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<h2>HL Paper 1</h2><div class="specification">
<p>An electric circuit has two power sources. The voltage, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mn>1</mn></msub></math>, provided by the first power source, at time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>, is modelled by</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mn>1</mn></msub><mo>=</mo><mtext>Re</mtext><mo>(</mo><mn>2</mn><msup><mtext>e</mtext><mrow><mn>3</mn><mi>t</mi><mtext>i</mtext></mrow></msup><mo>)</mo></math>.</p>
<p>The voltage, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mn>2</mn></msub></math>, provided by the second power source is modelled by</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mn>2</mn></msub><mo>=</mo><mtext>Re</mtext><mo>(</mo><mn>5</mn><msup><mtext>e</mtext><mrow><mo>(</mo><mn>3</mn><mi>t</mi><mo>+</mo><mn>4</mn><mo>)</mo><mtext>i</mtext></mrow></msup><mo>)</mo></math>.</p>
<p>The total voltage in the circuit, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mi>T</mi></msub></math>, is given by</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mi>T</mi></msub><mo>=</mo><msub><mi>V</mi><mn>1</mn></msub><mo>+</mo><msub><mi>V</mi><mn>2</mn></msub></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mi>T</mi></msub></math> in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo> </mo><mi>cos</mi><mo>(</mo><mi>B</mi><mi>t</mi><mo>+</mo><mi>C</mi><mo>)</mo></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>,</mo><mo> </mo><mi>B</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math> are real 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>Hence write down the maximum voltage in the circuit.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p>recognizing that the real part is distributive <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mi>T</mi></msub><mo>=</mo><mtext>Re</mtext><mfenced><mrow><mn>2</mn><msup><mtext>e</mtext><mrow><mn>3</mn><mi>t</mi><mtext>i</mtext></mrow></msup><mo>+</mo><mn>5</mn><msup><mtext>e</mtext><mrow><mn>3</mn><mi>t</mi><mtext>i</mtext><mo>+</mo><mn>4</mn><mtext>i</mtext></mrow></msup></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mtext>Re</mtext><mfenced><mrow><msup><mtext>e</mtext><mrow><mn>3</mn><mi>t</mi><mtext>i</mtext></mrow></msup><mfenced><mrow><mn>2</mn><mo>+</mo><mn>5</mn><msup><mtext>e</mtext><mrow><mn>4</mn><mtext>i</mtext></mrow></msup></mrow></mfenced></mrow></mfenced></math> <em><strong>(A1)</strong></em></p>
<p>(from the GDC) <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>+</mo><mn>5</mn><msup><mtext>e</mtext><mrow><mn>4</mn><mtext>i</mtext></mrow></msup><mo>=</mo><mn>3</mn><mo>.</mo><mn>99088</mn><mo>…</mo><msup><mtext>e</mtext><mrow><mo>-</mo><mn>1</mn><mo>.</mo><mn>89418</mn><mo>…</mo><mtext>i</mtext></mrow></msup></math> <em><strong>(A1)</strong></em></p>
<p><br><strong>Note:</strong> Accept arguments differing by <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi mathvariant="normal">π</mi></math> e.g. <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>.</mo><mn>38900</mn><mo>…</mo><mo>)</mo></math>.</p>
<p><br>therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mi>T</mi></msub><mo>=</mo><mn>3</mn><mo>.</mo><mn>99</mn><mo> </mo><mi>cos</mi><mfenced><mrow><mn>3</mn><mi>t</mi><mo>-</mo><mn>1</mn><mo>.</mo><mn>89</mn></mrow></mfenced><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mfenced><mrow><mn>3</mn><mo>.</mo><mn>99088</mn><mo>…</mo><mo> </mo><mi>cos</mi><mfenced><mrow><mn>3</mn><mi>t</mi><mo>-</mo><mn>1</mn><mo>.</mo><mn>89418</mn><mo>…</mo></mrow></mfenced></mrow></mfenced></math> <strong><em>A1</em></strong></p>
<p><br><strong>Note:</strong> Award the last <strong>A1</strong> for the correct values of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>,</mo><mo> </mo><mi>B</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi></math> seen either in the required form or not. If method used is unclear and answer is partially incorrect, assume Method 2 and award appropriate marks eg. <em><strong>(M1)A1A0A0</strong></em> if only <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math> value is correct.</p>
<p> </p>
<p><strong>METHOD</strong> <strong>2</strong></p>
<p>converting given expressions to cos form <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mi>T</mi></msub><mo>=</mo><mn>2</mn><mo> </mo><mi>cos</mi><mo> </mo><mn>3</mn><mi>t</mi><mo>+</mo><mn>5</mn><mo> </mo><mi>cos</mi><mfenced><mrow><mn>3</mn><mi>t</mi><mo>+</mo><mn>4</mn></mrow></mfenced></math></p>
<p>(from graph) <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mn>3</mn><mo>.</mo><mn>99</mn><mo> </mo><mo> </mo><mo> </mo><mfenced><mrow><mn>3</mn><mo>.</mo><mn>99088</mn><mo>…</mo></mrow></mfenced></math> <strong><em>A1</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mi>T</mi></msub><mo>=</mo><mn>3</mn><mo>.</mo><mn>99</mn><mo> </mo><mi>cos</mi><mfenced><mrow><mi>B</mi><mi>t</mi><mo>+</mo><mi>C</mi></mrow></mfenced></math></p>
<p>either by considering transformations or inserting points</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mo>=</mo><mn>3</mn></math> <strong><em>A1</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mo>=</mo><mo>-</mo><mn>1</mn><mo>.</mo><mn>89</mn><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mfenced><mrow><mo>-</mo><mn>1</mn><mo>.</mo><mn>89418</mn><mo>…</mo></mrow></mfenced></math> <strong><em>A1</em></strong></p>
<p><br><strong>Note:</strong> Accept arguments differing by <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi mathvariant="normal">π</mi></math> e.g. <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>.</mo><mn>38900</mn><mo>…</mo></math>.</p>
<p><br>(so, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mi>T</mi></msub><mo>=</mo><mn>3</mn><mo>.</mo><mn>99</mn><mo> </mo><mi>cos</mi><mfenced><mrow><mn>3</mn><mi>t</mi><mo>-</mo><mn>1</mn><mo>.</mo><mn>89</mn></mrow></mfenced><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mfenced><mrow><mn>3</mn><mo>.</mo><mn>99088</mn><mo>…</mo><mo> </mo><mi>cos</mi><mfenced><mrow><mn>3</mn><mi>t</mi><mo>-</mo><mn>1</mn><mo>.</mo><mn>89418</mn><mo>…</mo></mrow></mfenced></mrow></mfenced></math> )</p>
<p><br><strong>Note:</strong> It is possible to have <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mn>3</mn><mo>.</mo><mn>99</mn></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mo>=</mo><mo>-</mo><mn>3</mn></math> with <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mo>=</mo><mn>1</mn><mo>.</mo><mn>89</mn></math> <strong>OR </strong> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mo>-</mo><mn>3</mn><mo>.</mo><mn>99</mn></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mo>=</mo><mn>3</mn></math> with <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mo>=</mo><mn>1</mn><mo>.</mo><mn>25</mn></math> <strong>OR</strong> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mo>-</mo><mn>3</mn><mo>.</mo><mn>99</mn></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>B</mi><mo>=</mo><mo>-</mo><mn>3</mn></math> with <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mo>=</mo><mo>-</mo><mn>1</mn><mo>.</mo><mn>25</mn></math> due to properties of the cosine curve.</p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>maximum voltage is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>.</mo><mn>99</mn><mo> </mo><mo> </mo><mo> </mo><mfenced><mrow><mn>3</mn><mo>.</mo><mn>99088</mn><mo>…</mo></mrow></mfenced></math> (units) <strong><em>A1</em></strong></p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>The crucial step in this question was to realize that <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>Re</mtext><mo>(</mo><mn>2</mn><msup><mtext>e</mtext><mrow><mtext>3</mtext><mi>t</mi><mtext>i</mtext></mrow></msup><mtext>)+Re</mtext><mo>(</mo><mn>5</mn><msup><mtext>e</mtext><mrow><mtext>(3</mtext><mi>t</mi><mo>+</mo><mn>4</mn><mo>)</mo><mtext>i</mtext></mrow></msup><mtext>)=Re</mtext><mo>(</mo><mn>2</mn><msup><mtext>e</mtext><mrow><mtext>3</mtext><mi>t</mi><mtext>i</mtext></mrow></msup><mtext>+</mtext><mn>5</mn><msup><mtext>e</mtext><mrow><mtext>(3</mtext><mi>t</mi><mo>+</mo><mn>4</mn><mo>)</mo><mtext>i</mtext></mrow></msup><mo>)</mo></math>. Candidates who failed to do this step were usually unable to obtain the required result.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The following Argand diagram shows a circle centre <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn></math> with a radius of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math> units.</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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"></p>
<p>A set of points, <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="{" close="}"><msub><mi>z</mi><mi>θ</mi></msub></mfenced></math>, on the Argand plane are defined by the equation</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mi>θ</mi></msub><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>θ</mi><msup><mtext>e</mtext><mrow><mi>θ</mi><mtext>i</mtext></mrow></msup></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi><mo>≥</mo><mn>0</mn></math>.</p>
</div>
<div class="specification">
<p>Plot on the Argand diagram the points corresponding to</p>
</div>
<div class="specification">
