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<h2>HL Paper 3</h2><div class="specification">
<p>This question investigates some applications of differential equations to modeling population growth.</p>
<p>One model for population growth is to assume that the rate of change of the population is proportional to the population, i.e. <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}P}}{{{\text{d}}t}} = kP">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>P</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mi>k</mi>
<mi>P</mi>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k \in \mathbb{R}">
<mi>k</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> is the time (in years) and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P">
<mi>P</mi>
</math></span> is the population</p>
</div>
<div class="specification">
<p>The initial population is 1000.</p>
</div>
<div class="specification">
<p>Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k = 0.003">
<mi>k</mi>
<mo>=</mo>
<mn>0.003</mn>
</math></span>, use your answer from part (a) to find</p>
</div>
<div class="specification">
<p>Consider now the situation when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k">
<mi>k</mi>
</math></span> is not a constant, but a function of time.</p>
</div>
<div class="specification">
<p>Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k = 0.003 + 0.002t">
<mi>k</mi>
<mo>=</mo>
<mn>0.003</mn>
<mo>+</mo>
<mn>0.002</mn>
<mi>t</mi>
</math></span>, find</p>
</div>
<div class="specification">
<p>Another model for population growth assumes</p>
<ul>
<li>there is a maximum value for the population, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="L">
<mi>L</mi>
</math></span>.</li>
<li>that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k">
<mi>k</mi>
</math></span> is not a constant, but is proportional to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {1 - \frac{P}{L}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−<!-- − --></mo>
<mfrac>
<mi>P</mi>
<mi>L</mi>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</li>
</ul>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the general solution of this differential equation is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P = A{{\text{e}}^{kt}}"> <mi>P</mi> <mo>=</mo> <mi>A</mi> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mi>k</mi> <mi>t</mi> </mrow> </msup> </mrow> </math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A \in \mathbb{R}"> <mi>A</mi> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> </math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>the population after 10 years</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>the number of years it will take for the population to triple.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mathop {{\text{lim}}}\limits_{t \to \infty } P"> <munder> <mrow> <mrow> <mtext>lim</mtext> </mrow> </mrow> <mrow> <mi>t</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mo></mo> <mi>P</mi> </math></span></p>
<div class="marks">[1]</div>
<div class="question_part_label">b.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>the solution of the differential equation, giving your answer in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P = f\left( t \right)"> <mi>P</mi> <mo>=</mo> <mi>f</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>the number of years it will take for the population to triple.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}P}}{{{\text{d}}t}} = \frac{m}{L}P\left( {L - P} \right)"> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>P</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </mfrac> <mo>=</mo> <mfrac> <mi>m</mi> <mi>L</mi> </mfrac> <mi>P</mi> <mrow> <mo>(</mo> <mrow> <mi>L</mi> <mo>−</mo> <mi>P</mi> </mrow> <mo>)</mo> </mrow> </math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="m \in \mathbb{R}"> <mi>m</mi> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Solve the differential equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}P}}{{{\text{d}}t}} = \frac{m}{L}P\left( {L - P} \right)"> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>P</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>t</mi> </mrow> </mfrac> <mo>=</mo> <mfrac> <mi>m</mi> <mi>L</mi> </mfrac> <mi>P</mi> <mrow> <mo>(</mo> <mrow> <mi>L</mi> <mo>−</mo> <mi>P</mi> </mrow> <mo>)</mo> </mrow> </math></span>, giving your answer in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P = g\left( t \right)"> <mi>P</mi> <mo>=</mo> <mi>g</mi> <mrow> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </math></span>.</p>
<div class="marks">[10]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that the initial population is 1000, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="L = 10000"> <mi>L</mi> <mo>=</mo> <mn>10000</mn> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="m = 0.003"> <mi>m</mi> <mo>=</mo> <mn>0.003</mn> </math></span>, find the number of years it will take for the population to triple.</p>
<div class="marks">[4]</div>
<div class="question_part_label">f.</div>
</div>
<br><hr><br><div class="specification">
<p><strong>In this question you will explore some of the properties of special functions <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">f</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">g</mi></math> and their relationship with the trigonometric functions, sine and cosine.</strong></p>
<p><br>Functions <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math> are defined as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>z</mi></mfenced><mo>=</mo><mfrac><mrow><msup><mtext>e</mtext><mi>z</mi></msup><mo>+</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mi>z</mi></mrow></msup></mrow><mn>2</mn></mfrac></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>z</mi></mfenced><mo>=</mo><mfrac><mrow><msup><mtext>e</mtext><mi>z</mi></msup><mo>-</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mi>z</mi></mrow></msup></mrow><mn>2</mn></mfrac></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>∈</mo><mi mathvariant="normal">ℂ</mi></math>.</p>
<p>Consider <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi></math>, such that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>,</mo><mo> </mo><mi>u</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
</div>
<div class="specification">
<p>Using <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>e</mtext><mrow><mtext>i</mtext><mi>u</mi></mrow></msup><mo>=</mo><mi>cos</mi><mo> </mo><mi>u</mi><mo>+</mo><mtext>i</mtext><mo> </mo><mi>sin</mi><mo> </mo><mi>u</mi></math>, find expressions, in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mo> </mo><mi>u</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mo> </mo><mi>u</mi></math>, for</p>
</div>
<div class="specification">
<p>The functions <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mo> </mo><mi>x</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mo> </mo><mi>x</mi></math> are known as circular functions as the general point (<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mo> </mo><mi>θ</mi><mo>,</mo><mo> </mo><mi>sin</mi><mo> </mo><mi>θ</mi></math>) defines points on the unit circle with equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>1</mn></math>.</p>
<p>The functions <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> are known as hyperbolic functions, as the general point ( <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>θ</mi><mo>)</mo><mo>,</mo><mo> </mo><mi>g</mi><mo>(</mo><mi>θ</mi><mo>)</mo></math> ) defines points on a curve known as a hyperbola with equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>1</mn></math>. This hyperbola has two asymptotes.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Verify that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi><mo>=</mo><mi>f</mi><mfenced><mi>t</mi></mfenced></math> satisfies the differential equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>u</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mi>u</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>f</mi><mfenced><mi>t</mi></mfenced></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mi>g</mi><mfenced><mi>t</mi></mfenced></mrow></mfenced><mn>2</mn></msup><mo>=</mo><mi>f</mi><mfenced><mrow><mn>2</mn><mi>t</mi></mrow></mfenced></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><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mrow><mtext>i</mtext><mi>u</mi></mrow></mfenced></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><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mrow><mtext>i</mtext><mi>u</mi></mrow></mfenced></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find, and simplify, an expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>f</mi><mfenced><mrow><mtext>i</mtext><mi>u</mi></mrow></mfenced></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mi>g</mi><mfenced><mrow><mtext>i</mtext><mi>u</mi></mrow></mfenced></mrow></mfenced><mn>2</mn></msup></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>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>f</mi><mfenced><mi>t</mi></mfenced></mrow></mfenced><mn>2</mn></msup><mo>-</mo><msup><mfenced><mrow><mi>g</mi><mfenced><mi>t</mi></mfenced></mrow></mfenced><mn>2</mn></msup><mo>=</mo><msup><mfenced><mrow><mi>f</mi><mfenced><mrow><mtext>i</mtext><mi>u</mi></mrow></mfenced></mrow></mfenced><mn>2</mn></msup><mo>-</mo><msup><mfenced><mrow><mi>g</mi><mfenced><mrow><mtext>i</mtext><mi>u</mi></mrow></mfenced></mrow></mfenced><mn>2</mn></msup></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>1</mn></math>, stating the coordinates of any axis intercepts and the equation of each asymptote.</p>
<div class="marks">[4]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The hyperbola with equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><mn>1</mn></math> can be rotated to coincide with the curve defined by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mi>y</mi><mo>=</mo><mi>k</mi><mo>,</mo><mo> </mo><mi>k</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
<p>Find the possible values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">g.</div>
</div>
<br><hr><br><div class="specification">
<p>This question asks you to investigate some properties of the sequence of functions of the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f_n}(x) = {\text{cos}}\left( {n\,{\text{arccos}}\,x} \right)">
<mrow>
<msub>
<mi>f</mi>
<mi>n</mi>
</msub>
</mrow>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow>
<mtext>cos</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>n</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>arccos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, −1 ≤ <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> ≤ 1 and <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>
<p><strong>Important:</strong> When sketching graphs in this question, you are <strong>not</strong> required to find the coordinates of any axes intercepts or the coordinates of any stationary points unless requested.</p>
</div>
<div class="specification">
<p>For odd values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span> > 2, use your graphic display calculator to systematically vary the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span>. Hence suggest an expression for odd values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span> describing, in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span>, the number of</p>
</div>
<div class="specification">
<p>For even values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span> > 2, use your graphic display calculator to systematically vary the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span>. Hence suggest an expression for even values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span>describing, in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span>, the number of</p>
</div>
<div class="specification">
<p>The sequence of functions, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f_n}(x)">
<mrow>
<msub>
<mi>f</mi>
<mi>n</mi>
</msub>
</mrow>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span>, defined above can be expressed as a sequence of polynomials of degree <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span>.</p>
</div>
<div class="specification">
<p>Consider <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f_{n + 1}}(x) = {\text{cos}}\left( {\left( {n + 1} \right)\,{\text{arccos}}\,x} \right)">
<mrow>
<msub>
<mi>f</mi>
<mrow>
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msub>
</mrow>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow>
<mtext>cos</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>arccos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>On the same set of axes, sketch the graphs of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = {f_1}(x)"> <mi>y</mi> <mo>=</mo> <mrow> <msub> <mi>f</mi> <mn>1</mn> </msub> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = {f_3}(x)"> <mi>y</mi> <mo>=</mo> <mrow> <msub> <mi>f</mi> <mn>3</mn> </msub> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></span> for −1 ≤ <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> ≤ 1.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>local maximum points;</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>local minimum points;</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>On a new set of axes, sketch the graphs of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = {f_2}(x)"> <mi>y</mi> <mo>=</mo> <mrow> <msub> <mi>f</mi> <mn>2</mn> </msub> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = {f_4}(x)"> <mi>y</mi> <mo>=</mo> <mrow> <msub> <mi>f</mi> <mn>4</mn> </msub> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></span> for −1 ≤ <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> ≤ 1.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>local maximum points;</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>local minimum points.</p>
<div class="marks">[1]</div>
