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<h2>HL Paper 3</h2><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><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>Consider the functions <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g">
<mi>g</mi>
</math></span> : <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\mathbb{R} \times \mathbb{R} \to \mathbb{R} \times \mathbb{R}">
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
<mo>×<!-- × --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
<mo stretchy="false">→<!-- → --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
<mo>×<!-- × --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
</math></span> defined by</p>
<p style="text-align: center;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( {\left( {x{\text{,}}\,\,y} \right)} \right) = \left( {x + y,\,\,x - y} \right)">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>y</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mi>y</mi>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>−<!-- − --></mo>
<mi>y</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g\left( {\left( {x{\text{,}}\,\,y} \right)} \right) = \left( {xy,\,\,x + y} \right)">
<mi>g</mi>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>y</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mi>y</mi>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>+</mo>
<mi>y</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {f \circ g} \right)\left( {\left( {x{\text{,}}\,\,y} \right)} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mi>f</mi>
<mo>∘</mo>
<mi>g</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>y</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {g \circ f} \right)\left( {\left( {x{\text{,}}\,\,y} \right)} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mi>g</mi>
<mo>∘</mo>
<mi>f</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>y</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State with a reason whether or not <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g">
<mi>g</mi>
</math></span> commute.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the inverse of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</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 src="data:image/png;base64,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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><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,iVBORw0KGgoAAAANSUhEUgAAAqAAAAB3CAYAAADcmPJ7AAAgAElEQVR4Ae2di5Ekt7FF5ccLOaCgAwpaIFogWiBaIFpAeiBaIFogWiBaIHqwHqwH8+IM4yxzsdn/6mpU1UVEEwUUkJ+bF1k51cPZP72lBYEgEASCQBAIAkEgCASBFRH404q6oioIBIEgEASCQBAIAkEgCLx9UYD++c//95ZPMAgHwoFwIBwIB8KBcCAcWIoDY83dFqDjooyDQEUAMqYFgbUQCN/WQjp6QCB8Cw+CwPIIdOcqBejyOO9eYkek3TsdB1+GQPj2MugPqTh8O2TY4/STEejOVQrQJ4O+R/EdkfboZ3yaA4HwbY44HMWK8O0okY6fayLQnasUoGtGYCe6OiLtxLW4MSEC4duEQdmxSeHbjoMb116GQHeuUoC+LBzbVdwRabvexPLZEQjfZo/QvuwL3/YVz3gzBwLduUoBOkdsNmVFR6RNORBjN4VA+LapcG3e2PBt8yGMAxMi0J2rFKATBmp2kzoizW5z7NsuAuHbdmO3RcvDty1GLTbPjkB3rlKAzh61Ce3riDShmTFpJwiEbzsJ5EbcCN82EqiYuSkEunOVAnRTIZzD2I5Ic1gWK/aIQPi2x6jO61P4Nm9sYtl2EejOVQrQ7cbzZZZ3RBqN+eqrv7z/Qecff/zhs1s//fSv9/nvvvvHZ/PPGKAbW0cbnqHrFpnahW1ff/3XL7b+9tv/3u0Gw7WbccOGWVr49lgkwrfb8LvEN88n63755T+fCSevMU+ee3YzrlvLb0viMmO+Gv0zTvCiy/fj+r2Ou3OVAnSv0X6iXx2RRnUmBvqPHz9+un30ArQ+vMDxm2/+9gkbL1wDdms347bVAjR8+5wxcgmuhW+fY3NqdCm/VUzH83v0ArRic4pvp3C/Z/7efMVzaI2XIGvjcQ+Ga+3pzlUK0LXQ35GejkijeyYG1taDfvQClDcmYAI+p5pJ69yaU3sfnTdu2DBLC9/uj0T4djt2l/jm+WQdn/q28+gF6DV8uz0iy+7wGTT+8LCslt+lbQGPZ/jdyezOVQrQDqnMnUWgI9K4wULG/tdf//u+xMNvUcrbURN5LXrc55zj77//5/tbQ/aQQNj/88///iTj22///skUv/pgD/PqGb+yQgeyvI9t9a2t93y4YEu9/0nh29u7La5HHno/fPjwvsRkpB76+vBSjg849NTGWnFgL36NbVwz+jre5ysh8LMpX9ydf2WPr5eadtuf4psxYJ1txNux8VGmnAV37nlfOeHbl7/uUrECL85D5ZbY1jnxfFWPneea/MB2Ppwh84E5wnM9O9+wWy5rO/nrVCNXPJrfxAQ5nhm5IY7qH/PVmPNG/mgbcmvOH+Ohz/TEkxxd1zMPHqM92mW/BB7K2nsPpmNLAToikvFFBDoijZtMDBx81pvUHPswrwmwPoTc75xjZNUPyb+OuTbZ1OQ2ruEejcTTydZe1pjUlMH6rtVC2LX0rEePibfe09Yqrz7gnB8f5Mqodoqt9+z1tdPvGotksRB39b+yx8ZLTbvFQFwcyzcxqDEc8XYsNrXv+GaxG779fjbl2yk8wNBm3LbEN/mB7fqozxZxnuvZ+VbzrzynEOvaUvlNTNRXe88t+q/JeSN/xlxdZXNOO93E89Q+88Yz8ehk73GOWIwtBeiISMYXEeiING6qicGkTAIbC4KaAOtDqO5HtmOTo3KwxWTvT7AmDR8OPPAssJxDHs0kV3+yNhlZWDhWd7Wz+m1xoix8c682mQDVX/d7XR9wzDnGVzB0Tkz0fxxjP3tO6QIT7vPRJ2U41qZX9th3qVW7z/Gtw198xckxesVWmazhPrE13q6RW+Hb+R/QxJmY1rhdivFa9y/xTX7oB/HmmvMkT+TE7Hyr+Veb8a9r8v3R/CYm4Mw1rc6Rt8SYNdfkPG0239LjW3dO8RO5rLHJQ+1x/ly/FB7ndOzpXneuUoDuKcIr+dIRaVTtgSYxmEw4sB5+C7KaAE0iyKr769gEURMWMmgWAMp2TG+rRRf7TFj4NH5MyK4