<p>Consider the case where <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><msub><mi>z</mi><mi>θ</mi></msub></mfenced><mo>=</mo><mn>4</mn></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi><mo>=</mo><mi mathvariant="normal">π</mi></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi><mo>=</mo><mfrac><mrow><mn>3</mn><mi mathvariant="normal">π</mi></mrow><mn>2</mn></mfrac></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find this value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>For this value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi></math>, plot the approximate position of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mi>θ</mi></msub></math> on the Argand diagram.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p> <img src="data:image/png;base64,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"> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A1</strong> </em>for correct modulus and <em><strong>A1</strong> </em>for correct argument for part (a)(i), and <em><strong>A1</strong> </em>for other two points correct.<br>The points may not be labelled, and they may be shown by line segments.</p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p> <img 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"> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A1</strong> </em>for correct modulus and <em><strong>A1</strong> </em>for correct argument for part (a)(i), and <em><strong>A1</strong> </em>for other two points correct.<br>The points may not be labelled, and they may be shown by line segments.</p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p> <img 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fX9GlS3TyV7+v1q1KiB+vXro3DhwmI/Y8akL2hzcGeJQtUXVCrfu3eveI1ZzUJdG6tWrSqqMEqXLo2KFSvC1dVVOQrs2LEDP/zwQ7QScbVq1cTXoyqXxEqKrpDUcEuNvFS6P3v2LI4fPy6qm6jkr0FVRPQU0qBBA3z33Xci4KdPn145ypjhcHBnX+3FixeiuyItFNCol4oGlVrr1q0rerrUqlULBQoU+GKAffjwIdavXy/q5KtUqSISiX1tg2Vy9HOnaih6CqGeOtS2cOzYMfEEQ38vOkY/C7UP1KtXT1TfNGvWTDTaMmYIHNxZgtHFQNUQe/bsEV0UT548GXUhUcmUSuZUKm3UqJFoXEzKgKqPqQxioiqbmzdvYvfu3di/f78I+tRdk1DjL30Qtm/fXgR76n7JWGLpC9o6gvuX38gsGzUgUol67dq1okFRw97eXlQztG7dGk2aNBH9wk2NqQR3bVSNQ6V4+pDcsmWLKNlreuTQ35RK8s2bN0fjxo1FjyDGEiI+MZuDuxWjUiWVMpcvXy5KmhSQCAVwGnBEgYeCkKnXG5ticI+JqqG2bt2KjRs3ij7/mj791PjasmVL9OzZE3Xq1BFdQhn7kvjEbA7uVoi6/i1dulQEdSpdEuq3TYG8a9euosqFqhHMhTkEd23Ux56ekqiB+fz581GBngZN9enTB927dxd9+xnTJz4xm4O7FaE64Dlz5oi+6Jq6YOqn3atXL3Tp0sUkq1ziw9yCuzYK9KtWrcLKlSvFOqFRs3Q+fvnlFzFalrGY4hWzKbLLJQedCzN/Hz9+lPbs2SPVqlWLrgaxpEyZUmrdurV05MgRizjP7du3N/vfQ/6wlbZv3y7Vr18/6jzJN6rUokUL6dSpU3w/smg0MTrmoo0r+CwUJduaNGmS6HdNvVuoQY/qzmnA0IMHD/Dvv/+KLoz8dGYaqDuoHMjF+IEbN26IOniqj6eqm2+//VbUxx86dEhviY2xWESIp4iva2FmRw7q0oQJEyRnZ+eoEmC2bNmkadOmiWOWyBJK7rr4+vpKv/32W7RzKQd6SQ7yFvn7sgTQjtPaixYO7hbi3bt30vTp06WMGTNGBQK5ZC5t2rRJev/+vfIuy2SpwV2DPpT/+OOPaOe2RIkS0vLly0V1DrNCMWO1ZtEiGlTlCC9fLzrwI7tZ8Pb2Fo/0mv7pNDJy8eLFolHOGphzg2pCUG77zp07i3QNGtWrVxf96CmLJrMi8YjZXOduxiilrVyiE4mrKLDTCNIePXqIhFfWEtitCQ122rVrl+jtRCOFCSVsK1myJBYtWqQzbTGzXhzczRA9bNFE1pRnfMyYMSKLIY12pKRW1KWO8o0zy0SDnCgfD6VXPn36tEi8Rtk4+/XrJz7kDx8+LK4Pxji4mxkqlVNqWcpRQiloKb8Lpd2lXhU0exCzHpUrVxajXalahuajpV42lKiMMm/GNTEKsw4c3M0EpckdO3asSKdLpTN6RKd69YsXL4oujcw6UTsDTRxCCcsmT54s2luo2oY+6OfOnRuVUoJZH3Vwpzp4XQtLdlSPSvlIqIRONy+l3KU+0Hfv3sXPP/8s6tkZo2Rko0ePFm0vlKuG0hEPGjRI1M1Tlk9maXQFbFo+45K7CaMblQawdOzYUUxbR6UxSsFLOWE4kyDThSb53rZtm6imy5UrF86dOycmTZkwYUK0XPzM8nFwN0H0KD1z5kyUL19e3JxUKvv999/FZNMU7Bn7Empgpzp4SkRGeYTGjx8v6uhpQhFucLUOHNxNDM3sQ5M6DBs2TMziT1Uw1HBKJS+em5MlBKWboC6S1EaTL18+MS0gpTGg3jbUZ55ZNg7uJoS6N1KDKU1nR1PYUZ9mqoLJnDmz8g7GEo4a3Kne/ccffxTblFeIrjMqxTPLpR6hKkY06xK9gp4ZB03PNnjwYJFjndAj9YoVKywyqIcFPMMRr9PQmiNbYYfGrZrB0Tbh15y1jFA1BOpdNXDgQHh5eYk+89QDixZumDcz8RihysE9mVHqAHpMvnLliqh2obp2GpBiiYEqItgbXb8fhA+2YTj/KARVShbEyUOeyFOhPnI5Z8X8tcuQ1Z6Du7FRDyyq5vvzzz/FRCFUDbhhwwbxtMjMRDyCOzWuyCjhjK6FGdPOnTslJycnOktSwYIFpUuXLilHLNP1s2ekt5GfpBcXNkr9p66S90RK1fJD2n/n67JVWnriMGM5ePCgJAd0cf3lyJFDkkvzyhFm8mImDNMsWrjknkw8PT1Ff2QqOVHO9TNnzlhNyemc+xzsDMyOyf0boEw6F6z1/oQyWfVfa9S4THnO9aFG6KCgIC65J8KdO3dECgNqYKUBUNTeU6FCBeUoM1lcLWOaaFASNW5RXTt1T6OG1Jw5cypHLd/KP35GcO6uGNo6K7I4uuFEcCCKOOm/1ih3CiVJ04eqsejDkoN74tB4itatW4uBcTR+gka4Uq8aZsLiEdy5t0wSolI6NV7RoCQK7BTgaYYkawrsQAS2unsiQ84sUL0Jgh/skeoLbXnUsEyDc/QtNOcoSzwaHHf27Fk0adJElOAbNmwoUluIch8zWyK40ynUtTDDocyNHTp0ECkEqGcC5f2gbo7W1nd9x8Jx2HMrAOWK50bY20B5TzDuefti/5YtCOVgkmycnJzEqNYRI0aIBte+ffti6NChnJvGRMWM1ZpFG5fckwBl7qNZ7OlxlwaWUP916o5mfdUIKsyZPxcupeqiRBZbOGQtjHwuQONy2bHsv4dw4GrAZEWFjqlTp+Lvv/9G6tSpMXv2bNGTi54ymfkRde5SrJivZsM321ejASOdOnUSWR2p+mXPnj0oVaqUctT63Lh8DEhXBCULZhPbvs9u4MRlHzRr3Aj2iejjTrgrpOHRACfKNkkN1dTgSqX6DBkyKEdZctNXZaZ9D3DJ3YhoUBLlXafATnm2aXIFaw7spGS5WlGBnbjmLIl2zRonOrAz46BRrfTEScnHKFldzZo1Rc8aZj44uBvJ9OnTRUpeakSdNGmS6MpnXQ2nzNzR9H004xM1uFISMkpZ4O7uzg2tZoKDu4HRhT9u3DjRMEWPSAsXLhRT4XGVATNHVHKnvu+UNpgS2dEE3TSVIwd406cO7nSedC0sQeiCpwkTJk6cKBqn1qxZI3odMGbOMmXKhIMHD4pJ1+lJtHfv3qLKkQO8aRPBncqUuhYWf3Sh//bbb5gyZQpSpUolBipRKYcxS0C9vKiwMmvWLHGtU6GFA3wy0hWwadHC1TIGQlUx1I3M1tZW9NygLmSMWRrKXkpdJDUBftWqVRzgTRQHdwPYvn27aDQllG2PhnIzZqlobtZevXqJoE6dBmiGMGZ6OLh/JZp1XjMJAvW3poZUxiwdld5r1KghRrNSd9/nz58rR5ip4OD+FR4/fowGDRogMDAQzZo1w7p163jSA2YVKJ8PJWujXjQPHz5Eo0aNxH3ATAcH90Ty9/cXCZZevHghBnhs2rRJNKQyZi2okXX37t1R/eBpBrG4sneypKUEd13NrrQwXegCpgv53r17YmAHDc1OkyaNcpQx60HdJPft2yf6w9OAJ+ohxsnGkoKueE3LZ1xyTyDq50sXMKUSyJ07tyi5UEY9xqwVBXbKmeTi4oKtW7eKyVW4B03y4+CeQNSXnS5gZ2dncUFnz55dOcKY9aJUBZQkT5NNcsmSJRzgkxkH93iiEvvw4cMxbdo00Zd9y5YtKFGihHKUMUbJxmhgE6GU1keOHBHrLHlwcI+n33//HTNmzBDr8+bNEzPGM8ai69atW9SEH9RF8sGDB8oRltQ4uMcDPW7+8ccfYp1G5fXp00esM8Zio3uladOmomskDeijWchY0hPBXbLRvTD17PDdu3cX6zTpxpw5c8Q6Y0w3Guuxfv16MfvYtWvX8NNPP3H9u4HZyH9OXYs2LrnHISwsTMxG8/btWzH6dO3atdyXnbF4oB5k9MTr6OgocsBTJkmVSqUcZUmBg3sc+vXrh1u3bqFYsWJYsWIF52RnLAGow8GyZcvEOt0/NDk8l+CTDgd3PWiCAkpxShYtWiRKIIyxhOnQoQO+++47sU518dzAmnQ4uOtAc54OGDBArNMEBbVq1RLrjLGEo3YqGslK1TL9+/cX3YqZ8XFw12HkyJG4fv06ChUqJKbJY4wlHt1HNNk2Bfj9+/dj7ty5XD2TBERwp5pkXYs1oomsqR87NZxu2LCBq2MYMwBqt1q+fLlYp1HelGiMJZ6k5582LrlrCQ4ORo8ePUSpgmZW+uabb5QjjLGv1bJlS/Ts2VNUe3bt2hUfPnxQjjBj4OCuhaYQo0kHKleuzJNuMGYENAdrvnz5cPnyZfz5559cPWNEHNwVe/fuxerVq2Fvb4+///5b5I9hjBlWunTpsHLlSqRIkUIEdxrkxIyDg7uMBinRKDoyceJEFC1aVKwzxgyPEoxRCo/IyEhRTcODm4yDg7ts7Nix8PHxQYUKFUTVDGPMuKZMmSLmQ7hw4QIWLFig7GWGpAR3qvfStVi+ixcviouL8mHQaDqujmHM+Kh6Zv78+WKdMq5S4YrFn42ef9qsuuROgykoyyNNC0b5p8uVK6ccYYwZG01V2apVK1EtOmTIEG5cNTCrDu6rVq3CuXPnxGxKEyZMUPYyxpIKjV6lsSQ0+c2hQ4eUvcwQrDa4h4SEYPTo0WJ95syZ4jGRMZa0qN6dBjURau+iRlZmGFYb3ClL3evXr8XQ6B9++EHZyxhLalQlSimCb968iZ07dyp72deykURFl766LstMQvDmzRsxoS814lCfWxqVypJGWMAzHPE6jfBwZUcUOzRu1QyOtgm/5ijX/j///MMpmc0YNapOmjQJFStWhJeXl+jgwOIQj5CtDu76GjIs8GahRtQWLVpg165dKFu2rOiKxRdS0ogI9kbX7wfhg20Yzj8KQZWSBXHykCfyVKiPXM5ZMX/tMmS15+BujahRlcaXvHjxQuR06tixo3KE6RSf8jgFd+nTJ92LBXJ3d6c/i1gOHjyo7GVJ4frZM9LbyE/Siwsbpf5TV8l7IqVq+SHtvxOsfkMitW/fXr5cLfN6tSbLly8X92X+/Pml8PBwZS/TiS53XYsWq6pzp8YaGrBEGjRogHr16ol1ljRKVqwkql3u3niKjE4Z5T1vEPQCyOqcXv0GZtW6deuG4sWL4+HDh6JNjH0dqwrulDuGZoKhx3fKa8GSx5nzXnBOnx0I9MWDcFekc1AOMKtGAwgp/Qeh+5PmMGaJZzXBndKL0hyOZNSoUXBzcxPrLGlRvfvf7vuRu3guhIe9gaZdNcwvAPKDuLLFrBUNaqJ7k+rely5dquxliWE1wZ1K7U+fPhWT9mpKByzpTRrcH/feucCtYBZEvH8r7/HFnBnzMWXRUnDiB0bZIsePHy/WZ8yYgXfv3ol1lnDq4E4trLoWC0F17ZSoiIwZM4Z7xyQbFW7eu4yqLTohtwPglL8qujQph/nTR8O1eHXYcm8XJmvSpIkovfv6+nLdu15R/UJiLJ9ZRT/3devWiYmuCxcujFu3bnFwtzDcFdLybNu2Da1btxYjWG/fvo20adMqR5gQj+7rFl8tQ59d06dPF+s0uxIHdsZMH41FoZ4zVJVKT9uiDMoSxOKD+759+8RkvDly5ECnTp2UvYwxU0Z177/++qtYnzt3rnjiZglj8cF99uzZ4rV///5InTq1WGeMmT4apUoZW2lUOQV4Lr0njEXXub98+VKU2OlXpImv6UJhycPb21s0lMVH+fLl4e7uHu86dK5zt1xDhw4Vk2qnT59eJPqzs7NTjlg5fR90WveAReeW+euvv8SjHY1EPXjwoLKXJYeAgACsX79e2YobfQi3bds2KljT09fp06fFui50jOpmObhbnkuXLonpLylMUc73Nm3aKEesmyiT66B9DyjBXdmKyYzvFXqUo94xNCJ1+/btooGGWSYuuVs2aiujZGJ16tQRE3rweY5Xwd1y69yppE6BPU+ePGjatKmylzFmbmgyHZqt6ejRo7hz546yl32JxQb35cuXi9eePXty90cTo4oIwTFPT+zceRSh4SplL2O6ubq6igl1qJKBBzXFn0UGd2p4oRldKKh3795d2ctMge/dsyhdLB9qN2+OFi3qIHfuQjhw0Vs5yphuVEij6hhqt4mIiFD2srhYZHCnC4BSDjRu3Fj0lmGmY9aMBVjk8R8+yaWwGxcOIadtCLr1GoJIfZWIjMlohiaaPY0Kbrt371b2sriI4E63la7FXK1du1a8Un5oZkJUgWjTczRqVSgp2upLuNXFn6N/wstbD/BG/Q7GdKJSu+Z+pvtbX28R6xEzWmuWz9Qld7rTdC1m6Nq1a7h69SoyZMjADammxjYDKlUuqmyoOTq6IH/FssigbDOmT4cOHUTO971798Lf31/Za510hWtatFlctczGjRvFa7t27XjAgxnYvXcHRg4fxt3b2Bdly5YNdevWFXMzbN26VdnL9LGo4E6Pah4eHmKdPuWZafO9eRj37eqhZ5Myyh7G4qa5rzdt2sRVM19gUcH94sWLYv5F+oSvXr26speZIlWYHybPWYfVS8dzqZ3FW/PmzWFvb4///vsPr169UvYyXSwquB84cEC8UroByirHTNe4SRPw46QFcLbnwM7iz8XFRfSc+fjxI44cOaLsZbooETBmi6tmMS+HDx8Wrw0aNBCvzDQdXL0Yhep0QjlXR2WPCtduXlXWGYsbFd4I3e/WWjUTM1JrFm0it4y+5DI2sdpfTVdgYCCyZs0q1qkvLH3CM9NzcOWf+HHMelSqru4OSQKe3ES3aTvQuVZBZU/CcG4Z63L9+nWUKVMGmTNnxrNnz5AqVSrliPXQ95mmfQtYTN0FTcqhUqlQo0YNDuwmKsD7Ipbvv4LKWoGdZCxcD82rFVC2GIsbDWbKly+fKMSdO3dO2ctispjgvmfPHvEa35zhLOllLOgmejPFWtbNhZMtl7pZ/NATmmYMi+a+Z7FZRHCn9L779+8X69999514ZYxZLkotQui+5y6RullEcL98+bIYsZY3b14UKVJE2ctY4p3a9BfaDpwu6jYfnVwjlxaL4NUneSMiGHWruOHGy1DlnSw5UPVr2rRpxb3v5+en7GXa1MGdPvh0LWZCM8tS/fr1uVGNfTVV2HP0Hr0Kk34fKBqoshStCQ+POXCiS8vOGVOGdUaP/r9xiTEZUWCvVq2aeGrX9JKzLjGDtWb5TAR36hWj65+5oCT+hIYmMxYXVUQY/l2/AL27dEH79p0xacYC3PeNXgrfNHsqin7XFUUz2St7oqvYvBdSXf0H2y88U/aw5EAzMxHN/W9NqNCha9Fm9tUylGfi1KlTYr1mzZrilTHdItCzSQ38r8/v8A0NhUoViBm/D0PhkiVx3Tdc/RZVCGYvXom27dpHFW9e3zkufxAMQoimYGTrgI7tGuLvv9fp7ZLGjK9WrVpyQLMRo1WpBM+iM/vgThPohoWFoWjRomLGFsb0iXh1H9cfhWP9wbPYuXUrtm7djWNbZlBHexz/Tz192+ubR3ExMCfqVM4jtvWpVa06Dm7fgSCO7smmfPnycHBwwN27dzkVgQ5mH9w1pXaqf2PWJSQkREzMQvNqnjlz5ot14HZZS+Lig5toWSo7rl49g40