<div class="question_part_label">d.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Solve the equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f_n}^\prime (x) = 0"> <msup> <mrow> <msub> <mi>f</mi> <mi>n</mi> </msub> </mrow> <mi mathvariant="normal">′</mi> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </math></span> and hence show that the stationary points on the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = {f_n}(x)"> <mi>y</mi> <mo>=</mo> <mrow> <msub> <mi>f</mi> <mi>n</mi> </msub> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></span> occur at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = {\text{cos}}\frac{{k\pi }}{n}"> <mi>x</mi> <mo>=</mo> <mrow> <mtext>cos</mtext> </mrow> <mfrac> <mrow> <mi>k</mi> <mi>π</mi> </mrow> <mi>n</mi> </mfrac> </math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k \in {\mathbb{Z}^ + }"> <mi>k</mi> <mo>∈</mo> <mrow> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>+</mo> </msup> </mrow> </math></span> and 0 < <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k"> <mi>k</mi> </math></span> < <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n"> <mi>n</mi> </math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use an appropriate trigonometric identity to show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f_2}(x) = 2{x^2} - 1"> <mrow> <msub> <mi>f</mi> <mn>2</mn> </msub> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>1</mn> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use an appropriate trigonometric identity to show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f_{n + 1}}(x) = {\text{cos}}\left( {n\,{\text{arccos}}\,x} \right){\text{cos}}\left( {{\text{arccos}}\,x} \right) - {\text{sin}}\left( {n\,{\text{arccos}}\,x} \right){\text{sin}}\left( {{\text{arccos}}\,x} \right)"> <mrow> <msub> <mi>f</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mtext>cos</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>arccos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mtext>cos</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>arccos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mrow> <mtext>sin</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>arccos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mtext>sin</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>arccos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">g.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f_{n + 1}}(x) + {f_{n - 1}}(x) = 2x{f_n}\left( x \right)"> <mrow> <msub> <mi>f</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow> <msub> <mi>f</mi> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <mi>x</mi> <mrow> <msub> <mi>f</mi> <mi>n</mi> </msub> </mrow> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </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> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>+</mo> </msup> </mrow> </math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">h.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence express <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f_3}(x)"> <mrow> <msub> <mi>f</mi> <mn>3</mn> </msub> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </math></span> as a cubic polynomial.</p>
<div class="marks">[2]</div>
<div class="question_part_label">h.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the differential equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}} = 1 + \frac{y}{x}">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mn>1</mn>
<mo>+</mo>
<mfrac>
<mi>y</mi>
<mi>x</mi>
</mfrac>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x \ne 0">
<mi>x</mi>
<mo>≠<!-- ≠ --></mo>
<mn>0</mn>
</math></span>.</p>
</div>
<div class="specification">
<p>Consider the family of curves which satisfy the differential equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}} = 1 + \frac{y}{x}">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mn>1</mn>
<mo>+</mo>
<mfrac>
<mi>y</mi>
<mi>x</mi>
</mfrac>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x \ne 0">
<mi>x</mi>
<mo>≠<!-- ≠ --></mo>
<mn>0</mn>
</math></span>.</p>
</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="y\left( 1 \right) = 1">
<mi>y</mi>
<mrow>
<mo>(</mo>
<mn>1</mn>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>1</mn>
</math></span>, use Euler’s method with step length <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="h">
<mi>h</mi>
</math></span> = 0.25 to find an approximation for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y\left( 2 \right)">
<mi>y</mi>
<mrow>
<mo>(</mo>
<mn>2</mn>
<mo>)</mo>
</mrow>
</math></span>. Give your answer to two significant figures.</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>Solve the equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}} = 1 + \frac{y}{x}">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mn>1</mn>
<mo>+</mo>
<mfrac>
<mi>y</mi>
<mi>x</mi>
</mfrac>
</math></span> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y\left( 1 \right) = 1">
<mi>y</mi>
<mrow>
<mo>(</mo>
<mn>1</mn>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>1</mn>
</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>Find the percentage error when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y\left( 2 \right)">
<mi>y</mi>
<mrow>
<mo>(</mo>
<mn>2</mn>
<mo>)</mo>
</mrow>
</math></span> is approximated by the final rounded value found in part (a). Give your answer to two significant figures.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the equation of the isocline corresponding to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}} = k">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mi>k</mi>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k \ne 0">
<mi>k</mi>
<mo>≠</mo>
<mn>0</mn>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k \in \mathbb{R}">
<mi>k</mi>
<mo>∈</mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
</math></span>.</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>Show that such an isocline can never be a normal to any of the family of curves that satisfy the differential equation.</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>This question will investigate power series, as an extension to the Binomial Theorem for negative and fractional indices.</p>
<p>A power series in <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> is defined as a function of the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = {a_0} + {a_1}x + {a_2}{x^2} + {a_3}{x^3} + ...">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<msub>
<mi>a</mi>
<mn>0</mn>
</msub>
</mrow>
<mo>+</mo>
<mrow>
<msub>
<mi>a</mi>
<mn>1</mn>
</msub>
</mrow>
<mi>x</mi>
<mo>+</mo>
<mrow>
<msub>
<mi>a</mi>
<mn>2</mn>
</msub>
</mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msub>
<mi>a</mi>
<mn>3</mn>
</msub>
</mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>+</mo>
<mo>.</mo>
<mo>.</mo>
<mo>.</mo>
</math></span> where the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{a_i} \in \mathbb{R}">
<mrow>
<msub>
<mi>a</mi>
<mi>i</mi>
</msub>
</mrow>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
</math></span>.</p>
<p>It can be considered as an infinite polynomial.</p>
</div>
<div class="specification">
<p>This is an example of a power series, but is only a finite power series, since only a finite number of the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{a_i}">
<mrow>
<msub>
<mi>a</mi>
<mi>i</mi>
</msub>
</mrow>
</math></span> are non-zero.</p>
</div>
<div class="specification">
<p>We will now attempt to generalise further.</p>
<p>Suppose <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {1 + x} \right)^q}{\text{,}}\,\,q \in \mathbb{Q}">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mi>q</mi>
</msup>
</mrow>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>q</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">Q</mi>
</mrow>
</math></span> can be written as the power series <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{a_0} + {a_1}x + {a_2}{x^2} + {a_3}{x^3} + ...">
<mrow>
<msub>
<mi>a</mi>
<mn>0</mn>
</msub>
</mrow>
<mo>+</mo>
<mrow>
<msub>
<mi>a</mi>
<mn>1</mn>
</msub>
</mrow>
<mi>x</mi>
<mo>+</mo>
<mrow>
<msub>
<mi>a</mi>
<mn>2</mn>
</msub>
</mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msub>
<mi>a</mi>
<mn>3</mn>
</msub>
</mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>+</mo>
<mo>.</mo>
<mo>.</mo>
<mo>.</mo>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Expand <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {1 + x} \right)^5}">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mn>5</mn>
</msup>
</mrow>
</math></span> using the Binomial Theorem.</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>Consider the power series <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1 - x + {x^2} - {x^3} + {x^4} - ...">
<mn>1</mn>
<mo>−</mo>
<mi>x</mi>
<mo>+</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>4</mn>
</msup>
</mrow>
<mo>−</mo>
<mo>.</mo>
<mo>.</mo>
<mo>.</mo>
</math></span></p>
<p>By considering the ratio of consecutive terms, explain why this series is equal to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {1 + x} \right)^{ - 1}}">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</math></span> and state the values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> for which this equality is true.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Differentiate the equation obtained part (b) and hence, find the first four terms in a power series for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {1 + x} \right)^{ - 2}}">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
</msup>
</mrow>
</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>Repeat this process to find the first four terms in a power series for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {1 + x} \right)^{ - 3}}">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>−</mo>
<mn>3</mn>
</mrow>
</msup>
</mrow>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, by recognising the pattern, deduce the first four terms in a power series for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {1 + x} \right)^{ - n}}">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>−</mo>
<mi>n</mi>
</mrow>
</msup>
</mrow>
</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>
<msup>
<mrow>
<mi mathvariant="double-struck">Z</mi>
</mrow>
<mo>+</mo>
</msup>
</mrow>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By substituting <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 0">
<mi>x</mi>
<mo>=</mo>
<mn>0</mn>
</math></span>, find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{a_0}">
<mrow>
<msub>
<mi>a</mi>
<mn>0</mn>
</msub>
</mrow>
</math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By differentiating both sides of the expression and then substituting <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 0">
<mi>x</mi>
<mo>=</mo>
<mn>0</mn>
</math></span>, find the value of <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>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">g.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Repeat this procedure to find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{a_2}">
<mrow>
<msub>
<mi>a</mi>
<mn>2</mn>
</msub>
</mrow>
</math></span> and <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>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">h.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, write down the first four terms in what is called the Extended Binomial Theorem for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {1 + x} \right)^q}{\text{,}}\,\,q \in \mathbb{Q}">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mi>q</mi>
</msup>
</mrow>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>q</mi>
<mo>∈</mo>
<mrow>
<mi mathvariant="double-struck">Q</mi>
</mrow>
</math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the power series for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{{1 + {x^2}}}">
<mfrac>
<mn>1</mn>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">j.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, using integration, find the power series for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{arctan}}\,x">
<mrow>
<mtext>arctan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</math></span>, giving the first four non-zero terms.</p>
<div class="marks">[4]</div>
<div class="question_part_label">k.</div>
</div>
<br><hr><br><div class="specification">
<p style="text-align: left;">The function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> is defined by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x){\text{ }}={\text{ }}{(\arcsin{\text{ }}x)^2},{\text{ }} - 1 \leqslant x \leqslant 1">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mo>=</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<mo stretchy="false">(</mo>
<mi>arcsin</mi>
<mo><!-- --></mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>x</mi>
<msup>
<mo stretchy="false">)</mo>
<mn>2</mn>
</msup>
</mrow>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo>⩽<!-- ⩽ --></mo>
<mi>x</mi>
<mo>⩽<!-- ⩽ --></mo>
<mn>1</mn>
</math></span>.</p>
<p> </p>
</div>
<div class="specification">
<p>The function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> satisfies the equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {1 - {x^2}} \right)f''\left( x \right) - xf'\left( x \right) - 2 = 0">
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−<!-- − --></mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<msup>
<mi>f</mi>
<mo>″</mo>
</msup>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>−<!-- − --></mo>
<mi>x</mi>
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>−<!-- − --></mo>
<mn>2</mn>
<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="f'\left( 0 \right) = 0">
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
<mrow>
<mo>(</mo>
<mn>0</mn>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By differentiating the above equation twice, show that</p>
<p><span class="mjpage mjpage__block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" alttext="\left( {1 - {x^2}} \right){f^{\left( 4 \right)}}\left( x \right) - 5x{f^{\left( 3 \right)}}\left( x \right) - 4f''\left( x \right) = 0">
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<msup>
<mi>f</mi>
<mrow>
<mrow>
<mo>(</mo>
<mn>4</mn>
<mo>)</mo>
</mrow>
</mrow>
</msup>
</mrow>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mn>5</mn>
<mi>x</mi>
<mrow>
<msup>
<mi>f</mi>
<mrow>
<mrow>
<mo>(</mo>
<mn>3</mn>
<mo>)</mo>
</mrow>
</mrow>
</msup>
</mrow>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mn>4</mn>
<msup>
<mi>f</mi>
<mo>″</mo>
</msup>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span></p>
<p>where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f^{\left( 3 \right)}}\left( x \right)">
<mrow>
<msup>
<mi>f</mi>
<mrow>
<mrow>
<mo>(</mo>
<mn>3</mn>
<mo>)</mo>
</mrow>
</mrow>
</msup>
</mrow>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{f^{\left( 4 \right)}}\left( x \right)">
<mrow>
<msup>
<mi>f</mi>
<mrow>
<mrow>
<mo>(</mo>
<mn>4</mn>
<mo>)</mo>
</mrow>
</mrow>
</msup>
</mrow>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span> denote the 3rd and 4th derivative of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right)">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span> respectively.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence show that the Maclaurin series for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right)">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span> up to and including the term in <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^4}">
<mrow>
<msup>
<mi>x</mi>
<mn>4</mn>
</msup>
</mrow>
</math></span> is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^2} + \frac{1}{3}{x^4}">
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
<mrow>
<msup>
<mi>x</mi>
<mn>4</mn>
</msup>
</mrow>
