of $<em>P </em>at the end of every year in a savings account, paying an annual interest rate of 2% . He makes his first deposit at the end of the first year after taking out the loan.</p>
</div>
<div class="specification">
<p>David visits a different bank and makes a single deposit of $<em>Q </em>, the annual interest rate being 2.8%.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the amount Phil would owe the bank after 20 years. Give your answer to the nearest dollar.</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 total value of Phil’s savings after 20 years is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{({{1.02}^{20}} - 1)P}}{{(1.02 - 1)}}"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mrow> <msup> <mrow> <mn>1.02</mn> </mrow> <mrow> <mn>20</mn> </mrow> </msup> </mrow> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>P</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <mn>1.02</mn> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> </math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that Phil’s aim is to own the house after 20 years, find the value for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P"> <mi>P</mi> </math></span> to the nearest dollar.</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>David wishes to withdraw $5000 at the end of each year for a period of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n"> <mi>n</mi> </math></span> years. Show that an expression for the minimum value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="Q"> <mi>Q</mi> </math></span> is</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{5000}}{{1.028}} + \frac{{5000}}{{{{1.028}^2}}} + \ldots + \frac{{5000}}{{{{1.028}^n}}}"> <mfrac> <mrow> <mn>5000</mn> </mrow> <mrow> <mn>1.028</mn> </mrow> </mfrac> <mo>+</mo> <mfrac> <mrow> <mn>5000</mn> </mrow> <mrow> <mrow> <msup> <mrow> <mn>1.028</mn> </mrow> <mn>2</mn> </msup> </mrow> </mrow> </mfrac> <mo>+</mo> <mo>…</mo> <mo>+</mo> <mfrac> <mrow> <mn>5000</mn> </mrow> <mrow> <mrow> <msup> <mrow> <mn>1.028</mn> </mrow> <mi>n</mi> </msup> </mrow> </mrow> </mfrac> </math></span>.</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>Hence or otherwise, find the minimum value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="Q"> <mi>Q</mi> </math></span> that would permit David to withdraw annual amounts of $5000 indefinitely. Give your answer to the nearest dollar.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.ii.</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>A flying drone is programmed to complete a series of movements in a horizontal plane relative to an origin <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math> and a set of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-axes.</p>
<p>In each case, the drone moves to a new position represented by the following transformations:</p>
<ul>
<li>a rotation anticlockwise of <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></math> radians about <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math></li>
<li>a reflection in the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfrac><mi>x</mi><msqrt><mn>3</mn></msqrt></mfrac></math></li>
<li>a rotation clockwise of <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi mathvariant="normal">π</mi><mn>3</mn></mfrac></math> radians about <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math>.</li>
</ul>
<p>All the movements are performed in the listed order.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down each of the transformations in matrix form, clearly stating which matrix represents each transformation.</p>
<div class="marks">[6]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find a single matrix <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">P</mi></math> that defines a transformation that represents the overall change in position.</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>Find <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi mathvariant="bold-italic">P</mi><mn>2</mn></msup></math>.</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>Hence state what the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi mathvariant="bold-italic">P</mi><mn>2</mn></msup></math> indicates for the possible movement of the drone.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.iv.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Three drones are initially positioned at the points <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>C</mtext></math>. After performing the movements listed above, the drones are positioned at points <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext><mo>′</mo></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext><mo>′</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>C</mtext><mo>′</mo></math> respectively.</p>
<p>Show that the area of triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>ABC</mtext></math> is equal to the area of triangle <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext><mo>′</mo><mtext>B</mtext><mo>′</mo><mtext>C</mtext><mo>′</mo></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 a single transformation that is equivalent to the three transformations represented by matrix <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">P</mi></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</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 given by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = m{x^3} + n{x^2} + px + q">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>m</mi>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>+</mo>
<mi>n</mi>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mi>p</mi>
<mi>x</mi>
<mo>+</mo>
<mi>q</mi>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="m">
<mi>m</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
<mi>p</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q">
<mi>q</mi>
</math></span> are integers.</p>
<p>The graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> passes through the point (0, 0).</p>
</div>
<div class="specification">
<p>The graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> also passes through the point (3, 18).</p>
</div>
<div class="specification">