bV2PdZk/lqNqtK/dgnz8XMtsqO/TIW7wwIl5ew9lHPCtQcqHBS5UrVxbrXl5e4pV9ZvbBXXNSq1atKl6ZdaBgTj2jOnfuLHL40/mnG/3+/fvKO3RR4a8xHZDO0RFly1ZB94HDcPl29J4W/u9Ckc05/RfbnBwcMwA24fALClP2sOSgue9Pnz4tXtlnZh/caSgyKV26tHhllo9K6F26dIn1KE6jFfv166e3BO9zagd+/eMfDF32NwKDQhHu54cNC36NFcbfJWCOToc0uhtdWdLQ3Pc0QQ/3XopOCe70R9G1mLZ3796JFL80O0uxYsWUvczSnThxQm8J/dChQ3j69KmyFd1j74dAtooY3bMrXJwdqOsMVq9cK670sLAQ8Z5C2VzxKjBELuPHff2/fvlQDiaZkNs1jbKHJQcK7tSoevv2bQ7uMZh1yZ0GMFDqzxIlSoh+r8w60I2sD93g+o4Xr1wVGYLOoYxbDbRv316+bgphzk71e9+EBYjXohUrIM2j67gToBLb+lw/fxPZylVB6QxxV98w4ypQoACcnZ3x/PlzvHz5UtnLiFkHd5qcg1CrObMeNBI5LpRUSpeMRb7FfycOoXEV+bidHboN+BP3713CmB7t4JDaRpTV7bKXR4uqWbBr70n1f5I5ZMqDdu0awU4rjh86dhAt27ZF6pidi1mSonkbNPe/Jh4whVzSkaRPn3QvJq5nz550P0rz5s1T9liep/evS4fP3pDobIS+fiR5eOyR3otzEykd3Okh3X8aFPW+S7cfiXVLFxkZKck3tDj3MZcWLVrIl+7XXbsXdiyWCtXvovfrhPvfkLJnziXdev1e2cOS05AhQ8S5nzRpkrLHGtC1qWv5zKxL7prG1FKlSolXixMegLZt/gcbl+yi0e/1nWNiMM0bcTACo3q3x/7TT8RWZhdbdPu+PV6Gxl2dYAmojcXT0zPaoDXq7zxgwACsWUN5YL6uNO3WvDvKhJzHjou6R6D+s2ABWg76A0Uzc2OqKdA0qlI8kGOaWGcyEeI1JfWYiwlTqVSSfEOLT2x/f39lr2VZP6mf9L/hi5StL/t73E8Jer+5o5L13bt3pVq1akkhISHKXsPwu3NG+m3Wqti3QXiQNHTocOntB9O+P6zJ+fPnJfkDXSpRooT08eNHZa+lo+tP1/KZOrjrfBMtpuvy5csisGfNmlXZY17uXzkhtW9eQ3K2t5d/D3up1Dc1pK0nbihHJSky1Ecq7JJZOuPzVtkjSa9uHZXatRsoBYuI814a0audtNfrc1XMe98LUrp0uaRHwZHKHuvQvn17EeiZdQoNDZVSpEghpUqVSgoPD1f2WrZPev5pM9tqmbNnz4pXs0zxG/4Kzet/h/vBDpi3ahXc3RcjMwLRpWtfhCtPlae2rcfbvFXglkMzzygQ5v8Ymzfvg7oX9kcc9tyMBz7qLnzEPqsbGpe2wUbPz42BjFk6qpLLnTs3IiMj8eSJupqSmXFvGU1wL1gwcfNuJqfw8A/o0mMI3P/dis4dOqBDh25Yt+g3hD70VerTgTOnT6Bc5apI6ARFFSuUxf49B0XrImPWgrpEEm9vb/HKLCC4FypUSLyaE3vnXPh1/ADc3vUPxowZgtatm6B6897KUbWbd58ge46Mylb8Zc9fDA8eXgdHd2ZNNIW8Bw8eiFdmpsH9/fv3UQNV8uSJO3ufKVKF+aLmN8XQ5bffcPmyNxxd86Dfj12Uo2rhH1XI4OigbMWfrW0qqFQflS3GrAOX3GMzy+AeEBAQ1eUpV65c4tWcXNjmDq+AzPB5+BS7d+/E2kWLUKaAi3zkHcIj1L9X5rSOeBH0uT49vsLeBiKDEyW1UnYwZgU0A9t8fHy4O6RCBHdJstG5mKo3bzQ100D69OmVNfNBo+rw8hWOXHssto96rkGXAX/IaypoCt2FiueC75OEzw3pc/c+sucuxLGdWRUnJyfx+vbtW/Fq8Sg+61q0mGXJ3dyDe/kWP6BmydRoU0kOwjY2qNP8Z9T6vifs4Ysnr96J91Rwq4arl05ClcBCyJVbN1Cnbg1lizHrkC5dOvFqNcE9HswyuAcHBytrgIsLVWeYF1sHVxy6/BAnjuyEx86deOTriw0rFsNzpwdypYoU76nWshNcfM7h5IPPVTNZitYSEzWrP87sMGWZBxpW+dzmEPbsLA7eskWHZjxxCbMuVHKnghLFBq6WUbOhzu76/hammhNp48aN6NixI1KmTCnmUBXVHBZow+T+2BlcAJtmDlb2xG3FmG44FlkJ66f1UfZYh++//x7//PPPV6cdYObrxYsXyJkzJ7JmzYpnz56J2GDJ4hOzzTIqBgUFiVf6tLbUwE6+HzwGjw+uxtH76t83LuEBd7B470PMGNtL2cOY9aC0v5qSO1Mz62oZc6ySSQiqvtny7wZIQS++2G3dL0iFFRvWIpvjFyb/ZMwCpUmTBqlTp0Z4eLhYmCa428ihQ9diojQld0sP7iRXwZKoU7HEF3u/0PvKFY07zzljlopK7VR6J9ZQereR47OuRZtZV8tYQ3BnjMWPNQX3+DDrahnNyWSMMU1hT1P4M20q7Fw5TUz32Hfw7/DYvBEzp47BRk8vg/X24ZI7Y8wimFfJ3RbNfxyK11e2wsa1MNq364ABfQZgSt/6aPzzTHz49PUB3qyDO5fcGWMa5lYto3rzCHcffEL9Rg3Etp1TVtQtXwr7l03AHf+vzw/FJXcrQo97R44cwahRozB58mTcvXtXOcKY+Uvq4P7jjz9i+/bt+PgxcYH48uH9eJWhBKoXzaTsicBN7ydwcK0IV6foofnJ3UvYvHkzrvu8UPZ8mfgK1BND12KqNCePg3v8ffr0Cd27d0fdunUxdepUjB07FiVLlsTKlSuVdzBm3pI6uM+dO1cMmGratCkWLVqEsLAw5Uj8HD64B5Vq1IVLagrDKvw9ZyyuvHPFtqMeyGynDu4hr+5hoFwYe/0xE76rlBu1S1bGkzfyhwnV2uhatJhdyZ1Kn5rcMuaYVya57Ny5U0werU2lUqFv377w9/dX9jBmvjTxQDv3lDE5Ojqif//+4t6i2oQyZcpg//798W4QPX7aC5nehWDShPFoU+c77LvwCecvnEB9uSSvLlx/QL/W7dC861B8Uzw3It+9QbBkj1SpxMEvMrvgTp+OVAolHNzjb8OGDcpadJS+YcuWLcoWY+YruZKHpZKj7ejRo3H48GEcOHAAjRo1EqX6uH4OVeBtnLn2Bj9OnYY+Ldyw/dhpDJk2AXkyfp5Wc9/ff+BB9jqoUyQjIkJeocfwPzFxxlxks49fagWRWyZWeT7K58qZV69eYcCAAcrWl926dUvMjkKjxgyJRp95enqK9Ro1aohcEsZGJ+n169dREwKYo0OHDuntIkbz0JYuXVrZMj3Pnz/XWxqjD3qauKVt27bKHvPm6+srXl1dXcWrubty5QrKli2rbBnXo0ePcOHCBTGBT8WKFZW9hhMSEoLAwEDky5dP2aNbREQEjh49CltbW/j5+enMc+O1eQ7qDtyAV8/OIr0Uior5sqPG0DWYOaiN8o4I1C3siuwNR6J17dx4cu8J6rTugtKFs6sP63s60E4uQ8Gd5tLWvSRev379JPlCVbYMh2Y6px+bFvlxSNlrXJcvX5YmT56sbJmnDh06RP3dYi4rVqxQ3mV+5JtN/A5ykFf2mDcPDw+xWIp27dol2bmZP3++uBb69Omj7DGsM2fOSDNmzFC2Yrt27ZrUs2dPqVmzZtLatWslOcgrR2Kb2L2B9O33Q+W/jXp7+sDWUpkmP0kflR3hPmellClsJPeLL8R2LPQ+XYsWs6uWSZs2bdTTgLl0eTIFI0aMgL29vbL1Gc1BKwd+ZYsx86WJB0nZRVqOoaIqRv4Qw6pVq/Dbb7+JOvjOnTvrr7VQvYHnwZNo1LhxVEG7jfzkef3wDlx//V5sp7RLA7pbzx34L6qQ/t9Wd+w/cU+9EQ9mF9y1c0joq2ZgsVFjz+7du6NNKF6vXj1RXUMfmIyZu6QO7uvWrRM9ZajqadmyZZg9e/YXq2yoV8yeFXNx4dk7OLikjWp8zV+lIUrYv8avwyfBNzQCKTOXwohu9TFnVFfUad4Kbf/XDc8c86FBtcLi/eoqc13LZ2YX3ImmCySX3BOmTp06om/7mDFjcPDgQbHkzp1bOcqYeUvqwY0ZMmTAtm3bMHz48Hh3y7555hTCMhbFJg8P5Ix4hjuPlak0bTNg2fY96N20PE7s3QmfoPcYu3I3zpzcgcG9BmL1yuXo2KBKtCr1LzHr4M4l94SjJx/qZcQ9jZil0RT2kmr8S5MmTRLcYaRE5ZqiCkezFMuXRTkCVK7ZMGp/bpc08h5bVPq2IZo3rw1H+3j2f9TCwZ0xZhGSo87dlJllcOc6d8ZYTBzcozNacKe6XOrcbwxJXedODY7ZsmVTtswf9aF2cHBQtswb9SFOqsfwpEC/iyX9Pvnz51fWjM/YwZ3umaQYV2MoZjlBNjUI/vHHHyhVqhSuXbum7GXWiifIZiRjxoxikBENHMqUSZOMyzJJYohKbDZaPWa4WoYxZvYoMyONYtZ0GGBmGtyTOkEQY8y0aXJOURWqruH+1sisg3toaGhUEjHGmPWi/E9Uv8yl9s/MMrhrGpwosHPpnTGmaUx1cnLitheFWQd3wvXujDHK2Egld2qP4+CupgR3kUxNx2IYD66cwebNnnj8LEDZ83W0uzpxcGeMGaobZEjAc3hu3ow9e84iNDxS2WueRHCnzzldiyEsHPcjBk+cjvlThiJfobxYsfeiciTxtEvu1O2JMWbdaL4FQvleEltyv37cHf/r0ht/r1+CH5pWRr6SVXEvUJ2l0fTEjNaa5TOjVsvc8NqD4q1HYefWrfjv0i30r18IM6b9pRxNPOrPSlNcEZrDkDFm3R48eCBeEz1oKuI1jj9ICc9du7B1x2E8fewFZ/+LcqDfrbzB/Bg1uBet2gC1yxRUtmxRplIF5M37pZSYX5YiRQpUqFBBrHt7e4tXxpj10sQBmv0tUewy4Oce7aNK/c65q6BQgQwGiVfJRT3NXlIMUQ0Pxs/9BuHXqQtRMPPnoe9+3lcxedZfeOkfruz57H99J6BFrWLKVnQjR47EtGnT0KZNm2SbA/Tk9tVYtnUPwmP96M5YsHYZssSeG8NkifMwWz4PfrHPQ/5yjTBlVI8YD32mw5xHqJ7asQZL/92t4xpywrw1y+Caxnx+J/8H1+R7eSZe6LiGyjfqixHdaxk0pGijafXOnz+PkydP4ttvv1X2Jt6DQysxab8flk8fgVRaP/SpnWuxdMsunedr7uqlyJY2afqo6GsR1f7zJslPsn/9fFStWA57TlzC5btaJe0PQVi8YB0q126GJy+fiFSXLjZ+8PmYXqwXyat/CHGlSpXEa3KV3J9dOYz9V4JQp3xB+Nm4iJ/37llP5KzYQF5vBgdb5Y3mQBWGxQvl81BTfR7atm2H+qUcsfnkTbFer1o55Y3MkG4fWIl9lwJRx62QfA05i2vo3rldyPFNfXENOZrbNbRgLSrVbIqnvk/EzEIZUwbgaaSj+L2qlcurvNHwqHyqiQNfO8/xg6snMaBDQxRq0BNPHtzCI79Q5Qjw/NpR7LvoL5+vwvCTgzn9XvfP70Z2t3rq85XKxD6IqeROcxzqWgwnUrp84pBUo5yrBMdc0qPgSLE3IvCx9DqU1kOllk2biX0zBzeTBs9cK9bj8uLFC9Glx97eXlKpVMrepPPy1Uvx+uDo38rP+16qlBOS15P3Yr85iQx9Lb1+q5yHZs3kcy9J1zcPk3JVUq+buvbt2xv4ek0a9x/eFz/3w2NrpMEz1sh7wqUquW2kU4/fqd9gRiJD/eRr6IO8Fiq1kq8hmgt09q8tpV+mr1a/wYieP38uyU9tUoYMGQwSC/yfeUsLpgyWsqSH5NayX9S8pnTPi/N1fK00aPrf8p5w6du8KaUTD0PF8aREP5KuRZtScqdPHF2LodiibLW62LNzOxxDfbD/P/U8gKld8iCzXMRVvfbGyw/q2d5fPw6Ek+OXuzNRlsYcOXLIj0fhePjwobI34aaM7A05OOhdVu04prwzOtcs6p/36k1vZHTKKpdcAvH6mb38s9uJ/ebE1iEzMsvFRHEeIuTfRT71j648R3oHJ8NeBiyagvkKiqqkq7fuI4PmGvKxl//u5ngNZZKvoVRQ+T2QryGagMIGrx8Fqq8hI7t586YovZcoUUK0x32tjDkKoN/IWVgyaTgu7j2CgE/qShC65+l8Xb/lDZf0mvOVGukdk77+VTtKay/akqaCSOGQsxLkJxqksI3+bW+ePgl7J/WMJI8C/ZDGLn59VQsXVs8neO9e/CeNjWnU1GXw8PDQu/RoUUt5p24X5Z/dmU60/HM/grP8s8f8E5uPm2fk8yD/LvQbXHr+BPbxPA/s60RdQ0F+eCg/7qcxv9ge5daZU7BLl0XUrT+Uf5809sa/hrQbUw3Z7lKtdiWkSJUiVpC8KN8n6vPlj4eS6Z6vJA3uUIUgKDIX6leN3qK9zn0DshdW50sPD38nXuU3I8Av7nztmuB+//598ZrUwoO94bHnLLIXyab1cwNhfgGI3aRk+ta7u8vnQf1EEhYaIl5JoPz7MOOICH6AzbvPIJv8d48I/9ynOsxfvob0dXQwYes3boBrIfW9HKF1Lz9/4mvAYZHR0bzARHvyd0MI9guBW/1acEnx+QMjIuQBPHafFvd8RIRpny+jBvddq2dj6OhJeOYbAFV4GKb93he9J7gjr9PnlqIru1fgL4/TcCtZRmxHhIdiy8bFWD17PC6//ij26VO6dGnxevnyZfGa1CYN6of7711QvmBm8XMDvpgzYz6mLFoKc2oLI3QeZm7yQvlS6vPwLtRfLsnvwup16+C++7DYxwxv8uD+uBvmDLeCWeRr6C0g+WLuzAWYunApzC234dW9qzBz4ylxDVE4jHj/Flv/WYLVcyZgl3eE+k1GcPXqVfGqiQeJEXDvHLr++BOOel0WVTzPbh/HuEU7sHLGJKTQehr4Y8gA3HnrJN/z6vMlyedrnnzP0/lKYWqfxVTxrqtinpavdeeCp9S5c2epXbvO0riJU6TL99WNkNrWz+otOWYuKt32VzdEeiwYLdnbQ6rTebSkbnbV78yZM/TnlEqUKKHsSUqRUosqmaWq3w+XxJ8qMlTq0qScBHtHaYHHCfEOc7Jh1k/q8+CnPg/3j7pL+XKlkzLnLSddf/FW7DNV7c20QVWSIqRWVbNIVdoPU//88jXUtWl5cQ3N23RcfV2ZEfc5fSTHTEWkm6/VDcJbFo0V93LtTqOk90Y6P9SA6uTkJOLAy5ex40t8hQc/lkYM7SvHqnZSnz7DpFXrtkl+76iBWFuk1Lqaq1S57VB1I6t8vro1qyDO19yNx5L2fNE307VoUWZi0v2RY+r9hqkxlVJ8yjeGyC2hGbXKrAvPxGS9bt++LRpSc+XKhUePHhmkQdUs6KsC0roHxF+CNnUtps5eLhaULFlSzMJy4cIFZS9jzFqcPXtWVKO4ublZ14d7zGCtWbSY/cdc5cqVxeuZM2fEK2PMemju+ypVqvCTWwxmH9w1Q429vLzEK2PMepw6dUq8UnBn0RktuNetWxdp0qQxyEJpPPXRBPcTJ07wlHuJ1LNnT51/d13L7t3mmyXPlCxatEjn31fXMmnSJOV/MW3+/v6izp2qZ6laJrH27dsn5l7V9bdPzNKpUydRVZTcvpg47NiOVVi0YZ+yQ7cJC9aiWIwsWREREQYLtvS4RSdQF/rx8+XLhydPnuDSpUsoV85686DE61zNl89V1uh/yw8fPoh2i/hInTq1yU1AbI4NqiqVCpGR8ZsMwtbWFqlSpVK2mMb27dvRqlUr1KxZE0ePHk30+adrn+4BQ6H7g+4T49L34aH1N6DgLlG3Hl2LmejWrRv9ptJff/2l7GHWxHy7QrKvMXDgQHHfjxs3TtljRWLGas2iRamWoWivazEPderUEa+HD/NgG8asxaFDh8QrVQGz2JRqGWUrJiPF9ysn/8WyZTvgH26Lb6rUQN8+neBgn/gxnb6+viKRGNWbBQYGws7OjJNzJKNg35uYP2MZrvsEoVCJsuj8v44oWlCdjsCUmXs/96untmLZ0u14Izmizf86o0XDymb7uySVp0+fIm/evEiXLp2YYs+Y9/xVr21YtmSbOD+tO3ZCy0Ym0DMnjqr0KKL8TqV5XYsR3D+5RTxKaS91uvyqHE28AgUKiK/l5eWl7GEJEhEolcubMdp5ccxWUHocGKG8wXSZc7XM/VNbo/3NaVm444xylOmzceNG8beSn9qNeu69vbZJcryMdn4WbDeBGKNdFaO9aEnyrpBLV23DsQs3REPoq6c30atZTRxZuwDHHnxOVJUY1atXF6///fefeGUJs3vtMrQfvgTvIyMR7O+DGcO6QvXSG9OXbFLewYxh2aqtOHr+uuh8cOPCIZTKlgGTxv+BSH0lMyZo7nO6741Ziqbzc/jsNXF+bl48jNLZM2KyfH4+KGmATZoI8RTwdS2GFukvnTl9W9lQe+97QXKWf4wlO28qexLHw8NDfKpWq1ZN2cMS4syJU7FOeZeaBaQ6nUy/scpsS+4Rz2LdD7sWjpJgV1zy+2iGv08SoXOdO3ducb+fPXtW2WsEkQHy+bmlbKjtXjRaPj/FpNeqj8qeZKJdWtdetCRtyd02IypVLqpsqNk7usAWjqhQMZGzlisaNGgguh+dPn0afn5+yl4WX5WqVY3VxOLo4CKfl7LKFjO41Dli3Q+O8v2Qr2IZZOQqd72oy7OPjw+yZ8/+Vf3bv8g2g3x+os/h7OjojHzflEZGrTTApirZR6ie3bsD5bsOhFuMvtcJ5eTkJHrNUJ9VT09PZS9LrA9Bt3HqThgGdG+q7DGECIzp1BRp0tiIR+noiysehnJVxG75fhg5bJj4mzDdduzYIap1mzVrluSJwnbv24ER8vnRTgOcLOjb61q0JG9w/xCE+cuPYeGcscqOr0MDGsi///4rXlkiqcIw/Pse+HnOWuQ04CzNFw/uQuORS/D84k70nLxc3KD1Sthjxw1/ed0X+R1jXJ1WxvfmEdxLVRe9mvLTkj50zWjub7rfk/JD8NWto7iXsjZ6NSuv7DFxVDcTq65dsxgRTcrcp1VN6ciVR8qer/fq1SspZcqUUurUqaXAwEBlL0uoJeP7Sav2XTLaJXBl20Jp+JwN8lqwVNYZ0vlnMfNmJ4wlDGKiCab79ewqBb4379/D2K5fvy4mw86YMaMUEZF0Pbno/PTv2U0KeJfMde1RNEE65vJZ8pTc5ZLhiHEz0HvRbtQuk1fZ+fWyZMmC2rVri6HE27ZtU/ayhNi2Zh7S1uiF7g3LxXzKM5gLN68hS4Z8QFgAngc7I50Bnw7Mknw/jBw/HUOmL4GLvXU/vXwJzWssxy20bNky6VIyyOdn1PgZGDRtETKkSfaa7HhL+p9U/kNN6jcBPUb8hrKuDsquEKxasxIqsfV1fvjhB/G6YcMG8cri7+DqxQhMXx6da6un2oNKBa9dK3HtRZh62yAisNXdExlyZoHqTRD8YI9U5jafnCHJ98Pk/hPQbdgo5HdRtzsFPLkGj92HRKdq9hkF9Y0bN4p1us+TpEpGPj9/DJiILr+ORIEMacQuszk/ovyuq3RPi4FFvg+Sen1XVXRhirn0nbJOedfXoeoYOzs7KUWKFJKPj4+yl33J1iXjJTm0xDov2co1lyINWOWxfcEI+evaSZd9P0jBdw/I6/bS3ssvpH2bN0tvE/l9zLVaJiLUV+rd5Ntof2/1YicducfVijGdOnVK/H1y5MghyU/nyl7jiQwPln5qVj3GuVGfn0N3ApR3JRdNkI65fJak0+wF+3jj4Bndk1l/U+s75M2sLsl/rXbt2mHLli34448/8Ntvvyl7mX7h2LfZE2+VLW2uBUuherno3fUST4XaRdPhauo6eHV1F2zePEXhfEXxKCgcrQdMw5a5w+VrTnlrAphr+oFXt87jv5uPlS0tdi5o2awuUpnZ72NsP//8M5YuXYpff/0VM2bMUPYaT8izBzhw+pKypcXOWT4/9Uz+/Jj1HKr6UM7xpk2bolChQrh7967Z/h6W6MblY0C6IihZMJvY9n12Aycu+6BZ40awt03ceeI5VC1fWFgYcubMKeZKvnHjhpg3lcXNfFoHEqBhw4ZigMP9+/c5HYGJKVmuVlRgJ645S6Jds8aJDuzMOlD3RwrslSpVQvHixZW9LC4WGdxpcgNqTSebN28Wr4wx86W5j9u0acNPaPFkkcGdUL07oS6R8Z3xhjFmeii9L83VQKNRW7durexlX2KRde6Efieql6M5Fjdt2oT27dsrR1hyePv2LY4dO6ZsxY3mzK1atWrU9UcfzpSVT5///e9/omTHJTrLQ/cxBXSaUq9x48aiPY3Pc/wky2QdSWXhwoXo378/qlWrJibQZoZx8pAnXgaFK1uxOWQriMbflpNvQmWHjEpf/fr1U7biRnWqU6dOjbqJ58+fj3Pnzol1XejcPnr0iG96C0SJAGkSfApTlDOKOkowdZ9MXbTvAIsO7lRazJUrF0JCQnDhwgXjZpCzIokJ7sbEvWUsV4cOHcS5LViwIO7cuWNyk7MnF6sP7mTo0KGYNWuWuEjc3d2VvcyScHC3TI8fPxbdmVUqFebNm4cBAwYoR1h8QrbFNqhqDBw4UPSeoTrZJ0+eKHsZY6Zuzpw5IrDnyJED3bp1U/ay+LL44J4nTx7RmEoXCZXgWfJSRQRjzKAOyJImDeztM6PTj0PgF2qIrELMkvj7+2PlypViffny5WIibJYwFh/cyYgRI8QjO10kNFM6Sz5zRgxGkF1hLFi7FiMHtcG/q2ajTc/hkPQ9ZzKrRNUwoaGhaNSokVhYwongLtnoXixF6dKlxawt79+/x8yZM5W9LMmpApGhZBMsmDYB7du1w/ipSzB7RCec2L4fgXprEZm1oZGo1EOKUG4obkuJzUa+XXQt2qyi5E7GjlXP9rRo0SLRbY4lA9sM6NGzbbRGn7IlyyFdflekV7YZo7p2CvA0bSZ1Y2aJYzXBvUKFCqKPLCUgotGrPGrVNFy8cQmdO3SHLZfOmCwgIABz584V67///juX2r+C1QR3MmHCBHGxXLx4MaqxhiUfVdgzbDzhhz+G/c8Set0yA6BqUyq116tXDzVq1FD2ssSwquBevnz5qAtm9uzZYtQbSz7r/5yIwdOWwpmnlmMyahNbvHixWP/ll1+41P6VrCq4k0GDBonXe/fuYdeuXWKdJT3va8dwP0N5tP7WcHPoMvNGvdloNHmxYsVE2m72dZTgTiVYXYvladGiRdQ8q8OHD+e692Tg530J83dcw4QhP8mlM/W+kLAQ9QqzSkFBQZg8ebJY//PPP5Nu8muzpSte0/KZ1ZXc6VFv1apVKFCggMhVsWzZMuUISwoU2H9o/xsqFHHFti1bxMjhJTPH4J+9uqdfZNaBpsT08/MT1abNmzdX9rKvoU75GyPia9hYcDMX5XmnVKIZM2YUVTSUZpYZV4D3RVSpWhv3/WLM1mqXDfd9n6Cgc+JKa5xbxrzR/UdjUegp+uzZs6JnG/sCfe2FWveA1ZXcNWimJupHS12vxowZo+xlxpSxoBvuvX4jGrKjLeEvEh3YDe3Zgxu47u0j1u9ePomjXtdF0SfM/7H8lLEX4eKmUuGQ52Z4+wSL99H/uXSHx04kBp3/wYMHIyIiQuSP4cythmO1wZ1KedSflpKK0Yzq1D2SWbmIALRp3Qk2Di5ic+fqqRg/6x+x/vrOcbRvPwghosAUgd9++h77vB6LY5nlD6Ye8tPDi1Buv0moHTt2YM+ePXB2dhZVM/z0ZUDyJ6f0Sc8/azBkyBC6XSX5UVBSqVTKXmZO2rdvL3369PXX6/pJ/aWOwxcm6spfPf5nqeMw+f9ax21jEG/fvpVy584t7r958+Ype1m80IWma9GiDu7yTl2LNXjz5o2UM2dOvsDMFF2nbdu2/eL1un31DKli+eKSvXye7ewySY3b95AeBr5XjkpSZOgzqXCGzNLpp2+VPZK0Y9UU6feZ7iLYv7p9VGrXbqAUJL5PuDSyVztp76lH4n0k3PeClD5dLulh0AdlD/uSwYMHi/vOzc1NioyMVPay+JAjtM5/2qy2WkaDUolqJymi6eCY6fvw4YM4X1mzZsWWLVvEnKvUqCoHeeUdn93YvxYtuw1DydptscbDA3NmjMCVg+4YPWV5VFeCU9vX422eynDL6aDsAe5eOYljp2+I9TD/J9i8eR8ixH9Q4fCuLVF17sQuqxsal0mBjTtPKntYXM6fPy8yP9LMSkuWLBHVo8ywrD64E2pcbdOmjUgx2rt3b9HIw0xb165dMWXKFNF9jpw5c0bMtjV69OhY5y9VptwYJwfyFTPGi2yUPw/4FSN7f4fnj3yVdwBnT59A2UpVvyrHTcUKZbF/70H5+ys7mE7UeNqzZ098/PhRDCrkRlTj4OCuWLBggegOuX//ftEPnpkumg+XSum6UG6SV69eKVtqRdxq4ee2FbBy0Tz069cbTRpWw5i5W5WjajfvPkH2HBm/qvNv9vzF8PDhdXmNo3tcJk2ahGvXrol5UTX5npjhcXBXuLq6YtiwYWJ91KhRIs8FM02XL+sf8KRSqXD9OgXYz4KeeKFUuXLo1X8QFi1ajvtvUqJy2eLKUbXwjypkdXZSthLH1jYVIuXSKNPv+fPnmDFjhlinIO/g8LkajBkWB3ct/fv3FyNX6VF/4sSJyl5malKnTq2s6Rbz+MJJfyJTucY4f+e+GChz7/Rx1CyTS/4g+Nx10T6lLV4Ff10KBPp6ae3slC0WE1WXUQGK2kuojaRt27bKEWYMHNy1ODo6Ys2aNaKRZ/r06fjvv/+UI8yU1K9fH2nSpFG2oqMG1kqVKilbakF+AUhjlw1581CSso/YsXom5rrvR0TE56ezEkXy4MXzgK+qUHl69zpy5S4mr3E1gy7u7u7YuHGjuM9Wr17NjahGJoI7XYq6Fmv07bffYuTIkaLXRadOnURCI2ZasmfPLtpI6ENYG/V8oqBhb2+v7FHrMWwoHp9chcxpUiFVKnt0Gr4QtWo3xZNXD6g4Kd7zbfU6OO91BKqvaA09d/EyGjaurz0CnClokGDfvn3FOqXbLlSokFhniRMzVmuWaKg/pER9d3UtVkp+bJQqVqxId7nUunVr+U9hvX8LU3bu3Dnp559/luRgL82cOVMKCAhQjsQW7O8j7dzpIe04ckJ6+z5SCn39SPLw2CO915zbcF+pcAZn6cj9IPW27M6lE9KRU9dE7+FQP+33R0oH5a91/+nn94Y+O8v93PW4du2aJH/w8v1kUPQ31LV8JhKH6e27ZcVFkIcPH4rJPSi/NKUpGDhwoHKEmRpDJQ7bMHkAdgblwz9/DYldCvqClWO740jEN1g/rS+X3LW8fftWVJPdvn0befLkwaVLlzhJn0Hoe8L8fPFxnbse+fPnx4oVK8Q6NQJRtjpm2b4fPBqPD63BzZehyp74iQi4i8V7HmDG7704sGuhcuNPP/0kAjtVlVF9Owf2pMPBPQ7Umj9gwADRuk/rjx+rE0Uxy2Tr4Ip/t26AFJawdha/4Egs27AG2R15ggltNAJV04BK40cqV66sHGFJgatlvoACe+3ateHl5YUsWbLg+PHjKFq0qHKUmQLO52566D6hXk3U9ZTODZ0jZkjxrpahHboWRn2mabagbNmy4fXr12KaPpqdnTGmGz3htm/fXgR2qtKkdZb0uFomHqjr3b///ou0adOKWWOoFEIjIRlj0VEDKuVqooJQgwYNxHyo/ESVPDi4x1OVKlWiqmYOHDgges9QjRZjTI0KPB07dsTVq1dRpEgRUR3DA5WSDwf3BChTpoyYe9XOzg6LFy/GrFmzlCOMWTcq6NB0ebt27RI9Ynbu3AkXF/WMVix5cHBPIMqJMXbsWLE+fPjwOJNYMWYtKJjTqGFC40IKFy4s1lnyUYI7VS/oWpgu1EhUr149kaLghx9+iMopzpg1un//Pnr16iXWaZJrqpphRqYrXNOihUvuiUA9aGj2n7Jly4oG1u+++040JDFmbV68eIGGDRuKAg690mTzKVJwWDEFfBYSycnJCXv37hUjWWnyiObNm3MOeGZV/P39RY+YR48eoWLFiqLL8JfSMbOkw8H9K9AEH9RzhvrAHzt2TEzVR1OIMWbpaKxH48aNcfPmTZQoUQK7d+8WWTmZ6eDg/pVoco+DBw8iU6ZMoiRPAzZoVCtjloqS6VFgpydWmiqPUgvQ9c9MCwd3A6CSCwV4TRcwGuTEAZ5ZIk1gpwnJ8+bNi0OHDiFHjhzKUWZKRHDX1ehKC4s/alylKhrq27t9+3ZRRRMeHq4cZcz8aRpPT58+LdL3HjlyRLyypKcrXtOijUvuBuTm5iZKMlSCp8Ec1MgaFhamHGXMfFHBhZ5QKfV1vnz5cPToUfHKTBcHdwOjCT7owqc0BVRVQyUdnqqPmbOtW7eKggo1otLgJMr4yIHd9HFwN4LSpUuLybVz5cqFU6dOoWbNmmLCAsbMzbJly0QnAeoFRt0eKbDTdc1MHwd3I6HESSdPnkSxYsVw/fp1kZdm+fLlylHGTBvlihk3bpyYSenjx4+YMGEC9u3bJ7r/MvPAwd2IcufOjRMnToiMkpTbunfv3hg9erRIW8CYqaJSevfu3TFx4kSkTJlSjDqlfEqcute8cHA3sowZM4oSj2aKMcpvTTcNY6aISuxUWl+zZo3YnjZtmsgbw4Hd/Ihp9uiE6sIn1HCo3zvlgKdSEP1dx48fjzFjxlhdHo6wgGc44nUasXuJ2qFxq2ZwtE34NcfT7BkGXaM0ZzDVs9Pfkqpi6Brlv6vp0TeVhPap4uCehOjvPHv2bJFVkqpmqC/86tWrxQTC1iAi2Btdvx+ED7ZhOP8oBFVKFsTJQ57IU6E+cjlnxfy1y5DVnoN7cnj16pVoOKWOAGnSpMHKlStFxlP+m5omDu4mas+ePSItKo32K1mypOhqVqhQIeWo5bpx7izylq+It1c34c9D7zF/RGdUL5AKY/cEo0ERJ+VdsdGcnHHNW0t59WkIPF+viXPu3Dm0bdsWPj4+YkpJmpCGEoEx08XB3YTduXMHrVq1Eq/p06fH33//jdatWytHTdvUUb1x6YH+YJvVrTHmDe8e7ULTdmzNdBx7XxTjf66GkmkyYsPjTyiTVf+1RsEmrq6kGzduxLVr1/h6TSC672lGsSFDhohGVJqIhjI7UoBnpi0+wZ1OsPTp0yedCzMuueQutWnThk6TWH755RcpPDxcOWq5pvRrIc3ecF6SAm5K9nCVHrz9umutffv2fL0mEF17P/zwQ9S1169fP0kO8MpRZurocte1aOPeMsmISuxUUqK5WFOlSiWmJ6PSE81sY6mo3v1v9/3IXTwXwsPeQNOuGuYXAPljTdlixkTVMDSSmtopKE2vu7s75s+fz7nYLYw6uFNRXtfCjI6qEmhiYeoPT0O6L126JG68FStWiMdmSzNpcH/ce+cCt4JZEPGeZq/yxZwZ8zFl0VLwPPnG5evriz59+qBatWp48OCBSHZ3/vx5dOjQgau0zE3MWK1ZtHDJ3URUqlRJTLZNPRRCQ0NF3+KWLVuKXgyWQ4Wb9y6jaotOyO0AOOWvii5NymH+9NFwLV4dthxgjIYa7SktxpIlS6BSqfDLL7+I7I40kppZKKqb+aTnH0t6VHe8bt06ycnJSdSFZsqUSZo5c6YkB3zlHSwmrnPXz8/PT+rYsWNU3boczKV9+/bx38vM0dnTtWjjkruJocfjTp06id4f9erVE/NU/vrrr6LUdfjwYeVdjMVNvrexadMm0dWW6tQdHBwwb9483Lp1S2Qq5WoYy8fB3URRXhrKoU0jWjNnzoyHDx+KYN+tWzcx0zxj+tC4gGbNmokqPqrWo6ykV65cEaNPrW1EtDXjM23CqHRFycaePn2KyZMnw97eXuT8KFq0qAj6lK2PMQ3qq065izQTVjs7O4s6dpoxieY6ZdZFDGJSV8fpwo9upoTmrWzSpAkCAwPFNpXOaAAPPXJbM04/ALx+/VpMqEEzJRGa/o6e/GhyDWZ54jOIiUvuZoQyS9LjNQ0VJ56enqIunnpCiM9oZnXo6Y26zdJ1QIGdql0oqyN1qeXAbt04uJsZmgWHBj7RFH7FixcXdfGUgIzqValkz6wDfZhTKmnqq07dZqlunQbAUYCnqhiax5dZNw7uZooaV69evYoFCxaIBlfNpCCUn+bmzZvKu5iloUlfaEYv6gXTuHFj3LhxA3nz5hU9YmjmrwoVKijvZNaOg7sZs7W1Rb9+/US6glGjRiFt2rQiyRY9olPWSUpKxiwDBXVKD03TNlIjO3VppNL5jBkzRFI1HmXKYuLgbgGcnJxELwlvb2/07dtXBH1qaKVeE1TC9/Dw4J41Zop6wFBJnYI6TX1HaQNoVCnlIaJ1GgNBvagYi4V6y0SlFIu5MLP0+PFjSS7dSalTpxajEmkpVKiQtHTpUundu3fKuyyHJY5QDQoKkqZPny7lyJEj2jlcs2aNJJfilXcxa6WdSUD7nzZ1V8j49KthZocmX6BHecoV/+jRI7GP6uepNwWV8LNlyyb2mTtL6gpJT1/UjrJq1Sq8fUuJ1YBSpUqJard27dqJpzLG5KCtrEVno919XYT4mCV2zcIsApX0Nm7cKLm5uUWVAlOlSiXJQVE6evSofKrN+1ybe8ldpVJJnp6eUuPGjaUUKVJEnaM6depIe/bskT5+/Ki8kzE17dK69j9tXHK3InSqqVcN1dfu2LEjqh6eRrz++OOPIqeNq6ur2GdOzLHkLn8Y4fr162KMAj1Z0VMWofpzShtAWRvLlCnDjaRMp/iU3Dm4WykKJjTLPT3+v3jxQuyjR/5GjRqJIE/d7GgyEXNgTsH93r172LBhA9atWxdVVUYoPQD1V+/RowcyZcqk7GVMNw7u7IsotzflIaEgTxN30zahkY40qQPNiE9zvZryvJqmHtxpoBkNPKNeSzRyVIM+PJs2bSqemmrVqsVJvVi8xSe4q+vcGZPJQUjk/paDJF05UQvVA1P9/MWLF5V3mhZTrHOnn+fIkSOS/CQU7W9Ji/xBKXouffjwQXk3YwlEl7uuRYu65M6YFkoZu3fvXjG8ff/+/aKvtQb1nafkZVR9Q8Pd7ezslCPJx1RK7mFhYTh+/Lj429FTEJXYNRwdHUWiN/pZ6W9nCn83Zsb0RW2tW4CDO4vTmzdvRLXN9u3bRbCnbQ3KRlmjRg3Url1bVCuUK1cuWbrqJVdwp7z68tOMmIeU0up6eXlBLo0rR9XdTumDkKq16tevjzRp0ihHGPtKHNyZIVHgot42VDKlEj3lNdFGwZ4yV1IvD0qB8M0334jRlClTplTeYRxJEdxp+D/9vhTIKUEbzT8aM70D1Zm7ubmJmY6oQZrmxTX2786sFAd3ZkzUy+bo0aM4duyYqI6gHDcxUXUEBXoK+JTsiqp1aCg9lWoNFYwNGdzpdqDfi/K1UAI2mu6QFuq2qF09RSiXDw0wog8xenKhJxjOxsiSBAd3lpR8fX1FiZayVVIwvHDhgphFSheaJYi6/+XPnx/58uUTk0tQj5ysWbOKwE+v9CQQn4Ad3+BOlzpVK9HP+fLlSzx//lx0CaVgTutUR04fUFR3HhN97QIFCohATiVyekKhaqjUqVMr72AsCcU3uNNFr4uhSlbMelG9NE0wQqVfqtagbIZ3795FSEiI8g79aEBPxowZxQcBJUdLly6dKC3TfgqqqVKlEtUehw4dQt26dcWgLOrKSSXs8PBwEaRpCD99r6CgIAQEBIjqlS+h70kDu+gpg5426KmDFvoZGDMF8YnZHNxZkqPrjaaFozwqVFqm3jlUgn727JnY7+/vL17fv3+v/A/DoWqiLFmyiLw6OXLkQM6cOcUTA63TUwSVzim4M2bK4hOzObgzk0UlbyptBwcHi+oUKoXTPiqVU+MuldI108z17NlT9NShhboZUumeSvlU2qcSt4uLiwjanB6XWYL4xGwO7szsJVdXSMaSS3xiNo93ZowxC8TBnTHGLJA6uFNRXtfCGGPM9FB41rVo4ZI7Y4xZIA7ujDFmgTi4M4MJ8/NGh5Z1kc7GBvb2mdF1yFSE62nVZ4wZFwd3ZiARWLJ4BWYs3oS3UgT2rx+DbbNHYfG/Z5TjjLGkxMGdGUZEKNoMHI+c2WiKuNSo2fYXVC3rjGB/f/VxxliSUg9iUjZiitH4yli8qcJ8Ua/ZD9i48xCyOX7O8R4W8AxHvE4jPFzZEcUOjVs1g6Ntwq86HsTErE18ptkTJXfa1LUwlmAqFW4f2Yn/tfgOlVr3QWaHz/nMI4K98WPHn/H3kkUYMm6KmFf0l5+6YNbydfK6J8LU07cyxr6Agriuf9q4WoYZlPed87j01AeqFGkwfcAP+GnSmqgyxv17AVix2xMLJ/6Ell0HwMPDHQVcwjFh/jp5fTmy2nORgjFD4eDODKpgySr4X7d++PfAcYz5sQ42LFmMMKXir2TFSqLa5e6Np8joRJkX3yDoBZDVOb04zhgzHA7uzEhs0at7d0QEhSL6/EXAmfNecE6fHQj0xYNwV6RzUA4wxgyGgzszmvD3EfimSV1k0KoLpHr3v933I3fxXAgPewNNu2qYXwD3iWfMgDi4M4MIuHsGXXr8hGOnr6i3n93EuMWeWDVjYrQ0RZMG98e9dy5wK5gFEe/fynt8MWfGfExZtFQu6zPGDIWDOzMIx6yuyOZii0Wz/0T37gOw0fMy5m/YgpL5nJV3EBVu3ruMqi06IbcD4JS/Kro0KYf500fDtXh12HJXRsYMRvRzV9YZM0vcz52x2LjkzhhjFoiDO2OMWSAO7owxZoG4zp19tePHj6Nv377K1tcrV64c1q1bF1WHfurUKTx9+lSs6zJv3jx4eXlxnTtjWji4syjhr2+jS/9xypZujTr2QY+WtZUttbdv38YZfBMqbdq0yJcvn7IFnDt3Ds+fP1e2Yps5cyZOnjzJwZ0xLRzcmdnj3jKMxcZ17owxZoE4uDOju3JqK/p16YLOnftix77T4GdFxoyPgzszquOb/kLz7mPxOjwcl87uQsvGVTFi7mY9Uw0wxgyFgzszoggcPhOEe7duYLOHB27euo0udUpj3vSZUWmAGWPGwcGdGY9Khb6/jYS9Zuo8Wwf06d4VEZEf1duMMaPh4M6MRw7mrpkdlQ21ZwEvUP+7hnCIMSUYY8ywOLizJLVv93707zcA3GuRMePi4M6SjO/Nw/DP1hrNvsmq7GGMGQsHd5YkVGF+mDxnHVYvHc+DjRhLAhzcmfGpwjBi3AwMnbEEzvYc2BlLChzcmXHJgX1S/wnoPuI35HO2F7sCnlyFx+7DYp0xZhycW4YZjSoiGH3bNMXy3aeUPRp2OHrfF7UKak/Bl3ivXr1ClixZuLqHMS0c3JnRBPt44+CZy8qWFjsXtGpWl+dMZcyIOLgzxpgFSvLgLsWVVUTPIRs9A14kGz3/QdJTItRXUNT7I+k+oLe8mcCCqL5vq+/XUov9TcTbE/Qr6/kGcX1fXV8ojm+s7xTo+yY2ev+Dnh9Kz9slvedez/dVXuNL79eXf05dR3Tvpe+b0D+2bnHdT/rvG2UlGj1fJ44fU/9Pqe+I7i8W119CJz1fXu/Po+8b6Hly1BsS9bxf7/dNZtygyhhjFoiDO2OMWSAO7owxZoE4uDPGmAXi4M4YYxaIu0IyxpjFAf4PszMqAJ5l2ecAAAAASUVORK5CYII="> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A1</strong> </em>for correct modulus and <em><strong>A1</strong> </em>for correct argument for part (a)(i), and <em><strong>A1</strong> </em>for other two points correct.