</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>Use this series approximation for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right)">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span> with <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \frac{1}{2}">
<mi>x</mi>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</math></span> to find an approximate value for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\pi ^2}">
<mrow>
<msup>
<mi>π</mi>
<mn>2</mn>
</msup>
</mrow>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the differential equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}} = \frac{{4{x^2} + {y^2} - xy}}{{{x^2}}}">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mn>4</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−<!-- − --></mo>
<mi>x</mi>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</math></span>, with <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 2">
<mi>y</mi>
<mo>=</mo>
<mn>2</mn>
</math></span> when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 1">
<mi>x</mi>
<mo>=</mo>
<mn>1</mn>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use Euler’s method, with step length <span class="mjpage"><math alttext="h = 0.1" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>h</mi> <mo>=</mo> <mn>0.1</mn> </math></span>, to find an approximate value of <span class="mjpage"><math alttext="y" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>y</mi> </math></span> when <span class="mjpage"><math alttext="x = 1.4" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>x</mi> <mo>=</mo> <mn>1.4</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>Sketch the isoclines for <span class="mjpage"><math alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}} = 4" xmlns="http://www.w3.org/1998/Math/MathML"> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>y</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mfrac> <mo>=</mo> <mn>4</mn> </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>Express <span class="mjpage"><math alttext="{m^2} - 2m + 4" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <msup> <mi>m</mi> <mn>2</mn> </msup> </mrow> <mo>−</mo> <mn>2</mn> <mi>m</mi> <mo>+</mo> <mn>4</mn> </math></span> in the form <span class="mjpage"><math alttext="{\left( {m - a} \right)^2} + b" xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <msup> <mrow> <mo>(</mo> <mrow> <mi>m</mi> <mo>−</mo> <mi>a</mi> </mrow> <mo>)</mo> </mrow> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mi>b</mi> </math></span> , where <span class="mjpage"><math alttext="a{\text{, }}b \in \mathbb{Z}" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>a</mi> <mrow> <mtext>, </mtext> </mrow> <mi>b</mi> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> </math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Solve the differential equation, for <span class="mjpage"><math alttext="x > 0" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>x</mi> <mo>></mo> <mn>0</mn> </math></span>, giving your answer in the form <span class="mjpage"><math alttext="y = f\left( x \right)" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>y</mi> <mo>=</mo> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math></span>.</p>
<div class="marks">[10]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of <span class="mjpage"><math alttext="y = f\left( x \right)" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>y</mi> <mo>=</mo> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math></span> for <span class="mjpage"><math alttext="1 \leqslant x \leqslant 1.4" xmlns="http://www.w3.org/1998/Math/MathML"> <mn>1</mn> <mo>⩽</mo> <mi>x</mi> <mo>⩽</mo> <mn>1.4</mn> </math></span> .</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>With reference to the curvature of your sketch in part (c)(iii), and without further calculation, explain whether you conjecture <span class="mjpage"><math alttext="f\left( {1.4} \right)" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>f</mi> <mrow> <mo>(</mo> <mrow> <mn>1.4</mn> </mrow> <mo>)</mo> </mrow> </math></span> will be less than, equal to, or greater than your answer in part (a).</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.iv.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the differential equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2xy\frac{{{\text{d}}y}}{{{\text{d}}x}} = {y^2} - {x^2}">
<mn>2</mn>
<mi>x</mi>
<mi>y</mi>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mrow>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−<!-- − --></mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x > 0">
<mi>x</mi>
<mo>></mo>
<mn>0</mn>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Solve the differential equation and show that a general solution is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^2} + {y^2} = cx">
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mi>c</mi>
<mi>x</mi>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c">
<mi>c</mi>
</math></span> is a positive constant.</p>
<div class="marks">[11]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Prove that there are two horizontal tangents to the general solution curve and state their equations, in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="c">
<mi>c</mi>
</math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><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="\int\limits_4^\infty {\frac{1}{{{x^3}}}{\text{d}}x} ">
<munderover>
<mo>∫</mo>
<mn>4</mn>
<mi mathvariant="normal">∞</mi>
</munderover>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
</mrow>
</mfrac>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Illustrate graphically the inequality <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sum\limits_{n = 5}^\infty {\frac{1}{{{n^3}}}} < \int\limits_4^\infty {\frac{1}{{{x^3}}}{\text{d}}x} < \sum\limits_{n = 4}^\infty {\frac{1}{{{n^3}}}} ">
<munderover>
<mo movablelimits="false">∑</mo>
<mrow>
<mi>n</mi>
<mo>=</mo>
<mn>5</mn>
</mrow>
<mi mathvariant="normal">∞</mi>
</munderover>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<mrow>
<msup>
<mi>n</mi>
<mn>3</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</mrow>
<mo><</mo>
<munderover>
<mo>∫</mo>
<mn>4</mn>
<mi mathvariant="normal">∞</mi>
</munderover>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
</mrow>
</mfrac>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
<mo><</mo>
<munderover>
<mo movablelimits="false">∑</mo>
<mrow>
<mi>n</mi>
<mo>=</mo>
<mn>4</mn>
</mrow>
<mi mathvariant="normal">∞</mi>
</munderover>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<mrow>
<msup>
<mi>n</mi>
<mn>3</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</mrow>
</math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence write down a lower bound for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sum\limits_{n = 4}^\infty {\frac{1}{{{n^3}}}} ">
<munderover>
<mo movablelimits="false">∑</mo>
<mrow>
<mi>n</mi>
<mo>=</mo>
<mn>4</mn>
</mrow>
<mi mathvariant="normal">∞</mi>
</munderover>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<mrow>
<msup>
<mi>n</mi>
<mn>3</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</mrow>
</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>Find an upper bound for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sum\limits_{n = 4}^\infty {\frac{1}{{{n^3}}}} ">
<munderover>
<mo movablelimits="false">∑</mo>
<mrow>
<mi>n</mi>
<mo>=</mo>
<mn>4</mn>
</mrow>
<mi mathvariant="normal">∞</mi>
</munderover>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<mrow>
<msup>
<mi>n</mi>
<mn>3</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</mrow>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p><strong>This question asks you to explore cubic polynomials of the form</strong> <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>x</mi><mo>-</mo><mi>r</mi></mrow></mfenced><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>a</mi><mi>x</mi><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow></mfenced></math> <strong>for</strong> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math> <strong>and corresponding cubic equations with one real root and two complex roots of the form </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>z</mi><mo>-</mo><mi>r</mi><mo>)</mo><mo>(</mo><msup><mi>z</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>a</mi><mi>z</mi><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup><mo>)</mo><mo>=</mo><mn>0</mn></math> <strong>for</strong> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>∈</mo><mi mathvariant="normal">ℂ</mi></math>.</p>
<p> </p>
</div>
<div class="specification">
<p>In parts (a), (b) and (c), let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo> </mo><mi>a</mi><mo>=</mo><mn>4</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>=</mo><mn>1</mn></math>.</p>
<p>Consider the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>z</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><msup><mi>z</mi><mn>2</mn></msup><mo>-</mo><mn>8</mn><mi>z</mi><mo>+</mo><mn>17</mn></mrow></mfenced><mo>=</mo><mn>0</mn></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>∈</mo><mi mathvariant="normal">ℂ</mi></math>.</p>
</div>
<div class="specification">
<p>Consider the function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfenced><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>8</mn><mi>x</mi><mo>+</mo><mn>17</mn></mrow></mfenced></math> for <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>Consider the function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfenced><mrow><mi>x</mi><mo>-</mo><mi>r</mi></mrow></mfenced><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>a</mi><mi>x</mi><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow></mfenced></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>,</mo><mo> </mo><mi>a</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi><mo>,</mo><mo> </mo><mi>b</mi><mo>></mo><mn>0</mn></math>.</p>
</div>
<div class="specification">
<p>The equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>z</mi><mo>-</mo><mi>r</mi></mrow></mfenced><mfenced><mrow><msup><mi>z</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>a</mi><mi>z</mi><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow></mfenced><mo>=</mo><mn>0</mn></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>∈</mo><mi mathvariant="normal">ℂ</mi></math> has roots <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>±</mo><mi>b</mi><mtext>i</mtext></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>,</mo><mo> </mo><mi>a</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi><mo>,</mo><mo> </mo><mi>b</mi><mo>></mo><mn>0</mn></math>.</p>
</div>
<div class="specification">
<p>On the Cartesian plane, the points <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>C</mtext><mn>1</mn></msub><mfenced><mrow><mi>a</mi><mo>,</mo><mo> </mo><msqrt><mi>g</mi><mo>'</mo><mfenced><mi>a</mi></mfenced></msqrt></mrow></mfenced></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>C</mtext><mn>2</mn></msub><mfenced><mrow><mi>a</mi><mo>,</mo><mo> </mo><mo>-</mo><msqrt><mi>g</mi><mo>'</mo><mfenced><mi>a</mi></mfenced></msqrt></mrow></mfenced></math> represent the real and imaginary parts of the complex roots of the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>z</mi><mo>-</mo><mi>r</mi></mrow></mfenced><mfenced><mrow><msup><mi>z</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>a</mi><mi>z</mi><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow></mfenced><mo>=</mo><mn>0</mn></math>.</p>
<p><br>The following diagram shows a particular curve of the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfenced><mrow><mi>x</mi><mo>-</mo><mi>r</mi></mrow></mfenced><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>a</mi><mi>x</mi><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><mn>16</mn></mrow></mfenced></math> and the tangent to the curve at the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext><mfenced><mrow><mi>a</mi><mo>,</mo><mo> </mo><mn>80</mn></mrow></mfenced></math>. The curve and the tangent both intersect the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis at the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>R</mtext><mfenced><mrow><mo>-</mo><mn>2</mn><mo>,</mo><mo> </mo><mn>0</mn></mrow></mfenced></math>. The points <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>C</mtext><mn>1</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>C</mtext><mn>2</mn></msub></math> are also shown.</p>
<p style="text-align: center;"><img 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"></p>
</div>
<div class="specification">
<p>Consider the curve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>(</mo><mi>x</mi><mo>-</mo><mi>r</mi><mo>)</mo><mo>(</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>a</mi><mi>x</mi><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup><mo>)</mo></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>≠</mo><mi>r</mi><mo>,</mo><mo> </mo><mi>b</mi><mo>></mo><mn>0</mn></math>. The points <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext><mo>(</mo><mi>a</mi><mo>,</mo><mo> </mo><mi>g</mi><mo>(</mo><mi>a</mi><mo>)</mo><mo>)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>R</mtext><mo>(</mo><mi>r</mi><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math> are as defined in part (d)(ii). The curve has a point of inflexion at point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math>.</p>
</div>
<div class="specification">