xESqr9vqI3bWxB7n4TtPucV3dM8oSMwtg19bi2T3oEg/XjT148DGBskfsR9zHva8YY9+lVu0Wi45vHf6uNyaOK5ZyqT605IU8cQ297eh8w3/wEBswFWcwqnETs1f3l/gmP/TD80neebQA9eyK17P5hj785UOsTrXRZ9fdk988g9U35JmPkCmm53Iee0b+jGeSNb6Y8Jwqu+p3TiyIYz3H+mu/JB7K3HvfnasUoHuP+hP864g0qhkTgwnVeYvEmgA51DbXOeeY5EUziTFvU4eyHddEcktB4D6TmrrVV/slE9Ioy+R4Lhm7h9j4EKv2cS1myEFmh704i/so4xXjJfkmBpU345viDku5JLfAQV6AJc018oa5I/NNPMCMoqLDfot8kx+VQ3JBf+RE5/NMfKs54FTegMedz8w/UoD6QzlyaM8qQOEfOcSY0DOuBSj68RHOur5b87uly+KhzL33XR5PAbr3qD/Bv45IoxoTMYeaRnIzwbC/PshdOyYI1rnfNSR0WpfY6wOPNY7Rq5w6xxrfIlZ73hWU//hwUXe59dmlNnZfUTnX2f2ZkCbZ+8ACDxI+DX/U55xjfBzXUAj5JsD7JuIOZ/F6F/Ti/2Dfpabv2n2KbxVLrmk+cJBBQwY6+fhQljeVJ/JC3romfPv9bZox8dyMOIO1a4zbewBe/J9LfJMf8gVzK6/YLyfq/Ix8g98j10/Bb6zMZez1DDh3TX5zTcWpzoFTxc38Bu7a4Jxj+aM94o8v8s45dbH2VKs/OCp7XKtufb8Xj1HuXsfduUoButdoP9GvjkijOg9nPbwkAJNd9yD3Xu3drzySB80kwrzNAkDZjqs8r01GyHdu7NVtUlO3+sa++ldlYSMJjdbZPcrRpuqbxWOVy3VNoqf0iweJctwvrj4cHev7aNsrxth8qXV2VzzEgIeEazss0CP+3Gc9TS4phzl5IZdcM8pl7Joqe1wn5srdOt/qD5z6CvZ8bMZC351/ZY+t55oxrH6wHm7op/GenW/Yp81y/ZTv9Ty5hx4cbslv5sAqw+v6Lc81OW/kj2dH/GtcnBv1M+64ik3Mn2pL4XFK/t7mwXNsKUBHRDK+iEBHpHHTmBi8b4KoD3ISuvPs42CP+x37UDaJMG+zAFC2YwqvmsyYrw1Z6sc3rtXDOu/Vubq/XvOTueuRhV6TM+s6u+t+rk894CouyPYn77r/3BoeMBUH1ooR1zRx3lJBcM5uYyEnWEuM9JP7+A6eckn8mfOhLE5VjrLFzjXh2zuV3nnsg52+nrMt/8AjP+TL795+/hc15MTsfLulANUXeX9vfjMHIsczgyzOludNTMHRs8qaMed5z3ylbRV/5LK3ztU8yF5ytOtYqz01d2tT7ZfI91Xenq/BdGwpQEdEMr6IQEeki5uyIAjciUD4didw2XYXAuHbXbBdvakWoFdvysLNI9CdqxSgmw/r+g50RFrfimg8CgLh21EiPYef4dtz45AC9Ln4ziq9O1cpQGeN1sR2dUSa2NyYtnEEwreNB3Bj5odvzw1YCtDn4jur9O5cpQCdNVoT29URaWJzY9rGEQjfNh7AjZkfvm0sYDF3Ewh05yoF6CZCN5eRHZHmsjDW7AmB8G1P0Zzfl/Bt/hjFwu0h0J2rFKDbi+PLLe6I9HKjYsBuEQjfdhvaKR0L36YMS4zaOALduUoBuvGgvsL8jkivsCM6j4FA+HaMOM/iZfg2SyRix54Q6M5VCtA9RXglXzoiraT6ohr/llv9m28XN+1kwV59fxbf7sHr1N+A3AmFzrpxFN8v8e1eHMa/w8vftLyHg2eDVP4g/ow5cPy7nZ0v3d/y7NZl7g8EiDW8BTvavRz9Q+LyV925SgG6PM67l9gRaRann5HQZ/Htkh33+s7/lcofZh7/CPQlfWvdfxbf7sFrxsS+Vhwe8Z0/OM5nC+0S3+7FwSIB+XzA4x4OXsLwGTIv6bz2fleAjtxIAXotmn+sk1trFKDogmO3tu5cpQC9FcWsf0+egWEfCPgwJTkcrQDdRwTn98KCaC8F6L2I4z/njB/2bGLDQ/2ITf8rN1KA3s6EsQC9XcJ1Ox7RkwL0Ooyz6gICHZHqllrUQFh/6vWnJr56QgafMfHWe9wnWSOP5j3lMOc/9cc/iUYzoSnXMb3//Br2IJOP+0l6/rNrFGLap25k64dzjrHLpEnPfuxRRn3gvBtZ/lP/Jp4PKP0eC8KKJWvQW5u+6rvymPce+7imVT+11b1i7XyNQ9W5xjU2nGvEQzsrRvpZfdE/5ImJc5fwYo+6iH0do/8W3VXXu6C3t0//JKh2Iw+5cGr0gdhVPsOjU01uIs9r7PfMuA/flKneKnf0vfIH2Z4l/9lN5Oqn8RE35GqL95AxQ8Oec20JHNCBnJGD6CUu5hbWVVzMK2BsU4b5wHHddym25iH0ss+YwAfzovrs5aR6WOc+/6lV7UUOTb+w5xQ35AVy9WXEQRvs3aN+e+ZtVR/4Vf7rP+uVxXraJeyUX/vufOq7Z0o9+sh99l1zNipHkGMsuNZmMECmrdrEPWKCbzbtwe+aB4yvGIktfd2vnFM968eWN6AjIhlfRKAjUt0EKStJ67UPqTpnsqoJot432XKA3I8O13OAbR5mD43jKo9r5JgQvGeSRI9z9YC53jnHrrXXRsf02qOd9t2hdp/JhLUmGO/Z1zX6qi7xcW3tWVP99B7zp/YZB21fq8e2c+1evt2KFzaoy8TuWPxq3/Fg5Do22MCe/c45rjK9HmVrj7Jq74PFvbX3YUjxcIrPPqj1VV0df5TtmpFLzLPvlC58fnXDh3NtCRzQgZyRgxZs4mgPjjaLA7Ayf9SzOcq8JrbKUV/tzYvqt4fLrPN+td04mrccG3d877iBbPnq2mqL50cb7N1T13JtftSO8b787/zH5muw04baj/bgi/6os1tz7mx4DivOoz/6O3IU20Z97MUmdJ66r3xw7zBCz7UNWWNLAToikvFFBDoi1U2Sn3UmHpMihOc+pPch6poqg2sPGntsznGYmEeGB4g16lGmY9dV27hHYy22qgd5jPnUA8b9OufYBKwc1qjfh4W69MO+HmoTU53j4FebTULMqV9d+urYBM86/RjtqbLFUTn1oae9r+jB81yrPui7Puh7xzfXuOcWvJBLu1e3urDBhh346pxj5nzwembgPz4xz30+xlh59j545Cnz8sA5sWA88gCdNH3Vd9apW5s7e5Qtn5TDXnVp6ww9dp1r2n8rDl3MxUYOGl/PuQWQurCrzhlb+cH9Uabjc7GtOUdb5F/VPeLCPe/XIk9e6Q8201hbuaptcoM1+iTH4YhytG20o46rHfiFbnSiWzt8jmhn9V8sWat957CrurmuZwC5NLHEDuf0UxtOnd/Rd9Yjx33g4xwyaSNH9df7rBEnMdUeemR2uOtHlfOu8Ir/YPPYUoCOiGR8EYGOSHWT5GcdJKaZfCtxJbwHgHUceta6HhkmOHWYFLjnYR7vKdO1JjjsYV/da/JRT11Tk8KYPB1rg3KQPfqNHV1zT8WFdSYdEoeH3oSjHBMIPU1f9V0Mq25lOdfFqiZQfEEvskze6l+rx4ZzrfNB3yuuI98ewUuu3Ktb+4wD/o2xcdz5wH7aKa5WvPRbnnLPBxI8o8llH77MEW+w58O1vur7Kd3KYj1NnLWZOR+YyueevH3f9ML/XMu3W3HoYi42+F7xFJfa19jIDe5XXCve4mk86v4xtuYhfUKOeaDOjWHRfvbDJdbCN3p1yDH2Mo/N57ghX7W/82m0w7EYo0O+61vF0mt9c021tdp7Djt12xubem65h+xql35yFseGDHyp50Q8xFD/2Ot5Vud4Visu+m7vc0V71INc9TtHzz71jHafG7NvbClAR0QyvohAR6S6SfKz7lwhNhLeQ0JS40CZFEwS6nAf8tlTmwnRA+PYdTXJmwRHPd0adHjw3efYRDDKYU/1qdrptXvGxGeyAgcPvYnCvc8qQJEPBtiu/WCNvzxU1m7P4pvckCv6yrxN7J2T23LS8SNcP6VL3TXZy31tPsVVZdK7pz7ofGDJO7l87kGrr/p+SreyPCfiDL61YQNzPuTA0B+m6rq1r6/l2604dPwSG+JZ8cSG8WOeAQ/PPmsqP7hXZTI2Hudiax7SJ/aN8WZubPLIGNLLW22Uq+zVlnPckK913+jTaAdj8QWTipW+jXg6BnfXjFhq7znsRlv0f5RlTtc2/XSMHPIr6/jgD7a5Tjy0qe4zDuocY1ex0W9794x6sGfE/ZRvIwbdGH1jSwE6IpLxRQQ6ItVNkp91HCCaBwBC20bCjwdL8jNvU46HGR01ObjHw+qYfbSa5E2CJp+qR1uU48FDn/tcYyLo5Ghv9Vtf6N2DXHXVOXzjw30+JBoaNqjfOX1VTqdbP7Sn4mGsqn1ei7eynV+jx+9z7V6+3YOXuuSKY2wUvw73kevGAVzZx8c1xsY1zNtcYxzYJzewpWvuodfGWjCwp45dIz7YSNNXfT+lW15qj3jQn2o+QJV9at0a89fyTVtvxcH44osYG89rzpn5Af3d+lHmNbGtMsV4jLfztcd34w1u+FHzFXM1P7v2HDfkq5h0OFUbuJZj6MOX2nwTi2711vtc6z+6a7sGu7qe6+q/tuALtlX79NM17HWddmCvmIlHtYk9xEBZdR+62EvzfHGf9V1ThnpYM3LpFE6dvHEOe8aWAnREJOOLCHREqptMXKyT7CaImnxHwptM2eeHA+QhqgcbHR5WH5DYMB4Yxz78sEfZJiMPlXqQo22urb37WM+8CaST0/ldsXJPle81icZm0vGePXba9NUE0ukWM+NQ8UAm90/p4r6+q3ONHr3nGjaJxy18uwcvdckVx+i/Rfe5uBsbY1VjLC+NcY3fqdi4R4xqb3FQz1a9z7U/4Oirvp/S7bnQHnmILO6d06Xv5+L97HvYea49ikP1ceSgMR9jYI4Dc/MkayuW4j3KrGtGucZWPhpb/B/9PIVJzRfooskB7Xav89o6coN18lWOMzf6pDz6ysPRP8bcd/94n3ma/tezxvw12L0LGP6jD+rDb31HF801jpmzUHQfvfvEQ1vrGq+1v4udvHGtvXK1xzH2iJtzo27j/e7Qhf+gb2wpQEdEMr6IQEekuknys47DTzPReOCZGwnPPg8JPWR3DUT32sOADJOfX92NB8Yx+mk1WZkEPVQcdBv31Mc8Ok0E7nPMflonp/NbHXUPulwLbtgtdq6vNrBGn72vr+KjPOZt3FO+c65jftzLHB9w1k/3rdWj/1wjHtopZvpUfTee+ngPXuqSK47Rf4tu/NFG8XWszcYKu22jDx2fXWtf93jN+bL4cB2+eJ6wibU15vqq76d0ey5YT/NrRWSil4bcqot7+D9Dw5Zz7V4cxviiY+Qgc8RdDLEFnPzVF7/WZs428qSTeSm2Xe4a/VTf2KtfXnDf2I45Sr/OcUOOItfW+eS9ykPwGj+eS7HzfrVN/9E9tkvYjesZEy8xgPPI0HfPlH46Vk61E7+1rcacs6s85LhH+7vYVZvAgP0VY+2pcx3u+oUM46jt53rWjy0F6IhIxhcR6Ih0cVMWtAiYXEwc7aKDT4ZvjxHAB8v4oHtM6n53h2/7je0anln8wSN/qKpvUm8p2tawdy0d3blKAboW+jvS0xFpR+6t6koK0Mtwh2+XMTq3IgXoOXS+vBe+fYlJZm5DgLee8Gj8+A3AbdL2sbo7VylA9xHbVb3oiLSqATtSlgL0cjDDt8sYnVuRAvQcOl/eC9++xCQztyEwft0Np/jq2l+luE3aPlZ35yoF6D5iu6oXHZFWNSDKDoVA+HaocL/c2fDt5SGIATtEoDtXKUB3GOhnu9QR6dk6I/+4CIRvx439KzwP316BenTuHYHuXKUA3XvUn+BfR6QnqInIIPCOQPgWIqyJQPi2JtrRdRQEunOVAvQo0V/Qz45IC4qPqCDwGQLh22dwZPBkBMK3JwMc8YdEoDtXKUAPSYXHnO6I9JjE7A4CpxEI305jkzvLIxC+LY9pJAaB7lylAA0vbkagI9LNQrIhCFyJQPh2JVBZtggC4dsiMEZIEPgMge5ctQUoC/MJBuFAOBAOhAPhQDgQDoQDS3Dgs4r07e2tLUDHRRkHgYoAREwLAmshEL6thXT0gED4Fh4EgeUR6M5VCtDlcd69xI5Iu3c6Dr4MgfDtZdAfUnH4dsiwx+knI9CdqxSgTwZ9j+I7Iu3Rz/g0BwLh2xxxOIoV4dtRIh0/10SgO1cpQNeMwE50dUTaiWtxY0IEwrcJg7Jjk8K3HQc3rr0Mge5cpQB9WTi2q7gj0na9ieWzIxC+zR6hfdkXvu0rnvFmDgS6c5UCdI7YbMqKjkibciDGbgqB8G1T4dq8seHb5kMYByZEoDtXKUAnDNTsJnVEmt3m2LddBMK37cZui5aHb1uMWmyeHYHuXKUAnT1qE9rXEWlCM2PSThAI33YSyI24Eb5tJFAxc1MIdOcqBeimQjiHsR2R5rAsVuwRgfBtj1Gd16fwbd7YxLLtItCdqxSg243nyyzviPQyY6J49wiEb7sP8VQOhm9ThSPG7ASB7lxtogD9+PHj27ff/v3TPw/6448/vDQk6P/mm7+91IZR+Xff/eMTPgT6t9/+Ny5ZbNwRaTHhERQEBgTCtwGQDJ+KQPj2VHgj/KAIdOdqEwXo99//8+3Dhw/vYfv553+/F1r0r2qzFaAUmxSga7WOSGvpjp7jIRC+HS/mr/Q4fHsl+tG9VwS6c7WJAnQMyNdf//Xtp5/+NU6vNp6tAKX4fOYbzxHYjkjjmoyDwFIIhG9LIRk51yAQvl2DUtYEgdsQ6M7VZgtQvpa/plEs4rgfvjq3WEMG87xhZZ7rr776yxfFLcWu+/lVAAq+Wb6Cxxdtw3b8/eWX/1wDzd1r0JcWBNZCIHxbC+noAYHwLTwIAssj0J2rhwpQfy/TYswxBd0zGsUWOq79+p1ijLelFqt8jc8YGTQLUOYsSi1Y/cqf4pPCzqLOYlSfT/n566///VQYspcxcgiCsk/tvWceHdiOXfjzzNYR6Zn6IvvYCIRvx47/2t6Hb2sjHn1HQKA7Vw8VoIDG20CKMnoKQ687QCmMMOLU51zhSoFV91kQdnrOzWGnBZoFKDbbKA7Ro3zWjnZRwF4qQJFHUUvRSc969VnsqpMeHdW/8Vqb657uWpwoeJ/VsC0tCKyFQPi2FtLRAwLhW3gQBJZHoDtXDxegFFgUTxZsFGe1oFvaDWTf8paPYo83g74dBARsplkQans3x9rRH2Vd8o194OEbV9+CXtr36H1srj49Km/c3xFpXJNxEFgKgfBtKSQj5xoEwrdrUMqaIHAbAt25eqgA9W2bBZoFHf0zGwWgReQ5PdiF0xTI7OENLW9A3au9tVgb5x4pQCmU2S8e2MDn2Q291ael9XVEWlpH5AUBEQjfRCL9GgiEb2ugHB1HQ6A7Vw8VoBRT9athCj6Kn1ONtRhx6jN+1X1Kjm8WT913viseLQpZMxab3RzrR7uu+Qpe2fX3VbHnVGGIjlO4MF9x1r9TPXqe2bAnLQishUD4thbS0QMC4Vt4EASWR6A7Vw8VoBRntcBiTHFY55Z2g9/RRM81v+NI0ebX39hBwQwIFmgWibUoHOfwhT2ucYwN5xrra9HIGL3sR8dSDXn+Tily8Ze5Z7aOSM/UF9nHRiB8O3b81/Y+fFsb8eg7AgLdubq7ALVQq8UUCijM6twSwFK4IZsPBZYF1yXZrMMe9/o/TDGmkNUHi0vkdXMUdNqAfuRcKkDHr9v9dYCli0OLarGpvlzC59776EoLAmshEL6thXT0gED4Fh4EgeUR6M7V3QXo8uZF4lYQ6Ii0Fdtj5/YQCN+2F7MtWxy+bTl6sX1WBLpzlQJ01mhNbFdHpInNjWkbRyB823gAN2Z++LaxgMXcTSDQnasUoJsI3VxGdkSay8JYsycEwrc9RXN+X8K3+WMUC7eHQHeuUoBuL44vt7gj0suNigG7RSB8221op3QsfJsyLDFq4wh05yoF6MaD+grzOyK9wo7oPAYC4dsx4jyLl+HbLJGIHXtCoDtXKUD3FOGVfOmItJLqqDkgAuHbAYP+QpfDtxeCH9W7RaA7VylAdxvu5znWEel52iL56AiEb0dnwLr+h2/r4h1tx0CgO1cpQI8R+0W97Ii0qIIICwIFgfCtgJHLpyMQvj0d4ig4IALduUoBekAiPOpyR6RHZWZ/EDiFQPh2CpnMPwOB8O0ZqEbm0RHozlVbgLIwn2AQDoQD4UA4EA6EA+FAOLAEB8YivC1Ax0UZB4GKAERMCwJrIRC+rYV09IBA+BYeBIHlEejOVQrQ5XHevcSOSLt3Og6+DIHw7WXQH1Jx+HbIsMfpJyPQnasUoE8GfY/iOyLt0c/4NAcC4dsccTiKFeHbUSIdP9dEoDtXKUDXjMBOdHVE2olrcWNCBMK3CYOyY5PCtx0HN669DIHuXKUAfVk4tqu4I9J2vYnlsyMQvs0eoX3ZF77tK57xZg4EunOVAnSO2GzKio5Im3Igxm4KgfBtU+HavLHh2+ZDGAcmRKA7VylAJwzU7CZ1RJrd5ti3XQTCt+3GbouWh29bjFpsnh2B7lylAJ09ahPa1xFpQjNj0k4QCN92EsiNuBG+bSRQMXNTCHTnKgXopkI4h7EdkeawLFbsEYHwbY9Rnden8G3e2MSy7SLQnasUoNuN58ss74j0MmOiePcIhG+7D/FUDoZvU4UjxuwEge5cbaIA/fjx49u33/790z8P+t13/3hpSH788Ye3b77520ttOKV8Dds6Ip2yJ/NB4FEEwrdHEcz+WxAI325BK2uDwHUIdOdqEwXo99//8+2XX/7z7uXPP//7vRBl7lVtjSLvHt9+/fW/79g8uzjuiHSPvdkTBK5BIHy7BqWsWQqB8G0pJCMnCPyBQHeupi9AP3z48PbTT//6w4u3t/eeCq4AAATYSURBVDfegH711V8+m1tzMGsByltisEkBuiYbouvZCHSJ69k6I/+4CIRvx419PH8eAt25mr4A7eCgIL22AGXt11//9dPX9+zjLaqNYo2irX7FP75dRQbg8VmryNO+a3uKYt6ApgC9FrGs2woCXeLaiu2xc3sIhG/bi1ksnh+B7lw9VIBatPnGzfFYwC0NDfIptC41vrbHaQozGzbW4hXbWWNR6h7HFrv+CoDFqD4rd+z9OhzZ7GWMXsa81V2yIZsClJYCdElkI2sGBDgzaUFgLQTCt7WQjp4jIdCdq4cKUMCj4KEoo6do87oDtr6JxJjxc23hipx7izgLSPdTSFKU1oZ8CzquR7tYf6kARd5vv/3vveikZz3/MxU+Mx4b2I14jGNtHvdW+1OAjuhkvHUEOAdpQWAtBMK3tZCOniMh0J2rhwtQ3upRoPmGkGKIIu9ZDdm3yKfoo5jkg20WdRaBFIYWm9pc5/Bv1Mf6awpQ9qHTAtG3oOpZogd7fUFeCtAlUI2MmRDoEtdM9sWWfSEQvu0rnvFmDgS6c/VQAerX1RZovuGjf0arXzVfI5/1OE0BSNGInXyYs2irxaYy69wjBShy2C8e2DAWu+q8pxdv/Ok+xuUe2ef2oCstCKyFQPi2FtLRAwLhW3gQBJZHoDtXDxWgFFN8RW2j4Dn3ZpC1GHHqM37VrVx6CsZbizfeBvr2UVnIQP+1BSj+jHYh85yf6LI49HdJmaMY9U2x9tg/8hW8MujzBrSikes9INAlrj34FR/mRCB8mzMusWrbCHTn6qEClCKsFliMKULr3BKQ8buPYxHI3CU97KHo9Q0k6ykCbylA2cN6C0fHlwpQ1tfinDG62a89S2AzykgBOiKS8dYR6BLX1n2K/fMiEL7NG5tYtl0EunN1dwHqG75aTKGAwqzOPQoXbyotGpFfP6f+pxx1YgdvK93DtV/BUwjSsHd8szrOsVYbkHFNkYfMKnfUq41L99fY9qhO8EwLAmshEL6thXT0gED4Fh4EgeUR6M7V3QXo8uZF4lYQ6Ii0Fdtj5/YQCN+2F7MtWxy+bTl6sX1WBLpzlQJ01mhNbFdHpInNjWkbRyB823gAN2Z++LaxgMXcTSDQnasUoJsI3VxGdkSay8JYsycEwrc9RXN+X8K3+WMUC7eHQHeuUoBuL44vt7gj0suNigG7RSB8221op3QsfJsyLDFq4wh05yoF6MaD+grzOyK9wo7oPAYC4dsx4jyLl+HbLJGIHXtCoDtXKUD3FOGVfOmItJLqqDkgAuHbAYP+QpfDtxeCH9W7RaA7VylAdxvu5znWEel52iL56AiEb0dnwLr+h2/r4h1tx0CgO1cpQI8R+0W97Ii0qIIICwIFgfCtgJHLpyMQvj0d4ig4IALduUoBekAiPOpyR6RHZWZ/EDiFQPh2CpnMPwOB8O0ZqEbm0RHozlVbgLIwn2AQDoQD4UA4EA6EA+FAOLAEB8Yi/IsCdFyQcRAIAkEgCASBIBAEgkAQWBKBFKBLohlZQSAIBIEgEASCQBAIAhcRSAF6EaIsCAJBIAgEgSAQBIJAEFgSgRSgS6IZWUEgCASBIBAEgkAQCAIXEUgBehGiLAgCQSAIBIEgEASCQBBYEoEUoEuiGVlBIAgEgSAQBIJAEAgCFxH4f0pR7cP84OYpAAAAAElFTkSuQmCC"></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>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>
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<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>
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<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>
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<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>
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<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>
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<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 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>
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<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>
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<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>
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<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>
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<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><strong>This question asks you to explore some properties of polygonal numbers and to determine and prove interesting results involving these numbers.</strong></p>
<p><br>A polygonal number is an integer which can be represented as a series of dots arranged in the shape of a regular polygon. Triangular numbers, square numbers and pentagonal numbers are examples of polygonal numbers.</p>
<p>For example, a triangular number is a number that can be arranged in the shape of an equilateral triangle. The first five triangular numbers are <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>,</mo><mo> </mo><mn>3</mn><mo>,</mo><mo> </mo><mn>6</mn><mo>,</mo><mo> </mo><mn>10</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>15</mn></math>.</p>
<p>The following table illustrates the first five triangular, square and pentagonal numbers respectively. In each case the first polygonal number is one represented by a single dot.</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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"></p>
<p>For an <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math>-sided regular polygon, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>∈</mo><msup><mi mathvariant="normal">ℤ</mi><mo>+</mo></msup><mo>,</mo><mo> </mo><mi>r</mi><mo>≥</mo><mn>3</mn></math>, the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>th polygonal number <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mi>r</mi></msub><mfenced><mi>n</mi></mfenced></math> is given by</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mi>r</mi></msub><mfenced><mi>n</mi></mfenced><mo>=</mo><mfrac><mrow><mfenced><mrow><mi>r</mi><mo>-</mo><mn>2</mn></mrow></mfenced><msup><mi>n</mi><mn>2</mn></msup><mo>-</mo><mfenced><mrow><mi>r</mi><mo>-</mo><mn>4</mn></mrow></mfenced><mi>n</mi></mrow><mn>2</mn></mfrac></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>.</p>