<br>The points may not be labelled, and they may be shown by line segments.</p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p> <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>θ</mi><mo>=</mo><mn>4</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>θ</mi><mo>=</mo><mn>8</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p> <img 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"></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>8</mn></msub></math> is shown in the diagram above <em><strong>A1A1</strong></em></p>
<p><strong><br>Note:</strong> Award <em><strong>A1</strong> </em>for a point plotted on the circle and <em><strong>A1</strong> </em>for a point plotted in the second quadrant.</p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>This question was challenging to many candidates, and some left the answer blank. Those who attempted it often failed to gain any marks. It would have helped examiners credit responses if points that were plotted on the Argand diagram were labelled. Certainly, there was some confusion caused by the appearance of θ both in the modulus and argument of the complex numbers in Euler form. Better use of technology to help visualize the complex numbers by simply getting decimal approximations of values in terms of <em>π</em> or by converting from Euler to Cartesian form would have helped in this question.</p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>The sum of the first <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span> terms of a sequence <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\{ {u_n}\} ">
<mo fence="false" stretchy="false">{</mo>
<mrow>
<msub>
<mi>u</mi>
<mi>n</mi>
</msub>
</mrow>
<mo fence="false" stretchy="false">}</mo>
</math></span> is given by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{S_n} = 3{n^2} - 2n">
<mrow>
<msub>
<mi>S</mi>
<mi>n</mi>
</msub>
</mrow>
<mo>=</mo>
<mn>3</mn>
<mrow>
<msup>
<mi>n</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−<!-- − --></mo>
<mn>2</mn>
<mi>n</mi>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n \in {\mathbb{Z}^ + }">
<mi>n</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<msup>
<mrow>
<mi mathvariant="double-struck">Z</mi>
</mrow>
<mo>+</mo>
</msup>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_1}">
<mrow>
<msub>
<mi>u</mi>
<mn>1</mn>
</msub>
</mrow>
</math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_6}">
<mrow>
<msub>
<mi>u</mi>
<mn>6</mn>
</msub>
</mrow>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Prove that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\{ {u_n}\} ">
<mo fence="false" stretchy="false">{</mo>
<mrow>
<msub>
<mi>u</mi>
<mi>n</mi>
</msub>
</mrow>
<mo fence="false" stretchy="false">}</mo>
</math></span> is an arithmetic sequence, stating clearly its common difference.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_1} = 1">
<mrow>
<msub>
<mi>u</mi>
<mn>1</mn>
</msub>
</mrow>
<mo>=</mo>
<mn>1</mn>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_6} = {S_6} - {S_5} = 31">
<mrow>
<msub>
<mi>u</mi>
<mn>6</mn>
</msub>
</mrow>
<mo>=</mo>
<mrow>
<msub>
<mi>S</mi>
<mn>6</mn>
</msub>
</mrow>
<mo>−</mo>
<mrow>
<msub>
<mi>S</mi>
<mn>5</mn>
</msub>
</mrow>
<mo>=</mo>
<mn>31</mn>
</math></span> <strong><em>M1A1</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_n} = {S_n} - {S_{n - 1}}">
<mrow>
<msub>
<mi>u</mi>
<mi>n</mi>
</msub>
</mrow>
<mo>=</mo>
<mrow>
<msub>
<mi>S</mi>
<mi>n</mi>
</msub>
</mrow>
<mo>−</mo>
<mrow>
<msub>
<mi>S</mi>
<mrow>
<mi>n</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msub>
</mrow>
</math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = (3{n^2} - 2n) - \left( {3{{(n - 1)}^2} - 2(n - 1)} \right)">
<mo>=</mo>
<mo stretchy="false">(</mo>
<mn>3</mn>
<mrow>
<msup>
<mi>n</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>2</mn>
<mi>n</mi>
<mo stretchy="false">)</mo>
<mo>−</mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>3</mn>
<mrow>
<msup>
<mrow>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo>−</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>2</mn>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo>−</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = (3{n^2} - 2n) - (3{n^2} - 6n + 3 - 2n + 2)">
<mo>=</mo>
<mo stretchy="false">(</mo>
<mn>3</mn>
<mrow>
<msup>
<mi>n</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>2</mn>
<mi>n</mi>
<mo stretchy="false">)</mo>
<mo>−</mo>
<mo stretchy="false">(</mo>
<mn>3</mn>
<mrow>
<msup>
<mi>n</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>6</mn>
<mi>n</mi>
<mo>+</mo>
<mn>3</mn>
<mo>−</mo>
<mn>2</mn>
<mi>n</mi>
<mo>+</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 6n - 5">
<mo>=</mo>
<mn>6</mn>
<mi>n</mi>
<mo>−</mo>
<mn>5</mn>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d = {u_{n + 1}} - {u_n}">
<mi>d</mi>
<mo>=</mo>
<mrow>
<msub>
<mi>u</mi>
<mrow>
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msub>
</mrow>
<mo>−</mo>
<mrow>
<msub>
<mi>u</mi>
<mi>n</mi>
</msub>
</mrow>
</math></span> <strong><em>R1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 6n + 6 - 5 - 6n + 5">
<mo>=</mo>
<mn>6</mn>
<mi>n</mi>
<mo>+</mo>
<mn>6</mn>
<mo>−</mo>
<mn>5</mn>
<mo>−</mo>
<mn>6</mn>
<mi>n</mi>
<mo>+</mo>
<mn>5</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left( {6(n + 1) - 5} \right) - (6n - 5)">
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>6</mn>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>−</mo>
<mn>5</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mo stretchy="false">(</mo>
<mn>6</mn>
<mi>n</mi>
<mo>−</mo>
<mn>5</mn>
<mo stretchy="false">)</mo>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 6">
<mo>=</mo>
<mn>6</mn>
</math></span> (constant) <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Notes: </strong>Award <strong><em>R1 </em></strong>only if candidate provides a clear argument that proves that the difference between <strong>ANY </strong>two consecutive terms of the sequence is constant. Do not accept examples involving particular terms of the sequence nor circular reasoning arguments (<em>eg </em>use of formulas of APs to prove that it is an AP). Last <strong><em>A1 </em></strong>is independent of <strong><em>R1</em></strong>.</p>
<p> </p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Let <em>C</em> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} { - 2}&4 \\ 1&7 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>2</mn>
</mrow>
</mtd>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>7</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> and <strong><em>D</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 5&2 \\ { - 1}&a \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>5</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</mtd>
<mtd>
<mi>a</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
<p class="question" style="margin-top: 12.0pt; tab-stops: 1.0cm 36.0pt 72.0pt 108.0pt 144.0pt 180.0pt 216.0pt 252.0pt 288.0pt 324.0pt 360.0pt 396.0pt 432.0pt 468.0pt;">The 2 × 2 matrix <strong><em>Q</em></strong> is such that 3<strong><em>Q</em></strong> = 2<strong><em>C</em></strong> – <strong><em>D</em></strong></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <strong><em>Q</em></strong>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <strong><em>CD</em></strong>.</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>Find <strong><em>D</em></strong><sup>–1</sup>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p>3<strong><em>Q</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} { - 4}&8 \\ 2&{14} \end{array}} \right) - \left( {\begin{array}{*{20}{c}} 5&2 \\ { - 1}&a \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>4</mn>
</mrow>
</mtd>
<mtd>
<mn>8</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mrow>
<mn>14</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>5</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
<mtd>