<p class="indent1" style="margin-top: 12.0pt; tab-stops: 2.0cm 72.0pt 108.0pt 144.0pt 180.0pt 216.0pt 252.0pt 288.0pt 324.0pt 360.0pt 396.0pt 432.0pt 468.0pt 504.0pt;">The graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> also passes through the points (1, 0) and (–1, –10).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent1" style="margin-top:12.0pt;">Write down the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q">
<mi>q</mi>
</math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent1" style="margin-top:12.0pt;">Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="27m + 9n + 3p = 18">
<mn>27</mn>
<mi>m</mi>
<mo>+</mo>
<mn>9</mn>
<mi>n</mi>
<mo>+</mo>
<mn>3</mn>
<mi>p</mi>
<mo>=</mo>
<mn>18</mn>
</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 class="indent1" style="margin-top:12.0pt;">Write down the other two linear equations in <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="m">
<mi>m</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
<mi>p</mi>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent1a" style="margin-top:12.0pt;">Write down these three equations as a matrix equation.</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 class="indent1a" style="margin-top:12.0pt;">Solve this matrix equation.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent1a" style="margin-top:12.0pt;">The function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> can also be written <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = x\left( {x - 1} \right)\left( {rx - s} \right)">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>x</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>r</mi>
<mi>x</mi>
<mo>−</mo>
<mi>s</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r">
<mi>r</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="s">
<mi>s</mi>
</math></span> are integers. Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r">
<mi>r</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="s">
<mi>s</mi>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^5} - 3{x^4} + m{x^3} + n{x^2} + px + q = 0">
<mrow>
<msup>
<mi>x</mi>
<mn>5</mn>
</msup>
</mrow>
<mo>−<!-- − --></mo>
<mn>3</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>4</mn>
</msup>
</mrow>
<mo>+</mo>
<mi>m</mi>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>+</mo>
<mi>n</mi>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mi>p</mi>
<mi>x</mi>
<mo>+</mo>
<mi>q</mi>
<mo>=</mo>
<mn>0</mn>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="m">
<mi>m</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
<mi>p</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q \in \mathbb{R}">
<mi>q</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
</math></span>.</p>
<p>The equation has three distinct real roots which can be written as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{lo}}{{\text{g}}_2}\,a">
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>2</mn>
</msub>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>a</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{lo}}{{\text{g}}_2}\,b">
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>2</mn>
</msub>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>b</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{lo}}{{\text{g}}_2}\,c">
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>2</mn>
</msub>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>c</mi>
</math></span>.</p>
<p>The equation also has two imaginary roots, one of which is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d{\text{i}}">
<mi>d</mi>
<mrow>
<mtext>i</mtext>
</mrow>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d \in \mathbb{R}">
<mi>d</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
</math></span>.</p>
</div>
<div class="specification">
<p>The values <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>, and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c">
<mi>c</mi>
</math></span> are consecutive terms in a geometric sequence.</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="abc = 8"> <mi>a</mi> <mi>b</mi> <mi>c</mi> <mo>=</mo> <mn>8</mn> </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>Show that one of the real roots is equal to 1.</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>Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q = 8{d^2}"> <mi>q</mi> <mo>=</mo> <mn>8</mn> <mrow> <msup> <mi>d</mi> <mn>2</mn> </msup> </mrow> </math></span>, find the other two real roots.</p>
<div class="marks">[9]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>=</mo><mn>1</mn><mo>−</mo><mtext>i</mtext></math>.</p>
</div>
<div class="specification">
<p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>w</mi><mn>1</mn></msub><mo>=</mo><msup><mtext>e</mtext><mrow><mtext>i</mtext><mi>x</mi></mrow></msup></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>w</mi><mn>2</mn></msub><mo>=</mo><msup><mtext>e</mtext><mrow><mtext>i</mtext><mo>(</mo><mi>x</mi><mo>−</mo><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac><mo>)</mo></mrow></msup></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
</div>
<div class="specification">
<p>The current, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>I</mi></math>, in an AC circuit can be modelled by the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>I</mi><mo>=</mo><mi>a</mi><mo> </mo><mi>cos</mi><mo>(</mo><mi>b</mi><mi>t</mi><mo>−</mo><mi>c</mi><mo>)</mo></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math> is the frequency and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math> is the phase shift.</p>