<p>Consider the special case where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mi>r</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>></mo><mn>0</mn></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>+</mo><mtext>i</mtext></math> are roots of the equation, write down the third root.</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>Verify that the mean of the two complex roots is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math>.</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>Show that the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>x</mi><mo>-</mo><mn>1</mn></math> is tangent to the curve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mfenced><mi>x</mi></mfenced></math> at the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext><mfenced><mrow><mn>4</mn><mo>,</mo><mo> </mo><mn>3</mn></mrow></mfenced></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the curve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> and the tangent to the curve at point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math>, clearly showing where the tangent crosses the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>2</mn><mfenced><mrow><mi>x</mi><mo>-</mo><mi>r</mi></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>-</mo><mi>a</mi></mrow></mfenced><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>a</mi><mi>x</mi><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></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>Hence, or otherwise, prove that the tangent to the curve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>g</mi><mfenced><mi>x</mi></mfenced></math> at the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext><mfenced><mrow><mi>a</mi><mo>,</mo><mo> </mo><mi>g</mi><mfenced><mi>a</mi></mfenced></mrow></mfenced></math> intersects the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis at the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>R</mtext><mfenced><mrow><mi>r</mi><mo>,</mo><mo> </mo><mn>0</mn></mrow></mfenced></math>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">d.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Deduce from part (d)(i) that the complex roots of the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>z</mi><mo>-</mo><mi>r</mi></mrow></mfenced><mfenced><mrow><msup><mi>z</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>a</mi><mi>z</mi><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow></mfenced><mo>=</mo><mn>0</mn></math> can be expressed as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>±</mo><mtext>i</mtext><msqrt><mi>g</mi><mo>'</mo><mfenced><mi>a</mi></mfenced></msqrt></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use this diagram to determine the roots of the corresponding equation of the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>z</mi><mo>-</mo><mi>r</mi></mrow></mfenced><mfenced><mrow><msup><mi>z</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>a</mi><mi>z</mi><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><mn>16</mn></mrow></mfenced><mo>=</mo><mn>0</mn></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>∈</mo><mi mathvariant="normal">ℂ</mi></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">f.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State the coordinates of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mtext>C</mtext><mn>2</mn></msub></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">f.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-coordinate of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>3</mn></mfrac><mfenced><mrow><mn>2</mn><mi>a</mi><mo>+</mo><mi>r</mi></mrow></mfenced></math>.</p>
<p>You are <strong>not</strong> required to demonstrate a change in concavity.</p>
<div class="marks">[2]</div>
<div class="question_part_label">g.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence describe numerically the horizontal position of point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math> relative to the horizontal positions of the points <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>R</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">g.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the curve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfenced><mrow><mi>x</mi><mo>-</mo><mi>r</mi></mrow></mfenced><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>a</mi><mi>x</mi><mo>+</mo><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></mrow></mfenced></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mi>r</mi><mo>=</mo><mn>1</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>=</mo><mn>2</mn></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">h.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>For <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mi>r</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>></mo><mn>0</mn></math>, state in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math>, the coordinates of points <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">h.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>This question will investigate methods for finding definite integrals of powers of trigonometrical functions.</p>
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{I_n} = \int\limits_0^{\tfrac{\pi }{2}} {{\text{si}}{{\text{n}}^n}} x\,dx{\text{,}}\,\,n \in \mathbb{N}">
<mrow>
<msub>
<mi>I</mi>
<mi>n</mi>
</msub>
</mrow>
<mo>=</mo>
<munderover>
<mo>∫<!-- ∫ --></mo>
<mn>0</mn>
<mrow>
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mi>π<!-- π --></mi>
<mn>2</mn>
</mfrac>
</mstyle>
</mrow>
</munderover>
<mrow>
<mrow>
<mtext>si</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mi>n</mi>
</msup>
</mrow>
</mrow>
<mi>x</mi>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>x</mi>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>n</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">N</mi>
</mrow>
</math></span>.</p>
<p> </p>
</div>
<div class="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{J_n} = \int\limits_0^{\tfrac{\pi }{2}} {{\text{co}}{{\text{s}}^n}} x\,dx{\text{,}}\,\,n \in \mathbb{N}.">
<mrow>
<msub>
<mi>J</mi>
<mi>n</mi>
</msub>
</mrow>
<mo>=</mo>
<munderover>
<mo>∫<!-- ∫ --></mo>
<mn>0</mn>
<mrow>
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mi>π<!-- π --></mi>
<mn>2</mn>
</mfrac>
</mstyle>
</mrow>
</munderover>
<mrow>
<mrow>
<mtext>co</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>s</mtext>
</mrow>
<mi>n</mi>
</msup>
</mrow>
</mrow>
<mi>x</mi>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>x</mi>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>n</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">N</mi>
</mrow>
<mo>.</mo>
</math></span></p>
</div>
<div class="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{T_n} = \int\limits_0^{\tfrac{\pi }{4}} {{\text{ta}}{{\text{n}}^n}} x\,dx{\text{,}}\,\,n \in \mathbb{N}">
<mrow>
<msub>
<mi>T</mi>
<mi>n</mi>
</msub>
</mrow>
<mo>=</mo>
<munderover>
<mo>∫<!-- ∫ --></mo>
<mn>0</mn>
<mrow>
<mstyle displaystyle="false" scriptlevel="0">
<mfrac>
<mi>π<!-- π --></mi>
<mn>4</mn>
</mfrac>
</mstyle>
</mrow>
</munderover>
<mrow>
<mrow>
<mtext>ta</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mi>n</mi>
</msup>
</mrow>
</mrow>
<mi>x</mi>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>x</mi>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>n</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">N</mi>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the exact values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{I_0}">
<mrow>
<msub>
<mi>I</mi>
<mn>0</mn>
</msub>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{I_1}">
<mrow>
<msub>
<mi>I</mi>
<mn>1</mn>
</msub>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{I_2}">
<mrow>
<msub>
<mi>I</mi>
<mn>2</mn>
</msub>
</mrow>
</math></span>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use integration by parts to show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{I_n} = \frac{{n - 1}}{n}{I_{n - 2}}{\text{,}}\,\,n \geqslant 2">
<mrow>
<msub>
<mi>I</mi>
<mi>n</mi>
</msub>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mi>n</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mi>n</mi>
</mfrac>
<mrow>
<msub>
<mi>I</mi>
<mrow>
<mi>n</mi>
<mo>−</mo>
<mn>2</mn>
</mrow>
</msub>
</mrow>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>n</mi>
<mo>⩾</mo>
<mn>2</mn>
</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>Explain where the condition <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n \geqslant 2">
<mi>n</mi>
<mo>⩾</mo>
<mn>2</mn>
</math></span> was used in your proof.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, find the exact values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{I_3}">
<mrow>
<msub>
<mi>I</mi>
<mn>3</mn>
</msub>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{I_4}">
<mrow>
<msub>
<mi>I</mi>
<mn>4</mn>
</msub>
</mrow>
</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>Use the substitution <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \frac{\pi }{2} - u">
<mi>x</mi>
<mo>=</mo>
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
<mo>−</mo>
<mi>u</mi>
</math></span> to show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{J_n} = {I_n}">
<mrow>
<msub>
<mi>J</mi>
<mi>n</mi>
</msub>
</mrow>
<mo>=</mo>
<mrow>
<msub>
<mi>I</mi>
<mi>n</mi>
</msub>
</mrow>
</math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, find the exact values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{J_{5}}">
<mrow>
<msub>
<mi>J</mi>
<mrow>
<mn>5</mn>
</mrow>
</msub>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{J_{6}}">
<mrow>
<msub>
<mi>J</mi>
<mrow>
<mn>6</mn>
</mrow>
</msub>
</mrow>
</math></span></p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the exact values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{T_{0}}">
<mrow>
<msub>
<mi>T</mi>
<mrow>
<mn>0</mn>
</mrow>
</msub>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{T_{1}}">
<mrow>
<msub>
<mi>T</mi>
<mrow>
<mn>1</mn>
</mrow>
</msub>
</mrow>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use the fact that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ta}}{{\text{n}}^2}x = {\text{se}}{{\text{c}}^2}x - 1">
<mrow>
<mtext>ta</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mi>x</mi>
<mo>=</mo>
<mrow>
<mtext>se</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>c</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mi>x</mi>
<mo>−</mo>
<mn>1</mn>
</math></span> to show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{T_n} = \frac{1}{{n - 1}} - {T_{n - 2}}{\text{,}}\,\,n \geqslant 2">
<mrow>
<msub>
<mi>T</mi>
<mi>n</mi>
</msub>
</mrow>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mi>n</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mfrac>
<mo>−</mo>
<mrow>
<msub>
<mi>T</mi>
<mrow>
<mi>n</mi>
<mo>−</mo>
<mn>2</mn>
</mrow>
</msub>
</mrow>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>n</mi>
<mo>⩾</mo>
<mn>2</mn>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">g.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Explain where the condition <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n \geqslant 2">
<mi>n</mi>
<mo>⩾</mo>
<mn>2</mn>
</math></span> was used in your proof.</p>
<div class="marks">[1]</div>
<div class="question_part_label">g.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, find the exact values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{T_{2}}">
<mrow>
<msub>
<mi>T</mi>
<mrow>
<mn>2</mn>
</mrow>
</msub>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{T_{3}}">
<mrow>
<msub>
<mi>T</mi>
<mrow>
<mn>3</mn>
</mrow>
</msub>
</mrow>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">h.</div>
</div>
<br><hr><br><div class="specification">
<p>This question asks you to investigate regular <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span>-sided polygons inscribed and circumscribed in a circle, and the perimeter of these as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span> tends to infinity, to make an approximation for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\pi ">
<mi>π<!-- π --></mi>
</math></span>.</p>
</div>
<div class="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{P_i}\left( n \right)">
<mrow>
<msub>
<mi>P</mi>
<mi>i</mi>
</msub>
</mrow>
<mrow>
<mo>(</mo>
<mi>n</mi>
<mo>)</mo>
</mrow>
</math></span> represent the perimeter of any <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span>-sided regular polygon inscribed in a circle of radius 1 unit.</p>
</div>
<div class="specification">
<p>Consider an equilateral triangle ABC of side length, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> units, circumscribed about a circle of radius 1 unit and centre O as shown in the following diagram.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{P_c}\left( n \right)">
<mrow>
<msub>
<mi>P</mi>
<mi>c</mi>
</msub>
</mrow>
<mrow>
<mo>(</mo>
<mi>n</mi>
<mo>)</mo>
</mrow>
</math></span> represent the perimeter of any <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span>-sided regular polygon circumscribed about a circle of radius 1 unit.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Consider an equilateral triangle ABC of side length, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> units, inscribed in a circle of radius 1 unit and centre O as shown in the following diagram.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<p style="text-align: left;">The equilateral triangle ABC can be divided into three smaller isosceles triangles, each subtending an angle of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{2\pi }}{3}"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mn>3</mn> </mfrac> </math></span> at O, as shown in the following diagram.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<p style="text-align: left;">Using right-angled trigonometry or otherwise, show that the perimeter of the equilateral triangle ABC is equal to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3\sqrt 3 "> <mn>3</mn> <msqrt> <mn>3</mn> </msqrt> </math></span> units.</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>Consider a square of side length, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> units, inscribed in a circle of radius 1 unit. By dividing the inscribed square into four isosceles triangles, find the exact perimeter of the inscribed square.</p>
<p> </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 perimeter of a regular hexagon, of side length, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span> units, inscribed in a circle of radius 1 unit.</p>
<p> </p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{P_i}\left( n \right) = 2n\,{\text{sin}}\left( {\frac{\pi }{n}} \right)"> <mrow> <msub> <mi>P</mi> <mi>i</mi> </msub> </mrow> <mrow> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> <mo>=</mo> <mn>2</mn> <mi>n</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mi>π</mi> <mi>n</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use an appropriate Maclaurin series expansion to find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mathop {{\text{lim}}}\limits_{n \to \infty } {P_i}\left( n \right)"> <munder> <mrow> <mrow> <mtext>lim</mtext> </mrow> </mrow> <mrow> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mo></mo> <mrow> <msub> <mi>P</mi> <mi>i</mi> </msub> </mrow> <mrow> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> </math></span> and interpret this result geometrically.</p>
<div class="marks">[5]</div>
<div class="question_part_label">e.</div>
</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="{P_c}\left( n \right) = 2n\,{\text{tan}}\left( {\frac{\pi }{n}} \right)"> <mrow> <msub> <mi>P</mi> <mi>c</mi> </msub> </mrow> <mrow> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> <mo>=</mo> <mn>2</mn> <mi>n</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>tan</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mi>π</mi> <mi>n</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By writing <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{P_c}\left( n \right)"> <mrow> <msub> <mi>P</mi> <mi>c</mi> </msub> </mrow> <mrow> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> </math></span> in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{2\,{\text{tan}}\left( {\frac{\pi }{n}} \right)}}{{\frac{1}{n}}}"> <mfrac> <mrow> <mn>2</mn> <mspace width="thinmathspace"></mspace> <mrow> <mtext>tan</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mfrac> <mi>π</mi> <mi>n</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> </mfrac> </math></span>, find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mathop {{\text{lim}}}\limits_{n \to \infty } {P_c}\left( n \right)"> <munder> <mrow> <mrow> <mtext>lim</mtext> </mrow> </mrow> <mrow> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mo></mo> <mrow> <msub> <mi>P</mi> <mi>c</mi> </msub> </mrow> <mrow> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> </math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">g.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use the results from part (d) and part (f) to determine an inequality for the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\pi "> <mi>π</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">[2]</div>