<mi>a</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>(A1)</strong></em></p>
<p>3<strong><em>Q</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} { - 9}&6 \\ 3&{14 - a} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>9</mn>
</mrow>
</mtd>
<mtd>
<mn>6</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>3</mn>
</mtd>
<mtd>
<mrow>
<mn>14</mn>
<mo>−</mo>
<mi>a</mi>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>(A1)</strong></em></p>
<p><strong><em>Q</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} { - 3}&2 \\ 1&{\frac{{14 - a}}{3}} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>3</mn>
</mrow>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mrow>
<mfrac>
<mrow>
<mn>14</mn>
<mo>−</mo>
<mi>a</mi>
</mrow>
<mn>3</mn>
</mfrac>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>(A1) (N3)</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong><em>CD</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} { - 2}&4 \\ 1&7 \end{array}} \right)\left( {\begin{array}{*{20}{c}} 5&2 \\ { - 1}&a \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
</mtd>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>7</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>5</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
<mtd>
<mi>a</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left( {\begin{array}{*{20}{c}} { - 14}&{ - 4 + 4a} \\ { - 2}&{2 + 7a} \end{array}} \right)">
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>14</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>4</mn>
<mo>+</mo>
<mn>4</mn>
<mi>a</mi>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>2</mn>
<mo>+</mo>
<mn>7</mn>
<mi>a</mi>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>(A1)</strong></em><em><strong>(A1)(A1)(A1) (N4)</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>det <em><strong>D</strong></em> = 5<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span> + 2 (may be implied) <em><strong>(A1)</strong></em></p>
<p><strong><em>D</em></strong><sup>–1</sup><em><strong> </strong></em>= <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{{5a + 2}}\left( {\begin{array}{*{20}{c}} a&{ - 2} \\ 1&5 \end{array}} \right)">
<mfrac>
<mn>1</mn>
<mrow>
<mn>5</mn>
<mi>a</mi>
<mo>+</mo>
<mn>2</mn>
</mrow>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>a</mi>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>5</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span><em><strong> (A1) (N2)</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="w = a{{\text{e}}^{\frac{\pi }{4}{\text{i}}}}">
<mi>w</mi>
<mo>=</mo>
<mi>a</mi>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mfrac>
<mi>π<!-- π --></mi>
<mn>4</mn>
</mfrac>
<mrow>
<mtext>i</mtext>
</mrow>
</mrow>
</msup>
</mrow>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a \in {\mathbb{R}^ + }">
<mi>a</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<msup>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
<mo>+</mo>
</msup>
</mrow>
</math></span>.</p>
</div>
<div class="specification">
<p>for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span> = 2,</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>find the values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{w^2}"> <mrow> <msup> <mi>w</mi> <mn>2</mn> </msup> </mrow> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{w^3}"> <mrow> <msup> <mi>w</mi> <mn>3</mn> </msup> </mrow> </math></span>, and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{w^4}"> <mrow> <msup> <mi>w</mi> <mn>4</mn> </msup> </mrow> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>draw <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="w"> <mi>w</mi> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{w^2}"> <mrow> <msup> <mi>w</mi> <mn>2</mn> </msup> </mrow> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{w^3}"> <mrow> <msup> <mi>w</mi> <mn>3</mn> </msup> </mrow> </math></span>, and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{w^4}"> <mrow> <msup> <mi>w</mi> <mn>4</mn> </msup> </mrow> </math></span> on the following Argand diagram.</p>
<p><img src="data:image/png;base64,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"></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z = \frac{w}{{2 - {\text{i}}}}"> <mi>z</mi> <mo>=</mo> <mfrac> <mi>w</mi> <mrow> <mn>2</mn> <mo>−</mo> <mrow> <mtext>i</mtext> </mrow> </mrow> </mfrac> </math></span>.</p>
<p>Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a"> <mi>a</mi> </math></span> for which successive powers of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z"> <mi>z</mi> </math></span> lie on a circle.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="4{{\text{e}}^{\frac{\pi }{2}{\text{i}}}}"> <mn>4</mn> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> <mrow> <mtext>i</mtext> </mrow> </mrow> </msup> </mrow> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="8{{\text{e}}^{\frac{{3\pi }}{4}{\text{i}}}}"> <mn>8</mn> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>4</mn> </mfrac> <mrow> <mtext>i</mtext> </mrow> </mrow> </msup> </mrow> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="16{{\text{e}}^{\pi {\text{i}}}}"> <mn>16</mn> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mi>π</mi> <mrow> <mtext>i</mtext> </mrow> </mrow> </msup> </mrow> </math></span> (<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 4{\text{i}}"> <mo>=</mo> <mn>4</mn> <mrow> <mtext>i</mtext> </mrow> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - {\text{4}}\sqrt 2 + 4\sqrt 2 {\text{i}}"> <mo>−</mo> <mrow> <mtext>4</mtext> </mrow> <msqrt> <mn>2</mn> </msqrt> <mo>+</mo> <mn>4</mn> <msqrt> <mn>2</mn> </msqrt> <mrow> <mtext>i</mtext> </mrow> </math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 16"> <mo>−</mo> <mn>16</mn> </math></span>) <em><strong> (M1)A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img 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"> <em><strong> A3</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for correct arguments, award <em><strong>A1</strong></em> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="4{\text{i}}"> <mn>4</mn> <mrow> <mtext>i</mtext> </mrow> </math></span> and −16 clearly indicated, award <em><strong>A1</strong></em> for | <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="w"> <mi>w</mi> </math></span> | < 4 and 4 < | <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{w^3}"> <mrow> <msup> <mi>w</mi> <mn>3</mn> </msup> </mrow> </math></span> | < 16.</p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{2^2} + {1^2} = {a^2}"> <mrow> <msup> <mn>2</mn> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mn>1</mn> <mn>2</mn> </msup> </mrow> <mo>=</mo> <mrow> <msup> <mi>a</mi> <mn>2</mn> </msup> </mrow> </math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a = \sqrt 5 \,\,\left( { = 2.24} \right)"> <mi>a</mi> <mo>=</mo> <msqrt> <mn>5</mn> </msqrt> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mn>2.24</mn> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The geometric sequence <em>u</em><sub>1</sub>, <em>u</em><sub>2</sub>, <em>u</em><sub>3</sub>, … has common ratio <em>r.</em></p>
<p>Consider the sequence <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A = \left\{ {{a_n} = {\text{lo}}{{\text{g}}_2}\left| {{u_n}} \right|{\text{:}}\,n \in {\mathbb{Z}^ + }} \right\}">
<mi>A</mi>
<mo>=</mo>
<mrow>
<mo>{</mo>
<mrow>
<mrow>
<msub>
<mi>a</mi>
<mi>n</mi>
</msub>
</mrow>
<mo>=</mo>
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>2</mn>
</msub>
</mrow>
<mrow>
<mo>|</mo>
<mrow>
<mrow>
<msub>
<mi>u</mi>
<mi>n</mi>
</msub>
</mrow>
</mrow>
<mo>|</mo>
</mrow>
<mrow>
<mtext>:</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>n</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<msup>
<mrow>
<mi mathvariant="double-struck">Z</mi>
</mrow>
<mo>+</mo>
</msup>
</mrow>
</mrow>
<mo>}</mo>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <em>A</em> is an arithmetic sequence, stating its common difference<em> d</em> in terms of <em>r</em>.</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>A particular geometric sequence has <em>u</em><sub>1</sub> = 3 and a sum to infinity of 4.</p>
<p>Find the value of <em>d</em>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><strong>METHOD 1</strong></p>
<p>state that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_n} = {u_1}{r^{n - 1}}">
<mrow>
<msub>
<mi>u</mi>
<mi>n</mi>
</msub>
</mrow>
<mo>=</mo>
<mrow>
<msub>
<mi>u</mi>
<mn>1</mn>
</msub>
</mrow>
<mrow>
<msup>
<mi>r</mi>
<mrow>
<mi>n</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</math></span> (or equivalent) <em><strong>A1</strong></em></p>
<p>attempt to consider <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{{a_n}}">
<mrow>
<mrow>
<msub>
<mi>a</mi>
<mi>n</mi>
</msub>
</mrow>
</mrow>
</math></span> and use of at least one log rule <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{lo}}{{\text{g}}_2}\left| {{u_n}} \right| = {\text{lo}}{{\text{g}}_2}\left| {{u_1}} \right| + \left( {n - 1} \right){\text{lo}}{{\text{g}}_2}\left| r \right|">
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>2</mn>
</msub>
</mrow>
<mrow>
<mo>|</mo>
<mrow>
<mrow>
<msub>
<mi>u</mi>
<mi>n</mi>
</msub>
</mrow>
</mrow>
<mo>|</mo>
</mrow>
<mo>=</mo>
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>2</mn>
</msub>
</mrow>
<mrow>
<mo>|</mo>
