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</div><h2>HL Paper 3</h2><div class="specification">
<p>Consider the recurrence relation \(a{u_{n + 2}} + b{u_{n + 1}} + c{u_n} = 0,{\text{ }}n \in \mathbb{N}\) where \(a\), \(b\) and \(c\) are constants. Let \(\alpha \) and \(\beta \) denote the roots of the equation \(a{x^2} + bx + c = 0\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Verify that the recurrence relation is satisfied by</p>
<p>\[{u_n} = A{\alpha ^n} + B{\beta ^n},\]</p>
<p>where \(A\) and \(B\) are arbitrary constants.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Solve the recurrence relation</p>
<p>\({u_{n + 2}} - 2{u_{n + 1}} + 5{u_n} = 0\) given that \({u_0} = 0\) and \({u_1} = 4\).</p>
<div class="marks">[9]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that \({\text{gcd}}\left( {4k + 2,\,3k + 1} \right) = {\text{gcd}}\left( {k - 1,\,2} \right)\), where \(k \in {\mathbb{Z}^ + },\,k > 1\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State the value of \({\text{gcd}}\left( {4k + 2,\,3k + 1} \right)\) for odd positive integers \(k\).</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>State the value of \({\text{gcd}}\left( {4k + 2,\,3k + 1} \right)\) for even positive integers \(k\).</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>The weights of the edges in the complete graph \(G\) are given in the following table.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-10_om_09.18.27.png" alt="M17/5/MATHL/HP3/ENG/TZ0/DM/02"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Starting at A , use the nearest neighbour algorithm to find an upper bound for the travelling salesman problem for \(G\).</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>By first deleting vertex A , use the deleted vertex algorithm together with Kruskal’s algorithm to find a lower bound for the travelling salesman problem for \(G\).</p>
<div class="marks">[7]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Mathilde delivers books to five libraries, A, B, C, D and E. She starts her deliveries at library D and travels to each of the other libraries once, before returning to library D. Mathilde wishes to keep her travelling distance to a minimum.</p>
<p>The weighted graph \(H\), representing the distances, measured in kilometres, between the five libraries, has the following table.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2018-02-09_om_05.58.38.png" alt="N17/5/MATHL/HP3/ENG/TZ0/DM/01"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Draw the weighted graph \(H\).</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>Starting at library D use the nearest-neighbour algorithm, to find an upper bound for Mathilde’s minimum travelling distance. Indicate clearly the order in which the edges are selected.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By first removing library C, use the deleted vertex algorithm, to find a lower bound for Mathilde’s minimum travelling distance.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">A recurrence relation is given by \({u_{n + 1}} + 2{u_n} + 1 = 0,{\text{ }}{u_1} = 4\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Use the recurrence relation to find \({u_2}\).</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find an expression for \({u_n}\) in terms of \(n\).</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 class="p1">A second recurrence relation, where \({v_1} = {u_1}\) and \({v_2} = {u_2}\), is given by \({v_{n + 1}} + 2{v_n} + {v_{n - 1}} = 0,{\text{ }}n \ge 2\).</p>
<p class="p1">Find an expression for \({v_n}\) in terms of \(n\).</p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider \({\kappa _n}\), a complete graph with \(n\) vertices, \(n \geqslant 2\). Let \(T\) be a fixed spanning tree of \({\kappa _n}\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Draw the complete bipartite graph \({\kappa _{3,3}}\).</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>Prove that \({\kappa _{3,3}}\) is not planar.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>A connected graph \(G\) has \(v\) vertices. Prove, using Euler’s relation, that a spanning tree for \(G\) has \(v - 1\) edges.</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>If an edge \(E\) is chosen at random from the edges of \({\kappa _n}\), show that the probability that \(E\) belongs to \(T\) is equal to \(\frac{2}{n}\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The simple, complete graph \({\kappa _n}(n > 2)\) has vertices \({{\text{A}}_1},{\text{ }}{{\text{A}}_2},{\text{ }}{{\text{A}}_3},{\text{ }} \ldots ,{\text{ }}{{\text{A}}_n}\). The weight of the edge from \({{\text{A}}_i}\) to \({{\text{A}}_j}\) is given by the number \(i + j\).</p>
</div>
<div class="specification">
<p class="p1">Consider the general graph \({\kappa _n}\).</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>Draw the graph \({\kappa _4}\) </span>including the weights of all the edges.</p>
<p class="p2">(ii) <span class="Apple-converted-space"> </span>Use the nearest-neighbour algorithm, starting at vertex \({{\text{A}}_1}\), <span class="s2">to find a Hamiltonian cycle.</span></p>
<p class="p2">(iii) <span class="Apple-converted-space"> </span>Hence, find an upper bound to the travelling salesman problem for this weighted graph.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1"><span class="s1">Consider the graph \({\kappa _5}\). </span>Use the deleted vertex algorithm, with \({{\text{A}}_5}\) as the deleted vertex, to find a lower bound to the travelling salesman problem for this weighted graph.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Use the nearest-neighbour algorithm, starting at vertex \({{\text{A}}_1}\), <span class="s1">to find a Hamiltonian cycle.</span></p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Hence find and simplify an expression in \(n\), for an upper bound to the travelling salesman problem for this weighted graph.</p>
<div class="marks">[7]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1"><span class="s1">By splitting the weight of the edge \({{\text{A}}_i}{{\text{A}}_j}\) </span>into two parts or otherwise, show that all Hamiltonian cycles of \({\kappa _n}\) have the same total weight, equal to the answer found in (c)(ii).</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">In this question the notation \({({a_n}{a_{n - 1}} \ldots {a_2}{a_1}{a_0})_b}\) is used to represent a number in base \(b\), that has unit digit of \({a_0}\). For example \({(2234)_5}\) represents \(2 \times {5^3} + 2 \times {5^2} + 3 \times 5 + 4 = 319\) and it has a unit digit of 4.</p>
</div>
<div class="specification">
<p class="p1"><span class="s1">Let \(x\) </span>be the cube root of the base <span class="s2">7 </span>number \({(503231)_7}\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>By converting the base <span class="s1">7 </span>number to base <span class="s1">10</span>, find the value of \(x\), in base <span class="s1">10</span><span class="s2">.</span></p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Express \(x\) as a base <span class="s1">5 </span>number.</p>
<div class="marks">[7]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1"><span class="s1">Let \(y\) </span>be the base <span class="s2">9 </span>number \({({a_n}{a_{n - 1}} \ldots {a_1}{a_0})_9}\). Show that \(y\) is exactly divisible by <span class="s2">8 </span>if and only if the sum of its digits, \(\sum\limits_{i = 0}^n {{a_i}} \), is also exactly divisible by 8.</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">Using the method from part (b), find the unit digit when the base <span class="s1">9 </span><span class="s2">number \({(321321321)_9}\) is written as a base </span><span class="s1">8 </span><span class="s2">number.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the system of linear congruences</p>
<p>\[\begin{array}{*{20}{l}} {x \equiv 2(\bmod 5)} \\ {x \equiv 5(\bmod 8)} \\ {x \equiv 1(\bmod 3).} \end{array}\]</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>With reference to the integers 5, 8 and 3, state why the Chinese remainder theorem guarantees a unique solution modulo 120 to this system of linear congruences.</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>Hence or otherwise, find the general solution to the above system of linear congruences.</p>
<div class="marks">[7]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">In this question no graphs are required to be drawn. Use the handshaking lemma and other results about graphs to explain why,</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">a graph cannot exist with a degree sequence of \(1,{\text{ }}2,{\text{ }}3,{\text{ }}4,{\text{ }}5,{\text{ }}6,{\text{ }}7,{\text{ }}8,{\text{ }}9\)<span class="s1">;</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">a simple, connected, planar graph cannot exist with a degree sequence of \(4,{\text{ }}4,{\text{ }}4,{\text{ }}4,{\text{ }}5,{\text{ }}5\);</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">a tree cannot exist with a degree sequence of \(1,{\text{ }}1,{\text{ }}2,{\text{ }}2,{\text{ }}3,{\text{ }}3\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">The distances by road, in kilometres, between towns in Switzerland are shown in the following table.</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2016-01-25_om_10.43.43.png" alt></p>
<p class="p1">A cable television company wishes to connect the six towns placing cables along the road system.</p>
<p class="p1">Use Kruskal’s algorithm to find the minimum length of cable needed to connect the six towns.</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">Visitors to Switzerland can visit some principal locations for tourism by using a network of scenic railways as represented by the following graph:</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2016-01-25_om_10.44.46.png" alt></p>
<p class="p2">(i) <span class="Apple-converted-space"> </span>State whether the graph has any Hamiltonian paths or Hamiltonian cycles, justifying your answers.</p>
<p class="p2">(ii) <span class="Apple-converted-space"> </span>State whether the graph has any Eulerian trails or Eulerian circuits, justifying your answers.</p>
<p class="p2">(iii) <span class="Apple-converted-space"> </span>The tourist board would like to make it possible to arrive in Geneva, travel all the available scenic railways, exactly once, and depart from Zurich. Find which locations would need to be connected by a further scenic railway in order to make this possible.</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">In a computer game, Fibi, a magic dragon, is climbing a very large staircase. The steps are labelled <span class="s1">0, 1, 2, 3 … </span>.</p>
<p class="p1">She starts on step <span class="s1">0</span>. If Fibi is on a particular step then she can either jump up one step or fly up two steps. Let \({u_n}\) represent the number of different ways that Fibi can get to step \(n\). When counting the number of different ways, the order of Fibi’s moves matters, for example jump, fly, jump is considered different to jump, jump, fly. Let \({u_0} = 1\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the values of \({u_1},{\text{ }}{u_2},{\text{ }}{u_3}\).</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 class="p1">Show that \({u_{n + 2}} = {u_{n + 1}} + {u_n}\).</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">(i) <span class="Apple-converted-space"> </span>Write down the auxiliary equation for this recurrence relation.</p>
<p class="p2">(ii) <span class="Apple-converted-space"> </span>Hence find the solution to this recurrence relation, giving your answer in the form \({u_n} = A{\alpha ^n} + B{\beta ^n}\) where \(\alpha \) and \(\beta \) are to be determined exactly in surd form and \(\alpha > \beta \). The constants \(A\) and \(B\) do not have to be found at this stage.</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 class="p1">(i) <span class="Apple-converted-space"> </span>Given that \(A = \frac{1}{{\sqrt 5 }}\left( {\frac{{1 + \sqrt 5 }}{2}} \right)\), use the value of \({u_0}\) to determine \(B\).</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Hence find the explicit formula for \({u_n}\).</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 class="p1">Find the value of \({u_{20}}\).</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 class="p1">Find the smallest value of \(n\) <span class="s1">for which \({u_n} > 100\,000\).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">f.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Use the result \(2003 = 6 \times 333 + 5\) and Fermat’s little theorem to show that \({2^{2003}} \equiv 4(\bmod 7)\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: 'times new roman', times; font-size: medium;">Find \({2^{2003}}(\bmod 11)\) and \({2^{2003}}(\bmod 13)\).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Use the Chinese remainder theorem, or otherwise, to evaluate \({2^{2003}}(\bmod 1001)\), noting that \(1001 = 7 \times 11 \times 13\).</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Let the greatest common divisor of 861 and 957 be <em>h </em>.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Using the Euclidean algorithm, find <em>h </em>.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Hence find integers <em>A </em>and <em>B </em>such that 861<em>A </em>+ 957<em>B </em>= <em>h </em>.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Using part (b), solve \(287w \equiv 2(\bmod 319\)) , where \(w \in \mathbb{N},{\text{ }}w < 319\) .</span></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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the general solution to the diophantine equation \(861x + 957y = 6\) .</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the following weighted graph <em>G</em>.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State what feature of <em>G</em> ensures that <em>G</em> has an Eulerian trail.</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>State what feature of <em>G</em> ensures that <em>G</em> does not have an Eulerian circuit.</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>Write down an Eulerian trail in <em>G</em>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State the Chinese postman problem.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Starting and finishing at B, find a solution to the Chinese postman problem for<em> G</em>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate the total weight of the solution.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.iii.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The simple, connected graph \(G\) has <span class="s1"><em>e </em></span>edges and \(v\) vertices, where \(v \geqslant 3\).</p>
