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</div><h2>SL Paper 2</h2><div class="specification">
<p>Consider the matrix</p>
<p style="text-align: center;"><em><strong>A</strong></em> \( = \left[ {\begin{array}{*{20}{c}} \lambda &3&2 \\ 2&4&\lambda \\ 3&7&3 \end{array}} \right]\).</p>
</div>
<div class="specification">
<p>Suppose now that \(\lambda = 1\) so consider the matrix</p>
<p style="text-align: center;"><strong><em>B</em></strong> \( = \left[ {\begin{array}{*{20}{c}} 1&3&2 \\ 2&4&1 \\ 3&7&3 \end{array}} \right]\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an expression for det(<strong><em>A</em></strong>) in terms of \(\lambda \), simplifying your answer.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence show that <strong><em>A </em></strong>is singular when \(\lambda = 1\) and find the other value of \(\lambda \) for which <strong><em>A </em></strong>is singular.</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>Explain how it can be seen immediately that <strong><em>B </em></strong>is singular without calculating its determinant.</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>Determine the null space of <strong><em>B</em></strong>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Explain briefly how your results verify the rank-nullity theorem.</p>
<div class="marks">[[N/A]]</div>
<div class="question_part_label">b.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Prove, using mathematical induction, that</p>
<p style="text-align: center;"><strong><em>B</em></strong>\(^n = {8^{n - 2}}\)<strong><em>B</em></strong>\(^2\) for \(n \in {\mathbb{Z}^ + },{\text{ }}n \geqslant 3\).</p>
<div class="marks">[7]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>The hyperbola with equation \({x^2} - 4xy - 2{y^2} = 3\) is rotated through an acute anticlockwise angle \(\alpha \) about the origin.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The point \((x,{\text{ }}y)\) is rotated through an anticlockwise angle \(\alpha \) about the origin to become the point \((X,{\text{ }}Y)\). Assume that the rotation can be represented by</p>
<p>\[\left[ {\begin{array}{*{20}{c}} X \\ Y \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} a&b \\ c&d \end{array}} \right]\left[ {\begin{array}{*{20}{c}} x \\ y \end{array}} \right].\]</p>
<p>Show, by considering the images of the points \((1,{\text{ }}0)\) and \((0,{\text{ }}1)\) under this rotation that</p>
<p>\[\left[ {\begin{array}{*{20}{c}} a&b \\ c&d \end{array}} \right] = \left[ {\begin{array}{*{20}{l}} {\cos \alpha }&{ - \sin \alpha } \\ {\sin \alpha }&{\cos \alpha } \end{array}} \right].\]</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>By expressing \((x,{\text{ }}y)\) in terms of \((X,{\text{ }}Y)\), determine the equation of the rotated hyperbola in terms of \(X\) and \(Y\).</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>Verify that the coefficient of \(XY\) in the equation is zero when \(\tan \alpha = \frac{1}{2}\).</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>Determine the equation of the rotated hyperbola in this case, giving your answer in the form \(\frac{{{X^2}}}{{{A^2}}} - \frac{{{Y^2}}}{{{B^2}}} = 1\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find the coordinates of the foci of the hyperbola prior to rotation.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.iv.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">\(S\) is defined as the set of all \(2 \times 2\) <span class="s1">non-singular matrices. </span>\(A\) <span class="s1">and </span>\(B\) <span class="s1">are two elements of the set </span>\(S\)<span class="s1">.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Show that \({({A^T})^{ - 1}} = {({A^{ - 1}})^T}\).</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Show that \({(AB)^T} = {B^T}{A^T}\).</p>
<div class="marks">[8]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">A relation \(R\) is defined on \(S\) such that \(A\) is related to \(B\) if and only if there exists an element \(X\) of \(S\) such that \(XA{X^T} = B\). Show that \(R\) is an equivalence relation.</p>
<div class="marks">[8]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p><span style="font-family: 'times new roman', times; font-size: medium;">The matrix <em><strong>A</strong></em> is given by <em><strong>A</strong></em> = \(\left( {\begin{array}{*{20}{c}}1&2&3&4\\3&8&{11}&8\\1&3&4&\lambda \\\lambda &5&7&6\end{array}} \right)\).</span></p>