<div class="question_part_label">h.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The inequality found in part (h) can be used to determine lower and upper bound approximations for the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\pi "> <mi>π</mi> </math></span>.</p>
<p>Determine the least value for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n"> <mi>n</mi> </math></span> such that the lower bound and upper bound approximations are both within 0.005 of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\pi "> <mi>π</mi> </math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">i.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the differential equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x\frac{{{\text{d}}y}}{{{\text{d}}x}} - y = {x^p} + 1">
<mi>x</mi>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>−<!-- − --></mo>
<mi>y</mi>
<mo>=</mo>
<mrow>
<msup>
<mi>x</mi>
<mi>p</mi>
</msup>
</mrow>
<mo>+</mo>
<mn>1</mn>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x \in \mathbb{R},\,x \ne 0">
<mi>x</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>≠<!-- ≠ --></mo>
<mn>0</mn>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
<mi>p</mi>
</math></span> is a positive integer, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p > 1">
<mi>p</mi>
<mo>></mo>
<mn>1</mn>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Solve the differential equation given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = - 1">
<mi>y</mi>
<mo>=</mo>
<mo>−</mo>
<mn>1</mn>
</math></span> when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 1">
<mi>x</mi>
<mo>=</mo>
<mn>1</mn>
</math></span>. Give your answer in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f\left( x \right)">
<mi>y</mi>
<mo>=</mo>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span>.</p>
<div class="marks">[8]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>-coordinate(s) of the points on the curve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f\left( x \right)">
<mi>y</mi>
<mo>=</mo>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}} = 0">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mn>0</mn>
</math></span> satisfy the equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^{p - 1}} = \frac{1}{p}">
<mrow>
<msup>
<mi>x</mi>
<mrow>
<mi>p</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mi>p</mi>
</mfrac>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Deduce the set of values for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
<mi>p</mi>
</math></span> such that there are two points on the curve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f\left( x \right)">
<mi>y</mi>
<mo>=</mo>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}} = 0">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mn>0</mn>
</math></span>. Give a reason for your answer.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.ii.</div>
</div>
<br><hr><br><div class="specification">
<p><strong>This question asks you to explore properties of a family of curves of the type</strong> <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi></math> <strong>for various values of</strong> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> <strong>and</strong> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math>, <strong>where</strong> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>,</mo><mo> </mo><mi>b</mi><mo>∈</mo><mi mathvariant="normal">ℕ</mi></math>.</p>
</div>
<div class="specification">
<p>On the same set of axes, sketch the following curves for <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>2</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mn>2</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>2</mn><mo>≤</mo><mi>y</mi><mo>≤</mo><mn>2</mn></math>, clearly indicating any points of intersection with the coordinate axes.</p>
</div>
<div class="specification">
<p>Now, consider curves of the form <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mi>b</mi></math>, for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>≥</mo><mo>-</mo><mroot><mi>b</mi><mn>3</mn></mroot></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>∈</mo><msup><mi mathvariant="normal">ℤ</mi><mo>+</mo></msup></math>.</p>
</div>
<div class="specification">
<p>Next, consider the curve <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mi>x</mi><mo>,</mo><mo> </mo><mi>x</mi><mo>≥</mo><mn>0</mn></math>.</p>
</div>
<div class="specification">
<p>The curve <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mi>x</mi></math> has two points of inflexion. Due to the symmetry of the curve these points have the same <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-coordinate.</p>
</div>
<div class="specification">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mo>(</mo><mi>x</mi><mo>,</mo><mo> </mo><mi>y</mi><mo>)</mo></math> is defined to be a rational point on a curve if <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> are rational numbers.</p>
<p>The tangent to the curve <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi></math> at a rational point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math> intersects the curve at another rational point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Q</mtext></math>.</p>
<p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math> be the curve <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>2</mn></math>, for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>≥</mo><mo>-</mo><mroot><mn>2</mn><mn>3</mn></mroot></math>. The rational point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mo>(</mo><mo>-</mo><mn>1</mn><mo>,</mo><mo> </mo><mo>-</mo><mn>1</mn><mo>)</mo></math> lies on <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup><mo>,</mo><mo> </mo><mi>x</mi><mo>≥</mo><mn>0</mn></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><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>1</mn><mo>,</mo><mo> </mo><mi>x</mi><mo>≥</mo><mo>-</mo><mn>1</mn></math></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the coordinates of the two points of inflexion on the curve <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>1</mn></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By considering each curve from part (a), identify two key features that would distinguish one curve from the other.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By varying the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math>, suggest two key features common to these curves.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <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><mo>=</mo><mo>±</mo><mfrac><mrow><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow><mrow><mn>2</mn><msqrt><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mi>x</mi></msqrt></mrow></mfrac></math>, for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>></mo><mn>0</mn></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence deduce that the curve <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>y</mi><mn>2</mn></msup><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mi>x</mi><mo> </mo></math>has no local minimum or maximum points.</p>
<div class="marks">[1]</div>
<div class="question_part_label">d.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of this <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-coordinate, giving your answer in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><msqrt><mfrac><mrow><mi>p</mi><msqrt><mn>3</mn></msqrt><mo>+</mo><mi>q</mi></mrow><mi>r</mi></mfrac></msqrt></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>,</mo><mo> </mo><mi>q</mi><mo>,</mo><mo> </mo><mi>r</mi><mo>∈</mo><mi mathvariant="normal">ℤ</mi></math>.</p>
<div class="marks">[7]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the equation of the tangent to <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math> at <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">f.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, find the coordinates of the rational point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Q</mtext></math> where this tangent intersects <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math>, expressing each coordinate as a fraction.</p>
<div class="marks">[2]</div>
<div class="question_part_label">f.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>S</mtext><mo>(</mo><mo>-</mo><mn>1</mn><mo> </mo><mo>,</mo><mo> </mo><mn>1</mn><mo>)</mo></math> also lies on <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math>. The line <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>[QS]</mtext></math> intersects <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math> at a further point. Determine the coordinates of this point.</p>
<div class="marks">[5]</div>
<div class="question_part_label">g.</div>
</div>
<br><hr><br><div class="specification">
<p>The function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> is defined by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>ln</mi><mo> </mo><mfenced><mrow><mn>1</mn><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>1</mn><mo><</mo><mi>x</mi><mo><</mo><mn>1</mn></math>.</p>
</div>
<div class="specification">
<p>The seventh derivative of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> is given by <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>f</mi><mfenced><mn>7</mn></mfenced></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mrow><mn>1440</mn><mi>x</mi><mo> </mo><mfenced><mrow><msup><mi>x</mi><mn>6</mn></msup><mo>-</mo><mn>21</mn><msup><mi>x</mi><mn>4</mn></msup><mo>+</mo><mn>35</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>7</mn></mrow></mfenced></mrow><msup><mfenced><mrow><mn>1</mn><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced><mn>7</mn></msup></mfrac></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use the Maclaurin series for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>ln</mi><mo> </mo><mo>(</mo><mn>1</mn><mo>+</mo><mi>x</mi><mo>)</mo></math> to write down the first three non-zero terms of the Maclaurin series for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find the first three non-zero terms of the Maclaurin series for <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>x</mi><mrow><mn>1</mn><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></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>Use your answer to part (a)(i) to write down an estimate for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mrow><mn>0</mn><mo>.</mo><mn>4</mn></mrow></mfenced></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 the Lagrange form of the error term to find an upper bound for the absolute value of the error in calculating <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mn>0</mn><mo>.</mo><mn>4</mn><mo>)</mo></math>, using the first three non-zero terms of the Maclaurin series for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>With reference to the Lagrange form of the error term, explain whether your answer to part (b) is an overestimate or an underestimate for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>(</mo><mn>0</mn><mo>.</mo><mn>4</mn><mo>)</mo></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>The curve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> has a gradient function given by</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mtext>d</mtext><mi>y</mi></mrow><mrow><mtext>d</mtext><mi>x</mi></mrow></mfrac><mo>=</mo><mi>x</mi><mo>-</mo><mi>y</mi></math>.</p>
<p style="text-align: left;">The curve passes through the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>1</mn><mo>,</mo><mo> </mo><mn>1</mn><mo>)</mo></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>On the same set of axes, sketch and label isoclines for <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mtext>d</mtext><mi>y</mi></mrow><mrow><mtext>d</mtext><mi>x</mi></mrow></mfrac><mo>=</mo><mo>-</mo><mn>1</mn><mo>,</mo><mo> </mo><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math>, and clearly indicate the value of each <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-intercept.</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>Hence or otherwise, explain why the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>1</mn><mo>,</mo><mo> </mo><mn>1</mn><mo>)</mo></math> is a local minimum.</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 the solution of the differential equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mtext>d</mtext><mi>y</mi></mrow><mrow><mtext>d</mtext><mi>x</mi></mrow></mfrac><mo>=</mo><mi>x</mi><mo>-</mo><mi>y</mi></math>, which passes through the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>1</mn><mo>,</mo><mo> </mo><mn>1</mn><mo>)</mo></math>. Give your answer in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math>.</p>
<div class="marks">[8]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Explain why the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mfenced><mi>x</mi></mfenced></math> does not intersect the isocline <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mtext>d</mtext><mi>y</mi></mrow><mrow><mtext>d</mtext><mi>x</mi></mrow></mfrac><mo>=</mo><mn>1</mn></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mfenced><mi>x</mi></mfenced></math> on the same set of axes as part (a)(i).</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.ii.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Consider the differential equation</p>
<p><span class="mjpage mjpage__block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}} = f\left( {\frac{y}{x}} \right),{\text{ }}x > 0.">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mi>y</mi>
<mi>x</mi>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>x</mi>
<mo>></mo>
<mn>0.</mn>
</math></span></p>
<p>Use the substitution <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = vx">
<mi>y</mi>
<mo>=</mo>
<mi>v</mi>
<mi>x</mi>
</math></span> to show that the general solution of this differential equation is</p>
<p><span class="mjpage mjpage__block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" alttext="\int {\frac{{{\text{d}}v}}{{f(v) - v}} = \ln x + } {\text{ Constant.}}">
<mo>∫</mo>
<mrow>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>v</mi>
</mrow>
<mrow>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>v</mi>
<mo stretchy="false">)</mo>