<mrow>
<mrow>
<msub>
<mi>u</mi>
<mn>1</mn>
</msub>
</mrow>
</mrow>
<mo>|</mo>
</mrow>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>n</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>2</mn>
</msub>
</mrow>
<mrow>
<mo>|</mo>
<mi>r</mi>
<mo>|</mo>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p>(which is an AP) with <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d = {\text{lo}}{{\text{g}}_2}\left| r \right|">
<mi>d</mi>
<mo>=</mo>
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>2</mn>
</msub>
</mrow>
<mrow>
<mo>|</mo>
<mi>r</mi>
<mo>|</mo>
</mrow>
</math></span> (and 1<sup>st</sup> term <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{lo}}{{\text{g}}_2}\left| {{u_1}} \right|">
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>2</mn>
</msub>
</mrow>
<mrow>
<mo>|</mo>
<mrow>
<mrow>
<msub>
<mi>u</mi>
<mn>1</mn>
</msub>
</mrow>
</mrow>
<mo>|</mo>
</mrow>
</math></span>) <em><strong> A1</strong></em></p>
<p>so A is an arithmetic sequence <em><strong>AG</strong></em></p>
<p><strong>Note:</strong> Condone absence of modulus signs.</p>
<p><strong>Note:</strong> The final <em><strong>A</strong></em> mark may be awarded independently.</p>
<p><strong>Note:</strong> Consideration of the first two or three terms only will score <em><strong>M0</strong></em>.</p>
<p><em><strong>[4 marks]</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>consideration of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {d = } \right){a_{n + 1}} - {a_n}">
<mrow>
<mo>(</mo>
<mrow>
<mi>d</mi>
<mo>=</mo>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<msub>
<mi>a</mi>
<mrow>
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msub>
</mrow>
<mo>−</mo>
<mrow>
<msub>
<mi>a</mi>
<mi>n</mi>
</msub>
</mrow>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( d \right) = {\text{lo}}{{\text{g}}_2}\left| {{u_{n + 1}}} \right| - {\text{lo}}{{\text{g}}_2}\left| {{u_n}} \right|">
<mrow>
<mo>(</mo>
<mi>d</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>2</mn>
</msub>
</mrow>
<mrow>
<mo>|</mo>
<mrow>
<mrow>
<msub>
<mi>u</mi>
<mrow>
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msub>
</mrow>
</mrow>
<mo>|</mo>
</mrow>
<mo>−</mo>
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>2</mn>
</msub>
</mrow>
<mrow>
<mo>|</mo>
<mrow>
<mrow>
<msub>
<mi>u</mi>
<mi>n</mi>
</msub>
</mrow>
</mrow>
<mo>|</mo>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( d \right) = {\text{lo}}{{\text{g}}_2}\left| {\frac{{{u_{n + 1}}}}{{{u_n}}}} \right|">
<mrow>
<mo>(</mo>
<mi>d</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>2</mn>
</msub>
</mrow>
<mrow>
<mo>|</mo>
<mrow>
<mfrac>
<mrow>
<mrow>
<msub>
<mi>u</mi>
<mrow>
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msub>
</mrow>
</mrow>
<mrow>
<mrow>
<msub>
<mi>u</mi>
<mi>n</mi>
</msub>
</mrow>
</mrow>
</mfrac>
</mrow>
<mo>|</mo>
</mrow>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( d \right) = {\text{lo}}{{\text{g}}_2}\left| r \right|">
<mrow>
<mo>(</mo>
<mi>d</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>2</mn>
</msub>
</mrow>
<mrow>
<mo>|</mo>
<mi>r</mi>
<mo>|</mo>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p>which is constant <em><strong>R1</strong></em></p>
<p><strong>Note:</strong> Condone absence of modulus signs.</p>
<p><strong>Note:</strong> The final <em><strong>A</strong></em> mark may be awarded independently.</p>
<p><strong>Note:</strong> Consideration of the first two or three terms only will score <em><strong>M0</strong></em>.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempting to solve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{3}{{1 - r}} = 4">
<mfrac>
<mn>3</mn>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>r</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mn>4</mn>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r = \frac{1}{4}">
<mi>r</mi>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>4</mn>
</mfrac>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d = - \,2">
<mi>d</mi>
<mo>=</mo>
<mo>−</mo>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi><mo>=</mo><mtext>i</mtext><mi>z</mi><mo>+</mo><mn>1</mn></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi><mo>,</mo><mo> </mo><mi>z</mi><mo>∈</mo><mi mathvariant="normal">ℂ</mi></math>.</p>
</div>
<div class="specification">
<p>Find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi></math> when</p>
</div>
<div class="specification">
<p>Point <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi></math> on the Argand diagram can be transformed to point <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi></math> by two transformations.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>=</mo><mn>2</mn><mtext>i</mtext></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>=</mo><mn>1</mn><mo>+</mo><mtext>i</mtext></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Describe these two transformations and give the order in which they are applied.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence, or otherwise, find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi></math> when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi><mo>=</mo><mn>2</mn><mo>−</mo><mtext>i</mtext></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>i</mtext><mn>2</mn></msup><mo>=</mo><mo>-</mo><mn>1</mn></math> <em><strong>(M1)</strong></em></p>
<p><em><strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi><mo>=</mo><mo>-</mo><mn>2</mn><mo>+</mo><mn>1</mn><mo>=</mo><mo>-</mo><mn>1</mn></math> A1</strong></em></p>
<p><em><strong><br>[2 marks]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><em><strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>w</mi><mo>=</mo><mo>-</mo><mn>1</mn><mo>+</mo><mtext>i</mtext><mo>+</mo><mn>1</mn><mo>=</mo><mtext>i</mtext></math> A1</strong></em></p>
<p><em><strong><br>[1 mark]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p>rotation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>90</mn><mo>°</mo></math> (anticlockwise, centre at the origin) <em><strong> A1A1</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>A1</strong></em> for “rotation” and <em><strong>A1</strong></em> for “<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>90</mn><mo>°</mo></math>”.</p>
<p><br>followed by a translation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr></mtable></mfenced></math> <em><strong>A1</strong></em><br><br><strong>OR</strong><br>translation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd></mtr></mtable></mfenced></math> <em><strong>A1</strong></em><br><br>followed by rotation of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>90</mn><mo>°</mo></math> (anticlockwise, centre at the origin) <em><strong>A1</strong></em><em><strong>A1</strong></em><br><br><br><strong>Note:</strong> Award <em><strong>A1</strong></em> for “rotation” and A1 for “<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>90</mn><mo>°</mo></math>”.</p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p>move <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math> to left to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>-</mo><mtext>i</mtext></math> <em><strong>(M1)</strong></em></p>
<p>then rotate by <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>90</mn><mo>°</mo></math> to</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>1</mn><mo>-</mo><mtext>i</mtext></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>i</mtext><mi>z</mi><mo>+</mo><mn>1</mn><mo>=</mo><mn>2</mn><mo>-</mo><mtext>i</mtext></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>i</mtext><mi>z</mi><mo>=</mo><mn>1</mn><mo>-</mo><mtext>i</mtext></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>=</mo><mfrac><mrow><mn>1</mn><mo>-</mo><mtext>i</mtext></mrow><mtext>i</mtext></mfrac></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>1</mn><mo>-</mo><mtext>i</mtext></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} b&3 \\ 7&8 \end{array}} \right) + \left( {\begin{array}{*{20}{c}} 9&5 \\ { - 2}&7 \end{array}} \right) = \left( {\begin{array}{*{20}{c}} 4&8 \\ a&{15} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>b</mi>
</mtd>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>7</mn>
</mtd>
<mtd>
<mn>8</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>9</mn>
</mtd>
<mtd>
<mn>5</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>2</mn>
</mrow>
</mtd>
<mtd>
<mn>7</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>4</mn>
</mtd>
<mtd>
<mn>8</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>a</mi>
</mtd>
<mtd>
<mrow>
<mn>15</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent2">Write down the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent2">Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b">
<mi>b</mi>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent2">Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3\left( {\begin{array}{*{20}{c}} { - 4}&8 \\ 2&1 \end{array}} \right) - 5\left( {\begin{array}{*{20}{c}} 2&0 \\ q&{ - 4} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} { - 22}&{24} \\ 9&{23} \end{array}} \right)">
<mn>3</mn>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>4</mn>
</mrow>
</mtd>
<mtd>