<p>Two AC voltage sources of the same frequency are independently connected to the same circuit. If connected to the circuit alone they generate currents <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>I</mi><mtext>A</mtext></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>I</mi><mtext>B</mtext></msub></math>. The maximum value and the phase shift of each current is shown in the following table.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<p style="text-align: left;">When the two voltage sources are connected to the circuit at the same time, the total current <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>I</mi><mtext>T</mtext></msub></math> can be expressed as <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>I</mi><mtext>A</mtext></msub><mo>+</mo><msub><mi>I</mi><mtext>B</mtext></msub></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Plot the position of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi></math> on an Argand Diagram.</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>Express <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi></math> in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>=</mo><mi>a</mi><msup><mtext>e</mtext><mrow><mtext>i</mtext><mi>b</mi></mrow></msup></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>,</mo><mo> </mo><mi>b</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>, giving the exact value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> and the exact 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">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>w</mi><mn>1</mn></msub><mo>+</mo><msub><mi>w</mi><mn>2</mn></msub></math> in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>e</mtext><mrow><mtext>i</mtext><mi>x</mi></mrow></msup><mfenced><mrow><mi>c</mi><mo>+</mo><mtext>i</mtext><mi>d</mi></mrow></mfenced></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 find <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Re</mtext><mfenced><mrow><msub><mi>w</mi><mn>1</mn></msub><mo>+</mo><msub><mi>w</mi><mn>2</mn></msub></mrow></mfenced></math> in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo> </mo><mi>cos</mi><mfenced><mrow><mi>x</mi><mo>-</mo><mi>a</mi></mrow></mfenced></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>></mo><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo><</mo><mi>a</mi><mo>≤</mo><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the maximum value of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>I</mi><mtext>T</mtext></msub></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 phase shift of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>I</mi><mtext>T</mtext></msub></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>On the day of her birth, 1st January 1998, Mary’s grandparents invested <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\$ x">
<mi mathvariant="normal">$<!-- $ --></mi>
<mi>x</mi>
</math></span> in a savings account. They continued to deposit <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\$ x">
<mi mathvariant="normal">$<!-- $ --></mi>
<mi>x</mi>
</math></span> on the first day of each month thereafter.</p>
<p>The account paid a fixed rate of 0.4% interest per month. The interest was calculated on the last day of each month and added to the account.</p>
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\$ {A_n}">
<mi mathvariant="normal">$<!-- $ --></mi>
<mrow>
<msub>
<mi>A</mi>
<mi>n</mi>
</msub>
</mrow>
</math></span> be the amount in Mary’s account on the last day of the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n{\text{th}}">
<mi>n</mi>
<mrow>
<mtext>th</mtext>
</mrow>
</math></span> month, immediately after the interest had been added.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an expression for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{A_1}">
<mrow>
<msub>
<mi>A</mi>
<mn>1</mn>
</msub>
</mrow>
</math></span> and show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{A_2} = {1.004^2}x + 1.004x">
<mrow>
<msub>
<mi>A</mi>
<mn>2</mn>
</msub>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mn>1.004</mn>
<mn>2</mn>
</msup>
</mrow>
<mi>x</mi>
<mo>+</mo>
<mn>1.004</mn>
<mi>x</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>(i) Write down a similar expression for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{A_3}">
<mrow>
<msub>
<mi>A</mi>
<mn>3</mn>
</msub>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{A_4}">
<mrow>
<msub>
<mi>A</mi>
<mn>4</mn>
</msub>
</mrow>
</math></span>.</p>
<p>(ii) Hence show that the amount in Mary’s account the day before she turned 10 years old is given by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="251({1.004^{120}} - 1)x">
<mn>251</mn>
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mn>1.004</mn>
<mrow>
<mn>120</mn>
</mrow>
</msup>
</mrow>
<mo>−</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mi>x</mi>
</math></span>.</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>Write down an expression for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{A_n}">
<mrow>
<msub>
<mi>A</mi>
<mi>n</mi>
</msub>
</mrow>
</math></span> in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> on the day before Mary turned 18 years old showing clearly the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Mary’s grandparents wished for the amount in her account to be at least <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\$ 20\,000">
<mi mathvariant="normal">$</mi>
<mn>20</mn>
<mspace width="thinmathspace"></mspace>
<mn>000</mn>
</math></span> the day before she was 18. Determine the minimum value of the monthly deposit <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\$ x">
<mi mathvariant="normal">$</mi>
<mi>x</mi>
</math></span> required to achieve this. Give your answer correct to the nearest dollar.</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>As soon as Mary was 18 she decided to invest <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\$ 15\,000">
<mi mathvariant="normal">$</mi>
<mn>15</mn>
<mspace width="thinmathspace"></mspace>
<mn>000</mn>
</math></span> of this money in an account of the same type earning 0.4% interest per month. She withdraws <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\$ 1000">
<mi mathvariant="normal">$</mi>
<mn>1000</mn>
</math></span> every year on her birthday to buy herself a present. Determine how long it will take until there is no money in the account.</p>
<div class="marks">[5]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="specification">
<p>Let <strong><em>A </em></strong><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left( {\begin{array}{*{20}{c}} 3&1 \\ 4&3 \end{array}} \right)">
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>3</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>4</mn>
</mtd>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
</div>
<div class="specification">
<p>Let <strong><em>A</em></strong><sup>2</sup> + <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="m">
<mi>m</mi>
</math></span><em><strong>A </strong></em>+ <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span><em><strong>I</strong></em> = <strong><em>O</em></strong> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="m">