<p style="text-align: left;">Hence, for square numbers, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>4</mn></msub><mfenced><mi>n</mi></mfenced><mo>=</mo><mfrac><mrow><mfenced><mrow><mn>4</mn><mo>-</mo><mn>2</mn></mrow></mfenced><msup><mi>n</mi><mn>2</mn></msup><mo>-</mo><mfenced><mrow><mn>4</mn><mo>-</mo><mn>4</mn></mrow></mfenced><mi>n</mi></mrow><mn>2</mn></mfrac><mo>=</mo><msup><mi>n</mi><mn>2</mn></msup></math>.</p>
</div>
<div class="specification">
<p>The <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math>th pentagonal number can be represented by the arithmetic series</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>5</mn></msub><mfenced><mi>n</mi></mfenced><mo>=</mo><mn>1</mn><mo>+</mo><mn>4</mn><mo>+</mo><mn>7</mn><mo>+</mo><mo>…</mo><mo>+</mo><mfenced><mrow><mn>3</mn><mi>n</mi><mo>-</mo><mn>2</mn></mrow></mfenced></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>For triangular numbers, verify that <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>3</mn></msub><mfenced><mi>n</mi></mfenced><mo>=</mo><mfrac><mrow><mi>n</mi><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mrow><mn>2</mn></mfrac></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The number <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>351</mn></math> is a triangular number. Determine which one it is.</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>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mn>3</mn></msub><mfenced><mi>n</mi></mfenced><mo>+</mo><msub><mi>P</mi><mn>3</mn></msub><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>≡</mo><msup><mfenced><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State, in words, what the identity given in part (b)(i) shows for two consecutive triangular numbers.</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>For <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>4</mn></math>, sketch a diagram clearly showing your answer to part (b)(ii).</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>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn><msub><mi>P</mi><mn>3</mn></msub><mfenced><mi>n</mi></mfenced><mo>+</mo><mn>1</mn></math> is the square of an odd number for all <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">[3]</div>
<div class="question_part_label">c.</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"><msub><mi>P</mi><mn>5</mn></msub><mfenced><mi>n</mi></mfenced><mo>=</mo><mfrac><mrow><mi>n</mi><mfenced><mrow><mn>3</mn><mi>n</mi><mo>-</mo><mn>1</mn></mrow></mfenced></mrow><mn>2</mn></mfrac></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">[3]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By using a suitable table of values or otherwise, determine the smallest positive integer, greater than <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math>, that is both a triangular number and a pentagonal number.</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>A polygonal number, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mi>r</mi></msub><mfenced><mi>n</mi></mfenced></math>, can be represented by the series</p>
<p style="text-align:center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mtext>Σ</mtext><mrow><mi>m</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mfenced><mrow><mn>1</mn><mo>+</mo><mfenced><mrow><mi>m</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mfenced><mrow><mi>r</mi><mo>-</mo><mn>2</mn></mrow></mfenced></mrow></mfenced></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>∈</mo><msup><mi mathvariant="normal">ℤ</mi><mo>+</mo></msup><mo>,</mo><mo> </mo><mi>r</mi><mo>≥</mo><mn>3</mn></math>.</p>
<p>Use mathematical induction to prove that <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>P</mi><mi>r</mi></msub><mfenced><mi>n</mi></mfenced><mo>=</mo><mfrac><mrow><mfenced><mrow><mi>r</mi><mo>-</mo><mn>2</mn></mrow></mfenced><msup><mi>n</mi><mn>2</mn></msup><mo>-</mo><mfenced><mrow><mi>r</mi><mo>-</mo><mn>4</mn></mrow></mfenced><mi>n</mi></mrow><mn>2</mn></mfrac></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>.</p>
<div class="marks">[8]</div>
<div class="question_part_label">f.</div>
</div>
<br><hr><br><div class="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A">
<mi>A</mi>
</math></span> be the set <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\{ x|x \in \mathbb{R},{\text{ }}x \ne 0\} ">