<mo>−</mo>
<mi>v</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mi>ln</mi>
<mo></mo>
<mi>x</mi>
<mo>+</mo>
</mrow>
<mrow>
<mtext> Constant.</mtext>
</mrow>
</math></span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, or otherwise, solve the differential equation</p>
<p><span class="mjpage mjpage__block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}} = \frac{{{x^2} + 3xy + {y^2}}}{{{x^2}}},{\text{ }}x > 0,">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>3</mn>
<mi>x</mi>
<mi>y</mi>
<mo>+</mo>
<mrow>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>x</mi>
<mo>></mo>
<mn>0</mn>
<mo>,</mo>
</math></span></p>
<p>given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 1">
<mi>y</mi>
<mo>=</mo>
<mn>1</mn>
</math></span> when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 1">
<mi>x</mi>
<mo>=</mo>
<mn>1</mn>
</math></span>. Give your answer in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = g(x)">
<mi>y</mi>
<mo>=</mo>
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span>.</p>
<div class="marks">[10]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use L’Hôpital’s rule to determine the value of</p>
<p><span class="mjpage mjpage__block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" alttext="\mathop {{\text{lim}}}\limits_{x \to 0} \left( {\frac{{{{\text{e}}^{ - 3x}}^{^2} + 3\,{\text{cos}}\left( {2x} \right) - 4}}{{3{x^2}}}} \right)">
<munder>
<mrow>
<mrow>
<mtext>lim</mtext>
</mrow>
</mrow>
<mrow>
<mi>x</mi>
<mo stretchy="false">→</mo>
<mn>0</mn>
</mrow>
</munder>
<mo></mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<msup>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mn>3</mn>
<mi>x</mi>
</mrow>
</msup>
</mrow>
<mrow>
<msup>
<mi></mi>
<mn>2</mn>
</msup>
</mrow>
</msup>
<mo>+</mo>
<mn>3</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cos</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mn>4</mn>
</mrow>
<mrow>
<mn>3</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mathop {{\text{lim}}}\limits_{x \to 0} \left( {\frac{{\int_0^x {\left( {{{\text{e}}^{ - 3t}}^{^2} + 3\,{\text{cos}}\left( {2t} \right) - 4} \right)} \,{\text{d}}t}}{{\int_0^x {3{t^2}} \,{\text{d}}t}}} \right)">
<munder>
<mrow>
<mrow>
<mtext>lim</mtext>
</mrow>
</mrow>
<mrow>
<mi>x</mi>
<mo stretchy="false">→</mo>
<mn>0</mn>
</mrow>
</munder>
<mo></mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<msubsup>
<mo>∫</mo>
<mn>0</mn>
<mi>x</mi>
</msubsup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<msup>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mn>3</mn>
<mi>t</mi>
</mrow>
</msup>
</mrow>
<mrow>
<msup>
<mi></mi>
<mn>2</mn>
</msup>
</mrow>
</msup>
<mo>+</mo>
<mn>3</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cos</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mi>t</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mn>4</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
<mrow>
<msubsup>
<mo>∫</mo>
<mn>0</mn>
<mi>x</mi>
</msubsup>
<mrow>
<mn>3</mn>
<mrow>
<msup>
<mi>t</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the differential equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}} + \frac{x}{{{x^2} + 1}}y = x">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>+</mo>
<mfrac>
<mi>x</mi>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>1</mn>
</mrow>
</mfrac>
<mi>y</mi>
<mo>=</mo>
<mi>x</mi>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 1">
<mi>y</mi>
<mo>=</mo>
<mn>1</mn>
</math></span> when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 0">
<mi>x</mi>
<mo>=</mo>
<mn>0</mn>
</math></span>.</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="\sqrt {{x^2} + 1} ">
<msqrt>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>1</mn>
</msqrt>
</math></span> is an integrating factor for this differential equation.</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>Solve the differential equation giving your answer in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f(x)">
<mi>y</mi>
<mo>=</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p><strong>This question asks you to explore the behaviour and key features of cubic polynomials of the form</strong> <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mn>3</mn><mi>c</mi><mi>x</mi><mo>+</mo><mi>d</mi></math>.</p>
<p> </p>
<p>Consider the function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mn>3</mn><mi>c</mi><mi>x</mi><mo>+</mo><mn>2</mn></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math> and where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math> is a parameter, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
<p>The graphs of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>=</mo><mo>-</mo><mn>1</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>=</mo><mn>0</mn></math> are shown in the following diagrams.</p>
<p style="text-align: left;"><br> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>=</mo><mo>-</mo><mn>1</mn></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>=</mo><mn>0</mn></math></p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" 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"></p>
</div>
<div class="specification">
<p>On separate axes, sketch the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> showing the value of the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-intercept and the coordinates of any points with zero gradient, for</p>
</div>
<div class="specification">
<p>Hence, or otherwise, find the set of values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math> such that the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> has</p>
</div>
<div class="specification">
<p>Given that the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> has one local maximum point and one local minimum point, show that</p>
</div>
<div class="specification">
<p>Hence, for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>></mo><mn>0</mn></math>, find the set of values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math> such that the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> has</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>=</mo><mn>1</mn></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>=</mo><mn>2</mn></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>Write down an expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mo>(</mo><mi>x</mi><mo>)</mo></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>a point of inflexion with zero gradient.</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>one local maximum point and one local minimum point.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>no points where the gradient is equal to zero.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-coordinate of the local maximum point is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><msup><mi>c</mi><mfrac><mn>3</mn><mn>2</mn></mfrac></msup><mo>+</mo><mn>2</mn></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-coordinate of the local minimum point is <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>2</mn><msup><mi>c</mi><mfrac><mn>3</mn><mn>2</mn></mfrac></msup><mo>+</mo><mn>2</mn></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">d.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>exactly one <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis intercept.</p>
<div class="marks">[2]</div>
<div class="question_part_label">e.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>exactly two <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis intercepts.</p>
<div class="marks">[2]</div>
<div class="question_part_label">e.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>exactly three <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis intercepts.</p>
<div class="marks">[2]</div>
<div class="question_part_label">e.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Consider the function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mn>3</mn><mi>c</mi><mi>x</mi><mo>+</mo><mi>d</mi></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math> and where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo> </mo><mo>,</mo><mo> </mo><mi>d</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
<p>Find all conditions on <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math> such that the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> has exactly one <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis intercept, explaining your reasoning.</p>
<div class="marks">[6]</div>
<div class="question_part_label">f.</div>
</div>
<br><hr><br><div class="specification">
<p><strong>This question asks you to examine various polygons for which the numerical value of the area is the same as the numerical value of the perimeter. For example, a <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math> by <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn></math> rectangle has an area of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>18</mn></math> and a perimeter of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>18</mn></math>.</strong></p>
<p> </p>
<p>For each polygon in this question, let the numerical value of its area be <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> and let the numerical value of its perimeter be <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math>.</p>
</div>
<div class="specification">
<p>An <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>-sided regular polygon can be divided into <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> congruent isosceles triangles. Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> be the length of each of the two equal sides of one such isosceles triangle and let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> be the length of the third side. The included angle between the two equal sides has magnitude <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>2</mn><mi mathvariant="normal">π</mi></mrow><mi>n</mi></mfrac></math>.</p>
<p>Part of such an <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>-sided regular polygon is shown in the following diagram.</p>
<p style="text-align: center;"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAWAAAAE6CAYAAAAodIjdAAAgAElEQVR4Ae2d+dMdVZnH/QfmB3+eohxEZQrxB0fHpdApEUeQoaasmrJKqsYpHUeJAgMKQxxBAZWEfXFYlCVEQoRRDCBhkRjCEEkI2UPImxAgFchGNiALSdjO1PfCCZ37dvftvc/yOVVv3ff2cpbPOf3tp5/znL7vMyQIQAACEOiFwPt6KZVCIQABCEDAIMAMAghAAAI9EUCAewJPsRCAAAQQYMZAKoEXX3whdTsbIQCB5gggwM2xDCanm2++yRxxxOHmsMP++uDfZz7zafO1r/3Lwb8rr7zC2L+HHnrQzJs3b/C3cuXKYDjQEAi0TQABbpuwR/m/+uqr5tvf/ncjsT3vvHPND35w5sHaS1ityEpwrfjqMynMOjcp3Ml9yi95ns1PnyqbBIHYCCDAsfV4RnslghJPCbDEUIJ71FF/m3F0sc3KJymysqytAJ9//k8PEe6kaKvcpHDrWHve7373v4fkWawmHAUBNwkgwG72S6e1krhJ9CRuySRB7sOlIP+zFe5ha1s3CCvOxx//j4dY2/pu9+k4K9r6TLpJ8G8ne5n/+ySAAPdJv+eyJUQSLf2lCa11GfRczcLFW9HWp24mSQG2wqxP3WySFndyX9LalsWezBM3SeGu4MCCBBDggqBCO0wCJSGS4GQlWY0S55DTsJskKdq6ASXFOSnaejpI7kuel3STpN3YQuZJ28oRQIDL8fL+aAmOhEXiK+suL+lYiQ6W33hKElZrHQ+7SZLCPDwpmXST2CcMK942P33iJhnPPMQtCHCIvZrRJomGtdyKiqrEZNg3nJE9mwsQSIps3qRk0k2i/5OinnSTJK1t5U3yiwAC7Fd/Va6trCxZs7royyQdL0uN1B+B5KSkRNZazPrMm5S0N1uJd96kJG6S/voWAe6PfScly9LVBZg10TaqEro4ZYGR/COQdJPIUk4Kd9KiTlrbRx75YTN/PpZ0V72NAHdFuody5JvUxaVH1qIuh7RqypLCSkojE9Y2Pe184Qv/YD7wgcPMXXf9PqzGOdoaBNjRjqlTLYmtRFfiKxGum+xkUd18ON9tArrRarz8/ve/GyxFnzHjD25XOIDaIcABdGKyCbJU7Ux7UzPpuihDD0dLMozxf7koJMA2yQL+5Cf/zsyYMcNu4rMFAghwC1D7ylKPkLJ65etrMsmiJhytSaLu5SWf8HC0y6pVq8ynPvVJc889d7tX4UBqhAAH0JESSM1yt+mrTbtAA0BHE4wZxDPrxp2WxsZWDcbVvffem7abbTUJIMA1AfZ9usKSdPFIgOtMtI1qB+Foowj5u9+GqGW1YGxszHzuc8eY++77Y9YhbK9IAAGuCM6F0+xE2/CjYxt1IxytDar956l5giLupdWrVw8iJGbOvK//SgdUAwTYw87URaNJMf01NdFWBEObLo4i5XNM8wQU4VJ0oc2aNavNccd90dx//8zmKxJpjgiwZx0va1cuB1m/XSfC0bom3m55cllpLJW5ia9Zs8Z8+ctfGjdh125Nw80dAfakb+1Emy6Yvtb8E47myWApWE1Fy2hytWx65plnzAknHG8uvnhy2VM5fogAAjwExMWv8r/q8b/tibYibS/iLyySD8f0T0BjqurNXCJ80kn/ZC699JL+G+JxDRBgxztPVoqsXkUhuJB0E+hi0s+FtoZcB/Vh3cU1a9euNd/97nfN5ZdfFjKqVtuGALeKt3rm8svp8VAXiUvvYSAcrXqfunSmxlUTN9Ldu3eb731vgrniistdap43dUGAHewq+VrtRFubsb1Vmq4bg+pG8peA3A5yPzSVJMKnnvp9c9VVVzaVZTT5IMAOdbXEVpEGEjiJsKuJcDRXe6ZYvfRk1fRy9T179pjTTz/NXHPNNcUqwVEDAgiwIwMh+RId16zeYUQKgesjDG64HnwvT8A+wbQxxiTCZ55xhvmfX/6yfMUiPQMBdqDj5VdVdEHTVklbTSMcrS2y7edbZuFFldpIhM/64Q/NdddeW+X06M5BgHvsclkhehz08ZFeN4wyAfw9YqbodwlovHXRbxLh/zr7bHP5ZZeZvXv3wj+HAAKcAydr1/r16wcDWYN5woRTDh42adJFh3w/uCPlH/sSHVkkbTwOphTZ6CbC0RrF2UlmesLSeOsiSYQnnnOO+e8f/QgRzgGOAOfAGbVL4nv00UcdPGzu3McKCXCXL9E5WLmG/5HbRCJM8oeAJnerLryo0sq9e/YMBPi8c89FhDMAIsAZYIpsvv766waW8CuvvDI4fPr0243+spKdaOv6JTpZ9amz3U7m1MmDc7sjoJjfKsuO69ZQLoirr7rK/PQnP0GEU2AiwClQim6S2MoNIZeERHjixHMyT9UFIAvEl4m2zIYkdtRZyprIhn87IKC+amLhRdWq3nzTTebCCy5AhIcAIsBDQMp81btRJcByPcj/KyEeTvLv6lE9RLEiHG24t938rqgVjb++05RbbjE/u/BC89prr/VdFWfKR4BrdMWKFcsHAizLN+1F1XbFkQTYx4m2UWgIRxtFyI39cj248i6RW6dMMRf94heI8LtDAwGucY3YaAhZv8NJrga5HFwZ+MP1a+p7F2FNTdU1xnw076Bx6JIB8JupU83kSZMQYWMMAlzjqpQAD/t9NTmlSTbXXqJTo5m5pxKOloun950KO3Nx1eK0224zl15ySfQijACXvEQ08XbiiV8ZRDskY4CVjR7JZW24OOBLNrPw4YSjFUbV+YEyBlx+Qpl+++0Dn/DGDRs6Z+NKgQhwyZ6woWf6tEmPd7I0JL4S4ZgS4Wju9naXCy+qUvjt9Onm3B//2GzcuLFqFl6fhwDX7D752DTDrIkOl/xsNZtV6vQQIzxKAXDwYI1FGQQan66nO+64YxAnvGnTJter2nj9EOAaSPX4rUe80CfaRiEiHG0Uoe7397XwompL5z722CBOODYRRoArjBhZF7J4Y5loG4WIcLRRhLrfr6cS39xhixctMr/4+c/N5s2buwfWU4kIcEnwdqLN15folGxu4cNdnuwp3IhADpT168LCiyo4lyxePAhRi0WEEeASo0SP2jFOtBVBRDhaEUrdHKOnsz6XHddt5dIlS8zFkyebLRFYwghwgdFiX6Kjga1Zf9J4AoSjjWfSxxb7mlPfJ4SXLV06iBPesmVLHxg7KxMBHoFawiKrN6SX6IxocqXdhKNVwtb4SXKNhTJWly1bZq64/HLzUsAijABnXAKyIEJ9iU5Gk2tvJhytNsJaGdiFF75bv0kIy5cvN1ddeaV56qmnkpuD+R8BTulK+xgX6kt0UprcyCbC0RrBWDkT8ZcFHFpasWKFuezSS72IaS7LHgEeImZfouPzJMZQkzr7SjhaZ6jHFSSrV66yUOcoZAFfccXl5umn3V9YMq5zcjYgwO/C0cCN6SU6OWOi1i7C0Wrhq3yyDAdNEoectm59yegVAKtWrQqmmQiwMYOQHVkPeoQj1SNAOFo9flXPjsX/vnXrVvOrG24wY2NhiHDUAqzHNvsSnS5/rLDqRebDeYSjdd9LPi+8qEJLInzjjb82q8fGqpzu1DnRCrB9iQ4Tbc2OR8LRmuVZJDe5zmKbs9i2bZu55eabzZo1q4sgcvaYKAVY/jJeotPemIzlcbg9gsVzthE7xc8I50iJ8K23TjFr1qzxtlFRCbCsM16i0/5YJRytfca2BD3BhbLwwrapzOf27dvNb34z1TzzzDNlTnPm2GgE2L5ER+IQUqC6MyMpURHC0RIwWvzXuntiH88S4WnTbjNz585tkXY7WQcvwBqcEl1eotPOAMrKlXC0LDLNbdcEcogLL6oQ2rFju/ntb6ebxx9/vMrpvZ0TtADzEp3extVgGXdsE0Nd0pZhEfLCiyosd+zYYe684w4zf/68Kqf3ck6wAqxwKFlhMfvHehlR7xZKOFq79DWu5f8lHUrgwIED5oEH7jdPzJ9/6A5HvwUnwLIMNDA1E+/D72E5Oi5qV8v6J2tnRAapBGT9ErueisZIhB984AGzYMET6Qc4tDUoAbYhOcT2ujHCCEdrpx/k2lHsLymbgET4T396yCx88snsgxzYE4wA24k2/I4OjKp3q0A4Wjt9oRsb43w029dff93Mevhhs2jRotEH93SE9wKsR137Eh39T3KHgJ5IsNSa7Q8xlQCTihGQCM+e/Wej35pzMXktwLIC5Atjos3FofVOnZipb7ZvtJBIE5yk4gQkwnPmzDFLly41+t+l5KUA24k2JiJcGkrpdZE/nsfldDZlt9qJTY1/UjkCEt7/e/RRM/exx8wbDomwdwJsH8GYaCs3APs6WuKrviLVJ6BFF/Krk6oRkAhLgB//y1/MG2+8US2Ths/ySoDlapDVyyNYw6Ogxeys1dZiEVFkLY6sLqzf1RLhJxcsMPPnzXNChL0QYA0+XqJTf/D1lYMm4vR+CFJ1AjI+WHZcnd/wmSufeso88cQTvYuw8wLMS3SGh45/3wlHq9dndtmx3G+k5gg8vXLlwBru0x3hrABr0OmOL5cD1lNzg66PnKzfvo+yQyhTfvTQf++tr37S78stWriwN0vYSQFOvkSHGd++hmaz5RKOVp0nCy+qsyty5tjYmFm8eHEvIuycANuX6DDRVmTo+HOMIiHo0/L9pac/Fl6U51b2DP2qxrKlSzsXYWcEWJauHrM02HiJTtnh4/7xWeFo+plxze4fffRRZv369YO/E0/8ymCbvseedE0QR93NKFj7zDNm+fLlZv/+/d0UaIxxQoDlI9Qjqny+uBw66/tOC7JhVGmFvvLKK+aYYz5rJk48x5x88teNvpPM4G1nui64JrobDc+uXWsWLlxo