</div>
<div class="specification">
<p class="p1">Given that both \(G\) and \(G'\) are planar and connected,</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Show that the number of edges in \(G'\), the complement of \(G\), <span class="s1">is \(\frac{1}{2}{v^2} - \frac{1}{2}v - e\).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">show that the sum of the number of faces in \(G\) and the number of faces in \(G'\) is independent of \(e\);</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">show that \({v^2} - 13v + 24 \leqslant 0\) and hence determine the maximum possible value of \(v\).</p>
<div class="marks">[7]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>The Fibonacci sequence can be described by the recurrence relation \({f_{n + 2}} = {f_{n + 1}} + {f_n}\) where \({f_0} = 0,\,{f_1} = 1\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the auxiliary equation and use it to find an expression for \({f_n}\) in terms of \(n\).</p>
<div class="marks">[7]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>It is known that \({\alpha ^2} = \alpha + 1\) where \(\alpha = \frac{{1 + \sqrt 5 }}{2}\).</p>
<p>For integers \(n\) ≥ 3, use strong induction on the recurrence relation \({f_{n + 2}} = {f_{n + 1}} + {f_n}\) to prove that \({f_n} > {\alpha ^{n - 2}}\).</p>
<div class="marks">[8]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Use the Euclidean Algorithm to find the greatest common divisor of 7854 and 3315.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Hence state the number of solutions to the diophantine equation 7854<em>x </em>+ 3315<em>y </em>= 41 and justify your answer.</span></p>
</div>
<br><hr><br><div class="specification">
<p>Consider the recurrence relation</p>
<p style="text-align: center;">\({u_n} = 5{u_{n - 1}} - 6{u_{n - 2}},{\text{ }}{u_0} = 0\) and \({u_1} = 1\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an expression for \({u_n}\) in terms of \(n\).</p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>For every prime number \(p > 3\), show that \(p|{u_{p - 1}}\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The decimal number 1071 is equal to \(a\)060 in base \(b\), where \(a > 0\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Convert the decimal number 1071 to base 12.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write the decimal number 1071 as a product of its prime factors.</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>Using your answers to part (a) and (b), prove that there is only one possible value for \(b\) and state this value.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence state the value of \(a\).</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.ii.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use the Euclidean algorithm to find the greatest common divisor of 264 and 1365.</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, or otherwise, find the general solution of the Diophantine equation</p>
<p>\[264x - 1365y = 3.\]</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find the general solution of the Diophantine equation</p>
<p>\[264x - 1365y = 6.\]</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By expressing each of 264 and 1365 as a product of its prime factors, determine the lowest common multiple of 264 and 1365.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.5px Times; color: #2c2728;"><span style="font-family: 'times new roman', times; font-size: medium;">Prove that a graph containing a triangle cannot be bipartite.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.5px Times; color: #2c2728;"><span style="font-family: 'times new roman', times; font-size: medium;">Prove that the number of edges in a bipartite graph with <em>n </em>vertices is less than or equal to \(\frac{{{n^2}}}{4}\).</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The positive integer <em>N </em>is expressed in base <em>p </em>as \({({a_n}{a_{n - 1}}…{a_1}{a_0})_p}\) .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that when <em>p </em>= 2 , <em>N </em>is even if and only if its least significant digit, \({a_0}\) , is 0.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that when <em>p </em>= 3 , <em>N </em>is even if and only if the sum of its digits is even.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Given a sequence of non negative integers \(\{ {a_r}\} \) show that</p>
<p class="p1">(i) <span class="Apple-converted-space"> </span>\(\sum\limits_{r = 0}^n {{a_r}{{(x + 1)}^r}(\bmod x) = \sum\limits_{r = 0}^n {{a_r}(\bmod x)} } \) where \(x \in \{ 2,{\text{ }}3,{\text{ }}4 \ldots \} \).</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>\(\sum\limits_{r = 0}^n {(3{a_{2r + 1}} + {a_{2r}}){9^r} = \sum\limits_{r = 0}^{2n + 1} {{a_r}{3^r}} } \).</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">Hence determine whether the base \(3\) number \(22010112200201\) is divisible by \(8\)<span class="s1">.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the linear congruence \(ax \equiv b\left( {{\text{mod}}\,p} \right)\) where \(a,\,b,\,p,\,x \in {\mathbb{Z}^ + }\), \(p\) is prime and \(a\) is not a multiple of \(p\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State Fermat’s little theorem.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use Fermat’s little theorem to show that \(x \equiv {a^{p - 2}}b\left( {{\text{mod}}\,p} \right)\).</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 solve the linear congruence \(5x \equiv 7\left( {{\text{mod}}\,13} \right)\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.ii.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The following diagram shows the graph \(G\).</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2016-01-25_om_11.49.05.png" alt></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Show that \(G\) is bipartite.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">State which two vertices should be joined to make \(G\) equivalent to \({K_{3,{\text{ }}3}}\).</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 class="p1">In a planar graph the degree of a face is defined as the number of edges adjacent to that face.</p>
<p class="p1">(i) <span class="Apple-converted-space"> </span>Write down the degree of each of the four faces of \(G\).</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Explain why the sum of the degrees of all the faces is twice the number of edges.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">\(H\) is a simple connected planar bipartite graph with \(e\) edges, \(f\) faces, \(v\) vertices and \(v \ge 3\).</p>
<p class="p1">Explain why there can be no face in \(H\) of degree</p>
<p class="p1">(i) <span class="Apple-converted-space"> </span>one;</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>two;</p>
<p class="p1">(iii) <span class="Apple-converted-space"> </span>three.</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 class="p1">Hence prove that for \(H\)</p>
<p class="p1">(i) <span class="Apple-converted-space"> </span>\(e \ge 2f\);</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>\(e \le 2v - 4\).</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 class="p1">Hence prove that \({K_{3,{\text{ }}3}}\) is not planar.</p>
<div class="marks">[2]</div>
<div class="question_part_label">f.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Use the Euclidean algorithm to express gcd (123, 2347) in the form 123<em>p </em>+ 2347<em>q</em>, where \(p,{\text{ }}q \in \mathbb{Z}\).</span></p>
<div class="marks">[8]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the least positive solution of \(123x \equiv 1(\bmod 2347)\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the general solution of \(123z \equiv 5(\bmod 2347)\) .</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">State the solution set of \(123y \equiv 1(\bmod 2346)\) .</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The planar graph <em>G</em> and its complement \(G'\) are both simple and connected.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Given that <em>G</em> has 6 vertices and 10 edges, show that \(G'\) is a tree.</span></p>
</div>
<br><hr><br><div class="specification">
<p class="p1">The graph \({K_{2,{\text{ }}2}}\) is the complete bipartite graph whose vertex set is the disjoint union of two subsets each of order two.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Draw \({K_{2,{\text{ }}2}}\) as a planar graph.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Draw a spanning tree for \({K_{2,{\text{ }}2}}\).</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 class="p1">Draw the graph of the complement of \({K_{2,{\text{ }}2}}\).</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Show that the complement of any complete bipartite graph does not possess a spanning tree.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1"><span class="s1">Throughout this question, \({(abc \ldots )_n}\) </span>denotes the number \(abc \ldots \) written with number base \(n\). <span class="s1">For example \({(359)_n} = 3{n^2} + 5n + 9\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Given that \({(43)_n} \times {(56)_n} = {(3112)_n}\), show that \(3{n^3} - 19{n^2} - 38n - 16 = 0\).</p>
<p class="p2">(ii) <span class="Apple-converted-space"> </span>Hence determine the value of \(n\).</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 class="p1">Determine the set of values of \(n\) <span class="s1">satisfying \({(13)_n} \times {(21)_n} = {(273)_n}\).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Show that there are no possible values of \(n\) <span class="s1">satisfying \({(32)_n} \times {(61)_n} = {(1839)_n}\).</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Solve the recurrence relation \({v_n} + 4{v_{n - 1}} + 4{v_{n - 2}} = 0\) where \({v_1} = 0,{\text{ }}{v_2} = 1\).</p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Use strong induction to prove that the solution to the recurrence relation \({u_n} - 4{u_{n - 1}} + 4{u_{n - 2}} = 0\) where \({u_1} = 0,{\text{ }}{u_2} = 1\) is given by \({u_n} = {2^{n - 2}}(n - 1)\).</p>
<div class="marks">[8]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find a simplified expression for \({u_n} + {v_n}\) given that,</p>
<p class="p1">(i) <span class="Apple-converted-space"> </span>\(n\) is even.</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>\(n\) is odd.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Use the Euclidean algorithm to show that <span class="s1">1463 </span>and <span class="s1">389 </span>are relatively prime.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find positive integers \(a\) and \(b\) <span class="s1">such that \[1463a - 389b = 1\].</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>In the context of graph theory, explain briefly what is meant by a circuit;</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>In the context of graph theory, explain briefly what is meant by an Eulerian circuit.</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>The graph \(G\) has six vertices and an Eulerian circuit. Determine whether or not its complement \(G'\) can have an Eulerian circuit.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an example of a graph \(H\), with five vertices, such that \(H\) and its complement \(H'\) both have an Eulerian trail but neither has an Eulerian circuit. Draw \(H\) and \(H'\) as your solution.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">When numbers are written in base <em>n</em>, \({33^2} = 1331\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">By writing down an appropriate polynomial equation, determine the value of <em>n</em>.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Rewrite the above equation with numbers in base 7.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Let \(f(n) = {n^5} - n,{\text{ }}n \in {\mathbb{Z}^ + }\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the values of \(f(3)\), \(f(4)\) and \(f(5)\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Use the Euclidean algorithm to find</p>
<p class="p1">(i) <span class="Apple-converted-space"> </span>\(\gcd \left( {f(3),{\text{ }}f(4)} \right)\);</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>\(\gcd \left( {f(4),{\text{ }}f(5)} \right)\).</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 class="p1">Explain why \(f(n)\) is always exactly divisible by \(5\).</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">By factorizing \(f(n)\) explain why it is always exactly divisible by \(6\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Determine the values of <em>\(n\) </em>for which \(f(n)\) is exactly divisible by \(60\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) One version of Fermat’s little theorem states that, under certain conditions,</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[{a^{p - 1}} \equiv 1(\bmod p).\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that this result is not valid when <em>a</em> = 4, <em>p</em> = 9 and state which</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">condition is not satisfied.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Given that \({5^{64}} \equiv n(\bmod 7)\), where \(0 \leqslant n \leqslant 6\), find the value of <em>n</em>.</span></p>