<mi>m</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n \in \mathbb{Z}">
<mi>n</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">Z</mi>
</mrow>
</math></span> and <strong><em>O </em></strong>= <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 0&0 \\ 0&0 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\lambda ">
<mi>λ</mi>
</math></span> for which the matrix (<strong><em>A</em></strong> − <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\lambda ">
<mi>λ</mi>
</math></span><strong><em>I</em></strong>) is singular.</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 value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="m">
<mi>m</mi>
</math></span> and of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence show that <strong><em>I</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{5}">
<mfrac>
<mn>1</mn>
<mn>5</mn>
</mfrac>
</math></span><strong><em>A </em></strong>(6<strong><em>I</em></strong> – <strong><em>A</em></strong>).</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use the result from <strong>part (b) (ii)</strong> to explain why <strong><em>A</em></strong> is non-singular.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use the values from <strong>part (b) (i)</strong> to express <strong><em>A</em></strong><sup>4</sup> in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
<mi>p</mi>
</math></span><em><strong>A</strong></em>+ <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q">
<mi>q</mi>
</math></span><em><strong>I</strong></em> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
<mi>p</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q \in \mathbb{Z}">
<mi>q</mi>
<mo>∈</mo>
<mrow>
<mi mathvariant="double-struck">Z</mi>
</mrow>
</math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>The 3rd term of an arithmetic sequence is 1407 and the 10th term is 1183.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the first term and the common difference of the sequence.</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 number of positive terms in the sequence.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Let <strong><em>A</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 0&2 \\ 2&0 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
</div>
<div class="specification">
<p>Let <strong><em>B</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} p&2 \\ 0&q \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>p</mi>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mi>q</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent1" style="margin-top:12.0pt;">Find <strong><em>A</em></strong><sup>−1</sup>.</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 class="indent1" style="margin-top:12.0pt;">Find <strong><em>A</em></strong><sup>2</sup>.</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>Given that 2<strong><em>A</em></strong> + <em><strong>B </strong></em>=<em><strong> </strong></em><span style="background-color: #ffffff;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 2&6 \\ 4&3 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>6</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>4</mn>
</mtd>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
<mi>p</mi>
</math></span> and of <span style="display: inline !important;float: none;background-color: #ffffff;color: #000000;font-family: Verdana,Arial,Helvetica,sans-serif;font-size: 14px;font-style: normal;font-variant: normal;font-weight: 400;letter-spacing: normal;text-align: left;text-decoration: none;text-indent: 0px;white-space: normal;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q">
<mi>q</mi>
</math></span></span>.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent1" style="margin-top:12.0pt;">Hence find <strong><em>A</em></strong><sup>−1</sup><strong><em>B</em></strong>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent1" style="margin-top:12.0pt;">Let <strong><em>X</em></strong> be a 2 × 2 matrix such that <strong><em>AX</em></strong> = <strong><em>B</em></strong>. Find <strong><em>X</em></strong>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p><strong>In this question, give all answers to two decimal places.</strong></p>
<p>Bryan decides to purchase a new car with a price of €14 000, but cannot afford the full amount. The car dealership offers two options to finance a loan.</p>
<p><strong>Finance option A:</strong></p>
<p>A 6 year loan at a nominal annual interest rate of 14 % <strong>compounded quarterly</strong>. No deposit required and repayments are made each quarter.</p>
</div>
<div class="specification">
<p><strong>Finance option B:</strong></p>
<p>A 6 year loan at a nominal annual interest rate of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r">
<mi>r</mi>
</math></span> % <strong>compounded monthly</strong>. Terms of the loan require a 10 % deposit and monthly repayments of €250.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the repayment made each quarter.</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 total amount paid for the car.</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 interest paid on the loan.</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>Find the amount to be borrowed for this option.</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 annual interest rate, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r">
<mi>r</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>State which option Bryan should choose. Justify your answer.</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>Bryan chooses option B. The car dealership invests the money Bryan pays as soon as they receive it.</p>
<p>If they invest it in an account paying 0.4 % interest per month and inflation is 0.1 % per month, calculate the real amount of money the car dealership has received by the end of the 6 year period.</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>Matrices <strong><em>A</em></strong>, <strong><em>B</em></strong> and <strong><em>C</em></strong> are defined by</p>