9u7d20mhvQvwWWf9kIm2Trq6/0LywtEmTbrooBXcf03dqIEMEpbZd98Xzz777ODdEa+99lrrhfcqwHI1HH74B8yxx34Bt0PrXd1/AXnhaHPnPjZwO2D9vtNP9olBn6TuCWzcsMEsW7bMtC3CvQqwJmU0OaO7PL9e0f0g67rEvHA0uR7kB545876uq+VkebpZsfCi367Rj32uWLGiVRHuVYCTM+M29EyPqUzC9Tvw2iw9LRxN7gcJ74QJpxj9r8m36dNvb7MaTuctn684cR303036nTn1w759+1qpTK8CnDbIeN9DK/3sTKbJm66NgNCnkkRXT0KyhmNOejJk4YU7I2Dnzp1GCzbaEOHeBFh3laz4Ru2TJaxBiA/MnYHYRE2ywtGayDuUPHRdsOzYrd6UCGvBRtMi3JsA6y4/ysclP5isZAL43RqMdWpjJ5fq5BHyubpBZRkmIbfbh7a9/PLLRgs2mowT7k2A9ShaJMDcTtxgDfswRIvVMS8crVgO4R7Fwgu3+1ZROgpT27NnTyMV7U2A0yZjslqkSQms4Sw6/m3PC0fzrzXN1dguSGouR3Jqg4D0SCK8e/fu2tn3IsB5/t+8FmEN59HxZ5/tR39q3E1NbUhmN6VRSh0Cu3btMs8//3xtEe5FgIv4f7PgYA1nkfFre5knIL9aVq221jeu8U3yg4CWK7/wwgu13BG9CHBR/29eN1grSj4zBm0eKTf3JcPR3Kxht7XShPSoSelua0RpRQgcOHDAbNiwofK7I3oR4Kasn6Q1zEvbiwwXd44hHO29vtA4buqaeC9X/uuKgH5nbtOmTZVWzHUuwFX9v3kw7eSFrCqs4TxS7uyzj9zu1Ki/mmjxkZ7kSP4SkAhv3ry5dJxw5wJcx/+b1z0SXgmwLAms4TxS7uwjHO2dvmDhhTtjsk5NJMJbt24tJcKdC3AT/t88SBJfiTDWcB4lN/YRjmYGsfC6EZHCIKAf+Ny2bVvhxRqdC3AXvi6sYT8Gs51I9aO27dRS1m+RBUntlE6ubRCQCOslPkVe6t6pALfh/80DiDWcR8eNfV3ckN1o6fhacAMazySULRJhLV0e9T7hTgVYkw1dh9pgDbs9pOUqivVdH5p40zVBCpPAm2++abRgI+8FPp0KcJ/r3K01rBsAkRLuDPhYw9EUBSLrn7HozlhsoyZvvfXWwBWR9QKfTgVY73rVwOsrWWuYWee+emB8ubGGo7HwYvxYCHXL22+/PXBFaNHGcOpMgF3yd8nqkvWhWXgskOEh0f332MLRNOb6Nka67+W4S5QIywqWbziZOhPgPvy/yYYO/y/LSy4RrOFhMt1/jy0czbVrofsej7NEifDrBw4Y+YZt6kyA+/T/2samfWoCCGs4jUx321x6Ouqi1RpvajMpTgISYImxUmcC7PIjF9Zw/xeCRKnP+YGuCMj9JWOEFDcBTc4pdSLAvlg4WMP9XRSxhKOx8KK/MeZiyZ0IsHxe8vP5kJLWsBaOkLohEEM4mkIhJcAkCFgCnQiwHrk0+HxK1homUL6bXtONT26qkJOug1gXnYTcr3Xa1okA68LyMdzLWsMKk8IarjPMip0bcjiaxo/83D5eB8V6j6OqEGhdgOX/9f1tT7KCdRPBGq4yxIqf45Orqnir3jlSCy8YP2WphX986wIcykUlC0Y3Eqzh9i4KMQ7RR2rdK/okQSBJoHUB9tH/mwQ0/L9uKFjDw1Sa+x5iOJrGTNcvoWquR8ipTQKtC7Cv/t886FjDeXTq7ZNQhTRRJZ+vbioaMyQIDBNoVYBD8P8OA0t+xxpO0mjm/9DC0XQzYeFFM2MjxFxaFeBQ/L95HZ+0hvHx5ZEqts++qKbY0e4fJZ+2byGY7lMNp4atCnBo/t+8btfNRo+aIT0+57W3zX2hhKPJmg9xUrHNvo8t71YFOET/b94AsbP4uvFgDeeRyt8XyqSVxoFEmASBLAKtCXDo/t8soHqE1rJrrOEsQqO32xvZ6CPdPULjX2OAhRfu9pELNWtNgGN7x+twZ+oC1OMn1vAwmWLffY8cYOFFsX6O/ajWBDgUP16dAYI1XJ2ez+FoduEF1m/1/o/lzFYEWAMvNv9v3oDBGs6jk75PvlM9PfiY9PTHwgsfe677OrciwAq7kQVMeo9A0hpmYuY9Lln/+XoTV73lPmESNqtn2Z4k0IoAx+7/TQIe/t9aw3oBuS5WUjYBH91YiuDw1XLP7gn2tEWgFQH28cJpC3BavklrmCD9NELvbPMxHE0Tr7rJkiBQhEDjAuzro2MRWE0fI/HV4yrWcDpZ38LRWHiR3o9szSbQuADj/82GnbZHNywJsIQYa3g8IZ/C0fTkh39/fB+yJZtA4wIs/68eHUnlCGANp/PyJRzN+vbTW8FWCKQTaFyAZQXgA0uHPWor1vB4Qr6Eo+kpBsNjfP+xJZ9AowJs/b/5Rcazd+7cxwbx0IqJnjnzvkHDTz7564Nt06ffngkCa/g9NHZM6dPVpJAzuUpcrqOr7GKvV6MCLOEgBGf8kJow4RRz4olfMRMnnmNWrFg+/oCULbqYrW849icK16Nq5CZh4UXKIGbTSAKNCjD+33Tesn6TVnD6UelbrTUstrFaWC6Ho6lP1LcsvEgfv2zNJ9CoAOP/zYZdVYCVo7WGY40xdTkcTTcHPamQIFCFQGMCbC2BKpUI/ZxJky4yxxzzWaPPOkkve5evMUZr2NVwNNUrdhdRnTEd+7mNCTD+3/ShJPeDhPf6668b+IF1lP6vmvSoKz97bNawi+FoitDQUx8JAlUJNCbA+H8P7QIbAaGoB6X169cPfIWajNP/dVNs1rCL4Wi6CbLwou5Ijvv8xgQY/2/3Aykma9i6uPTpQmLhhQu94H8dGhFge3H4j8PPFsRiDbsUjiY3ED/A6uf14lKtGxFg/L/9d6m1hiVSihoIMbkSjibWLLwIcYR136ZGBFgTJCzD7L7z0kq01nCI/eFKOJrGu+Y8SBCoS6ARAY5tRr4u9LbPl1DJEg7RGu47HE3WLwsv2h7B8eRfW4DtgIwHmT8tlRUssQjJGu47HE0sVQcSBJogUFuAXQwPagJMKHmEZg33Od402czCi1CuDDfaUVuA8f+60ZGjahGKNWwjbvoIR+tT/Ef1L/v9JFBbgPH/+tPxoVjD8m3nvc6zrR7RWFfEDwkCTRGoJcD4f5vqhm7zsdawb3GsEj89cR1xxOHmpJNO7BSaypYAkyDQJIFaAswjWZNd0W1e1hrWggLdSF1NqptCvuR7lQDqpjF79uzOxVCcNN5JEGiSQC0B7ntGukkQseYla1ji5po1LOHV+FIUhz51w0imLsPRtOxY5fXhd062mf/DI1BLgGWRDF8Y4SEKv0X2vQYuWMMSuaTwZlnnXd78VZZuVCQINE2gsgDrwpBVQAqDgITPPur3ZQ2rXI0pCV6W8FraXbm/VA8WXljqfDZNoLIA6wLglwCa7o7+8+vDGpbI2ZV7Kr9I0g1Dwti2W0A3Jd0QSBBog0BlAe7yEbCNhpNnNoEurWHdyGX1VnnEl2i3GRYmDqobbrbsscKeegQqCzD+33rgfTi7bWtYN3EJXFGrd5iZXBZtWqfKX35xEgTaIlBJgPH/ttUd7uWbtIabsjaVp3U56P+qSZZpm7G5yrvqzaFqmzgvLgKVBBj/b1yDRK211rD8/nVE04qvLNc6+dgeaOtJTGO8TXG39eczbgKVBBj/b5yDRoJp3QZVrOGk+DZFsK2xKAtdIlwmLV261Bx77OfN+9//V4O/q6++sszpHBshgUoC3JbVESF/L5ss8ZXvtow13Ib4Cp7q0rSfVta+2lc2ffWr/2zWrVs3OO3MM/9zIMJTp04pmw3HR0SgtADj/41odOQ0VYIqAZZQjbKG2xJfVU95Nx2OpnaVjcqQ0MoCTqaPf/xjRqJMgkAWgdICjP83C2Wc24tYwxI0/bWVmnxPg114IWGvm775zX87RIAl0tY9MfyJUNel7ef5pQW4LZ+bn/iotQjkWcOyJOVPbULQsmg3GY6m8a2/JpIsYOsHlmvCiqz8xPp+zz0zzOGHH9ZEUeThKYHSAqxHTgLTPe3tlqs9bA1bX2rZ8bJlyxYza9aswrVV/lV8tsMF6CahfGQF1012Qi4tHwnzyy+/bGQR639SvARKCXBTAz1e3OG33FrDn/7035sjj/xw6UgCia+s5smTJ5mZM+8rDKyJiWGV25SrRO6HYZ+wbYzcD0qyjq1VbPfxGReBUgKsR72mBmhcmONrrazhj3zkQ4MX/JRxP6xYsXwgvhJgH/7SelbCKvdCWpLrwVq9CHAaobi2lRJgia9EmASBIgT0KK8JMlmnZVaUSYTLWL+qiwRfvuaqSZPLdc635UpU80LPJMzyASshwJZavJ+lBBj/b7wDpU7LddPW2NGbxcpYw2XKVL51wtF0kyi78GK4fhJXiWoyadujjz5ycJPE94ILfjr4rmM1CSerWHHDpPgIFBZg/L/xDY4mW1zVGi5Th6rhaHaZdZmyho+VmA6Hlum7dTfoeImxtllB1kSc9mtblr94uBy+h0WgsADj/w2r4/tqTZvWsPKuEkIm4S678KIvfpQbFoHCAoz/N6yO77M1bVnDVZ7SVBe5R9pyjfTJmbLdJ1BYgJuKj3QfCTXsioC1hpu0PsuGo8lilm+aBIE+CBQSYFkWGtgkCDRNwFrDikDQOKubJKhFBd1O3KkOJAj0QaCQAFf1rfXRIMr0k4DGmKIYiopnVivLhKOprCo+46yy2Q6BsgQKCbD8v3VDdMpWjOPjIyALWJZwHWvYWrWjfLraL7damfjk+HqEFrdNoJAA4/9tuxvIP0lAlmkda7hIOJoMCh3XREoLP6uzrYk6kYcfBEYKMP5fPzoytFrWsYaLuMyaWHgRGnPa0z2BkQJcZDB3X21KjIVAFWtY4q2ntqwkPzGTyll02N4lgZECjP+3y+6grDQCVazhvHA0uR5kWJAg0DeBkQKM/7fvLqJ8S0DWsMZjEfHMCkez1vGoSTpbJp8QaJNArgBrsPKo1iZ+8i5LwFrDsmLz4nezwtGyhLlsPTgeAk0QyBVg/L9NICaPpgnIetXqtTxrWMcMvx1Ngq1tecLddF3JDwJ5BHIFGP9vHjr29U3AvsUsyxrW9mT8Ogsv+u4xyh8mkCvA+H+HcfHdNQJ51nDyCU7HaTzLhUGCgCsEMgXYWheuVJR6QCCPgB2vSWvYTrjpPImx9pEg4BKBTAHmcc2lbqIuRQikWcM2HE2fmpgjQcAlApkCPOw/c6nS1AUCeQSS1vBpp506+CHZPqJ59IvHWpJsfwPOfufnh/J6L659mQLMbHFcAyG01lpr+EMf+qA54ojDD5mM67Kt+nl6/VyRPvVjnfb/LutAWe4SSBVga0G4W21qFjsBCazGqf7k37UuMz256U8GhP6OPPLD5hOf+HhvuPSjm7J49XtwSrKCJcIkCIhAqgDbwQwiCPRBQJNnElaFkGks6s8KqyIZrLjabXZxhYTYirIEuu9kf4TTCq5+hFMuCX2SICACqQKsgZ2MnwQVBJogoAUQViCtsCrWXONN7wC2wip/rbZpnz1OE2g616cwMv38fPJXkSXE1h/cBE/y8J9AqgDrQmC1kP+d22ULrLBad4BWqlkL1QqrrFe7TfslrrrR23O7rG8XZUls5fe1Sd8lwsltdh+fcRIYJ8C6GGSNkCAgAtYdIAvUWqNWRGWpWnHVmNF26w7QsVZYY7yZp7kbbEQELgiuLUtgnADrwuFXYi2ecD+Tk1hWWCWeVlytsFp3gLbb46zV6pM7INyepGU+ExgnwLrQCFj3uUvNQcvT90ksv3uB2kNgNIFxAizLx4UZ5NFVj+8IO4mVdAekTWJZd0DaJFaM7oD4Rgot9oXAIQKM/7efbku6A5jE6qcPKBUCfRA4RIDx/zbfBU1MYvFE0ny/kCMEXCBwiADj/y3eJUmr1U5O2UmstJjW5CSWjzGtxclwJAQgUJTAIQKM//cdbDZ8yk5iJWNa01Zi2ZhW11ZiFR0EHAcBCPRD4KAAx+D/zZvESotpTU5iWVFmEqufgUqpEAiRwEEB9tn/m3QH2Eks6w7Qo7+NaU2uxLJuAxvTKoElQQACEOiSwEEBdtX/ayexrDtAwqm66i9ptVAfexsAAAmgSURBVNptaSuxmMTqckhRFgQgUJTAQQHu2v9r3QGyPK01aq1WJrGKdh/HQQACPhMYCLBm5SV6TSXrL7VWa9okVtIdkDaJ1VRdyAcCEICAqwQGAiwB1N+oZN0BaSuxku6AtJVYVpSZxBpFmf0QgEAsBAYC/KUvfdFcc83Vg3cIVJ3E4sUssQwZ2gkBCDRFYCDAH/vYR81JJ514yOsEiWltCjH5QAACEEgnMBBgTYKdffZZ6UewFQIQgAAEWiEwEGD5ZT/4wb8xGzZsaKUQMoUABCAAgfEEBgKszbKAJ048Z/wRbIEABCAAgVYIHBTgF198cfAT3hs3YgW3QppMIQABCAwROCjA2i4L+Nxzfzx0CF8hAAEIQKANAocIsHzARx/9UbNx48Y2yiJPCEAAAhBIEDhEgLVdFvCvfnVD4hD+hQAEIACBNgiME2D5gD//+c+ZXbt2tVEeeUIAAhCAwLsExgmwtuulODfddCOQIAABCECgRQKpArxw4UJz3HFfNLt3726xaLKGAAQgEDeBVAEWkgkTJpgpU26Jmw6thwAEINAigUwBXrx4sTnhhOOxgluET9YQgEDcBDIFWFhOP/00M3Xq1LgJ0XoIQAACLRHIFeAlS5aYk0/+ektFky0EIACBuAnkCrDQKCLi4Yf/FDclWg8BCECgBQIjBXjZsqXmG9/41xaKJksIQAACcRMYKcDCIwGeNWtW3KRoPQQgAIGGCRQSYLkgvvWtbzZcNNlBAAIQiJtAIQEWIgnw7Nmz46ZF6yEAAQg0SKCwAMsF8d3vfKfBoskKAhCAQNwECguwMEmA58yZEzcxWg8BCECgIQKlBFguiAvOP7+hoskGAhCAQNwESgmwUJ36/e+bVU8/HTc1Wg8BCECgAQKlBfj+mTPNz3/2swaKJgsIQAACcRMoLcB79uwx//Htb5uxsbG4ydF6CEAAAjUJlBZglXfXXXeZyZMm1Sya0yEAAQjETaCSAO/du9dMOOUUs3r16rjp0XoIQAACNQhUEmCVd/eMGebSSy6pUTSnQgACEIibQGUBlhV82qmnmu3bt8dNkNZDAAIQqEigsgCrvHvvuYcXtlcEz2kQgAAEagmwrOAfnHmm2bFjByQhAAEIQKAkgVoCrLL0k0XTpk0rWSyHQwACEIBAbQGWD/j0007DCmYsQQACEChJoLYAqzxZwNOnTy9ZNIdDAAIQiJtAIwIsH/CZZ5xhdu7cGTdNWg8BCECgBIFGBFjlyQK+8847SxTNoRCAAATiJtCYAMsK/sl555l9+/bFTZTWQwACEChIoDEBVnmygP/Mj3cWRM9hEIBA7AQaFWD5gC+84AKs4NhHFe2HAAQKEWhUgFXijb/+tXnkkUcKFc5BEIAABGIm0LgAP/fcc2bSRRdhBcc8qmg7BCBQiEDjAqxSp9xyi3mUH+8s1AEcBAEIxEugFQFe9/zz5uLJk7GC4x1XtBwCEChAoBUBVrlTb73VPLlgQYEqcAgEIACBOAm0JsDr1q0z1117bZxUaTUEIACBAgRaE2CVffu0aeapp54qUA0OgQAEIBAfgVYFePny5eZXN9wQH1VaDAEIQKAAgVYFWOVLgFeuXFmgKhwCAQhAIC4CrQuwXBA33XRjXFRpLQQgAIECBFoXYNVBArxq1aoC1eEQCEAAAvEQ6ESA5YK4dcqUeKjSUghAAAIFCHQiwKqHBHjDhg0FqsQhEIAABOIg0JkAywVx990z4qBKKyEAAQgUINCZAKsut932G7Nx48YC1eIQCEAAAuET6FSAn3xygfnjH+8NnyothAAEIFCAQKcCvH//fnPrrVPMpk2bClSNQyAAAQiETaBTARbKRYsWmvvvnxk2VVoHAQhAoACBzgX4wIEDZtq028zmzZsLVI9DIAABCIRLoHMBFsolSxabRx6ZHS5VWgYBCECgAIFeBFhW8B13/Nbs2bOnQBU5BAIQgECYBHoRYKFctmypmT9/XphUaRUEIACBAgR6E+Ddu3eb6dNvxwou0EkcAgEIhEmgNwEWTlnATzzxRJhkaRUEIACBEQR6FWD5gO+88w6zd+/eEdVkNwQgAIHwCPQqwMIpC3jRwoXhkaVFEIAABEYQ6F2AZQX/4a67sIJHdBS7IQCB8Aj0LsBCKgt49dhYeHRpEQQgAIEcAk4IsHzAD9x/v3n99ddzqsouCEAAAmERcEKAhXTJ4sVmzZo1YdGlNRCAAARyCDgjwNu2bTMPPfigeQMrOKe72AUBCIREwBkBFtT58+aZtWvXhsSXtkAAAhDIJOCUAG/fvt38edYs88Ybb2RWmB0QgAAEQiHglAAL6pMLFpjnnnsuFL60AwIQgEAmAecEeMeOHWbOnDmZFWYHBCAAgVAIOCfAArto0SJe2B7KCKMdEIBAJgEnBXjnzp1m/vz5mZVmBwQgAIEQCDgpwAIrAd6yZUsIjGkDBCAAgVQCzgqwfjNOE3IkCEAAAqEScFaABVwCvHXr1lDZ0y4IQCByAk4LsFwQixcvjryLaD4EIBAqAacFWNAlwFqmTIIABCAQGgHnBVguiDFeVRnauKM9EICAMcZ5AVYvLVu2zOzatYsOgwAEIBAUAS8EWC4IXlUZ1LijMRCAgC8W8JtvvjmwgvVT9iQIQAACoRDwwgIW7E2bNplnn302FO60AwIQgIAfPmD1k6zglStXGv2IJwkCEIBACAS8sYAFW3HB69atC4E7bYAABCDgjwWsvpIVrJC0ffv20XUQgAAEvCfglQUs2i+99NLAH+w9eRoAAQhET8A7AZYVrJA0fsI++rELAAh4T8A7ARZxRUTwqkrvxx4NgED0BLwUYFm/+vVkfrwz+vELAAh4TcBLARZxWcC8qtLrsUflIRA9AW8FWFbw888/jxUc/RAGAAT8JeCtAAu5LGD9ijIJAhCAgI8EvBZg+YA3bNhg3nrrLR/ZU2cIQCByAl4LsPpOFjCvqox8FNN8CHhKwHsBlhWsH/B8++23Pe0Cqg0BCMRKwHsBVsdt376dl/TEOoJpNwQ8JhCEAB84cGAwIYcV7PFIpOoQiJBAEAKsftu5c6fZu3dvhF1IkyEAAV8JBCPAsoI1IYcV7OtQpN4QiI9AMAKsrnv11VfN/v374+tFWgwBCHhJICgB1uo4iTAJAhCAgA8EghJgAdcPd/KqSh+GHnWEAASCE2CJL78bx8CGAAR8IBCcAAu6BFgvbidBAAIQcJlAkAIsK5jfjXN52FE3CEBABIIUYDVMAsxLehjkEICAywSCFWC5IJiMc3noUTcIQCBYAVbXSoBZmMEghwAEXCXw/1bA2+wvz46GAAAAAElFTkSuQmCC"></p>
</div>
<div class="specification">
<p>Consider a <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>-sided regular polygon such that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mi>P</mi></math>.</p>
</div>
<div class="specification">
<p>The Maclaurin series for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tan</mi><mo> </mo><mi>x</mi></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>+</mo><mfrac><msup><mi>x</mi><mn>3</mn></msup><mn>3</mn></mfrac><mo>+</mo><mfrac><mrow><mn>2</mn><msup><mi>x</mi><mn>5</mn></msup></mrow><mn>15</mn></mfrac><mo>+</mo><mo>…</mo></math></p>
</div>
<div class="specification">
<p>Consider a right-angled triangle with side lengths <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>,</mo><mo> </mo><mi>b</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><msup><mi>b</mi><mn>2</mn></msup></msqrt></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>≥</mo><mi>b</mi></math>, such that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mi>P</mi></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the side length, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mo>></mo><mn>0</mn></math>, of a square such that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mi>P</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down, in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>, an expression for the area, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>A</mi><mi>T</mi></msub></math>, of one of these isosceles triangles.</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>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>2</mn><mi>x</mi><mo> </mo><mi>sin</mi><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use the results from parts (b) and (c) to show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mi>P</mi><mo>=</mo><mn>4</mn><mi>n</mi><mo> </mo><mi>tan</mi><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac></math>.</p>
<div class="marks">[7]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use the Maclaurin series for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>tan</mi><mo> </mo><mi>x</mi></math> to find <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi>lim</mi><mrow><mi>n</mi><mo>→</mo><mo>∞</mo></mrow></munder><mfenced><mrow><mn>4</mn><mi>n</mi><mo> </mo><mi>tan</mi><mfrac><mi mathvariant="normal">π</mi><mi>n</mi></mfrac></mrow></mfenced></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">e.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Interpret your answer to part (e)(i) geometrically.</p>
<div class="marks">[1]</div>
<div class="question_part_label">e.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mfrac><mn>8</mn><mrow><mi>b</mi><mo>-</mo><mn>4</mn></mrow></mfrac><mo>+</mo><mn>4</mn></math>.</p>
<div class="marks">[7]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By using the result of part (f) or otherwise, determine the three side lengths of the only two right-angled triangles for which <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>,</mo><mo> </mo><mi>b</mi><mo>,</mo><mo> </mo><mi>A</mi><mo>,</mo><mo> </mo><mi>P</mi><mo>∈</mo><mi mathvariant="normal">ℤ</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">g.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the area and perimeter of these two right-angled triangles.</p>
<div class="marks">[1]</div>
<div class="question_part_label">g.ii.</div>
</div>
<br><hr><br><div class="specification">
<p><strong>This question asks you to explore the behaviour and some key features of the function</strong> <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mi>n</mi></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><msup><mi>x</mi><mi>n</mi></msup><mo>(</mo><mi>a</mi><mo>-</mo><mi>x</mi><msup><mo>)</mo><mi>n</mi></msup><mo> </mo></math><strong>, where</strong> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>∈</mo><msup><mi mathvariant="normal">ℝ</mi><mo>+</mo></msup></math> <strong>and</strong> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>∈</mo><msup><mi mathvariant="normal">ℤ</mi><mo>+</mo></msup></math><strong>.</strong></p>
<p>In parts (a) and (b), <strong>only</strong> consider the case where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>2</mn></math>.</p>
<p>Consider <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mn>1</mn></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>x</mi><mo>(</mo><mn>2</mn><mo>-</mo><mi>x</mi><mo>)</mo></math>.</p>
</div>
<div class="specification">
<p>Consider <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mi>n</mi></msub><mfenced><mi>x</mi></mfenced><mo>=</mo><msup><mi>x</mi><mi>n</mi></msup><msup><mfenced><mrow><mn>2</mn><mo>-</mo><mi>x</mi></mrow></mfenced><mi>n</mi></msup></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>∈</mo><msup><mi mathvariant="normal">ℤ</mi><mo>+</mo></msup><mo>,</mo><mo> </mo><mi>n</mi><mo>></mo><mn>1</mn></math>.</p>
</div>
<div class="specification">
<p>Now consider <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mi>n</mi></msub><mfenced><mi>x</mi></mfenced><mo>=</mo><msup><mi>x</mi><mi>n</mi></msup><msup><mfenced><mrow><mi>a</mi><mo>-</mo><mi>x</mi></mrow></mfenced><mi>n</mi></msup></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>∈</mo><msup><mi mathvariant="normal">ℝ</mi><mo>+</mo></msup></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>∈</mo><msup><mi mathvariant="normal">ℤ</mi><mo>+</mo></msup><mo>,</mo><mo> </mo><mi>n</mi><mo>></mo><mn>1</mn></math>.</p>
</div>
<div class="specification">