<mo fence="false" stretchy="false">{</mo>
<mi>x</mi>
<mrow>
<mo stretchy="false">|</mo>
</mrow>
<mi>x</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>x</mi>
<mo>≠<!-- ≠ --></mo>
<mn>0</mn>
<mo fence="false" stretchy="false">}</mo>
</math></span>. Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="B">
<mi>B</mi>
</math></span> be the set <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\{ x|x \in ] - 1,{\text{ }} + 1[,{\text{ }}x \ne 0\} ">
<mo fence="false" stretchy="false">{</mo>
<mi>x</mi>
<mrow>
<mo stretchy="false">|</mo>
</mrow>
<mi>x</mi>
<mo>∈<!-- ∈ --></mo>
<mo stretchy="false">]</mo>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mo>+</mo>
<mn>1</mn>
<mo stretchy="false">[</mo>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>x</mi>
<mo>≠<!-- ≠ --></mo>
<mn>0</mn>
<mo fence="false" stretchy="false">}</mo>
</math></span>.</p>
<p>A function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f:A \to B">
<mi>f</mi>
<mo>:</mo>
<mi>A</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi>B</mi>
</math></span> is defined by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = \frac{2}{\pi }\arctan (x)">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mn>2</mn>
<mi>π<!-- π --></mi>
</mfrac>
<mi>arctan</mi>
<mo><!-- --></mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span>.</p>
</div>
<div class="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="D">
<mi>D</mi>
</math></span> be the set <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\{ x|x \in \mathbb{R},{\text{ }}x > 0\} ">
<mo fence="false" stretchy="false">{</mo>
<mi>x</mi>
<mrow>
<mo stretchy="false">|</mo>
</mrow>
<mi>x</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>x</mi>
<mo>></mo>
<mn>0</mn>
<mo fence="false" stretchy="false">}</mo>
</math></span>.</p>
<p>A function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g:\mathbb{R} \to D">
<mi>g</mi>
<mo>:</mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
<mo stretchy="false">→<!-- → --></mo>
<mi>D</mi>
</math></span> is defined by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g(x) = {{\text{e}}^x}">
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mi>x</mi>
</msup>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>(i) Sketch the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f(x)">
<mi>y</mi>
<mo>=</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span> and hence justify whether or not <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> is a bijection.</p>
<p>(ii) Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A">
<mi>A</mi>
</math></span> is a group under the binary operation of multiplication.</p>
<p>(iii) Give a reason why <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="B">
<mi>B</mi>
</math></span> is not a group under the binary operation of multiplication.</p>
<p>(iv) Find an example to show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(a \times b) = f(a) \times f(b)">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>a</mi>
<mo>×</mo>
<mi>b</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>a</mi>
<mo stretchy="false">)</mo>
<mo>×</mo>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>b</mi>
<mo stretchy="false">)</mo>
</math></span> is not satisfied for all <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a,{\text{ }}b \in A">
<mi>a</mi>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>b</mi>
<mo>∈</mo>
<mi>A</mi>
</math></span>.</p>
<div class="marks">[13]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>(i) Sketch the graph of <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> and hence justify whether or not <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g">
<mi>g</mi>
</math></span> is a bijection.</p>
<p>(ii) Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g(a + b) = g(a) \times g(b)">
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>a</mi>
<mo>+</mo>
<mi>b</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>a</mi>
<mo stretchy="false">)</mo>
<mo>×</mo>
<mi>g</mi>
<mo stretchy="false">(</mo>
<mi>b</mi>
<mo stretchy="false">)</mo>
</math></span> for all <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a,{\text{ }}b \in \mathbb{R}">
<mi>a</mi>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>b</mi>
<mo>∈</mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
</math></span>.</p>
<p>(iii) Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\{ \mathbb{R},{\text{ }} + \} ">
<mo fence="false" stretchy="false">{</mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mo>+</mo>