<p><span style="font-family: 'times new roman', times; font-size: medium;">(a) Given that \(\lambda = 2\), </span><em style="font-family: 'times new roman', times; font-size: medium;"><strong>B = </strong></em><span style="font-family: 'times new roman', times; font-size: medium;">\(\left( \begin{array}{l}2\\4\\\mu \\3\end{array} \right)\) and <em><strong>X</strong></em> = \(\left( \begin{array}{l}x\\y\\z\\t\end{array} \right)\),</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(i) find the value of \(\mu \) for which the equations defined by <strong><em>AX </em></strong>= <strong><em>B </em></strong>are consistent and solve the equations in this case;</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) define the rank of a matrix and state the rank o</span><span style="font-family: 'times new roman', times; font-size: medium;">f <em><strong>A</strong></em>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Given that \(\lambda = 1\),</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(i) show that the four column vectors in <strong><em>A </em></strong>form a basis for the space of four-dimensional column vectors;</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) express the vector \(\left( \begin{array}{c}6\\28\\12\\15\end{array} \right)\) as a linear combination of these basis vectors.</span></p>
</div>
<br><hr><br><div class="specification">
<p class="p1">The set of all permutations of the list of the integers \(1,{\text{ }}2,{\text{ }}3{\text{ }} \ldots {\text{ }}n\) is a group, \({S_n}\), under the operation of composition of permutations.</p>
</div>
<div class="specification">
<p class="p1"><span class="s1">Each element of \({S_4}\) </span>can be represented by a \(4 \times 4\) matrix. For example, the cycle \({\text{(1 2 3 4)}}\) is represented by the matrix</p>
<p class="p1">\(\left( {\begin{array}{*{20}{c}} 0&1&0&0 \\ 0&0&1&0 \\ 0&0&0&1 \\ 1&0&0&0 \end{array}} \right)\) acting on the column vector \(\left( {\begin{array}{*{20}{c}} 1 \\ 2 \\ 3 \\ 4 \end{array}} \right)\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1"><span class="s1">(i) <span class="Apple-converted-space"> </span></span>Show that the order of \({S_n}\) is \(n!\);</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>List the <span class="s2">6 </span>elements of \({S_3}\) <span class="s1">in cycle form;</span></p>
<p class="p1">(iii) <span class="Apple-converted-space"> </span>Show that \({S_3}\) <span class="s1">is not Abelian;</span></p>
<p class="p1">(iv) <span class="Apple-converted-space"> </span>Deduce that \({S_n}\) is not Abelian for \(n \geqslant 3\).</p>
<div class="marks">[9]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Write down the matrices <span class="s1"><strong><em>M</em></strong>\(_1\), <strong><em>M</em></strong>\(_2\)</span> representing the permutations \((1{\text{ }}2),{\text{ }}(2{\text{ }}3)\)<span class="s1">, </span><span class="s3">respectively;</span></p>
<p class="p2"><span class="s4">(ii) <span class="Apple-converted-space"> </span>Find </span><span class="s1"><strong><em>M</em></strong>\(_1\)<strong><em>M</em></strong>\(_2\)</span> and state the permutation represented by this matrix;</p>
<p class="p1"><span class="s3">(</span>iii) <span class="Apple-converted-space"> </span>Find \(\det (\)<span class="s1"><strong><em>M</em></strong></span><span class="s5">\(_1)\)</span><span class="s1">, \(\det (\)<strong><em>M</em></strong></span><span class="s5">\(_2)\) </span>and deduce the value of \(\det (\)<span class="s1"><strong><em>M</em></strong>\(_1\)<strong><em>M</em></strong></span>\(_2)\)<span class="s5">.</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1"><span class="s1">(i) <span class="Apple-converted-space"> </span></span>Use mathematical induction to prove that</p>
<p class="p1"><span class="s1">\((1{\text{ }}n)(1{\text{ }}n{\text{ }} - 1)(1{\text{ }}n - 2) \ldots (1{\text{ }}2) = (1{\text{ }}2{\text{ }}3 \ldots n){\text{ }}n \in {\mathbb{Z}^ + },{\text{ }}n > 1\).</span></p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Deduce that every permutation can be written as a product of cycles of length 2.</p>
<div class="marks">[8]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Given that the elements of a \(2 \times 2\) symmetric matrix are real, show that</span></p>
<p style="margin-left: 30px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"> (i) the eigenvalues are real;</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"> (ii) the eigenvectors are orthogonal if the eigenvalues are distinct.</span></p>
<div class="marks">[11]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">The matrix \(\boldsymbol{A}\) is given by\[\boldsymbol{A} = \left( {\begin{array}{*{20}{c}}<br>{11}&{\sqrt 3 }\\<br>{\sqrt 3 }&9<br>\end{array}} \right) .\]Find the eigenvalues and eigenvectors of \(\boldsymbol{A}\).</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">The ellipse \(E\) has equation \({{\boldsymbol{X}}^T}{\boldsymbol{AX}} = 24\) where \(\boldsymbol{X} = \left( \begin{array}{l}<br>x\\<br>y<br>\end{array} \right)\) and \(\boldsymbol{A}\) is as defined in </span><span style="font-family: times new roman,times; font-size: medium;">part (b).</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"> (i) Show that \(E\) can be rotated about the origin onto the ellipse \(E'\) having </span><span style="font-family: times new roman,times; font-size: medium;">equation \(2{x^2} + 3{y^2} = 6\) .</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"> (ii) Find the acute angle through which \(E\) has to be rotated to coincide with \(E'\) .</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">The function \(f:\mathbb{R} \times \mathbb{R} \to \mathbb{R} \times \mathbb{R}\)</span><span style="font-family: times new roman,times; font-size: medium;"> is defined by \(\boldsymbol{X} \mapsto \boldsymbol{AX}\) , where \(\boldsymbol{X} = \left[ \begin{array}{l}<br>x\\<br>y<br>\end{array} \right]\) and \(\boldsymbol{A} = \left[ \begin{array}{l}<br>a\\<br>c<br>\end{array} \right.\left. \begin{array}{l}<br>b\\<br>d<br>\end{array} \right]\) </span><span style="font-family: times new roman,times; font-size: medium;">where \(a\) , \(b\) , \(c\) , \(d\) are all non-zero.</span></p>