<div class="marks">[8]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the general solution to the simultaneous congruences</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[x \equiv 3(\bmod 4)\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[3x \equiv 2(\bmod 5).\]</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<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;">(a) (i) Write down the general solution of the recurrence relation \({u_n} + 2{u_{n - 1}} = 0,{\text{ }}n \geqslant 1\).</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;"> (ii) Find a particular solution of the recurrence relation \({u_n} + 2{u_{n - 1}} = 3n - 2,{\text{ }}n \geqslant 1\), in the</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;"> form \({u_n} = An + B\), where \(A,{\text{ }}B \in \mathbb{Z}\).</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;"> (iii) Hence, find the solution to \({u_n} + 2{u_{n - 1}} = 3n - 2,{\text{ }}n \geqslant 1\) where \({u_1} = 7\).</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) Find the solution of the recurrence relation \({u_n} = 2{u_{n - 1}} - 2{u_{n - 2}},{\text{ }}n \geqslant 2\), where \({u_0} = 2,{\text{ }}{u_1} = 2\). Express your solution in the form \({2^{f(n)}}\cos \left( {g(n)\pi } \right)\), where the functions <em>f </em>and <em>g </em>map \(\mathbb{N}\) to \(\mathbb{R}\).</span></p>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A version of Fermat’s little theorem states that when <em>p</em> is prime, <em>a</em> is a positive integer and <em>a</em> and <em>p</em> are relatively prime, then \({a^{p - 1}} \equiv 1(\bmod p)\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Use the above result to show that if <em>p</em> is prime then \({a^p} \equiv a(\bmod p)\) where <em>a</em> is any positive integer.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that \({2^{341}} \equiv 2(\bmod 341)\).</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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) State the converse of the result in part (a).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Show that this converse is not true.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<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;">The weighted graph <em>K</em>, representing the travelling costs between five customers, has the following adjacency table.</span></p>
<p style="font: normal normal normal 21px/normal 'Times New Roman'; text-align: center; margin: 0px;"><br><span style="font-family: 'times new roman', times; font-size: medium;"><img src="images/Schermafbeelding_2014-09-10_om_11.10.37.png" alt></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<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;">Draw the graph \(K\).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 18.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">Starting from customer D, use the nearest-neighbour algorithm, to determine an upper bound to the travelling salesman problem for <em>K</em>.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: 'times new roman', times; font-size: medium;">By removing customer A, use the method of vertex deletion, to determine a lower bound to the travelling salesman problem for <em>K</em>.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: justify; font: 26.0px Times; color: #2c2728;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Given that \(a \equiv d(\bmod n)\) and \(b \equiv c(\bmod n)\) prove that</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.5px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[(a + b) \equiv (c + d)(\bmod n){\text{ .}}\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: justify; font: 26.0px Times; color: #2c2728;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Hence solve the system</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: justify; font: 26.0px Times; color: #2c2728;"><span style="font-family: 'times new roman', times; font-size: medium;">\[\left\{ {\begin{array}{*{20}{r}}<br> {2x + 5y \equiv 1(\bmod 6)} \\ <br> {x + y \equiv 5(\bmod 6)} <br>\end{array}} \right.\]</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 style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: justify; line-height: 12.1px; font: 25.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><span style="font-family: 'times new roman', times;">Show that \({x^{97}} - x + 1 \equiv 0(\bmod 97)\) has no solution.</span></span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">The sequence \(\{ {u_n}\} ,{\text{ }}n \in \mathbb{N}\), satisfies the recurrence relation \({u_{n + 1}} = 7{u_n} - 6\).</p>
<p class="p1">Given that \({u_0} = 5\), find an expression for \({u_n}\) in terms of \(n\).</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">The sequence \(\{ {v_n}\} ,{\text{ }}n \in \mathbb{N}\), satisfies the recurrence relation \({v_{n + 2}} = 10{v_{n + 1}} + 11{v_n}\).</p>
<p class="p1">Given that \({v_0} = 4\) and \({v_1} = 44\), find an expression for \({v_n}\) in terms of \(n\).</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">The sequence \(\{ {v_n}\} ,{\text{ }}n \in \mathbb{N}\), satisfies the recurrence relation \({v_{n + 2}} = 10{v_{n + 1}} + 11{v_n}\).</p>
<p class="p1">Show that \({v_n} - {u_n} \equiv 15(\bmod 16),{\text{ }}n \in \mathbb{N}\).</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="font-family: times new roman,times; font-size: medium;">The diagram shows a weighted graph with vertices A, B, C, D, E, F, G. The weight of each edge is marked on the diagram.</span></p>
<p><span style="font-family: times new roman,times; font-size: medium;"> </span></p>
<p style="font: normal normal normal 27px/normal Helvetica; text-align: center; margin: 0px;"><span style="font-family: times new roman,times; font-size: medium;"><img src="data:image/png;base64,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" alt> <br></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Explain briefly why the graph contains an Eulerian trail but not an </span><span style="font-family: 'times new roman', times; font-size: medium;">Eulerian circuit.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Write down an Eulerian trail.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Use Dijkstra’s algorithm to find the path of minimum total weight joining A to D.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) State the minimum total weight.</span></p>
<div class="marks">[8]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Use the Euclidean algorithm to find gcd(\(12\,306\), 2976) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Hence give the general solution to the diophantine equation \(12\,306\)<em>x</em> + 2976<em>y</em> = 996 .</span></p>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Use the Euclidean algorithm to find the greatest common divisor of the numbers 56 and 315.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Find the general solution to the diophantine equation \(56x + 315y = 21\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Hence or otherwise find the smallest positive solution to the congruence \(315x \equiv 21\) (modulo 56) .</span></p>
<div class="marks">[9]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Given that <em>a</em> , \(b \in \mathbb{N}\) and \(c \in {\mathbb{Z}^ + }\), show that if \(a \equiv 1(\bmod c)\) , then \(ab \equiv b(\bmod c)\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Using mathematical induction, show that \({9^n} \equiv 1(\bmod 4)\) , for \(n \in \mathbb{N}\) .</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The positive integer <em>M</em> is expressed in base 9. Show that <em>M</em> is divisible by 4 if the sum of its digits is divisible by 4.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Two mathematicians are planning their wedding celebration and are trying to arrange the seating plan for the guests. The only restriction is that all tables must seat the same number of guests and each table must have more than one guest. There are fewer than 350 guests, but they have forgotten the exact number. However they remember that when they try to seat them with two at each table there is one guest left over. The same happens with tables of 3, 4, 5 and 6 guests. When there were 7 guests per table there were none left over. Find the number of guests.</span></p>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that a positive integer, written in base 10, is divisible by 9 if the sum of its digits is divisible by 9.</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The representation of the positive integer <em>N</em> in base <em>p</em> is denoted by \({(N)_p}\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">If \({({5^{{{(126)}_7}}})_7} = {({a_n}{a_{n - 1}} \ldots {a_1}{a_0})_7}\) , find \({a_0}\) .</span></p>
<div class="marks">[9]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The positive integer <em>N</em> is expressed in base 9 as \({({a_n}{a_{n - 1}} \ldots {a_0})_9}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Show that <em>N</em> is divisible by 3 if the least significant digit, \({a_0}\), is divisible by 3.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Show that <em>N</em> is divisible by 2 if the sum of its digits is even.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 23.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) Without using a conversion to base 10, determine whether or not \({(464860583)_9}\) is divisible by 12.</span></p>
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<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Explaining your method fully, determine whether or not 1189 is a prime number.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) State the fundamental theorem of arithmetic.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) The positive integers <em>M</em> and <em>N</em> have greatest common divisor <em>G</em> and least common multiple <em>L</em> . Show that <em>GL</em> = <em>MN</em> .</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">Andy and Roger are playing tennis with the rule that if one of them wins a game when serving then he carries on serving, and if he loses then the other player takes over the serve.</p>
<p class="p1">The probability Andy wins a game when serving is \(\frac{1}{2}\) and the probability he wins a game when not serving is \(\frac{1}{4}\). Andy serves in the first game. Let \({u_n}\) denote the probability that Andy wins the \({n^{{\text{th}}}}\) game.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">State the value of \({u_1}\).</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Show that \({u_n}\) satisfies the recurrence relation</p>
<p class="p1">\[{u_n} = \frac{1}{4}{u_{n - 1}} + \frac{1}{4}.\]</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 class="p1">Solve this recurrence relation to find the probability that Andy wins the \({n^{{\text{th}}}}\) game.</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">After they have played many games, Pat comes to watch. Use your answer from part (c) to estimate the probability that Andy will win the game they are playing when Pat arrives.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Use the pigeon-hole principle to prove that for any simple graph that has two or more vertices and in which every vertex is connected to at least one other vertex, there must be at least two vertices with the same degree.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Seventeen people attend a meeting.</p>
<p class="p1">Each person shakes hands with at least one other person and no-one shakes hands with</p>
<p class="p1">the same person more than once. Use the result from part (a) to show that there must be at least two people who shake hands with the same number of people.</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 class="p1">Explain why each person cannot have shaken hands with exactly nine other people.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A graph has <em>n</em> vertices with degrees 1, 2, 3, …, <em>n</em>. Prove that \(n \equiv 0(\bmod 4)\) or \(n \equiv 3(\bmod 4)\).</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Let <em>G</em> be a simple graph with <em>n</em> vertices, \(n \geqslant 2\). Prove, by contradiction, that at least two of the vertices of <em>G</em> must have the same degree.</span></p>
<div class="marks">[8]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Use the Euclidean algorithm to find \(\gcd (752,{\text{ }}352)\).</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A farmer spends £8128 buying cows of two different breeds, A and B, for her farm. A cow of breed A costs £752 and a cow of breed B costs £352.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Set up a diophantine equation to show this.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Using your working from part (a), find the general solution to this equation.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) <strong>Hence</strong> find the number of cows of each breed bought by the farmer.</span></p>
<div class="marks">[10]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<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;">The following diagram shows a weighted graph <em>G </em>with vertices A, B, C, D and E.</span></p>
<p style="font: normal normal normal 21px/normal 'Times New Roman'; text-align: center; margin: 0px;"><br><span style="font-family: 'times new roman', times; font-size: medium;"><img src="images/Schermafbeelding_2014-09-17_om_14.54.24.png" alt></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<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;">Show that graph \(G\) is Hamiltonian. Find the total number of Hamiltonian cycles in \(G\), giving reasons for your answer.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p 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;">State an upper bound for the travelling salesman problem for graph \(G\).</span></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 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;">Hence find a lower bound for the travelling salesman problem for \(G\).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="font-family: 'times new roman', times; font-size: medium;">Show that the lower bound found in (d) cannot be the length of an optimal solution for the travelling salesman problem for the graph \(G\).</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Define what is meant by the statement \(x \equiv y(\bmod n){\text{ where }}x{\text{, }}y{\text{, }}n \in {\mathbb{Z}^ + }\) .</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Hence prove that if \(x \equiv y(\bmod n)\) then \({x^2} \equiv {y^2}(\bmod n)\) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Determine whether or not \({x^2} \equiv {y^2}(\bmod n)\) implies that \(x \equiv y(\bmod n)\) .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Convert the decimal number 51966 to base 16.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Using the Euclidean algorithm, find the greatest common divisor, <em>d </em>, of 901 and 612.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Find integers <em>p </em>and <em>q </em>such that 901<em>p </em>+ 612<em>q </em>= <em>d </em>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) Find the least possible positive integers <em>s </em>and <em>t </em>such that 901<em>s </em>− 612<em>t </em>= 85.</span></p>