<p><strong><em>A </em></strong>= <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 5&1 \\ 7&2 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>5</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>7</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>B</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 2&4 \\ { - 3}&{15} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>3</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>15</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>C</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 9&{ - 7} \\ 8&2 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>9</mn>
</mtd>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>7</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>8</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
<p>Let <strong><em>X</em></strong> be an unknown 2 × 2 matrix satisfying the equation</p>
<p style="text-align: center;"><strong><em>AX</em> </strong>+<strong> <em>B</em> </strong>=<strong> <em>C</em></strong>.</p>
<p>This equation may be solved for <strong><em>X</em></strong> by rewriting it in the form</p>
<p style="text-align: center;"><strong><em>X</em> </strong>=<strong> <em>A</em></strong><sup>−1</sup><strong> <em>D</em></strong>.</p>
<p>where <strong><em>D</em></strong> is a 2 × 2 matrix.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down <strong><em>A</em></strong><sup>−1</sup>.</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 <strong><em>D</em></strong>.</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 <strong><em>X</em></strong>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="S">
<mi>S</mi>
</math></span><sub><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span></sub> be the sum of the first <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span> terms of the arithmetic series 2 + 4 + 6 + ….</p>
</div>
<div class="specification">
<p>Let <strong><em>M </em></strong>= <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 1&2 \\ 0&1 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
</div>
<div class="specification">
<p>It may now be assumed that <strong><em>M</em></strong><sup><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span></sup> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 1&{2n} \\ 0&1 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mrow>
<mn>2</mn>
<mi>n</mi>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span> ≥ 4. The sum <strong><em>T</em></strong><sub><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span></sub> is defined by</p>
<p style="text-align: center;"><strong><em>T</em></strong><sub><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span></sub> = <strong><em>M</em></strong><sup>1</sup> + <strong><em>M</em></strong><sup>2</sup> + <strong><em>M</em></strong><sup>3</sup> + ... + <strong><em>M</em></strong><sup><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span></sup>.</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="S">
<mi>S</mi>
</math></span><sub>4</sub>.</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 <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="S">
<mi>S</mi>
</math></span><sub>100</sub>.</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>Find <strong><em>M</em></strong><sup>2</sup>.</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>Show that <strong><em>M</em></strong><sup>3</sup> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 1&6 \\ 0&1 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>6</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</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>Write down <strong><em>M</em></strong><sup>4</sup>.</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 <strong><em>T</em></strong><sub>4</sub>.</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>Using your results from part (a) (ii), find <strong><em>T</em></strong><sub>100</sub>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>Let <strong><em>M</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 2&1 \\ 2&{ - 1} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent1" style="margin-top:12.0pt;">Write down the determinant of <strong><em>M</em></strong>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent1" style="margin-top:12.0pt;"> Write down <strong><em>M</em></strong><sup>−1</sup>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent1" style="margin-top:12.0pt;"><strong>Hence</strong> solve <strong><em>M</em></strong><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} x \\ y \end{array}} \right) = \left( {\begin{array}{*{20}{c}} 4 \\ 8 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>x</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>y</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>8</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
<p class="indent1" style="margin-top:12.0pt;"> </p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Let <strong><em>A</em> </strong>= <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} a&b \\ c&0 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>a</mi>
</mtd>
<mtd>
<mi>b</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>c</mi>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> and <strong><em>B</em> </strong>= <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 1&0 \\ d&e \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>d</mi>
</mtd>
<mtd>
<mi>e</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>. Giving your answers in terms 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>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d">
<mi>d</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="e">
<mi>e</mi>
</math></span>,</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>write down <strong><em>A</em> </strong>+<strong> <em>B</em></strong>.</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 <strong><em>AB</em></strong>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z = a + b{\text{i}}">
<mi>z</mi>
<mo>=</mo>
<mi>a</mi>
<mo>+</mo>
<mi>b</mi>
<mrow>
<mtext>i</mtext>
</mrow>
</math></span>, <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="{\text{b}} \in {\mathbb{R}^ + }">
<mrow>
<mtext>b</mtext>
</mrow>
<mo>∈<!-- ∈ --></mo>
<mrow>
<msup>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
<mo>+</mo>
</msup>
</mrow>
</math></span> and let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{arg}}\,z = \theta ">
<mrow>
<mtext>arg</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>z</mi>
<mo>=</mo>
<mi>θ<!-- θ --></mi>