<mo fence="false" stretchy="false">}</mo>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\{ D,{\text{ }} \times \} ">
<mo fence="false" stretchy="false">{</mo>
<mi>D</mi>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mo>×</mo>
<mo fence="false" stretchy="false">}</mo>
</math></span> are both groups, explain whether or not they are isomorphic.</p>
<div class="marks">[8]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p><strong>This question asks you to investigate conditions for the existence of complex roots of polynomial equations of degree <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext mathvariant="bold">3</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext mathvariant="bold">4</mtext></math>.</strong></p>
<p> <br>The cubic equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mi>p</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>q</mi><mi>x</mi><mo>+</mo><mi>r</mi><mo>=</mo><mn>0</mn></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><mo>∈</mo><mo> </mo><mi mathvariant="normal">ℝ</mi></math>, has roots <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>,</mo><mo> </mo><mi>β</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>γ</mi></math>.</p>
</div>
<div class="specification">
<p>Consider the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mn>7</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>q</mi><mi>x</mi><mo>+</mo><mn>1</mn><mo>=</mo><mn>0</mn></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
</div>
<div class="specification">
<p>Noah believes that if <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>p</mi><mn>2</mn></msup><mo>≥</mo><mn>3</mn><mi>q</mi></math> then <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>,</mo><mo> </mo><mi>β</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>γ</mi></math> are all real.</p>
</div>
<div class="specification">
<p>Now consider polynomial equations of degree <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math>.</p>
<p>The equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>4</mn></msup><mo>+</mo><mi>p</mi><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mi>q</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>r</mi><mi>x</mi><mo>+</mo><mi>s</mi><mo>=</mo><mn>0</mn></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><mo> </mo><mi>s</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>, has roots <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>,</mo><mo> </mo><mi>β</mi><mo>,</mo><mo> </mo><mi>γ</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>δ</mi></math>.</p>
<p>In a similar way to the cubic equation, it can be shown that:</p>
<p style="padding-left: 30px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mo>-</mo><mo>(</mo><mi>α</mi><mo>+</mo><mi>β</mi><mo>+</mo><mi>γ</mi><mo>+</mo><mi>δ</mi><mo>)</mo></math></p>
<p style="padding-left: 30px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi><mo>=</mo><mi>α</mi><mi>β</mi><mo>+</mo><mi>α</mi><mi>γ</mi><mo>+</mo><mi>α</mi><mi>δ</mi><mo>+</mo><mi>β</mi><mi>γ</mi><mo>+</mo><mi>β</mi><mi>δ</mi><mo>+</mo><mi>γ</mi><mi>δ</mi></math></p>
<p style="padding-left: 30px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mo>-</mo><mo>(</mo><mi>α</mi><mi>β</mi><mi>γ</mi><mo>+</mo><mi>α</mi><mi>β</mi><mi>δ</mi><mo>+</mo><mi>α</mi><mi>γ</mi><mi>δ</mi><mo>+</mo><mi>β</mi><mi>γ</mi><mi>δ</mi><mo>)</mo></math></p>
<p style="padding-left: 30px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mo>=</mo><mi>α</mi><mi>β</mi><mi>γ</mi><mi>δ</mi></math>.</p>
</div>
<div class="specification">
<p>The equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>4</mn></msup><mo>-</mo><mn>9</mn><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>24</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>22</mn><mi>x</mi><mo>-</mo><mn>12</mn><mo>=</mo><mn>0</mn></math>, has one integer root.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By expanding <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>x</mi><mo>-</mo><mi>α</mi></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>-</mo><mi>β</mi></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>-</mo><mi>γ</mi></mrow></mfenced></math> show that:</p>
<p style="padding-left:30px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mo>-</mo><mfenced><mrow><mi>α</mi><mo>+</mo><mi>β</mi><mo>+</mo><mi>γ</mi></mrow></mfenced></math></p>
<p style="padding-left:30px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi><mo>=</mo><mi>α</mi><mi>β</mi><mo>+</mo><mi>β</mi><mi>γ</mi><mo>+</mo><mi>γ</mi><mi>α</mi></math></p>