<div class="marks">[10]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">In each of the following cases find the solutions, if any, of the given linear congruence.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) \(9x \equiv 3(\bmod 18)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) \(9x \equiv 3(\bmod 15)\)</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Given that \(a,{\text{ }}b,{\text{ }}c,{\text{ }}d \in \mathbb{Z}\), show that</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[(a - b)(a - c)(a - d)(b - c)(b - d)(c - d) \equiv 0(\bmod 3).\]</span></p>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The sequence \(\{ {u_n}\} \) , \(n \in {\mathbb{Z}^ + }\) , satisfies the recurrence relation \({u_{n + 2}} = 5{u_{n + 1}} - 6{u_n}\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Given that \({u_1} = {u_2} = 3\) , obtain an expression for \({u_n}\) in terms of <em>n</em> .</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The sequence \(\{ {v_n}\} \) , \(n \in {\mathbb{Z}^ + }\) , satisfies the recurrence relation \({v_{n + 2}} = 4{v_{n + 1}} - 4{v_n}\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Given that \({v_1} = 2\) and \({v_2} = 12\) , use the principle of strong mathematical induction to show that \({v_n} = {2^n}(2n - 1)\) .</span></p>
<div class="marks">[8]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">State the Fundamental theorem of arithmetic for positive whole numbers greater than \(1\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Use the Fundamental theorem of arithmetic, applied to \(5577\) and <span class="s1">\(99\,099\)</span>, to calculate \(\gcd (5577,{\text{ }}99\,099)\) and \({\text{lcm}}(5577,{\text{ }}99\,099)\), expressing each of your answers as a product of prime numbers.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Prove that \(\gcd (n,{\text{ }}m) \times {\text{lcm}}(n,{\text{ }}m) = n \times m\) for all \(n,{\text{ }}m \in {\mathbb{Z}^ + }\).</p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<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;">The following figure shows the floor plan of a museum.</span></p>
<p style="font: normal normal normal 21px/normal 'Times New Roman'; text-align: center; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;"><br><img src="images/Schermafbeelding_2014-09-17_om_11.01.03.png" alt></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) (i) Draw a graph <em>G </em>that represents the plan of the museum where each exhibition room is represented by a vertex labelled with the exhibition room number and each door between exhibition rooms is represented by an edge. Do not consider the entrance foyer as a museum exhibition room.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Write down the degrees of the vertices that represent each exhibition room.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) Virginia enters the museum through the entrance foyer. Use your answers to </span><span style="font-family: 'times new roman', times; font-size: medium;">(i) and (ii) to justify why it is possible for her to visit the thirteen exhibition rooms going through each internal doorway exactly once.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Let <em>G </em>and <em>H </em>be two graphs whose adjacency matrices are represented below.</span></p>
<p style="font: normal normal normal 21px/normal Helvetica; text-align: center; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;"><br><img src="images/Schermafbeelding_2014-09-17_om_11.02.25.png" alt></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;"> </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;">Using the adjacency matrices,</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 number of edges of each graph;</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) show that exactly one of the graphs has a Eulerian trail;</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;">(iii) show that neither of the graphs has a Eulerian circuit.</span></p>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<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;">Show that \(30\) is a factor of \({n^5} - n\) for all \(n \in \mathbb{N}\).</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Show that \({3^{{3^m}}} \equiv 3({\text{mod}}4)\) for all \(m \in \mathbb{N}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Hence show that there is exactly one pair \((m,{\text{ }}n)\) where \(m,{\text{ }}n \in \mathbb{N}\), satisfying the equation \({3^{{3^m}}} = {2^{{2^n}}} + {5^2}\).</span></p>
<div class="marks">[8]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Draw a spanning tree for</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"> (i) the complete graph, \({K_4}\);</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;"> (ii) the complete bipartite graph, \({K_{4,4}}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Prove that a simple connected graph with <em>n </em>vertices, where \(n > 1\), must have two vertices</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">of the same degree.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 16.0px 'Times New Roman'; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) Prove that every simple connected graph has at least one spanning tree.</span></p>
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<br><hr><br><div class="specification">
<p class="p1">The weights of the edges in the complete graph \(G\) are shown in the following table.</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2017-02-06_om_05.29.01.png" alt="M16/5/MATHL/HP3/ENG/TZ0/DM/02"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1"><span class="s1">Starting at A</span>, use the nearest neighbour algorithm to find an upper bound for the travelling salesman problem for \(G\).</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"><span class="s1">By first removing A </span>, use the deleted vertex algorithm to find a lower bound for the travelling salesman problem for \(G\).</p>
<div class="marks">[7]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">The following graph represents the cost in dollars of travelling by bus between 10 towns in a particular province.</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2015-12-11_om_14.37.25.png" alt></p>
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<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Use Dijkstra’s algorithm to find the cheapest route between \(A\) and \(J\), and state its cost.</p>
<div class="marks">[7]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">For the remainder of the question you may find the cheapest route between any two towns by inspection.</p>
<p class="p1">It is given that the total cost of travelling on all the roads without repeating any is \(\$ 139\).</p>
<p class="p1">A tourist decides to go over all the roads at least once, starting and finishing at town \(A\).</p>
<p class="p1">Find the lowest possible cost of his journey, stating clearly which roads need to be travelled more than once. You must fully justify your answer.</p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
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<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Use the Euclidean algorithm to find the greatest common divisor of 259 and 581.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Hence, or otherwise, find the general solution to the diophantine equation 259<em>x</em> + 581<em>y</em> = 7 .</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">An arithmetic sequence has first term 2 and common difference 4. Another arithmetic sequence has first term 7 and common difference 5. Find the set of all numbers which are members of both sequences.</span></p>
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<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Using the Euclidean algorithm, show that \(\gcd (99,{\text{ }}332) = 1\).</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Find the general solution to the diophantine equation \(332x - 99y = 1\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Hence, or otherwise, find the smallest positive integer satisfying the congruence \(17z \equiv 1(\bmod 57)\).</span></p>
<div class="marks">[11]</div>
<div class="question_part_label">b.</div>
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<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Solve, by any method, the following system of linear congruences</p>
<p class="p1"><span class="Apple-converted-space"> </span>\(x \equiv 9(\bmod 11)\)</p>
<p class="p1"><span class="Apple-converted-space"> </span>\(x \equiv 1(\bmod 5)\)</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 class="p1">Find the remainder when \({41^{82}}\) is divided by 11.</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 class="p1">Using your answers to parts (a) and (b) find the remainder when \({41^{82}}\) is divided by \(55\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
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<br><hr><br><div class="question">
<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;">Consider an integer \(a\) with \((n + 1)\) digits written as \(a = {10^n}{a_n} + {10^{n - 1}}{a_{n - 1}} + \ldots + 10{a_1} + {a_0}\), where \(0 \leqslant {a_i} \leqslant 9\) for \(0 \leqslant i \leqslant n\), and \({a_n} \ne 0\).</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;">(a) Show that \(a \equiv ({a_n} + {a_{n - 1}} + \ldots + {a_0}) ({\text{mod9}})\).</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) Hence find all pairs of values of the single digits \(x\) and \(y\) such that the number \(a = 476x212y\) is a multiple of \(45\).</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;">(c) (i) If \(b = 34\,390\) in base 10, show that \(b\) is \(52\,151\) written in base 9.</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) Hence find \({b^2}\) in base 9, by performing all the calculations without changing base.</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Find the general solution for the following system of congruences.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> \(N \equiv 3(\bmod 11)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> \(N \equiv 4(\bmod 9)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> \(N \equiv 0(\bmod 7)\)</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Find all values of <em>N</em> such that \(2000 \leqslant N \leqslant 4000\).</span></p>
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<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Anna is playing with some cars and divides them into three sets of equal size. However, when she tries to divide them into five sets of equal size, there are four left over. Given that she has fewer than 50 cars, what are the possible numbers of cars she can have?</span></p>
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<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<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;">Consider the integers \(a = 871\) and \(b= 1157\), given in base \(10\).</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) Express \(a\) and \(b\) in base \(13\).</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) Hence show that \({\text{gcd}}(a,{\text{ }}b) = 13\).</span></p>
<div class="marks">[7]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p 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;">A list \(L\) contains \(n+ 1\) distinct positive integers. Prove that at least two members of \(L\)leave the same remainder on division by \(n\).</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p 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;">Consider the simultaneous equations</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;"> \(4x + y + 5z = a\)</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;"> \(2x + z = b\)</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;"> \(3x + 2y + 4z = c\)</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;">where \(x,{\text{ }}y,{\text{ }}z \in \mathbb{Z}\).</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 7 divides \(2a + b - c\).</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) Given that <em>a </em>= 3, <em>b </em>= 0 and <em>c </em>= −1, find the solution to the system of equations modulo 2.</span></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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-size: medium; font-family: 'times new roman', times;">Consider the set \(P\) of numbers of the form \({n^2} - n + 41,{\text{ }}n \in \mathbb{N}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-size: medium; font-family: 'times new roman', times;">(i) Prove that all elements of <em>P </em>are odd.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-size: medium; font-family: 'times new roman', times;">(ii) List the first six elements of <em>P </em>for <em>n </em>= 0, 1, 2, 3, 4, 5.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 21.0px 'Times New Roman';"><span style="font-size: medium; font-family: 'times new roman', times;">(iii) Show that not all elements of <em>P </em>are prime.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">One version of Fermat’s little theorem states that, under certain conditions, \({a^{p - 1}} \equiv 1(\bmod p)\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Show that this result is not true when <em>a</em> = 2, <em>p</em> = 9 and state which of the conditions is not satisfied.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Find the smallest positive value of <em>k</em> satisfying the congruence \({2^{45}} \equiv k(\bmod 9)\) .</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find all the integers between 100 and 200 satisfying the simultaneous congruences \(3x \equiv 4(\bmod 5)\) and \(5x \equiv 6(\bmod 7)\) .</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">A simple connected planar graph, has \(e\) edges, \(v\) vertices and \(f\) faces.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>Show that \(2e \ge 3f{\text{ if }}v > 2\).</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Hence show that \({K_5}\), the complete graph on five vertices, is not planar.</p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) <span class="Apple-converted-space"> </span>State the handshaking lemma.</p>