</math></span>.</p>
</div>
<div class="question">
<p>Show the points represented by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z">
<mi>z</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z - 2a">
<mi>z</mi>
<mo>−</mo>
<mn>2</mn>
<mi>a</mi>
</math></span> on the following Argand diagram.</p>
<p><img src="data:image/png;base64,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"></p>
</div>
<br><hr><br><div class="specification">
<p>Consider a geometric sequence with a first term of 4 and a fourth term of −2.916.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the common ratio of this sequence.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the sum to infinity of this sequence.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="M = \left( {\begin{array}{*{20}{c}} a&2 \\ 2&{ - 1} \end{array}} \right)">
<mi>M</mi>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>a</mi>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a \in \mathbb{Z}">
<mi>a</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">Z</mi>
</mrow>
</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="{M^2}">
<mrow>
<msup>
<mi>M</mi>
<mn>2</mn>
</msup>
</mrow>
</math></span> in terms 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>If <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{M^2}">
<mrow>
<msup>
<mi>M</mi>
<mn>2</mn>
</msup>
</mrow>
</math></span> is equal to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 5&{ - 4} \\ { - 4}&5 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>5</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>4</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>4</mn>
</mrow>
</mtd>
<mtd>
<mn>5</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, find the value 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>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Using this value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span>, find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{M^{ - 1}}">
<mrow>
<msup>
<mi>M</mi>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</math></span> and <strong>hence </strong>solve the system of equations:</p>
<p style="padding-left:180px;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - x + 2y = - 3">
<mo>−</mo>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
<mi>y</mi>
<mo>=</mo>
<mo>−</mo>
<mn>3</mn>
</math></span></p>
<p style="padding-left:180px;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2x - y = 3">
<mn>2</mn>
<mi>x</mi>
<mo>−</mo>
<mi>y</mi>
<mo>=</mo>
<mn>3</mn>
</math></span></p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\gamma = \frac{{1 + {\text{i}}\sqrt 3 }}{2}">
<mi>γ<!-- γ --></mi>
<mo>=</mo>
<mfrac>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mrow>
<mtext>i</mtext>
</mrow>
<msqrt>
<mn>3</mn>
</msqrt>
</mrow>
<mn>2</mn>
</mfrac>
</math></span>.</p>
</div>
<div class="specification">
<p>The matrix <strong><em>A</em> </strong>is defined by <strong><em>A </em></strong>= <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} \gamma &1 \\ 0&{\frac{1}{\gamma }} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>γ<!-- γ --></mi>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mrow>
<mfrac>
<mn>1</mn>
<mi>γ<!-- γ --></mi>
</mfrac>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
</div>
<div class="specification">
<p>Deduce that</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent1" style="margin-top:12.0pt;">Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\gamma ^2} = \gamma - 1">
<mrow>
<msup>
<mi>γ</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mi>γ</mi>
<mo>−</mo>
<mn>1</mn>
</math></span>.</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 class="indent1" style="margin-top:12.0pt;">Hence find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {1 - \gamma } \right)^6}">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>γ</mi>
</mrow>
<mo>)</mo>
</mrow>
<mn>6</mn>
</msup>
</mrow>
</math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent1" style="margin-top:12.0pt;"><strong><em>A</em></strong><sup>3</sup><strong> </strong>= –<strong><em>I</em></strong>.</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 class="indent1" style="margin-top:12.0pt;"><strong><em>A</em></strong><sup>–1</sup> = <strong><em>I</em></strong> – <strong><em>A</em></strong>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>Long term experience shows that if it is sunny on a particular day in Vokram, then the probability that it will be sunny the following day is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>8</mn></math>. If it is not sunny, then the probability that it will be sunny the following day is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>3</mn></math>.</p>
<p>The transition matrix <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">T</mi></math> is used to model this information, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">T</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mn>0</mn><mo>.</mo><mn>8</mn></mtd><mtd><mo> </mo><mn>0</mn><mo>.</mo><mn>3</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo>.</mo><mn>2</mn></mtd><mtd><mo> </mo><mn>0</mn><mo>.</mo><mn>7</mn></mtd></mtr></mtable></mfenced></math>.</p>
</div>
<div class="specification">
<p>The matrix <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">T</mi></math> can be written as a product of three matrices, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">P</mi><mi mathvariant="bold-italic">D</mi><mo mathvariant="bold"> </mo><msup><mi mathvariant="bold-italic">P</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup></math> , where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">D</mi></math> is a diagonal matrix.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>It is sunny today. Find the probability that it will be sunny in three days’ time.</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 and eigenvectors of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">T</mi></math>.</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>Write down the matrix <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">P</mi></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>Write down the matrix <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">D</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>Hence find the long-term percentage of sunny days in Vokram.</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>The matrix <em><strong>M</strong></em> is given by <em><strong>M</strong></em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left[ {\begin{array}{*{20}{c}} 1&2&2 \\ 3&1&1 \\ 2&3&1 \end{array}} \right]">