<p style="padding-left:30px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mo>-</mo><mi>α</mi><mi>β</mi><mi>γ</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>p</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>q</mi><mo>=</mo><msup><mi>α</mi><mn>2</mn></msup><mo>+</mo><msup><mi>β</mi><mn>2</mn></msup><mo>+</mo><msup><mi>γ</mi><mn>2</mn></msup></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>Hence show that <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mi>α</mi><mo>-</mo><mi>β</mi></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mi>β</mi><mo>-</mo><mi>γ</mi></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mi>γ</mi><mo>-</mo><mi>α</mi></mrow></mfenced><mn>2</mn></msup><mo>=</mo><mn>2</mn><msup><mi>p</mi><mn>2</mn></msup><mo>-</mo><mn>6</mn><mi>q</mi></math>.</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>Given that <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>p</mi><mn>2</mn></msup><mo><</mo><mn>3</mn><mi>q</mi></math>, deduce that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>,</mo><mo> </mo><mi>β</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>γ</mi></math> cannot all be real.</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>Using the result from part (c), show that when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi><mo>=</mo><mn>17</mn></math>, this equation has at least one complex root.</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>By varying the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi></math> in the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>3</mn></msup><mo>-</mo><mn>7</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>q</mi><mi>x</mi><mo>+</mo><mn>1</mn><mo>=</mo><mn>0</mn></math>, determine the smallest positive integer value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi></math> required to show that Noah is incorrect.</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>Explain why the equation will have at least one real root for all values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi></math>.</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>Find an expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>α</mi><mn>2</mn></msup><mo>+</mo><msup><mi>β</mi><mn>2</mn></msup><mo>+</mo><msup><mi>γ</mi><mn>2</mn></msup><mo>+</mo><msup><mi>δ</mi><mn>2</mn></msup></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">f.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence state a condition in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi></math> that would imply <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>4</mn></msup><mo>+</mo><mi>p</mi><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mi>q</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>r</mi><mi>x</mi><mo>+</mo><mi>s</mi><mo>=</mo><mn>0</mn></math> has at least one complex root.</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>Use your result from part (f)(ii) to show that the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>4</mn></msup><mo>-</mo><mn>2</mn><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>4</mn><mi>x</mi><mo>+</mo><mn>5</mn><mo>=</mo><mn>0</mn></math> has at least one complex root.</p>
<div class="marks">[1]</div>
<div class="question_part_label">g.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State what the result in part (f)(ii) tells us when considering this equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>4</mn></msup><mo>-</mo><mn>9</mn><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>24</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>22</mn><mi>x</mi><mo>-</mo><mn>12</mn><mo>=</mo><mn>0</mn></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">h.i.</div>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the integer root of this equation.</p>
<div class="marks">[1]</div>
<div class="question_part_label">h.ii.</div>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By writing <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>4</mn></msup><mo>-</mo><mn>9</mn><msup><mi>x</mi><mn>3</mn></msup><mo>+</mo><mn>24</mn><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>22</mn><mi>x</mi><mo>-</mo><mn>12</mn></math> as a product of one linear and one cubic factor, prove that the equation has at least one complex root.</p>
<div class="marks">[4]</div>
<div class="question_part_label">h.iii.</div>
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