<p class="p1">(ii) <span class="Apple-converted-space"> </span>Determine the value of \(f\), if each vertex has degree \(2\).</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 class="p1">Draw an example of a simple connected planar graph on \(6\) vertices each of degree \(3\).</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The weights of the edges of a graph \(H\) are given in the following table.</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2016-01-07_om_13.41.35.png" alt></p>
<p>(i) Draw the weighted graph \(H\).</p>
<p>(ii) Use Kruskal’s algorithm to find the minimum spanning tree of \(H\). Your solution should indicate the order in which the edges are added.</p>
<p>(iii) State the weight of the minimum spanning tree.</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">Consider the following weighted graph.</p>
<p class="p1" style="text-align: center;"><img src="images/Schermafbeelding_2016-01-07_om_15.46.10.png" alt></p>
<p class="p1">(i) Write down a solution to the Chinese postman problem for this graph.</p>
<p class="p1">(ii) Calculate the total weight of the solution.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">(i) State the travelling salesman problem.</p>
<p class="p1">(ii) Explain why there is no solution to the travelling salesman problem for this graph.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Show that there are exactly two solutions to the equation \(1982 = 36a + 74b\)<span class="s1">, with \(a,{\text{ }}b \in \mathbb{N}\).</span></p>
<div class="marks">[8]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Hence, or otherwise, find the remainder when \({1982^{1982}}\) is divided by \(37\)<span class="s1">.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Using Fermat’s little theorem, show that, in base 10, the last digit of <em>n</em> is always equal to the last digit of \({n^5}\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Show that this result is also true in base 30.</span></p>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Write 57128 as a product of primes.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Prove that \(\left. {22} \right|{5^{11}} + {17^{11}}\).</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The diagram below shows the graph G with vertices A, B, C, D, E and F.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="font: normal normal normal 26px/normal Helvetica; text-align: center; margin: 0px;"><img src="data:image/png;base64,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" alt></p>
</div>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Determine if any Hamiltonian cycles exist in <em>G</em> . If so, write one down.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Otherwise, explain what feature of G makes it impossible for a Hamiltonian cycle to exist.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Determine if any Eulerian circuits exist in <em>G</em> . If so, write one down.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Otherwise, explain what feature of <em>G</em> makes it impossible for an Eulerian circuit to exist.</span></p>
</div>
<br><hr><br><div class="question">
<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;">The following diagram shows a weighted graph.</span></p>
<p style="font: normal normal normal 21px/normal 'Times New Roman'; text-align: center; margin: 0px;"><span style="font-family: 'times new roman', times; font-size: medium;"><br><img src="images/Schermafbeelding_2014-09-17_om_10.52.09.png" alt></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;"> </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;">(a) Use Kruskal’s algorithm to find a minimum spanning tree, clearly showing the order in which the edges are added.</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) Sketch the minimum spanning tree found, and write down its weight.</span></p>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: justify; line-height: 12.1px; font: 24.5px Times; color: #2c2728;"><span style="font-family: 'times new roman', times; font-size: medium;">Let <span style="color: #000000;"><em>G </em></span>be a simple, connected, planar graph. </span></p>
</div>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) (i) Show that Euler’s relation \(f - e + v = 2\) is valid for a spanning tree of G.</span></p>
<p style="margin: 0px 0px 0px 30px; font: 22px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (ii) By considering the effect of adding an edge on the values of <em>f</em>, <em>e</em> and <em>v</em> show that Euler’s relation remains true.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Show that <em>K</em><sub>5</sub> is not planar.</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Write down Fermat’s little theorem.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) In base 5 the representation of a natural number <em>X</em> is \({\left( {k00013(5 - k)} \right)_5}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">This means that \(X = k \times {5^6} + 1 \times {5^2} + 3 \times 5 + (5 - k)\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">In base 7 the representation of <em>X</em> is \({({a_n}{a_{n - 1}} \ldots {a_2}{a_1}{a_0})_7}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find \({a_0}\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 33.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) Given that <em>k</em> = 2, find <em>X</em> in base 7.</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Show that, for a connected planar graph,</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[v + f - e = 2.\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Assuming that \(v \geqslant 3\), explain why, for a simple connected planar graph, \(3f \leqslant 2e\) and hence deduce that \(e \leqslant 3v - 6\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) The graph <em>G</em> and its complement \({G'}\) are simple connected graphs, each having 12 vertices. Show that \({G}\) and \({G'}\) cannot both be planar.</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">State Fermat’s little theorem.</span></p>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The positive integer <em>p</em> is an odd prime number.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 34.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that \(\sum\limits_{k = 1}^p {{k^p} \equiv 0(\bmod p)} \).</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Given that \(\sum\limits_{k = 1}^p {{k^{p - 1}} \equiv n(\bmod p)} \) where \(0 \leqslant n \leqslant p - 1\), find the value of <em>n</em>.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: justify; font: 22.0px Times; color: #2c2728;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Use the Euclidean algorithm to find the gcd of 324 and 129.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Hence show that \(324x + 129y = 12\) has a solution and find both a particular solution and the general solution.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) Show that there are no integers <em>x</em> and <em>y</em> such that \(82x + 140y = 3\) .</span></p>
<div> </div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The weights of the edges of a graph <em>G</em> with vertices A, B, C, D and E are given in the following table.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAgkAAACXCAIAAAAQ81+1AAAYDklEQVR4nO2dzWsbWbrG859IUCsJBAneiLvIqrm4vRDGZBHwxkEWOAsPWTT0QkJhQnYDEha5NDQ0JRICzUBSwk2mIbgpDIO5OBQzF7IQBT1ggqiFF6aohRHm8NxFScdlfTi9aL/noHoezqYlNVZ+es95zle97z2nUGRjY2NjY8u2e06hiNyLEEgAhEACAAgBAEBvmIgQSACEQAIACAEAvUGLEEgAhEACAAgBAL1BixBIAIRAAgAIAQC9QYsQSACEQAIACAEAvUGLEEgAhEACAAgBAL1ByzAENezXHmy6Q2XuK4gT+DKoP1h0c67a6HhBNJb9MhOZC4M49L1+szaBUNrtvjsOk6v4n8f/vhL9HtIEIq+x6PZkpd55fxhEhvqDNISroHt/+V3S+51ANgZS0RsmMgtBhe5moejU3NCcOZggoGK/XSkUncL+ILoCVBJ+7NarTqHolNp+bICFmTBIPvfrVadQ3mi+nZriOArettbL8uOCCQKXobvtFIpO4UHD+wJARSe9etUpFCv1Hz6ZmCUY7QsTCACAODzqNar0BqMyCuEi6Gw5haJT2OoGF6a+hBECV0Fn7dobsq9ke4icDEDQxtA+mp0mJ6fdrV4OvOEq8vadmR89Oe2ul51CsbLnjcQnCUb7wkzkj0cffhVeO6aiN0xkEkLsP699+9/pUrrpx4a+hWXe8LDln8t/H3EI45H3rFIoOoXtfng5965a/T0lYLE3XC8mDESCFd5w9X+DDwamR1r0honMQRiPvPbzo+N+rWxwLwVWeMN0I6VQbRycGNlrloaghpPf3eh2YlbWeMN0o7VQXBPfU7HAG+Kh++ypiaWzFr1hImMQ1LC/9cKPL6fzx7KpE2mj/SHbaq3Xp3k5hNQnsXUvEv3DS2WPNxiEY0dfMLOtqkVvmMgUBBW+edo5TQDEfqtUNDiFtGPd4E3Ootdf+nk4hKQ3APSGVDfWDSr6dLDPdYMVMgTh3G/uTzeaz/3mQ4Mn0hZ4AwCokfe0ZOzoRRqC6QnBvOzxhnzvKfG8wRqZgaCHhpnL3XkYFgEs8obrLXgT17rFIUwnBKVng5GZRzpmZI036DudeT2LvtZl6P+v/LKS3jCRCQjjkdd+no17PSyaOJG2xRumfpkTg5yuk8ob6dbizff818c52E5Z5A3qbLBX5R1WAEhO/8f9l/wtVnrDRPKXFyP/5casB+gbjcVK/c0wEe0TRp/3mdzg1A89OaW9/tDAbV4jEJLhm0YpfSD8Yzj50cdR4HX3vpOHYMOzb0iGg2Ytr8++ZWdF6QncN6ae9aE3ANLRkASdb+duI2RfTFtmNn33Eg+DZTkzai33Q+5yZiSh7/3UWi87N3Jm5ODJ8MU5M8obzZ8Gfji7kJKSNITbc2aYe9aH3gCYzplhg0gAhEACAAgBAL1BixBIAIRAAgAIAQC9QYsQSACEQAIACAEAvUGLEEgAhEACAAgBAL1BixBIAIRAAgAIAQC9QYsQSACEQAIACAGA9gY2NjY2NrZsu+fQJDlTIAEAhEACAAgBAPeUtAiBBEAIJACAEADQG7QIgQRACCQAgBAA0Bu0CIEEQAgkAIAQANAbtAiBBEAIJACAEADQG7QIgQRACCQAgBAA2OENl6G7bbzulWkI5iVNIAl97+duvTaXY1Il4fHgXadxf+WLWMShf/i+s7s2X6kiCX/r7E6ztfd+C+UyddsSBtcEyhvNt8J5eQ2OBio6fdOsOYWiU9rtHhnLRAsrvGFS0GZ7WhrTjOgNogTUsL+13dipzucfVqH7aGf3SWnlCxxdhu7ek/rjynwVI3U22KtOC3gsrPNxh7IiDNTZLwc/po44Kemx3gsEM5YbGw2S025tr3caqUlVj+pT78zUnNm4N6gk6G2ks4P5uleCojfIE1Chu7k4N/1l6G6vujcAmEyMZrxhFosa9mvfiGXwtyEM1O8nx5kKqcvj5K5kaDS4DN3tbGGfkfesYqIXpDLuDed++/HGN8YKYWrRG2wYFKbKtTcg9lulzFQp9luC22s2hcFUsd8q5cAb1LBfe7DpDvUvLW+KWRn2BjXy/tL+8GlSEdAYBdAb7BoU8u0NuAg6W06h2nA/J2rk/7XdO43EQNgUBlPFfuv+s8FI7sjBzGgQ+61SOesNC14RlFlvuAzd/ZZ/Pi2nXjR4Ik1vsGlQyLk3ACr6dLCbraQtJpvCYPJ+7L/YlN1wNrduKN8YA8UXTFkZ9QY17O+9ChIFnPvNh46JnqBFb7BpUMi9NyAOvVfdzvcbhaKz/tIXvKVjUxgAAJLTbv2V5EE0jI0GKvbblXS9CEBFwevmhtEh0ZQ3qNh/8WiyXEqhmDyRFodwGU520hY1E+snmwaFnHtDPHS/+4t3pgAVHT1fL0ve0rEpDABcBAcv+0O5K7ypzM0U4/Co1ygVnUK10Xnbb9bMbqWY8ga9VrjZDJ1Ic91g06CQa29QobuZnS0mp931B2KbzjaFwXh0+OMbcWOAJaOB7P20eRnzBjXy/tLOdgk9jzazv2ZFNBiVTYNCnr0hXUNndxLO/eY3+fOGceT/2PNH0391PHz99nh1L2vNSo389pbxa/0GvEFFR8/XZy3x+kS6tJenVaQtsmZQQL69IV0olKfPvqlk+KYheEvHjjCIQ6+9YW6j1ehoEIf+37v1bxsHJ3K30xbJgDdcBZ01/XvXvWj+xULRKTxoeF8kvxW9QZbAzR3F2bsZOgykL/AJQtBnbMX5WxjXiRNWPGfGwjAYj7xnldndZtFIMLrNXt5our7gL75Mxp99s0WEQAIgBBIAQAgA6A1ahEACIAQSAEAIAOgNWoRAAiAEEgBACADoDVqEQAIgBBIAQAgA6A1ahEACIAQSAEAIAOgNWoRAAiAEEgBACAC0N7CxsbGxsWXbPYcmyZkCCQAgBBIAQAgAuKekRQgkAEIgAQCEAIDeoEUIJABCIAEAhACA3qBFCCQAQiABAIQAgN6gRQgkAEIgAQCEAIDeoEUIJABCIAEAhADAkDd8GdQfLLoyVW10vECw/GFWshDi0D9839ldmy8GmYS+93O3XpMvYsH+AFvCYFHZK6kM1dJhsDTgx9HpD41S0SmUN5peuMo1QVUSHg/edRr3lyelV2eDvYfyOYmNrBt0guL9QXQFqCT82K1XnUIe6r5dhu7ek/rjSqE4m7hfDftb242dqpECR/QGWBIGN7KUp00uQ7VoGCwNeJX86/DNaaQAFZ306tVFJbXvUJIQVOg+2tl9Urpl6EuTlhvIV29mT2lasCH1huwr0pUbUklDWFpE/mul1e9M9AZYEQYq9l+9uK53BuDcb+6LFZSXD4MFAa/+c3z8ZUpAxX67IjtlFIdwSzErlQSvGt9/3yjRG3JSE5TeYKUsCINxFJ5lK0Gq0N1c6ZJnXwv4y9DdEa6OaZE3JKfd+qtgdNTKqzeMo+Bta73sFKqmKuFZMChM36E3mJM9YTDVZejuPfXOVrjk2W0Bn4S+2260j4THBGu84SLofNcNLhD7OfSGbKu1Xp+aKpFqz6BAbzAoe8Lg+gNbL1Z6O2VZwGdqpq63B7I1Mu3wBpUErxrpgimX3qDXDd7kLHr9pW/iqpI9gwK9waDsCYPJ+6H7aHWPYVPdGvDTHYXSs8FIbliwwhuS0177cHLulGNvAAA18p6Wis78tQ0R2TMo5Mgb1LBfKy/JASndE1LZEwYAgMvQ3ReOBMu84Q994E+XBd5wERz87doOc+4N1yPF/U5wdev/fAeyZ1DIkTfYJ3vCYPKu7IYSrPQGqGG/9k2+vGHBPea0bUveWLPGG6Y4uG6gN5iSPWEAExtKsNMbYr91P397Slnlad2gD5omNpg+4eIUik5prz+UtwaLBgV6g0HZEwZGNpRghzeokfe0NL2ZokZ+e7vhfs7pHdZUufGGZTkzai33Qw5yZmQuYMwuEm8mSxC81Q56AwBbwiB9f9jfexXI5oqAdBgsCfjkcz+dKRaKlXpnEEjfXhSFcGP7aIkB5MYbbBQhkAAIgQQAEAIAeoMWIZAACIEEABACAHqDFiGQAAiBBAAQAgB6gxYhkAAIgQQAEAIAeoMWIZAACIEEABACAHqDFiGQAAiBBAAQAgDtDWxsbGxsbNl2z6FJcqZAAgAIgQQAEAIA7ilpEQIJgBBIAAAhAKA3aBECCYAQSAAAIQCgN2gRAgmAEEgAACEAoDdoEQIJgBBIAAAhAKA3aBECCYAQSAAAIQAw7Q1x6Hv9Zm1yZaq02313HCZX8T+P/73atX2S0Pd+7tZr8zVyk/DjpDZqodrofAwF03DKhoFKwuPBu07j/myN3JvZSdMmWs9E5g9ZK2kCS/vCRCoKfnnXaZSKt6Ww/rNlR1/AdUnUQrVxcCKcjdacN0zS8JY3mm+nebmnIFa77psa9re2GztVZ65Igxp5T0vVSap6NfLbNckyR5JhoEL30c7ukwUd/mbS5rQJ5iqnN4gSWN4XAADj6PSHRqna6PzdD0VrutjRF1QS9DbWX/rRGGrkt7c2OqfCRSxMeIM2hvbRrBkmp92t3ip7A4DFBXxm63uo0N1c2bkSFtcziY+7f82GhIr9F48E09bTG+QJLClmpZKgt1Ha7Z1KF2+AJX1hphJq7LdKcgtoGPKG8ch7Vlm6V6BWf09pcX9It1O2usGF/s+1FV03AFjYH1T0++/JTPfYEe4PYn/LTtniDclpd/3hU+9M3hhgSV8I3c3Zwl/fSJb3MeENativlYX3Cr4qC7wh7Q/ltCqqio5eNH/8JFgFz4b+MCMVuo/2vBGL3wnKDm+4DN1tZ73ZT08aVvzsDYv6wvwr537zofD+qrg3RF4j3Ueue5HoH75NVnhDtm62uHFa0B/mPyBdMJneYIU3qGG/9mB6EqmS4ZtGqSy5225BX5h3AnqDIVniDUiGg06n26wtPoy5S1nQH25KDftbL8SOW1LRG6zwhthvlbKvXIbu9kpfV6M3pNK1s7mnNLun9Lm/1xyMxsA48l9uFPI2V7ohFbqPBI9bUhkaGZclwhQ9e0xlgzf8kVfuVBb0hXzuKel7iqVng5HcfvrtssAbLkN3O/PbqyTobeRrrjTzrvSGErhusMMbEPutUjl77jp3MHu3sqEvzF5TVMN+7cGqn0VPLvIXnYXzYjXyXx/L7zVZ4A1z8wLZW2s29IdrmdhQAr3BEm/Aud98WLm+hqBiv11Z5cdceIf1Wun50swNhHEUeN297/pD4Y0EwApvSBcK02ffEA/dvTXBWzpW9IepjGwogd5gizfceA5URSe9+s70breE7OgL+Xz2LVUS+t5PrfXyzZwZZo4gBCHcfPT3xmxIPyK/6vf29JlToegUynMr5cvQbUqOBVr0BlkCt/SFTP6Y9eabQPQJOGv6QvpkeNEp1FqvpR8CNJtPySIRAgmAEEgAACEAoDdoEQIJgBBIAAAhAKA3aBECCYAQSAAAIQCgN2gRAgmAEEgAACEAoDdoEQIJgBBIAAAhAKA3aBECCYAQSAAAIQDQ3sDGxsbGxpZt9xyaJGcKJACAEEgAACEA4J6SFiGQAAiBBAAQAgB6gxYhkAAIgQQAEAIAeoMWIZAACIEEABACAHqDFiGQAAiBBAAQAgB6gxYhkAAIgQQAEAIAE97wZVB/sOC+VGm3++7XIDJW6kcUQhL63s/dem2udk0cHvUaaVLG1c49CZWEx4N3ncb9uRzdKvp0sFspFJ1CreUNJZMSQxpCHPqH7zu7a8uykasoOPy5W686q1zyzEZZFAY3srTmoH7DdR3EtGS0Hg5Kuz3pTLQTyUFQw/7WdmNnvsOPR4c/9I7CBBrI1qrmrFeh+2hn90mpOJez/uLf3rtP0XianVhuTEwlCOEydPee1B9XCsXKokFBRSe9erVS77z3Q+Gs/YJ/zVJZEwbjkfesoifQskWUDe0pRV4j6w0AcBF0tpyCsUKhwhAW1DNR/zk+/pJJXz/s18oLR407kngYLKqD+PvJ8fWvf+43H0oSgDyEZb9yctpdf9g4OJGfKNEbYFUY1AxUP0xljzdcLyaEh4NU5r1hVtIjow3ecENq2K89Fq7wY8egcBF0tiqCJf+yojfAljBQsd+uFMobzZ8GsmvHVBZ5QwrIKRSd+53gSvobWekNtafemWTpcKk/leoWb1BJ6Peb+8996eHRhkFBhe5modZy36ZVzyr13m+h3GSJ3gA7wuB6PCwUnUJ5o+kJ18S0yRuuj6n3B5G0OVjnDbHfqvWCVa0JCiz3Bl0n0kx/kPxziwaFy9Ddvr6JkHzu16vOulwk0BtgRRjot6LgMC2cXM5HvWh6w1e84SLofLfi2ym37ympKHjd3CgUhbdWLBgUZvcSVehuLiipfVeiN8CKMJj5wMhv127bgL0D2eQN3FOavpkEP7bczyt9fRNfP2/4+gf+fJkfFP7IK3cpA6PBjc2TmSZnilmZD4N5xX5L9tqeTd4wvcmb87NoNfq191raGGCjN0CF7mbevCHdUsveVlTDfu0B1w2SsiAMFn5mZ/Wfb1jkDdObvLm9w5q+Hvm9A396c1Elw3f941V96Omr3qBiv72Wuz0lFfvtSrYXxH6rJPfQE70BVoTBnGL/eVv6PrcFz74hDr32Rk6efQOw2BtUEnqt9ZnF9QoPCvPeMB55zyrTY1gVHT2v7feHomtIKwYFdTbYq1bqb4aJgoo+HextCh5C0htgRxioKPjlcJIaQUUnvWbHl00bYU3OjPVm3zsWvpSSlSAEfQ/nxrOOauQ9Lc1hEXwSUjQMbmQC0HvKKhm+maQMKVQbHU8+h4oghPT2+pJJQBL+1pkmDnktOl2iN8COMFDR0fN0plja7b4zMDYa2lOyT4RAAiAEEgBACADoDVqEQAIgBBIAQAgA6A1ahEACIAQSAEAIAOgNWoRAAiAEEgBACADoDVqEQAIgBBIAQAgA6A1ahEACIAQSAEAIALQ3sLGxsbGxZds9hybJmQIJACAEEgBACAC4p6RFCCQAQiABAIQAgN6gRQgkAEIgAQCEAIDeoEUIJABCIAEAhACA3qBFCCQAQiABAIQAgN6gRQgkAEIgAQCEAMCIN1wFnbXlt6bWDNR8AwxGw7T4pVOoNjof85GJFoBKwuPBu07j/vL6DepssPdQuOyXLIQ49A/fd3bX5hP366gQz1ov3RGS0Pd+7tZri+pcZTMWixaAozfA3LpB/+qZ0tBJ+Ftn97/y5Q0XQWencXASqUnVeOHyyFlJElCh+2hn90mpuLy2T1rrSbokpCCEy9Dde1J/XFlQ6PAi6DzeaB9FKk3U/FiycrhoR1DD/tZ2Y6fqLKxzdSNrvVwhE9AbAJjzhiTofDvrDQDU2T8OP+fHG2ZqXqqR91S2JGxW4gRuqfumkuBV4/vvG6UV9gYAS4q63IgKFfvtyqqW8QCwtAbiRdDZzU1fsFG2eMNV8OEfkRFTmMgEhMvQ3Z6rDCxXNX5GFnlDctqtvwpGR608esNcVKx6TdDF3pBWfzJU8oveAFu8Ifncr38/yJ03zFWNX/CKnKzxhoug8103uEDs59Eb5l+R5WCHN1yG7vb1MeR6exCudGlYK2XcG3Tbz583pB0gOyU895sP871uUEnwqpGWR86nN8z/q/PoDanGUfCh36w5haKz3gsEVw/0BljgDfuD6EpFJ708rhsmC+dJ1XiMo+Bta/2B8GioZYU3JKe99uHkNJ7esOyVu5RN3pBqHPkvN277wJ8vegMs8Qbk9LwBgErCj9161SkUK/XOe7e5IXsfIysLvOEiOPjbYDSe/Fc+vYF7SgskvZ42BGHZzX4zY4It3nAt9bt/PBL/PjbMFC5Dd9vUhhJs8Ib0+NFo3zDvDXNhoEJ3c4UJ/CFvuAzdHT7fICzbvEElwZt+kIh/H+PRMI78lxuym6ozMu8NM8rnuoF3WBfo3G//bWmc3IFMjwZWyPizb5mYUFHgdRolM4fS5qJBJeHx+87uWv2HT9H46x+/M9EbYIc35OjZNwALvUFFweGvQdodVPTpoP3CF30klN4AC3Nm3DZY3KVMRIOK/XalUHTWm30/NLBWuilRAje2j5YYwIp7w/TXX7hvpqJPB7tpbLwJVjhnRjYrRvH69rYa+e2aM0kk83df9gIr6A0AmGtPixBIAIRAAgAIAQC9QYsQSACEQAIACAEAvUGLEEgAhEACAAgBAL1BixBIAIRAAgAIAQC9QYsQSACEQAIACAEAvUGLEEgAhEACAAgBgPYGNjY2Nja2bPt/EIOFc6Fb8hMAAAAASUVORK5CYII=" alt></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Starting at A, use the nearest neighbour algorithm to find an upper bound for the travelling salesman problem for <em>G</em> .</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Use Kruskal’s algorithm to find and draw a minimum spanning tree for the subgraph obtained by removing the vertex A from <em>G</em> .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Hence use the deleted vertex algorithm to find a lower bound for the travelling salesman problem for <em>G</em> .</span></p>