<mo>=</mo>
<mrow>
<mo>[</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>3</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>3</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>]</mo>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that <em><strong>M</strong></em><sup>3</sup> can be written as a quadratic expression in <em><strong>M</strong></em> in the form <em>a<strong>M</strong></em><sup>2</sup> + <em>b<strong>M</strong></em> + <em>c<strong>I</strong></em> , determine the values of the constants <em>a</em>, <em>b</em> and <em>c</em>.</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>Show that <em><strong>M</strong></em><sup>4</sup> = 19<em><strong>M</strong></em><sup>2</sup> + 40<em><strong>M</strong></em> + 30<em><strong>I</strong></em>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Using mathematical induction, prove that <em><strong>M</strong></em><sup>n</sup> can be written as a quadratic expression in <em><strong>M</strong></em> for all positive integers <em>n </em>≥ 3.</p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find a quadratic expression in <em><strong>M</strong></em> for the inverse matrix <em><strong>M</strong></em><sup>–1</sup>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent1a" style="margin-top:12.0pt;">Write down the inverse of the matrix <strong><em>A</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 1&{ - 3}&0 \\ 2&0&1 \\ 4&1&3 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>3</mn>
</mrow>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>4</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</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 class="indent1a" style="margin-top:12.0pt;">Hence or otherwise solve</p>
<p class="indent1a" style="margin-top:12.0pt;"><span class="mjpage mjpage__block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" alttext="x - 3y = 1">
<mi>x</mi>
<mo>−</mo>
<mn>3</mn>
<mi>y</mi>
<mo>=</mo>
<mn>1</mn>
</math></span></p>
<p class="indent1a" style="margin-top:12.0pt;"><span class="mjpage mjpage__block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" alttext="2x + z = 2">
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mi>z</mi>
<mo>=</mo>
<mn>2</mn>
</math></span></p>
<p class="indent1a" style="margin-top:12.0pt;"><span class="mjpage mjpage__block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" alttext="4x + y + 3z = - 1">
<mn>4</mn>
<mi>x</mi>
<mo>+</mo>
<mi>y</mi>
<mo>+</mo>
<mn>3</mn>
<mi>z</mi>
<mo>=</mo>
<mo>−</mo>
<mn>1</mn>
</math></span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Let <strong><em>A</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 1&1&1 \\ 0&1&1 \\ 0&0&1 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> and <strong><em>B</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 1&0&0 \\ 1&1&0 \\ 1&1&1 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
</div>
<div class="specification">
<p>Given that <strong><em>X</em> </strong>= <strong><em>B</em> </strong>– <strong><em>A</em></strong><sup>–1</sup> and <strong><em>Y</em> </strong>= <strong><em>B</em></strong><sup>–1</sup> – <strong><em>A</em></strong>,</p>
</div>
<div class="specification">
<p>You are told that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{A^n} = \left( {\begin{array}{*{20}{c}} 1&n&{\frac{{n\left( {n + 1} \right)}}{2}} \\ 0&1&n \\ 0&0&1 \end{array}} \right)">
<mrow>
<msup>
<mi>A</mi>
<mi>n</mi>
</msup>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mi>n</mi>
</mtd>
<mtd>
<mrow>
<mfrac>
<mrow>
<mi>n</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</mfrac>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mi>n</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n \in {\mathbb{Z}^ + }">
<mi>n</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<msup>
<mrow>
<mi mathvariant="double-struck">Z</mi>
</mrow>
<mo>+</mo>
</msup>
</mrow>
</math></span>.</p>
</div>
<div class="specification">
<p>Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {{A^n}} \right)^{ - 1}} = \left( {\begin{array}{*{20}{c}} 1&x&y \\ 0&1&x \\ 0&0&1 \end{array}} \right)">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msup>
<mi>A</mi>
<mi>n</mi>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mi>x</mi>
</mtd>
<mtd>
<mi>y</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mi>x</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n \in {\mathbb{Z}^ + }">
<mi>n</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<msup>
<mrow>
<mi mathvariant="double-struck">Z</mi>
</mrow>
<mo>+</mo>
</msup>
</mrow>
</math></span>,</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent1" style="margin-top:12.0pt;">find<strong><em> X </em></strong>and<strong> <em>Y</em></strong>.</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 class="indent1" style="margin-top:12.0pt;">does <strong><em>X</em></strong><sup>–1</sup> + <strong><em>Y</em></strong><sup>–1</sup> have an inverse? Justify your conclusion.</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 class="indent1" style="margin-top:12.0pt;">find <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> in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent1" style="margin-top:12.0pt;">and hence find an expression for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{A^n} + {\left( {{A^n}} \right)^{ - 1}}">
<mrow>
<msup>
<mi>A</mi>
<mi>n</mi>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msup>
<mi>A</mi>
<mi>n</mi>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.ii.</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="question">
<p>The matrices <strong><em>A</em></strong>, <strong><em>B</em></strong>, <strong><em>X</em></strong> are given by</p>
<p><em><strong>A</strong></em> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 3&1 \\ { - 5}&6 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>3</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>5</mn>
</mrow>
</mtd>
<mtd>
<mn>6</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, <em><strong>B</strong></em> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 4&8 \\ 0&{ - 3} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>4</mn>