</div>
<div class="specification">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Consider the group \(\left\{ {S,{ + _m}} \right\}\) where \(S = \left\{ {0,1,2 \ldots m - 1} \right\}\) , \(m \in \mathbb{N}\) , \(m \ge 3\) and \({ + _m}\) </span><span style="font-family: times new roman,times; font-size: medium;">denotes addition modulo \(m\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Show that \(f\) is a bijection if \(\boldsymbol{A}\) is non-singular.</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">A.a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Suppose now that \(\boldsymbol{A}\) is singular.</span></p>
<p style="margin-left: 30px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"> (i) Write down the relationship between \(a\) , \(b\) , \(c\) , \(d\) .</span></p>
<p style="margin-left: 30px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"> (ii) Deduce that the second row of \(\boldsymbol{A}\) is a multiple of the first row of \(\boldsymbol{A}\) .</span></p>
<p style="margin-left: 30px;"><span style="font-family: times new roman,times; font-size: medium;"> (iii) Hence show that \(f\) is not a bijection.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">A.b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: times new roman,times; font-size: medium;">Show that \(\left\{ {S,{ + _m}} \right\}\) is cyclic for all <strong><em>m</em></strong> .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">B.a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Given that \(m\) is prime,</span></p>
<p style="margin-left: 30px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"> (i) explain why all elements except the identity are generators of \(\left\{ {S,{ + _m}} \right\}\) ;</span></p>
<p style="margin-left: 30px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"> (ii) find the inverse of \(x\) , where <strong><em>x</em></strong> is any element of \(\left\{ {S,{ + _m}} \right\}\) apart from the </span><span style="font-family: times new roman,times; font-size: medium;">identity;</span></p>
<p style="margin-left: 30px;" align="LEFT"><span style="font-family: times new roman,times; font-size: medium;"> (iii) determine the number of sets of two distinct elements where each element </span><span style="font-family: times new roman,times; font-size: medium;">is the inverse of the other.</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">B.b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p align="LEFT"><span style="font-family: times new roman,times; font-size: medium;">Suppose now that \(m = ab\) where \(a\) , \(b\) are unequal prime numbers. Show that </span><span style="font-family: times new roman,times; font-size: medium;">\(\left\{ {S,{ + _m}} \right\}\) has two proper subgroups and identify them.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">B.c.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">By considering the points \((1,{\text{ }}0)\) and \((0,{\text{ }}1)\) determine the \(2 \times 2\) <span class="s1">matrix which represents</span></p>
<p class="p2">(i) <span class="Apple-converted-space"> </span>an anticlockwise rotation of \(\theta \) about the origin;</p>
<p class="p2">(ii) <span class="Apple-converted-space"> </span>a reflection in the line \(y = (\tan \theta )x\).</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Determine the matrix \(A\) which represents a rotation from the direction \(\left( {\begin{array}{*{20}{c}} 1 \\ 0 \end{array}} \right)\) to the direction \(\left( {\begin{array}{*{20}{c}} 1 \\ 3 \end{array}} \right)\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1"><span class="s1">A triangle whose vertices have coordinates \((0,{\text{ }}0)\), \((3,{\text{ }}1)\) and \((1,{\text{ }}5)\) </span>undergoes a transformation represented by the matrix \({A^{ - 1}}XA\), where \(X\) is the matrix representing a reflection in the <span class="s1">\(x\)-axis</span>. Find the coordinates of the vertices of the transformed triangle.</p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">The matrix \(B = {A^{ - 1}}XA\) represents a reflection in the line \(y = mx\). Find the value of \(m\).</p>
<div class="marks">[6]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br>