<div class="marks">[8]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-size: medium; font-family: times new roman,times;">The graph <em>G </em>has adjacency matrix <strong><em>M </em></strong>given below.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><br><span style="font-size: medium; font-family: times new roman,times;"><img src="data:image/png;base64,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" alt></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Draw the graph <em>G </em>.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">What information about <em>G </em>is contained in the diagonal elements of <em><strong>M</strong></em>\(^2\)<span style="font: 7.0px Times;"> </span>?</span></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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">List the trails of length 4 starting at A and ending at C.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 29.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that a graph is bipartite if and only if it contains only cycles of even length.</span></p>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The weighted graph <em>H </em>is shown below.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><br><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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" alt></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"> </p>
<p style="margin: 0px; font: 22px Helvetica; text-align: left;"><span style="font-family: 'times new roman', times; font-size: medium;">Use Kruskal’s Algorithm, indicating the order in which the edges are added, to find and draw the minimum spanning tree for <em>H</em>.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) A tree has <em>v </em>vertices. State the number of edges in the tree, justifying your answer.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) We will call a graph with <em>v </em>vertices a “forest” if it consists of <em>c </em>components each of which is a tree.</span><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Here is an example of a forest with 4 components.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Helvetica; min-height: 23.0px;"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAlsAAAClCAIAAADkoK9mAAAKR0lEQVR4nO3dPXJURxiFYe0EqtgGZRXLwSHejQjFWiZiBcSTkpJ+DlRWydJo5v707T7n9PtktgNG9+3bB7sscVcA3vjx+Hg6nUZ/CgBd3Y3+AICi0+n06cNHRhGYCosIXMYoArNhEYF3MYrAVFhE4BpGEZgHiwjcwCgCk2ARgdsYRWAGLCKwCKMIxGMRgaUYRSAbiwiswCgCwVhEYB1GEUjFIgKrMYpAJBYR2IJRBPKwiMBGjCIQhkUEtmMUgSQsIrALowjEYBGBvRhFIAOLCDTAKAIBWESgDUYRcHdXVX/+/Bn9MSZyPp+5NFMxioCp0+l0Op3ufv/+/eX+/nw+j/48Xf399euQa+t8Pn+5v+/8S//69avnLze57w8P/3z71u2XI25Pp9Ppx+Pj6E+B9p5/L3v39BezjeKQ38sPmcOn3/Hwby19dH6ViNvThPfkJF7Owd3z35otdudRHDKHw3/pqQx5iYjbx4Q35CReDcHdy38wW/Juozj82hr+AeINfH2Ie7QJ78ZJvJ2Au1f/eLbwHUZR5MIS+RiRhr84xD3O8Lg4yMXL//V3X0yY/9BRlLqqpD5MDJFXhrhHEImL5t679i98P+KEh+CgURS8pAQ/kjWpl4W4bUnFRUNXLvzL36E/4VFoPoqy15PsB7Mj+JoQtxXBuGji+lX/7s+smfBANBxF8YtJ/ONZkH1BiLufbFzsdPOSv/ZT3CY8Fk1G0eJKsviQssRfDeLuIR4Xmy253m/8XNMJD8fOUTS6jIw+qhSLl4K421jExQYLL/bbP+l7wiOyeRTtriG7Dzyc0etA3LWM4mKV5Vf6oj/7YsKDsmEUTS8g0489hN2LQNzl7OJioVWX+dI/DWrC47LqOVpfPdYfvhvTV4C4S5jGxU1r/91mxZ+POOGhWfg0Ay6dgC/hUNaHn7jXWcfFFRv+U9+6PzF4wqNz85nGXDcxX0hzAceeuO8JiIuLtv3vIOsWsaY8QFeebNhFE/blNBFz4In7VkxcvLL5/45cvYg15TG6+Hwjr5jIL2qzsKNO3JfC4uLZnu+g27KINeVhevWUgy+X4C9tlchDTtwnkXFRu7+hfOMi1pRH6vlZx18r8V/gTcHHm7jBcSe3/4eObV/EmvJgPT3xvz5/jr9QZr434w82cYPjTqvJz+DctYg13/E6n88zzOGTOe/NSY40cZGkyRzW/kWsmQ7ZhJfIbF/yPIe5iIsUreawmixizXHUZrs+ns3zhc9wjF8hLtw1nMNqtYiVfuDmuTgumuHLzz7AVxAXvtrOYTVcxMo9djNcGTdlP4TUo7sQceGo+RxW20WsxMOXfVmskvoo8g7tBsSFlyPmsJovYmUdwdRrYrO8B5J0XHciLlwcNId1xCJWykHMuyCaSHosGQe1IeJC33FzWActYvkfx6SrobmMh+N+RA9CXCg7dA7ruEUs50OZcSkcyv0R+R7ODogLTUfPYR26iOV5NN2vg258H5TjseyMuFDTYQ7r6EUstwPqexEM4fi4vA7kQMSFjj5zWB0WsXyOqeMVMJzXQ3M5iiKICwXd5rD6LGI5HFavl1+Ky6PTP4SCiIuxes5hdVvE0j6yLq+9LP0HqHz8xBEXo3Sew+q5iKV6cPVfeAvKj1Hz4BkhLvrrP4fVeRFL7/gqv+p2NB+m2pEzRVz0NGQOq/8iltIh1nzJrak9Up3DFoC46GPUHNaQRSyNo6z2esfQebAKxywMcXG0gXNYoxaxRh9onRc7ksLj5cY8CHFxnLFzWAMXscYda4VXOt7Yh8yNeSji4gjD57DGLmKNONzMYTejHjU3ZgfERVsKc1jDF7H6HnHmsLP+D5wbsxviohWROSyFRaxeB505HKLnY+fG7Iy42E9nDktkEev4484cDtTn4XNjDkFc7CE1h6WziHXkoWcOhzs6ATfmQMTFNmpzWFKLWMccfeZQxHEhuDGHIy7WEpzDUlvEav0CMIdSjsjBjSmCuFhOcw5LcBGr3WvAHApqG4UbUwpxsYTsHJbmIlaLl4E5lNUqDTemIOLiOuU5LNlFrH2vBHMobn8gbkxZxMV7xOewlBextr4YzKGFPZm4McURF2/pz2GJL2Ktfz2YQyPbYnFjWiAuXrKYw9JfxFrzkjCHdtYm48Y0Qlw8cZnDsljEWvaqMIemlofjxrRDXBjNYbksYt16YZhDa0vycWOaIu7MvOawjBax3n9tmMMA1yNyY1oj7pzs5rC8FrEuvTzMYYz3UnJjBiDubBznsOwWsf7/CjGHYd4G5caMQdx5mM5hOS5i/fci/fz5kznM8/Le5MYMQ9wZ+M5hmS5imT90XPd0b35/eODGzEPceNY3s+siVtWnDx9HfwQc5Xw+f/rwkRszEnGzWd/MLCJE0TcYcYNZx2URIYq+wYgbzDouiwhR9A1G3GDWcVlEiKJvMOIGs47LIkIUfYMRN5h1XBYRougbjLjBrOOyiBBF32DEDWYdl0WEKPoGI24w67gsIkTRNxhxg1nHZREhir7BiBvMOi6LCFH0DUbcYNZxWUSIom8w4gazjssiQhR9gxE3mHVcFhGi6BuMuMGs47KIEEXfYMQNZh2XRYQo+gYjbjDruCwiRNE3GHGDWcdlESGKvsGIG8w6LosIUfQNRtxg1nFZRIiibzDiBrOOyyJCFH2DETeYdVwWEaLoG4y4wazjsogQRd9gxA1mHZdFhCj6BiNuMOu4LCJE0TcYcYNZx2URIYq+wYgbzDouiwhR9A1G3GDWcVlEiKJvMOIGs47LIkIUfYMRN5h1XBYRougbjLjBrOOyiBBF32DEDWYdl0WEKPoGI24w67gsIkTRNxhxg1nHZREhir7BiBvMOi6LCFH0DUbcYNZxWUSIom8w4gazjssiQhR9gxE3mHVcFhGi6BuMuMGs47KIEEXfYMQNZh2XRYQo+gYjbjDruCwiRNE3GHGDWcdlESGKvsGIG8w6LosIUfQNRtxg1nFZRIiibzDiBrOOyyJCFH2DETeYdVwWEaLoG4y4wazjsogQRd9gxA1mHZdFhCj6BiNuMOu4LCJE0TcYcYNZx2URIYq+wYgbzDouiwhR9A1G3GDWcVlEiKJvMOIGs47LIkIUfYMRN5h1XBYRougbjLjBrOOyiBBF32DEDWYdl0WEKPoGI24w67gsIkTRNxhxg1nHZREhir7BiBvMOi6LCFH0DUbcYNZxWUSIom8w4gazjssiQhR9gxE3mHVcFhGi6BuMuMGs47KIEEXfYMQNZh2XRYQo+gYjbjDruCwiRNE3GHGDWcdlESGKvsGIG8w6LosIUfQNRtxg1nFZRIiibzDiBrOOyyJCFH2DETeYdVwWEaLoG4y4wazjsogQRd9gxA1mHZdFhCj6BiNuMOu4LCJE0TcYcYNZx2URIYq+wYgbzDouiwhR9A1G3GDWcVlEiKJvMOIGs47LIkIUfYMRN5h1XBYRougbjLjBrOOyiBBF32DEDWYdl0WEKPoGI24w67gsIkTRNxhxg1nHNV5EZLN+r3AdcaHpXw2jFRrsDZCXAAAAAElFTkSuQmCC" alt></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 19.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">How many edges will a forest with <em>v </em>vertices and <em>c </em>components have?</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0px; font: 29px Helvetica; text-align: left;"><span style="font-family: times new roman,times; font-size: medium;">The graph <em>G </em>has the following cost adjacency table.</span></p>
<p style="margin: 0px; font: 29px Helvetica; text-align: left;"><span style="font-family: times new roman,times; font-size: medium;">\[\begin{array}{*{20}{c|ccccc}}<br> {}&{\text{A}}&{\text{B}}&{\text{C}}&{\text{D}}&{\text{E}} \\ <br>\hline<br> {\text{A}}& {\text{ - }}&9&{\text{ - }}&8&4 \\ <br> {\text{B}}& 9&{\text{ - }}&7&{\text{ - }}&2 \\ <br> {\text{C}}& {\text{ - }}&7&{\text{ - }}&7&3 \\ <br> {\text{D}}& 8&{\text{ - }}&7&{\text{ - }}&5 \\ <br> {\text{E}}& 4&2&3&5&{\text{ - }} <br>\end{array}\]</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Draw <em>G </em>in a planar form.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Giving a reason, determine the maximum number of edges that could be added to <em>G </em>while keeping the graph both simple and planar.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">List all the distinct Hamiltonian cycles in <em>G </em>beginning and ending at A, noting that two cycles each of which is the reverse of the other are to be regarded as identical. Hence determine the Hamiltonian cycle of least weight.</span></p>
<div class="marks">[10]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The complete graph <em>H</em> has the following cost adjacency matrix.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><img src="data:image/png;base64,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" alt></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"> </p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Consider the travelling salesman problem for <em>H</em> .</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">By first finding a minimum spanning tree on the subgraph of <em>H</em> formed by deleting vertex A and all edges connected to A, find a lower bound for this problem.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the total weight of the cycle ADCBEA.</span></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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">What do you conclude from your results?</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Sameer is trying to design a road system to connect six towns, A, B, C, D, E and F. The possible roads and the costs of building them are shown in the graph below. Each vertex represents a town, each edge represents a road and the weight of each edge is the cost of building that road. He needs to design the lowest cost road system that will connect the six towns.</span></p>
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" alt></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Name an algorithm which will allow Sameer to find the lowest cost road system.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Find the lowest cost road system and state the cost of building it. Show clearly the steps of the algorithm.</span></p>
</div>
<br><hr><br><div class="specification">
<p>Consider the complete bipartite graph \({\kappa _{3,\,3}}\).</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Draw \({\kappa _{3,\,3}}\).</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>Show that \({\kappa _{3,\,3}}\) has a Hamiltonian cycle.</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>Draw \({\kappa _{3,\,2}}\) and explain why it does not have a Hamiltonian cycle.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>In the context of graph theory, state the handshaking lemma.</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>Hence show that a graph <em>G</em> with degree sequence 2, 3, 3, 4, 4, 5 cannot exist.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Let <em>T</em> be a tree with \(v\) where \(v\) ≥ 2.</p>
<p>Use the handshaking lemma to prove that <em>T</em> has at least two vertices of degree one.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Draw the complement of the following graph as a planar graph.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><br><img src="data:image/png;base64,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" alt></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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">A simple graph <em>G </em>has <em>v </em>vertices and <em>e </em>edges. The complement \(G'\) of <em>G </em>has \({e'}\) edges.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Prove that \(e \leqslant \frac{1}{2}v(v - 1)\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Find an expression for \(e + e'\) in terms of <em>v </em>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) Given that \({G'}\) is isomorphic to <em>G </em>, prove that <em>v </em>is of the form 4<em>n </em>or 4<em>n </em>+ 1 for \(n \in {\mathbb{Z}^ + }\)<span style="font: 7.0px Times;"> </span>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(iv) Prove that there is a unique simple graph with 4 vertices which is isomorphic to its complement.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">(v) Prove that if \(v \geqslant 11\) , then <em>G </em>and \({G'}\) cannot both be planar.</span></p>
<div class="marks">[14]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) A connected planar graph <em>G </em>has <em>e </em>edges and <em>v </em>vertices.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Prove that \(e \geqslant v - 1\).</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Prove that <em>e </em>= <em>v </em>−1 if and only if <em>G </em>is a tree.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) A tree has <em>k </em>vertices of degree 1, two of degree 2, one of degree 3 and one of degree 4. Determine <em>k </em>and hence draw a tree that satisfies these conditions.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) The graph <em>H </em>has the adjacency table given below.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[\left( {\begin{array}{*{20}{c}}<br> 0&1&1&0&0&0 \\ <br> 1&0&0&1&1&0 \\ <br> 1&0&0&0&1&1 \\ <br> 0&1&0&0&0&0 \\ <br> 0&1&1&0&0&0 \\ <br> 0&0&1&0&0&0 <br>\end{array}} \right)\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Explain why <em>H </em>cannot be a tree.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Draw the graph of <em>H </em>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(d) Prove that a tree is a bipartite graph.</span></p>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The diagram below shows the weighted graph <em>G</em> .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><br><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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" alt></p>
</div>
<div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) (i) What feature of the graph enables you to deduce that <em>G</em> contains an Eulerian circuit?</span></p>