</mtd>
<mtd>
<mn>8</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>3</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, <em><strong>X</strong></em> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} a&b \\ c&d \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>a</mi>
</mtd>
<mtd>
<mi>b</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>c</mi>
</mtd>
<mtd>
<mi>d</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, <span style="background-color: #ffffff;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{where}}">
<mrow>
<mtext>where</mtext>
</mrow>
</math></span></span> <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 \in \mathbb{Q}">
<mi>d</mi>
<mo>∈</mo>
<mrow>
<mi mathvariant="double-struck">Q</mi>
</mrow>
</math></span>.</p>
<p>Given that <strong><em>AX</em></strong> + <strong><em>X</em></strong> = <strong><em>Β</em></strong>, find the <strong>exact</strong> values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span><em>,</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b">
<mi>b</mi>
</math></span><em>,</em> <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><em>.</em></p>
</div>
<br><hr><br><div class="specification">
<p>Let <strong><em>M</em></strong><sup>2</sup> = <strong><em>M</em></strong> where <strong><em>M</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} a&b \\ c&d \end{array}} \right){\text{,}}\,\,bc \ne 0">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>a</mi>
</mtd>
<mtd>
<mi>b</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>c</mi>
</mtd>
<mtd>
<mi>d</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>b</mi>
<mi>c</mi>
<mo>≠<!-- ≠ --></mo>
<mn>0</mn>
</math></span>. </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="a + d = 1">
<mi>a</mi>
<mo>+</mo>
<mi>d</mi>
<mo>=</mo>
<mn>1</mn>
</math></span>.</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 class="indent2" style="margin-top:12.0pt;">Find an expression for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="bc">
<mi>b</mi>
<mi>c</mi>
</math></span> in terms 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">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent2" style="margin-top:12.0pt;"><strong>Hence</strong> show that <strong><em>M</em></strong> is a singular matrix.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent2" style="margin-top:12.0pt;">If all of the elements of <strong><em>M</em></strong> are positive, find the range of possible values for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent2" style="margin-top:12.0pt;">Show that (<strong><em>I</em></strong> − <strong><em>M</em></strong>)<sup>2</sup> = <strong><em>I</em></strong> − <strong><em>M</em></strong> where <strong><em>I</em></strong> is the identity matrix.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>It is known that the number of fish in a given lake will decrease by 7% each year unless some new fish are added. At the end of each year, 250 new fish are added to the lake.</p>
<p>At the start of 2018, there are 2500 fish in the lake.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that there will be approximately 2645 fish in the lake at the start of 2020.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the approximate number of fish in the lake at the start of 2042.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</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>In a small village there are two doctors’ clinics, one owned by Doctor Black and the other owned by Doctor Green. It was noted after each year that <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>.</mo><mn>5</mn><mo>%</mo></math> of Doctor Black’s patients moved to Doctor Green’s clinic and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mo>%</mo></math> of Doctor Green’s patients moved to Doctor Black’s clinic. All additional losses and gains of patients by the clinics may be ignored.</p>
<p>At the start of a particular year, it was noted that Doctor Black had <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2100</mn></math> patients on their register, compared to Doctor Green’s <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3500</mn></math> patients.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down a transition matrix <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">T</mi></math> indicating the annual population movement between clinics.</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 a prediction for the ratio of the number of patients Doctor Black will have, compared to Doctor Green, after two years.</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 a matrix <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">P</mi></math>, with integer elements, such that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">T</mi><mo>=</mo><mi mathvariant="bold-italic">P</mi><mi mathvariant="bold-italic">D</mi><mo mathvariant="bold"> </mo><msup><mi mathvariant="bold-italic">P</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">D</mi></math> is a diagonal matrix.</p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, show that the long-term transition matrix <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi mathvariant="bold-italic">T</mi><mo>∞</mo></msup></math> is given by <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi mathvariant="bold-italic">T</mi><mo>∞</mo></msup><mo>=</mo><mfenced><mtable><mtr><mtd><mfrac><mn>10</mn><mn>17</mn></mfrac></mtd><mtd><mfrac><mn>10</mn><mn>17</mn></mfrac></mtd></mtr><mtr><mtd><mfrac><mn>7</mn><mn>17</mn></mfrac></mtd><mtd><mfrac><mn>7</mn><mn>17</mn></mfrac></mtd></mtr></mtable></mfenced></math>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, or otherwise, determine the expected ratio of the number of patients Doctor Black would have compared to Doctor Green in the long term.</p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="question">
<p>Boxes of mixed fruit are on sale at a local supermarket.</p>
<p>Box A contains 2 bananas, 3 kiwifruit and 4 melons, and costs $6.58.</p>
<p>Box B contains 5 bananas, 2 kiwifruit and 8 melons and costs $12.32.</p>
<p>Box C contains 5 bananas and 4 kiwifruit and costs $3.00.</p>
<p>Find the cost of each type of fruit.</p>
</div>
<br><hr><br>