<p>By using the result from part (f) and considering the sign of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><msub><mi>f</mi><mi>n</mi></msub><mo>'</mo></msup><mfenced><mrow><mo>-</mo><mn>1</mn></mrow></mfenced></math>, show that the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>0</mn><mo>,</mo><mo> </mo><mn>0</mn></mrow></mfenced></math> on the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><msub><mi>f</mi><mi>n</mi></msub><mfenced><mi>x</mi></mfenced></math> is</p>
</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>y</mi><mo>=</mo><msub><mi>f</mi><mn>1</mn></msub><mo>(</mo><mi>x</mi><mo>)</mo></math>, stating the values of any axes intercepts and the coordinates of any local maximum or minimum points.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use your graphic display calculator to explore the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><msub><mi>f</mi><mi>n</mi></msub><mo>(</mo><mi>x</mi><mo>)</mo></math> for</p>
<p>• the odd values <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>3</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>5</mn></math>;</p>
<p>• the even values <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>2</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>4</mn></math>.</p>
<p>Hence, copy and complete the following table.</p>
<p><img src="data:image/png;base64,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"></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>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><msub><mi>f</mi><mi>n</mi></msub><mo>'</mo></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><mi>n</mi><msup><mi>x</mi><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><mfenced><mrow><mi>a</mi><mo>-</mo><mn>2</mn><mi>x</mi></mrow></mfenced><msup><mfenced><mrow><mi>a</mi><mo>-</mo><mi>x</mi></mrow></mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup></math>.</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>State the three solutions to the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><msub><mi>f</mi><mi>n</mi></msub><mo>'</mo></msup><mfenced><mi>x</mi></mfenced><mo>=</mo><mn>0</mn></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mfrac><mi>a</mi><mn>2</mn></mfrac><mo>,</mo><mo> </mo><msub><mi>f</mi><mi>n</mi></msub><mfenced><mfrac><mi>a</mi><mn>2</mn></mfrac></mfenced></mrow></mfenced></math> on the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><msub><mi>f</mi><mi>n</mi></msub><mfenced><mi>x</mi></mfenced></math> is always above the horizontal axis.</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>Hence, or otherwise, show that <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><msub><mi>f</mi><mi>n</mi></msub><mo>'</mo></msup><mfenced><mfrac><mi>a</mi><mn>4</mn></mfrac></mfenced><mo>></mo><mn>0</mn></math>, for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>∈</mo><msup><mi mathvariant="normal">ℤ</mi><mo>+</mo></msup></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>a local minimum point for even values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>></mo><mn>1</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>∈</mo><msup><mi mathvariant="normal">ℝ</mi><mo>+</mo></msup></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">g.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>a point of inflexion with zero gradient for odd values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>></mo><mn>1</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>∈</mo><msup><mi mathvariant="normal">ℝ</mi><mo>+</mo></msup></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">g.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Consider the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><msup><mi>x</mi><mi>n</mi></msup><msup><mfenced><mrow><mi>a</mi><mo>-</mo><mi>x</mi></mrow></mfenced><mi>n</mi></msup><mo>-</mo><mi>k</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>∈</mo><msup><mi mathvariant="normal">ℤ</mi><mo>+</mo></msup></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>∈</mo><msup><mi mathvariant="normal">ℝ</mi><mo>+</mo></msup></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
<p>State the conditions on <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> such that the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mi>n</mi></msup><msup><mfenced><mrow><mi>a</mi><mo>-</mo><mi>x</mi></mrow></mfenced><mi>n</mi></msup><mo>=</mo><mi>k</mi></math> has four solutions for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">h.</div>
</div>
<br><hr><br><div class="specification">
<p><strong>In this question you will be exploring the strategies required to solve a system of linear differential equations.</strong></p>
<p> </p>
<p>Consider the system of linear differential equations of the form:</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>x</mi><mo>-</mo><mi>y</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mi>a</mi><mi>x</mi><mo>+</mo><mi>y</mi></math>,</p>
<p>where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>,</mo><mo> </mo><mi>y</mi><mo>,</mo><mo> </mo><mi>t</mi><mo>∈</mo><msup><mi mathvariant="normal">ℝ</mi><mo>+</mo></msup></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> is a parameter.</p>
<p>First consider the case where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>0</mn></math>.</p>
</div>
<div class="specification">
<p>Now consider the case where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mo>-</mo><mn>1</mn></math>.</p>
</div>
<div class="specification">
<p>Now consider the case where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mo>-</mo><mn>4</mn></math>.</p>
</div>
<div class="specification">
<p>From previous cases, we might conjecture that a solution to this differential equation is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>F</mi><msup><mtext>e</mtext><mrow><mi>λ</mi><mi>t</mi></mrow></msup></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>F</mi></math> is a constant.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By solving the differential equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mi>y</mi></math>, show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>A</mi><msup><mtext>e</mtext><mi>t</mi></msup></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> is a constant.</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>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>-</mo><mi>x</mi><mo>=</mo><mo>-</mo><mi>A</mi><msup><mtext>e</mtext><mi>t</mi></msup></math>.</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>Solve the differential equation in part (a)(ii) to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> as a function of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>.</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>By differentiating <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><mi>x</mi><mo>+</mo><mi>y</mi></math> with respect to <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>, show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>y</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mn>2</mn><mfrac><mstyle displaystyle="true"><mo>d</mo><mi>y</mi></mstyle><mstyle displaystyle="true"><mo>d</mo><mi>t</mi></mstyle></mfrac></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By substituting <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi><mo>=</mo><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></math>, show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi><mo>=</mo><mi>B</mi><msup><mtext>e</mtext><mrow><mn>2</mn><mi>t</mi></mrow></msup></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi></math> is a constant.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> as a function of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mo>-</mo><mfrac><mi>B</mi><mn>2</mn></mfrac><msup><mtext>e</mtext><mrow><mn>2</mn><mi>t</mi></mrow></msup><mo>+</mo><mi>C</mi></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math> is a constant.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.iv.</div>
</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><msup><mo>d</mo><mn>2</mn></msup><mi>y</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>-</mo><mn>2</mn><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>-</mo><mn>3</mn><mi>y</mi><mo>=</mo><mn>0</mn></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 two values for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi></math> that satisfy <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>y</mi></mrow><mrow><mo>d</mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>-</mo><mn>2</mn><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>-</mo><mn>3</mn><mi>y</mi><mo>=</mo><mn>0</mn></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Let the two values found in part (c)(ii) be <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>λ</mi><mn>1</mn></msub></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>λ</mi><mn>2</mn></msub></math>.</p>
<p>Verify that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>F</mi><msup><mtext>e</mtext><mrow><msub><mi>λ</mi><mn>1</mn></msub><mi>t</mi></mrow></msup><mo>+</mo><mi>G</mi><msup><mtext>e</mtext><mrow><msub><mi>λ</mi><mn>2</mn></msub><mi>t</mi></mrow></msup></math> is a solution to the differential equation in (c)(i),where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi></math> is a constant.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.iii.</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 defined by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = {\text{arcsin}}\left( {2x} \right)">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mtext>arcsin</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - \frac{1}{2} \leqslant x \leqslant \frac{1}{2}">
<mo>−<!-- − --></mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mo>⩽<!-- ⩽ --></mo>
<mi>x</mi>
<mo>⩽<!-- ⩽ --></mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By finding a suitable number of derivatives of <span class="mjpage"><math alttext="f" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>f</mi> </math></span>, find the first two non-zero terms in the Maclaurin series for <span class="mjpage"><math alttext="f" xmlns="http://www.w3.org/1998/Math/MathML"> <mi>f</mi> </math></span>.</p>
<div class="marks">[8]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence or otherwise, find <span class="mjpage"><math alttext="\mathop {{\text{lim}}}\limits_{x \to 0} \frac{{{\text{arcsin}}\left( {2x} \right) - 2x}}{{{{\left( {2x} \right)}^3}}}" xmlns="http://www.w3.org/1998/Math/MathML"> <munder> <mrow> <mrow> <mtext>lim</mtext> </mrow> </mrow> <mrow> <mi>x</mi> <mo stretchy="false">→</mo> <mn>0</mn> </mrow> </munder> <mo></mo> <mfrac> <mrow> <mrow> <mtext>arcsin</mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mn>2</mn> <mi>x</mi> </mrow> <mrow> <mrow> <msup> <mrow> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mn>3</mn> </msup> </mrow> </mrow> </mfrac> </math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>Using L’Hôpital’s rule, find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mathop {{\text{lim}}}\limits_{x \to 0} \left( {\frac{{{\text{tan}}\,3x - 3\,{\text{tan}}\,x}}{{{\text{sin}}\,3x - 3\,{\text{sin}}\,x}}} \right)">
<munder>
<mrow>
<mrow>
<mtext>lim</mtext>
</mrow>
</mrow>
<mrow>
<mi>x</mi>
<mo stretchy="false">→</mo>
<mn>0</mn>
</mrow>
</munder>
<mo></mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mrow>
<mtext>tan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>3</mn>
<mi>x</mi>
<mo>−</mo>
<mn>3</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>tan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mrow>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>3</mn>
<mi>x</mi>
<mo>−</mo>
<mn>3</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
</div>
<br><hr><br><div class="specification">
<p>A simple model to predict the population of the world is set up as follows. At time <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> years the population of the world is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>, which can be assumed to be a continuous variable. The rate of increase of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> due to births is 0.056<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> and the rate of decrease of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> due to deaths is 0.035<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</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="\frac{{{\text{d}}x}}{{{\text{d}}t}} = 0.021x">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mn>0.021</mn>
<mi>x</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>Find a prediction for the number of years it will take for the population of the world to double.</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>Use l’Hôpital’s rule to determine the value of</p>
<p><span class="mjpage mjpage__block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" alttext="\mathop {\lim }\limits_{x \to 0} \frac{{{{\sin }^2}x}}{{x\ln (1 + x)}}.">
<munder>
<mrow>
<mo form="prefix">lim</mo>
</mrow>
<mrow>
<mi>x</mi>
<mo stretchy="false">→</mo>
<mn>0</mn>
</mrow>
</munder>
<mo></mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mrow>
<mi>sin</mi>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mi>x</mi>
</mrow>
<mrow>
<mi>x</mi>
<mi>ln</mi>
<mo></mo>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>+</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</mrow>
</mfrac>
<mo>.</mo>
</math></span></p>
</div>
<br><hr><br><div class="question">
<p>Use l’Hôpital’s rule to find</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mi>lim</mi><mrow><mi>x</mi><mo>→</mo><mn>1</mn></mrow></munder><mfrac><mrow><mi>cos</mi><mo> </mo><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow></mfenced><mo>-</mo><mn>1</mn></mrow><mrow><msup><mtext>e</mtext><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></msup><mo>-</mo><mi>x</mi></mrow></mfrac></math>.</p>
</div>
<br><hr><br><div class="specification">
<p>Consider the differential equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}} + \left( {\frac{{2x}}{{1 + {x^2}}}} \right)y = {x^2}">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mn>2</mn>
<mi>x</mi>
</mrow>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mi>y</mi>
<mo>=</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</math></span>, given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 2">
<mi>y</mi>
<mo>=</mo>
<mn>2</mn>
</math></span> when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 0">
<mi>x</mi>
<mo>=</mo>
<mn>0</mn>
</math></span>.</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="1 + {x^2}">
<mn>1</mn>
<mo>+</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</math></span> is an integrating factor for this differential equation.</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>Hence solve this differential equation. Give the answer in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f(x)">
<mi>y</mi>
<mo>=</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br>