<p style="margin: 0px 0px 0px 30px; font: 31px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (ii) Find an Eulerian circuit.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) Use Kruskal’s Algorithm to find the minimum spanning tree for <em>G</em> , showing the order in which the edges are added.</span></p>
</div>
<br><hr><br><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The graph <em>G</em> has vertices <em>P</em> , <em>Q</em> , <em>R</em> , <em>S</em> , <em>T</em> and the following table shows the number of edges joining each pair of vertices.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Helvetica;"><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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" alt></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Draw the graph <em>G</em> as a planar graph.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Giving a reason, state whether or not <em>G</em> is</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) simple;</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) connected;</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) bipartite.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Explain what feature of <em>G</em> enables you to state that it has an Eulerian trail and write down a trail.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-size: medium; font-family: 'times new roman', times;">Explain what feature of <em>G</em> enables you to state it does not have an Eulerian circuit.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Find the maximum number of edges that can be added to the graph <em>G</em> (not including any loops or further multiple edges) whilst still keeping it planar.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) A graph is simple, planar and connected. Write down the inequality connecting <em>v</em> and <em>e</em>, and give the condition on <em>v</em> for this inequality to hold.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Sketch a simple, connected, planar graph with <em>v</em> = 2 where the inequality from part (b)(i) is not true.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) Sketch a simple, connected, planar graph with <em>v</em> =1 where the inequality from part (b)(i) is not true.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iv) Given a connected, planar graph with <em>v</em> vertices, \({v^2}\) edges and 8 faces, find <em>v</em>. Sketch a graph that fulfils all of these conditions.</span></p>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; line-height: 12.1px; font: 22.5px Times; color: #2c2728;"><span style="font-family: 'times new roman', times; font-size: medium;">The table below shows the distances between towns A, B, C, D and E.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; line-height: 12.1px; font: 22.5px Times; color: #2c2728;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; line-height: 12.1px; font: 22.5px Times;"><span style="font-family: 'times new roman', times; font-size: medium;"><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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" alt></span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; line-height: 12.1px; font: 22.5px Times; color: #2c2728;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; line-height: 12.1px; font: 22.5px Times; color: #2c2728;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Draw the graph, in its planar form, that is represented by the table.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; line-height: 12.1px; font: 22.5px Times; color: #2c2728;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Write down with reasons whether or not it is possible to find an Eulerian trail in this graph.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; line-height: 12.1px; font: 22.5px Times; color: #2c2728;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) Solve the Chinese postman problem with reference to this graph if A is to be the starting and finishing point. Write down the walk and determine the length of the walk.</span></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 style="margin: 0.0px 0.0px 0.0px 0.0px; text-align: justify; line-height: 12.1px; font: 25.5px Times; color: #2c2728;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that a graph cannot have exactly one vertex of odd degree.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">A graph <em>G</em> with vertices A, B, C, D, E has the following cost adjacency table.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">\[\begin{array}{*{20}{c|ccccc}}<br> {}&{\text{A}}&{\text{B}}&{\text{C}}&{\text{D}}&{\text{E}} \\ <br>\hline<br> {\text{A}}& - &{12}&{10}&{17}&{19} \\ <br> {\text{B}}&{12}& - &{13}&{20}&{11} \\ <br> {\text{C}}&{10}&{13}& - &{16}&{14} \\ <br> {\text{D}}&{17}&{20}&{16}& - &{15} \\ <br> {\text{E}}&{19}&{11}&{14}&{15}& - <br>\end{array}\]</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) (i) Use Kruskal’s algorithm to find and draw the minimum spanning tree for <em>G</em>.</span></p>
<p style="margin: 0px 0px 0px 30px; font: 24px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (ii) The graph <em>H</em> is formed from <em>G</em> by removing the vertex D and all the edges connected to D. Draw the minimum spanning tree for <em>H</em> and use it to find a lower bound for the travelling salesman problem for <em>G</em>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 24.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Show that 80 is an upper bound for this travelling salesman problem.</span></p>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Use Kruskal’s algorithm to find the minimum spanning tree for the following weighted graph and state its length.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"><br><img style="display: block; margin-left: auto; margin-right: auto;" 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" alt></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 22.0px Helvetica;"> </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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Use Dijkstra’s algorithm to find the shortest path from A to D in the following weighted graph and state its length.</span></p>
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<p style="font: normal normal normal 25px/normal Helvetica; text-align: center; margin: 0px;"><img 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" alt></p>
<div class="marks">[7]</div>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The adjacency table of the graph <em>G</em> , with vertices P, Q, R, S, T is given by:</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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" alt></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) Draw the graph <em>G</em> .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) (i) Define an Eulerian circuit.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (ii) Write down an Eulerian circuit in <em>G</em> starting at P.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(c) (i) Define a Hamiltonian cycle.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (ii) Explain why it is not possible to have a Hamiltonian cycle in <em>G</em> .</span></p>
</div>
<br><hr><br><div class="specification">
<p><img 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" alt></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">In the graph given above, the numbers shown represent the distance along that edge.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Using Dijkstra’s algorithm, find the length of the shortest path from vertex <em>S</em> to vertex <em>T </em>. Write down this shortest path.</span></p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Does this graph have an Eulerian circuit? Justify your answer.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Does this graph have an Eulerian trail? Justify your answer.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 12.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">The graph above is now to be considered with the edges representing roads in a town and with the distances being the length of that road in kilometres. Huan is a postman and he has to travel along every road in the town to deliver letters to all the houses in that road. He has to start at the sorting office at <em>S</em> and also finish his route at <em>S </em>. Find the shortest total distance of such a route. Fully explain the reasoning behind your answer.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Consider the following weighted graph.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="font: normal normal normal 27px/normal Helvetica; text-align: center; margin: 0px;"><img 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" alt></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(a) (i) Use Kruskal’s algorithm to find the minimum spanning tree. Indicate the order in which you select the edges and draw the final spanning tree.</span></p>
<p style="margin: 0px 0px 0px 30px; font: 27px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> (ii) Write down the total weight of this minimum spanning tree.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(b) Sketch a spanning tree of maximum total weight and write down its weight.</span></p>
</div>
<br><hr><br><div class="question">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">The vertices of a graph <em>L</em> are A, B, C, D, E, F, G and H. The weights of the edges in <em>L</em> are given in the following table.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><img style="display: block; margin-left: auto; margin-right: auto;" 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7cKybS0K2HJML8TpAga397b+T6XdXKrk0q9jrufz+32a+OK8PWDjW/S7rwU/dl6/i4URCJ8/3gnL3/wdQHmhiKg61VzE1cqWvVkFO7gphsKiMg/3Mxk1crmLutKWDXMr6x2GTTurlAqTRafuwwad2fWoluY4fLjybukDuZKNqQTXe/Jw2EtUCI69yp7ynX9TaXSyAXU6rwfrBqQSvE49yrrI8Waz73KtsVQEEFja+VaivfC4Gz0rlkEjS29CoiGUmnY6uDOqnX0tmJAKsVjcJ2eLx21o8vQq1WVthH0DP3RDbxGtfTTVJv71AAGhgnAZdDY1Wk1YjCVko3wW2X3kyxCEmB4WAcoGJBKIyPZ2VTfy9A1DDdWilW3vdJtMkW7uf1QeWeNkEpMAAoGpNLIEFFwXK/9XN1YczLFB2MbHKkZiIhEeHpUKeo0JVvmVRgMETS+Vd3XSwzLeAdnvTNfPxTmfeNZpX/nEqaS/VUgIiIRtY/u/8XtCCLR8Q6KTma77l8o/VOGrhTmtH5ZDIDJKox69vT7/ZhKpfiBhdX6LW3NwCCVDA+Nz+Bujdy4Xfi1baudl9L/ENDk0AVo376RwVQSHfd+Dk8GpGdAKg1G16vmRreTRNc7yNtMpXOvUlzZOzj92zcy9BSliAK3urGGpyjTNCCV4nHh17ad+ClKij40S3dSDQVx1tpd36y88MMeUS/0Hm2Vjtqr9WRAMgzcvpG5b5wUqw1XrUUlLf1KpG+Y3wly7BFbuQ+krB8EdYAI/eeVQftcC9846bYbu/nk/+6qfvFKHWByaAFEu7n7RPlZrVEDvp1rH8DBAAAHAwCEVGIC4GAAgIMBAEIqMQFwMADAwQAAIZWYADgYAOBgAICQSkwAHAwAcDAAQEglJgAOBgA4GACgJJUwMTEx+cwbju10BICDAQAOBgAId3BMABwMAHAwAEBIJSYADgYAOBgAIKQSEwAHAwAcDAAQUokJgIMBAA4GAAipxATAwQAABwMApJJKV3795vyP9G7W/Ksv/xvTCOmfGQz1poyGAMaGqmGkQeNG5Ujj++rqByEK3tR2+l+az5cO3yxt6QhJw2fOvV7ov1Cr7WP9IFgHkOq1kuh6B/lMdrxEdzd4fVgupJpKOk0ZjQAMDiVDr3P89LBfuKPfHzH94rDirLVbyA8qCvVC79Gm6los1yrMP/dE5B9ubjzywh6JjnewvVk7/frKKtYPgnUAKd/BXfm125OpRES9zu9//1eq10qkXOTMHMDMUDGIf5+cfBq+7LQrRhNNF58T7WZRsXXtEq7CrHNv4gh0vWpOoo679YNgHUDGUunqz9bvcn1NJhDKP7vSqTQ5zr3KetoVo7teNbc2vBzoelXVu+klXIUZ554IGltj7STOvco3X19+z/pBsA4gQ6nUbTf270t2W5pAKP8sUmlkWKkYfeHXtp1Modz4EImO9z8Hh6eKe1tLuArT5970n5x7lfWv725g/SBYB5B2KiVTugfcBEL5Z5FKw9H1qkUbvaQHW1pZtZZTBgDmhnYqTWcQUknFoH2tJML3j/dwraQ5tA0Xfu0H5a1uPUA3cJ/Uaz9uZrJOf5c3bYCxgVSyDiDsKykDrHepnOBE/i/VxgflKvoagG678cN/uWeCSISvH2ysORuK12tLuAoM7uCGDYemp0p/c/lVMNwmk0x/BkdEl4H3LpRHyALG/o8rf60kOn8/fK4VScqAyc3d6LS+cUvhzaAMMDvM7HaP/slkE02zAPPDOoDMp1J0+r+Nf8o+G4BU0jGI0Dt87MU7zCJqv2qepPbBfP/JNfWPnLQBhgeeDLAOIO2nKLMjn0P3Qt+tl75R2PZGKqkakv6guncuqoD+xdFa/BSliNpH5Zv7rY7K1tISrsLMcw9PURowGP3Gyexurl9GyP7IYGg0ZTQDMDdUdlUGXdQN3MmrAQaM8PSoUnRW7Rsnnzv3euHp03Iu62SK1edyD0pYPwjWAYRv5zIBcDAAwMEAACGVmAA4GADgYACAkEpMABwMAHAwAEBIJSYADgYAOBgAIKQSEwAHAwAcDAAQUokJgIMBAA4GAChJJUxMTEw+84ZjOx0B4GAAgIMBAMIdHBMABwMAHAwAEFKJCYCDAQAOBgAIqcQEwMEAAAcDAIRUYgLgYACAgwEAQioxAXAwAMDBAAApptIXygbstUK5CksGDoQ4a+2u26kZEAWe+7JeKqq1GzJjMDHUAcO63cWq21YuPicJ6Abe8a+1ndvTPV1E6D+vbGayTm5HtrXBEq9Cf4jQP35ZLxXUqncYAJgYytdKwxJLt4eNKfstQ/dTT6Vex93PW6lkItrN7bvle+ongQGDoaEKuPiX++p92IvLd6gfBxnAZdDY/b70XX6swtfA49e+2zx4HYp+xd7vpGqZL+0qEBGJ8O1hqZAv1X71grR+NyxkqN/BJW1Obo+3yxXBP05STSUR+U/KP/5YzlmrrzTZqdGGQX+oAcTHtyfDGm/pNqSb1ZVzvECt6HoHeZmCU0u6CkT98nvrsg3ETQLMDYOppFKxO0Eo/RwREUWn9dITv/O6ilTSGybuo9vNoty1iRZgRipdBo27Y3XvVqVA7YVf287vuh29SNIAmBymUqkXnj4t77qpp1LcbqjrIZU0h+4Va+A1K3sPPPX3hYFUmv4TyRNjSVdBBI2tTLHaeFEvFZxrURFUN5WGs5RyKonIf1LuF0VGKmkPDcC5V1l3Mlkns7ZZcYPU2mR+TQatRCpdBo27zkblyA8FEUUfmqVCmv2vjI9lvlaKTg8Pjge/mpFK2sPApz/PK5uZrPJ9BFJJFTC5nSeCxpbqhz/WjwAt877Shf/452EvjRVMJfvtCaeHVr+ZpbyD47AKX3McFghg06WS5n8GpzBUAGMdJkZn+n1rca00HJM9GhcKmLPbPXXVcO13u6fa80q2xtQGGB5Lm0oTYwWvlUwPEwDR9Q5uW7yDW9EnA0TXO8jn9sdvHdJtC2h0KKfSzEaV6gi9fwCpZGAoAXoddz8f77OK8PWD4l6zndanP7PvU1byKUpx1totDHqFivD9490tmdaYBgBGh1IqzfjGyZ26r97mfplTKfn4Sf0uWttgZqj9coraR+XBrXShXHP9UKVrrjxg+Etxxm178g2Y5GOphRgWMtQBUfCmFn/vR7I1phmAuaF+B2cWseIADgYAOBgAIKQSEwAHAwAcDAAQUokJgIMBAA4GAAipxATAwQAABwMAhFRiAuBgAICDAQBCKjEBcDAAwMEAACWphImJicln3nBspyMAHAwAcDAAQLiDYwLgYACAgwEAQioxAXAwAMDBAAAhlZgAOBgA4GAAgJBKTAAcDABwMABASCUmAA4GADgYACCtVIoCz31W3YgL8eV26q9Oguiq6738Lb3OSyIKTlqvauWbipXGtAHGhnUDAPoGEfq/vaqVc9n0CnKaHqqA8eIZ/alaQkMtlfr9CLNObqf+Ou6HJ0LfrZVz2TR754qg8e29ne81TgJNgMFh3QCAnqH/piiUa3/1VPuL6AHMDEXAjNqwWpWFZFNJRP7hZibrZLan62mJjns/l3JHb61a0SYAZoZ1AwAaBhH5h5vyDcTNAYwNJYDoek8ejjXdOvcqe2rFMEkllZLy6bMvz3od91mKd3CEVALAviE6rW+s33fPtHtE2j8ISoBeGJyNVn0UQWNLrwKiXCqJoLGld3k2E6Hx00glAOwaBu3Ymv0dpUyhXPsjvaZ4pocJwGXQ2NXJaNlUugrdPSeTdTK3yu4n1f/pDITGTyOVALBqEO1m8dZm5YUf9uKSwWubS1s22wBAtJvbDzXfj0gl+6cCBwMAioauV82N9pLo90db1hYj+gARNL7V6zAim0pJFffRO7gkqpIpl1nLmErxnezMmUpPOvv9EQ23J7QOUDLM6HCj0/NmCU+DiXEZNPb0+/1I7nbHHwFOdrwZHh3p9+QyppLxYd0AgKJhqr+ObGtMXYDRoQvQvn0jpScDeqH3aHNG+nxqlW45mVSfVyIipBIAtg3nXmU9P+zNKd0aUxtgcmgC9G/fSP0pyn4wbVSaXvIUZXytJB8QSCUOBgCUDaLj3s8Vyo0PEZEI3x6W7km1xtQHGBzab0bd2zfS+cbJ8OH6+CZ2s/KsdSzZFTBGKACIJp4otdY718iwbgBAwyCi4I96qeBkVFpjmgAYG1oA0W7uPvFVn4oYNeDbufYBHAwAcDAAQEglJgAOBgA4GAAgpBITAAcDABwMABBSiQmAgwEADgYACKnEBMDBAAAHAwCUpBImJiYmn3nDsZ2OAHAwAMDBAADhDo4JgIMBAA4GAAipxATAwQAABwMAhFRiAuBgAICDAQBCKjEBcDAAwMEAACGVmAA4GADgYACAkEpMABwMAHAwAEDSqXTl129OPFmw1wr/HVdWSuadui9RtljrQIyVDUin1FY38I5/re3cnqojI8K3h4Mvjh+0JJuCWT8bFAHmGoFZPwIcDFgFUrpWGhbJHS2ZnlSMnaxR+XUIyZ9IRq/j7ueTlUij1NZl0Nj9vvRdfvqVRn+2nr8LBZEI3z/eyUuWfLJ+NigBkpNhdC5rGUYOBqwCqd3BXfm125nsZHHu0C1nsk4me7vmy1Wi1DkQ0Wm9qFuOUwUg2s3i2ngqXX48eTds0zdZYX4BBtNDCXDu1Q69sDf8g65X3V7WMowcDFgFWvJU6v+KWNusPGslJTGVholUGv/7oLG1cShV/sr62aACEP8XfBw9BpdB4x5q76UNuI6rsLSpNNbdYW2z4qbXGvBzqdQNvEa19NPY765FGEwPAwDRbm7/0OrIvXCTAO1h3YBVIL1Umj1TvYMjIhH6x8+qG2sT+1yLBcxLpeGmY7HqtqUw1s8GfYBmJXnrR4CDAatAy32tNDpExzsoKvcUMHsHJ8LTo0rRyRSkmhpbPxu0AbqV5K0fAQ4GrAJdn1QilQ1mdcCX9pW+/B8YMsy5YrXRntBEH2fJn7DfpdJ+s9JJ0DVZheuSSqLdLN5L6dPQL4fOZdC4u9hUMj00AUb6OOsAjAzrBqwCXatU6noPDhTXYwGpdO5Viqt0B2emj7POjxsZ1g1YBVrqpyhF6P8Wt58T4dvDSk32Yy91wHQqibPW7vpm5YUf9vpdPLdKR+1r/2RAMgz1cdb5cSPDugGrQEv9jRMRvn6w0e/Wu1N/daL8WIAkYOI52mTvoNtu7A7+PLdTd6UbFVo/G3QAIji6r/oBqBGAqWHdgFUgfDuXCYCDAQAOBgAIqcQEwMEAAAcDAIRUYgLgYACAgwEAQioxAXAwAMDBAAAhlZgAOBgA4GAAgJJUwsTExOQz/x9WB4RzmnkAAgAAAABJRU5ErkJggg==" alt></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;"> </span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 26.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Draw the graph <em>L</em>.</span></p>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">In any graph, show that</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) the sum of the degrees of all the vertices is even;</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) there is an even number of vertices of odd degree.</span></p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">Consider the following graph, <em>M</em>.</span></p>
<p style="font: normal normal normal 25px/normal Helvetica; text-align: center; margin: 0px;"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAWIAAAFjCAIAAACIYpaIAAAgAElEQVR4nO2dz4sj6XnH/Z9IIEjoIW1mWRyPTRjRkMPuxDTLHkTDJmDS2zBq2PXi5CQNs5g4AfWhh0xg+mR6SE/ABOJq7EOMtSCCQQeDQHj30NTNbDU6DMI0DbUMg3hzeEbv1pSk0lv1/n7r+0EHu3emp2ak/tRT7/s83/c7DAAACvmO7QsAQBMvR/37rfeeTm4Xtq/Ee6AJECaL66i702w17j8avbR9Ld4DTYAg+SY+//GDwx8/aDR3+6Mb21fjO9AECJGb0aOdj55fXV0+vNfa+ezy+pXtC/IbaAKEx+Jm9Pjdh9H1YnEzerzbuPPB+RXWJ2SAJkBwLK6e7//tmyWJm9GjnWZr/zyGJySAJkBgLG4nTx9864U/T04/xEKmJNAECIyXo/79VqOZe2EhUwZoAgTFIj7/YOfx6CbzjLG4er5/p5X7IigDNAEC4var54f33j2dvH7rq69n0Setxp0H/ShGq1UloAkQCq8nT+4unzIOo9nyq7Pok8zTxyeXs9eF3wWsAZoAAGwBmgAAbAGaAH6TpumLi4u9drvVaO6125dRZPuKAgSaAH5zMhjk9j5fXFzYvqjQgCaAxyRJstoi0Wo05/O57UsLCmgCeMxlFK3VRBzHti8tKKAJ4DHD4RCaMAA0ATxmPp+vOuLu7nfTNLV9aUEBTQC/GY/Hq6bo93owhUKgCeA9cRyfPTvb/9GPWo3ml3/8khYsjrtdmEIV0AQIhOl02mo0x+MxW5YYx90utjyUAE2AcNhrt8+endH/JlPstdtJkti9qgCAJkA4nAwGe+02/79Jkuy12zCFPNAECAeqIKbTKf8KNwU9jIBqQBMgHNI0bTWa/LmDSJLkoNPhyxagAtAECIqTweCg08l9MU3T424XpqgMNAGCgvoyVxcjuClytQYQAZoAQUF9mWuHRNM0PXt2BlNUAJoAoXHc7a4+d3DIFCeDAZqvxIEmQGjQc0dBY9WLiwu0aZYCmgChQSEUw+Gw4NfwNk2YQgRoAgTIQadz3O0W/xq0aYoDTYAAoceKrQMd0+kUbZoiQBMgQOi5Q6RLgrdpZns3QQ5oAoTJQadzMhiI/EoyBZqvCoAmQJjQxqfgCiXaNIuBJkCYZOMnROCmQH7/KtAECJZs/IQIaZrSqR9o08wBTYBgOXt2lo2fEP9dZAq0VHCgCRAsq/ETgpAp0HzFgSZAsKyNnxAEbZpZoAkQMrnYu1KQKQ46HTRfQRMgZOhHvfLP+Xg8RpsmgyZA2BTETwiC3F0GTYDgKY6fEAFtmtAECJxNsXelmM/ndW7ThCZA4NBzR3H8hAi8TfMyipRcmEdAEyB8ROInREjTtN/r1bBNE5oA4SMYPyFIDXN3oQkQPiKxd6WoW5smNAFqgXj8hCCXUVQfU0AToBaUip8QhDd0B99SAU2AWlA2fkKQmuTuQhOgLuy122qfO4g6tGlCE6AuVIufEIGbItTmK2gC1IXK8RMiJEkScJsmNAHqgkz8hOD3J1Mo3Hl1BGgC1AiZ+AkRuCkCa76CJkCNkIyfECS8Nk1oAtQI+fgJQcgU/V4vjOYraALUC/n4CUFokCSMNk1oAtQLJfETggSTuwtNgHqhKn5CkDDaNKEJUDtUxU8IEsex722a0ASoHWrjJ0TgbZqamrt0A02A2qE8fkLwDz3odDxt04QmQB1RHj8hAm++8s4U0ASoIzriJ0TwtE0TmgB1RFP8hAhpmp4MBn6ZApoANUVT/IQgVM6cDAZetFRAE6Cm6IufEL8AX5qvoAlQU7TGT5S6BvdNAU2AmqI7fkIQMsVBp+Ny8xU0AeqL7vgJQabTqeNtmtAEqC9m4idE4G2acRzbvpY1QBOgvhiLnxCBTOFm85VXmljMJr/+xaP37rQazVaj2Xqv/3x0dT365W9mr21fGfAVY/ETIjjbpumNJhazLz5/705r5+Mn0WS2oK/84aK/32q8/2Rya/vqgK+YjJ8QgZvCkRqH8EQTt189P7zX2nn4/Ormra8v/nT58ACaAJUxHD8hQpqm/V7PhV0YjheaeHUdfbbbaO72Rzf5/7S4nfziyeiljasCgXDc7ZqMnxDEqdxdLzTx9eXhO63G/UfQAdCA+fgJQdxp0/RBE68nT+42V9YgyB3NN6/DaGbt+oDfWImfEISWTqybwgdNLK6e799pNd45ir7O/YfbydMH6x9GACiBlfgJQXhDt8V1Vh80wb6Jzz9qNZq7D6Prxdv/ZRYdNZrvnk6wIwpkoOcO67X9Jsbjsd02TS80wRbXUXen2Wrsfz56WxTQBFCBxfgJQXibphVT+KEJxha3VxdHO81W497RaTSZvWKMMXYTR48fQBNABXbjJ0Sw2KbpiyYYY4zdxqMo04VJjZjR/y6tAUB1aFvB9lVsIUkSK22aXmkCAG3Qc4f7Afm8TfMyioz9oYFoIo7js2dn/V7v7NmZmzN2wHEciZ8QwXzubgiaoB2j7MvltSjgLI7ETwhisk3Te01QT37utdduO7u5BZyF7jceVaNkin6vp/vT7r0mVksJern/kAlcg547nBrN3MplFBlo0/ReE9TNuvrCcweogFPxE4LwNk19Yynea4Ia8ldfv//9721fGvAP1+InBCFT6Gu+8l4TbPmEln3d3f0unZXi4NgfcBla6jK516gKrW2aIWiCMUanrZFQL6MoTVPq0t9rt92c/APO4mb8hAjcFCWfuN8etl6+dg9PfzWKaSg7EE2sPcEpjmPaXrY7XQf8wtn4CRGSJDnodCqszS3i8w++HcJe3Ma/e3J4r9W4143+tAhGEwUN+cPhkDrhX1xcYJcUbMXl+AkRKubuzqKjt7MaFvH5B41ma/88XgShia3jffP5nJ5KDjodbJSCrRx0Ov1ez/ZVVKdKm+aKJtjN6NFOQJoQDAuYTqdUj2FpExTjePyECGmalmvTzGni9uqyvx/UQ4e4+7NLmz6uZgMzuB8/IQiZ4mQw2K68WXS0uoq5zIXzXhMVniSTJKGA8+Nu16POXGAS9+MnBKH74vY2zbericVscnn68W6juXt4cXW78F4T1A9T4SECS5ugAC/iJwThbZpf/vHLk8Gg3+v1e718rbS6NsFejvr3W407H5xfea8Jme7aNE1paXOv3cbSJsjiS/yEIGtHn95atghYE0qOio3jmJY2+70eljYB4VH8hCA/+Ovvr5pi+dBNIfUZTSxmk//sP2g0WzufXV6/8lsTqg6eT9OUJu08bdQFOvArfqKYOI7Xjj5dRtGmLsxW497R6X+P4hvm+xKm2jeSL20edDpY2gTexU8UUKiJ7XisCU1lIZ2JQN8ZS5t1xsf4ibWMx2Pqtqocy+KxJvQtMvHWlPJTNCAofIyfyDIej2ndjfZ3c44Q7zT1WBO6t6yyS5uYHKsnnsZPsIwgDjqd4XCYpmmapj/4/ptVzINOhwapBb+bx5ow0wCTXdrEM0jd8DF+IiuIbC0ss0jvqyZoScbME8F8PsfSZm3xJX6CduvWCoL+K624VVuU9VUT5odzptMp/UMLdciDUHA/foIEQR/OVUEQVErQ3a7Cp9dXTRx0OuYdj1CsGuJy/ERWEP1eb9NyPj06UY92tRVZLzVBf21b7xwPxcLSZk1wMH4iJ4ji5wha7I/jeK/drtZA4KUmXFh/5m8SJseCx6n4ifl8zjOitwqCLe+pZ8/OZMoiLzXhyG42QrFqgiPxE2UFQdBvmc/nMk2l/mnCtd44hGLVAbvxE1lBnD07E/+Y8VKCLWuiahfgnyYc7LTH0mbw2IqfSJKkmiAIXkowxo673corLP5pwtm5vSRJeN6/UxYD8piPn4jjmPYvqwmCvV1KMMZkanD/NOF4CgBCsYLEZPyEvCAI+ib026kdsbLmPNOEF5lCfGkToVghYaCM5YLgh9fJfKus1ypnQRKeaYKetby4SyMUKzC0LopNp1NVgiD6vd5eu80/eJKO80wTfuUd81As5P0HgKYttuyst6rxQiolsh+5g05H5gfHJ0243DZbAEKxgkFtw052lFPt/DGVEvwbkuBkblQ+acL9IZwC+NImQrH8RVX776ZZbyWslhK0oidzi/JJE76M9G4CoVi+Ixk/kabpcDjUJwgiV0qw5XiozM3JG034GBCyFixtek21e5XIrLcSVksJxli/15O8v3qjCRfGvRSCvH9PoTdO3O+5UU7dVeRBp5MrJRhjlQdDOd5oovKovLMgFMtHxNfRS816K4G2bHM3HiUL/35owrVxL4XwvH+EYvnC1viJ+Xz+4uLCpCD4ha3eSpUceeWHJuivGmpHIybH/KIgfqLarLcS6Gdk9aFGydCaH5pwdtxLIQjF8oW1cc25WW/zT5FrSwkmNxjK8UMT8mswvsCfZpH37zLZbuDKYRAK2VRKMLnBUI4HmnAkO8gY2aXNUJ+zfIe8MJ1O+VFatgRBbColJAdDOR5owqkkQmNk8/7RXuEatD3vgiD4xRTk7stfngeacDDX2AxY2nSQbBjEcbdr3eB0Ts+mXgFVi3qua8LTcS+FZEOxsLRpkVwYBPVE276oN/XCpscKycFQjuuaUFU1+Q5CsSyydtbbhU16KiU21doK5xtc14Tv414KQSiWeQrCIFxo+aOb6KbNV/nBUI7TmiAdBtl8WRme94/JMa3kZr3XVnB2BwiKSwmmYjCU47QmAhv3UgVCsbQiHgZh9/NZXEowFYOhHKc1cTIYBDbupRAeioW8fyVUmPW2WO1uLSWY0gx6dzVhMvLcX7C0KY9MGISttTN+evCmX6B2i9BdTbiwkuwFaZrypc369KoqQX7W28pOHFUxxaWEksFQjruaOHt25sK+tC8gFKsU8/lcSRiElb4eKiWKFaD2NEN3NeFX1r4jIBRrK8pnvQ86HZPPHbkj/zahZDCU46gm1s7qAhEQirWJrCBOBgNV/ziGZ46ypwcXoPZW4agm6jnupRAeioW8f6Z51tvkLU2wlFA1GMpxVBOGC7kgyeb913YoJo5jA2EQxh6QBUsJ5QurLmoC414KyS5t1qpRTdW53iKYOdqWfi5EWgSUNxy5qAnJ05PBKrUKxcqOcpoJg6DpCd2b9/SXEvnrqBoM5bioCbUnNQKiDkubuVlvk0LU3QpIyw0iHZ86Dr5yThMY99JKqKFY4/HYliAI3anOq0f+bULhYCjHOU2o7R4DqwQWilUw6234MvQ9d6w98m8TtH6p9gKc00QdsvZdgOf9+xuKlR3lHA6HdtdctMZPiJcSTOlgKMc5TWDcyySeTo6Jz3qbRFP8RKlSgun5CXJLE2ZWjEEWHorlft4/TWo5KAhCU/xEqVKCNk2V/8u4pQkz+89gFR6K5ebSpsystzF0rL7TjVO8lNCkKrc0gXEvi2SXNt2ZHMvNerspCI7y+ImDTke8lGCqB0M5DmlCU70ESuFOKJZ8GIR51HZJ0+5Jqd0otYOhHIc0QbcyByveGmJ3aZMmtfwSBKF2zmDTkX+boN0WHZWgQ5rAuJdTZEOxjC1tKg+DMI+qj3HB6cGbUD4YynFFE7T8E0C3T2AYC8XKzXp72srB1GUglC0lmM7EPVc0gax9Z+F5/5oKWq1hEOZREj9RoZRgOpPoXdEExr0chy9tKpwcMznrbRLJDbs0TSuUEvJ/bgFOaMKFc9aACKpCsUIVBCHZ/kO1W9lSQsdgKMcJTVCJ5eN6VQ3JhmJVKK0tznobQ6aZmM7pqVBK6BgM5TihCYx7eUeFUKzpdBq8IDiVByu2Hvm3CVo6rfAniuCEJihlyPZVgNJklzYLfuwdmfU2SbU7n8iRf5vQMRjKsa8JKpbQfOkpxaFYNRQEUS1+onIpwTSPVtvXBMa9AiAbikVvpZuz3saosCovU0roHnSwrwnl8Z7ACtnJsToLglO2i0GmlNDddmRZE8jaD4bspBbJoubNcqV+dOfzeeVSguk/cNeyJjDuFQBZQRx3u+PxmP9fv0Kx1FIqfoIevStvZ2oaDOVY1oTy+XxgkoJZb49CsfQh+PEmoVR+9NY3GMqxqQmtfWNAK/P5/MXFxdZZb8dDsXQjeDCV4JF/m9A3GMqxqQmMe/lI2VnvwPL+SyGy9CZ4enABtPap9eHOpiY0JREDTeRGOUs9SCdJwvP+a9WVvzV+QrKUYDoHQznWNIFxL49QNevtad6/DMWL9PKlBDOSIGtNE1pPSQKqiONYbRgEX9o0GYplkeLGJ/lSwkyekzVN6N7pBZJonfU2ForlApvu9rT0KFlKaB0M5VjTBLL2nSU76/3i4kLTjzEPxXIq718Hm8YR6B9Z8p9X62Aox44mMO7lJubDIHSEYrnG2k972SP/NqF1MJRjRxOqYkWBKsbjMe1EWBnl5EubkqFYzrIalVDqyL8CzOwD2NHEQaejtbcUiOPIrLdkKJbj5OInVJUSSuJ5RbCgCYx7OUJu1tuF23ioS5u5fT1VpYSxBkULmtB3mgAQxPEwCK15/1agLiF67lBVSjCD24UWNIFxL1t4ca43URyK5SO8V/K421VSSjCDD++mNaHjcHewFY8EkYXn/fNQLH+hBwRavFdSShgYDOWY1gQ9pGHcyxi5WW/vGh+DmRyjGyT9RZQoz8BgKMe0JpC1b4yCMAjviOOY9mvF8/4dhP4Kqko5A4OhHKOayC7kAH0EcK73Wrj1PA3ppmVjVYv3JgesjWpC5jQkIEKoguBklzb9+iDR47bCVgCT4w5GNUGfYJN/Yn0I7FzvYrJ5/778Ten04K3xE4KYGQzlGNUExr10kCRJfQTB8Wtpk0qJ8XisKiPa8LG75jRhrLG0PoR9rrcI2VAsZ5c2s6cHqzp3x8xgKMecJjDupZDsKGc9BZHF8VAs2pLgalBSU5sZDOWY04Sqp7KaY37W2wucDcVaPfJPyWmYhnsUDWkC417yjMdjCKIYnvfvzuTY6pF/8mEr5p/fDWlC8MACsBZHZr29wKlQrE2nB6/GT5TC/E+TIU0cd7vI2q9AdpQTghCHh2LZzfvfdHqwZC+y+RxZE5pA1n4FHJ/19gK7S5ubSgkmHStvPtXJhCYM7/F6TZqmw+EQglBFmqZ8adPwP+amUoLJTS2YHAzlmNAExr1E8HTW2wvMh2JRl2TBPb/yUV20fmn4pmtCExj3KiY3yglBaMJkKBbtehb8MFfOpzM5GMrRrgmMexUQ0qy3F5gJxRI58q9yPpOVk3e1a0JJM0l40KQWBGEFHoqlKe9f8Mi/att/kpup1dCuCYx75Qh+1tsLsnn/arv+xE8PrtD+YHgwlKNXE6oGXcIgN+vt7KhSfcgubap6O8RPD67Qmmxr01CvJlSNzfpOrcIgvENhKJZ4KUGUHXQyPBjK0asJjHvFcUz79hCEy6ha2ix7enDZ++hxt2vluDyNmrD1HOUICIPwDslQLOpoKLW+WPap3FY3s0ZNGDu5zDVys94QhEfIhGLRkX9l327xNX6LwU4aNWFlg9cuCIMIA573Lx6KVfnIP/GOAYtj1ro0UbdxL8x6h0epybHKpweLx09YHHrQpQnJGTiPgCAChodiFef9S54eLNgxZX4wlKNLE3UY98Ksd03goVibljYrlxKESH6ElcFQji5NWGkpNUOaphBE3cgubeZ+ViVLCSZWetOzia2eXS2akE/7cxPMetectaFYB52O5OnBIvETVgZDOVo0Ed64FwQBONmlTdp9kH8W2Bo/YXffUIsmDjqdYMa9MOsNVuGhWHTbkP+G9NxRsPlq9ylevSaCydqfz+cQBCiA59zIh2IVx09Y/5lSrwn6t/O69RCz3kAEOjpYVShWQfyE9TRZ9Zo47nb9HffKCuJkMIAgwCb46cEss7QpMzlW0GRpazCUo1gTlaO7rINZb1AKKiWyX5EMxSoYlbQ1GMpRrAkfx73iOIYgQCnoc7664ZUNxaqwHbYpeMH6rVexJirHilsBs96gAnROT8HnvHIo1tr4CWrfsjv3oFITMoeUGCY7yvni4gKCAOLQmuXWn9vs0qbgM8jaHQ0Xzt9VqQkvxr0w6w1kKDjyb5UKoVirPUcujEep1IT5E1BLMR6PKUQAggCVKTjybxPZUKytn7rVDmYXmhVVasLZrH3MegMllColcr9RMBQrNw9ldzCUo0wTbo57ZUc5h8MhBAFkqFBKZOGhWMVLm9nbrd3BUI4yTZAs3fk5xKw3UEvlUiIHnwDYFIqVfXi3OxjKUaYJi9E6WWhSC4IAytl6erA4xaFY2a2Afq/nQk+zGk1YH01hmPUGOqEWSbU3wk2hWNnGAkfindRowu7Wbm7W2/EdWeAjVEooby/etLRJm6Au3H0JNZqodrayPAiDAAYoe+RfWZIk4Xn/9Bn+7W9/22o0/+mnP201ml999ZWmP1ccBZqwMu6FWW9gDPHTg2XgoVj/8fTp3/zwh/TZppf1J2gFmtgavKMWCAKYRHcpkfuzeChW7uV9s7axZtLcrLdfc6jAU8yUEhwa9Fp92V2hkNWEmXEvhEEAK9Aiosm9hk2asNuIKasJ6hLTt7mAWW9gEfrsmfzU0TPO6mv5I7a4jf/vV6cf7775+v6j81F8/cXnj0c3Oq9KVhN0k1dyKTkgCGAXurGbz4P5h4/+PueIZYfVq9no5w8adx70z0fxDWOMLWaT6PRop9k6jGY6L0lWEzrGvTDrDVxA8si/atB99+N/POSO6Pd6aZoytridPH3QaO4+jK4X2d+xuJ08feCyJki3CndrptMpBAFcQP7IvwpQqxVfConjOPMj8HLUv99q3H80epn/bYs/Xf7sl/Ei/2WFSGlC4bgXZr2BU5gvJaixYGM/+M3o0U6ztfN4dKPTBxuQ0sSmhM9SZEc5IQjgAuZLCXLEcbe78fM/i44azdbd08lrYxf1LdU1UZAXLghmvYGbGC4ltjuCLTXhXTVROWs/TdPhcAhBADehPX5jpYSQIxhji6vn+3dajY+ex9+YubAs1TVRYdwLs97AfQ46HWOlBD3dbHcEY4y9uo4+212z08HY4nr07/81udVYZVTUBDVfim8pZwVx3O1CEMBN6N5upjM6SZK9dnuv3RbtCVpcjx7vtxp3HvR/8ZvJbMEYY4vb+HdPHv7z8yut3VVVNSF+9ilmvYFHrB75pwnuiJKP7a9mk+jJ4b1lV8W9o9NoMnul6yqXVNSEyLjXfD5/cXEBQQBfyJ4erJWqjrBGRU0Uz8Ng1hv4iJlSwjtHsGqaKBj3yo1yQhDAF8yUEmmaUlCVR45g1TSxejARw6w38Jk0TQ2UEtwR3i3hl9NEkiSXUfR377//bz//V/7FOI555A4EAXyEjsPQ+tPrryOYuCbSNOVj3a1G83vvvjudTjHrDQKAzunRWkp47QgmromsI+j1Vzs7fFILggD+InnknwheO4IJamJTos6//OxnmNQCXqPqyL8CaM3O+nHBMghpws18PgDk0V1KkCNcOLlLhu/Mok/WKiA7ZLKpmnDhPCIAKqO7lAjDEexNNUHDZ9lR9sX16PHDJ5Nb/utW1ybu/MVfunAeEQCV0VpKBOMItnzo+Pry8J3ixIvcTsf3v/c9Wr/0q0sEAM58PtdXSpCAwnAEE9cEkSTJcDikiD4fe04B4NDdXkcpwSMklH9nW6xqYnF7dXH0UCioF6YAnkLLbcpD4Zl4zIxXZDSRXXoQzvPmpvB3TxjUEE1H/gXpCLb2oWNxHXXFqgkiSRJKrIMpgBdoOj04VEewsmsTm/C9FxXUCh2lBJ02GqQjmCpNMJgCeIKOUiL4RbpM34R0tjc3RTD7QCA8lJcSwTuCrXZhvit3XEiapiF1lYDAoLEDhR/OOjiCyR81vBaYArgJtQiqKiWoQSt4RzBNmmBLU5wMBkGu6AAfUXvkH3/ErkOSoy5NsKUpQl37Bd6h8Mi/ui3Ya9QEC3onGfiFwlKibo5gujXBYArgBqpKiRo6ghnQBFua4qDTCX6lB7iJwlKCFkFr5QhmRhOMsfF4XJM1YeAgx92uklKCltvEj84NBkOaYLXZYQauQcWsfClR521+c5pgMAWwwUGnI19K1NkRzLAm2NIUNXy6A1ZQcuRfzR3BzGuCMZYkSQ3XioEV5I/8Gw6HNXcEs6IJVtddJWAY+VIC2/mEHU2wjCmQ4g80IVlKwBEca5pgGDwHOpEsJeCILDY1QWB9CChH8vRgOCKHfU0wmAKoho7JqFZK0GbcQacDR3Cc0ARbmqLf6+G9AZLIHPmH1p61uKIJxtiLiwtUekCeykf+wRGbcEgTDM+EQJrKpQQcUYBbmmAwBZCjWilBcoEjNuGcJtjSFHjPQFmqlRJ8Y346nWq6MN9xUROMsel0CruDslQoJdAQLIKjmmB4VgQloXN6SpUScIQg7mqCZUxRh/BiIAntqYt/VOAIcZzWBMPgORCjwpF/J4MBPleCuK4Jxth8Pof1QTFlj/xD428pPNAEQ30ICilbSsARZfFDEyxjClWnNoFgKFVKwBEV8EYTjLE0TSn+HO8x4JQqJeCIavikCQLvNMgifnowte2dDAYGriow/NMEgynAEjqnR+STgCEAGbzUBMM5xoAxtjzyb2spAUdI4qsm2LIzF+99bRE88g+OkMdjTTB8AuqNyOnBpBJ8QiTxWxMsYwrx1hoQACKlBG/2hyMk8V4TDIPntWRrKYHRQYWEoAnGWBzH+EzUh62lBByhlkA0wfDJqBPFpwfjk6CccDTBMp8PxBAFDD1jbiol0jQlicARCglKEwyD5zWg4Mg/jAhqIjRNMHxWgqbgyD+87/oIUBMMn5hw2VRK4B3XSpiaYJnPzYuLC9vXAtSwqZSAI3QTrCYYY2maUpAZhsTCYFMpQQM+w+HQ/CXVhJA1QWCcNAyGw+HaegHvrwHC1wTLfJLQtOspdE7PaikBR5ihFppgGDz3HJoGzrXDwBHGqIsm2LJqhSm8Y+2Rf3R+PRxhhhppgmHw3E9Wj/yj97HCseOgGvXSBMuYAs28XrBaSsD15qmdJhgGz70iV0rAEe0RkW0AAAKDSURBVFaooyYYhgg9IVdKwBG2qKkmGEzhA9nTg5MkgSNsUV9NsIwp0OTrIHROD5US/J1ClKEVaq0JxliSJAedDsYBHIRKiSRJUPdZp+6aYBgcchJ+5B8c4QLQBGMwhXtQKTGdTuEIF4Am3sBNgcY+61ApcTIY0DsCR1gHmviWNE0xJuAC9C4grNAdoIk83BTD4bDf6/V7vZPBADc03czn87NnZ/1e7yefftpqNOkFRzgCNLEGMkXuhbRuffCg4+xrNBrZvi7wBmhiDdPpdFUTm9KcgTwUMpZ7IY3KHaCJNdAcwerrJ59+So8heKl9rf3XxgCoO0ATa4AmoAmQBZpYA40Y5V577bbt6wqWtaYoPm0cmASaWA/t2GdfWHXXB411ZV8Y8XIKaGI9aZpeRhGNexx3u9jm0E2SJLSQudduI9zYNaAJAMAWoAkAwBZqr4lZdLRumb3VaLbunk5e2768OvJ6Fn2y7h3Zf3T+P7+ZzBa2r6+G1F4TjDH2TXz+Uavx/pPJ7bdfu/3Dkw+fQhPWuBk92mm2DqMZ/d/FbBKdHu00W417R+df3Rb/XqAaaIItb19vawJY5uvLw3e+1QRjjLHF7IvP37vTanz4ZPJna9dVS6AJBk04yRpNMLa4GT3ebTR3+6MbS5dVT6AJtqqJxeyLz/u/nhX/JqCXtZpYPozsPB7dYI3CHNAEW79mtvoBBUbZoInXkyd3m63GJ5czrBuZA5pga6qJ66j7EJqwS2E1gU0os0ATDGsTToK1CYeAJtgmTSyuo08f4+NoC+x0OAQ0wdb2TSxm46eH+49GLy1eVq3Z2Dex//noGquXhqm9Jgq6MLGcboeCLsxoFKO8s8D/A/0SGxj3AXTtAAAAAElFTkSuQmCC" alt></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) Show that <em>M</em> is planar.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Explain why <em>M</em> is not Eulerian.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iii) By adding one edge to <em>M</em> it is possible to make it Eulerian. State which edge must be added.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">This new graph is called <em>N</em>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(iv) Starting at A, write down a possible Eulerian circuit for <em>N</em>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(v) Define a Hamiltonian cycle. If possible, write down a Hamiltonian cycle for <em>N</em>, and if not possible, give a reason.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(vi) Write down the adjacency matrix for <em>N</em>.</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 25.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(vii) Which pair of distinct vertices has exactly 30 walks of length 4 between them?</span></p>
<div class="marks">[12]</div>
<div class="question_part_label">b.</div>
</div>
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