<mn>8</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mn>5</mn>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>q</mi>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>4</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>22</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>24</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>9</mn>
</mtd>
<mtd>
<mrow>
<mn>23</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
<p class="indent2">Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q">
<mi>q</mi>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span> = 5<em><strong> A1 N1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b">
<mi>b</mi>
</math></span> + 9 = 4 <em><strong> (M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b">
<mi>b</mi>
</math></span> = −5 <em><strong> A1 N2</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Comparing elements 3(2) − 5(<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q">
<mi>q</mi>
</math></span>) = −9 <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="q">
<mi>q</mi>
</math></span> = 3 <em><strong>A2 N2</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The diagram shows a sector, <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>OAB</mtext></math>, of a circle with centre <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math> and radius <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math>, such that <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>AÔB</mtext><mo>=</mo><mi>θ</mi></math>.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<p>Sam measured the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math> to be <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo> </mo><mtext>cm</mtext></math> and the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi></math> to be <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>30</mn><mo>°</mo></math>.</p>
</div>
<div class="specification">
<p>It is found that Sam’s measurements are accurate to only one significant figure.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Use Sam’s measurements to calculate the area of the sector. Give your answer to four significant figures.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the upper bound and lower bound of the area of the sector.</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, with justification, the largest possible percentage error if the answer to part (a) is recorded as the area of the sector.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">π</mi><mo>×</mo><msup><mn>2</mn><mn>2</mn></msup><mo>×</mo><mfrac><mn>30</mn><mn>360</mn></mfrac></math> <strong><em>(M1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>1</mn><mo>.</mo><mn>047</mn><mo> </mo><msup><mtext>cm</mtext><mn>2</mn></msup></math> <strong><em>A1</em></strong></p>
<p><br><strong>Note:</strong> Do not award the final mark if the answer is not correct to 4 sf.</p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to substitute any two values from <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>.</mo><mn>5</mn><mo>,</mo><mo> </mo><mn>2</mn><mo>.</mo><mn>5</mn><mo>,</mo><mo> </mo><mn>25</mn></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>35</mn></math> into area of sector formula <strong><em>(M1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mtext>upper bound</mtext><mo>=</mo><mi mathvariant="normal">π</mi><mo>×</mo><mn>2</mn><mo>.</mo><msup><mn>5</mn><mn>2</mn></msup><mo>×</mo><mfrac><mn>35</mn><mn>360</mn></mfrac><mo>=</mo></mrow></mfenced><mo> </mo><mo> </mo><mn>1</mn><mo>.</mo><mn>91</mn><mo> </mo><msup><mtext>cm</mtext><mn>2</mn></msup><mo> </mo><mo> </mo><mfenced><mrow><mn>1</mn><mo>.</mo><mn>90895</mn><mo>…</mo></mrow></mfenced></math> <strong><em>A1</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mtext>lower bound</mtext><mo>=</mo><mi mathvariant="normal">π</mi><mo>×</mo><mn>1</mn><mo>.</mo><msup><mn>5</mn><mn>2</mn></msup><mo>×</mo><mfrac><mn>25</mn><mn>360</mn></mfrac><mo>=</mo></mrow></mfenced><mo> </mo><mo> </mo><mn>0</mn><mo>.</mo><mn>491</mn><mo> </mo><msup><mtext>cm</mtext><mn>2</mn></msup><mo> </mo><mo> </mo><mfenced><mrow><mn>0</mn><mo>.</mo><mn>490873</mn><mo>…</mo></mrow></mfenced></math> <strong><em>A1</em></strong></p>
<p><br><strong>Note:</strong> Given the nature of the question, accept correctly rounded <strong>OR</strong> correctly truncated <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math> significant figure answers.</p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mfenced open="|" close="|"><mfrac><mrow><mn>1</mn><mo>.</mo><mn>047</mn><mo>-</mo><mn>1</mn><mo>.</mo><mn>90895</mn><mo>…</mo></mrow><mrow><mn>1</mn><mo>.</mo><mn>90895</mn><mo>…</mo></mrow></mfrac></mfenced><mo>×</mo><mn>100</mn><mo>=</mo></mrow></mfenced><mo> </mo><mo> </mo><mn>45</mn><mo>.</mo><mn>2</mn><mo> </mo><mo> </mo><mfenced><mo>%</mo></mfenced><mo> </mo><mo> </mo><mfenced><mrow><mn>45</mn><mo>.</mo><mn>1532</mn><mo>…</mo></mrow></mfenced></math> <strong><em>A1</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mfenced open="|" close="|"><mfrac><mrow><mn>1</mn><mo>.</mo><mn>047</mn><mo>-</mo><mn>0</mn><mo>.</mo><mn>490873</mn><mo>…</mo></mrow><mrow><mn>0</mn><mo>.</mo><mn>490873</mn><mo>…</mo></mrow></mfrac></mfenced><mo>×</mo><mn>100</mn><mo>=</mo></mrow></mfenced><mo> </mo><mo> </mo><mn>113</mn><mo> </mo><mo> </mo><mfenced><mo>%</mo></mfenced><mo> </mo><mo> </mo><mfenced><mrow><mn>113</mn><mo>.</mo><mn>293</mn><mo>…</mo></mrow></mfenced></math> <strong><em>A1</em></strong></p>
<p>so the largest percentage error is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>113</mn><mo> </mo><mo>%</mo></math> <strong><em>A1</em></strong></p>
<p><strong><br>Note:</strong> Accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>45</mn><mo>.</mo><mn>1</mn><mo> </mo><mo>(</mo><mo>%</mo><mo>)</mo></math> (<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>45</mn><mo>.</mo><mn>1428</mn></math>), from use of full accuracy answers. Given the nature of the question, accept correctly rounded <strong>OR</strong> correctly truncated <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math> significant figure answers. Award <em><strong>A0A1A0</strong></em> if <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>113</mn><mo>%</mo></math> is the only value found.</p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>In part (a), the area was almost always found correctly although some candidates gave the answer 1.0472 which is correct to 4 decimal places, not 4 significant figures as required. In part (b), many candidates failed to realize that the upper bounds for <em>r</em> and <em>θ</em> were 2.5 and 35° and lower bounds were 1.5 and 25°. Consequently, the bounds for the area were incorrect. In many cases, the incorrect values in part (b) were followed through into part (c) although in the percentage error calculations, many candidates had 1.047 in the denominator instead of the appropriate bound.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p>In an arithmetic sequence, the sum of the 3rd and 8th terms is 1.</p>
<p>Given that the sum of the first seven terms is 35, determine the first term and the common difference.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p>attempting to form two equations involving <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_1}">
<mrow>
<msub>
<mi>u</mi>
<mn>1</mn>
</msub>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d">
<mi>d</mi>
</math></span><em><strong> M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {{u_1} + 2d} \right) + \left( {{u_1} + 7d} \right) = 1">
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msub>
<mi>u</mi>
<mn>1</mn>
</msub>
</mrow>
<mo>+</mo>
<mn>2</mn>
<mi>d</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msub>
<mi>u</mi>
<mn>1</mn>
</msub>
</mrow>
<mo>+</mo>
<mn>7</mn>
<mi>d</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>1</mn>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{7}{2}\left[ {2{u_1} + 6d} \right] = 35">
<mfrac>
<mn>7</mn>
<mn>2</mn>
</mfrac>
<mrow>
<mo>[</mo>
<mrow>
<mn>2</mn>
<mrow>
<msub>
<mi>u</mi>
<mn>1</mn>
</msub>
</mrow>
<mo>+</mo>
<mn>6</mn>
<mi>d</mi>
</mrow>
<mo>]</mo>
</mrow>
<mo>=</mo>
<mn>35</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2{u_1} + 9d = 1">
<mn>2</mn>
<mrow>
<msub>
<mi>u</mi>
<mn>1</mn>
</msub>
</mrow>
<mo>+</mo>
<mn>9</mn>
<mi>d</mi>
<mo>=</mo>
<mn>1</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="14{u_1} + 42d = 70\,\,\,\left( {2{u_1} + 6d = 10} \right)">
<mn>14</mn>
<mrow>
<msub>
<mi>u</mi>
<mn>1</mn>
</msub>
</mrow>
<mo>+</mo>
<mn>42</mn>
<mi>d</mi>
<mo>=</mo>
<mn>70</mn>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mrow>
<msub>
<mi>u</mi>
<mn>1</mn>
</msub>
</mrow>
<mo>+</mo>
<mn>6</mn>
<mi>d</mi>
<mo>=</mo>
<mn>10</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for any two correct equations</p>
<p>attempting to solve their equations:<em><strong> M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_1} = 14">
<mrow>
<msub>
<mi>u</mi>
<mn>1</mn>
</msub>
</mrow>
<mo>=</mo>
<mn>14</mn>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d = - 3">
<mi>d</mi>
<mo>=</mo>
<mo>−</mo>
<mn>3</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>Let <strong><em>A</em></strong>, <strong><em>B</em> </strong>and <strong><em>C</em> </strong>be non-singular 2×2 matrices, <strong><em>I</em> </strong>the 2×2 identity matrix and <em>k</em> a scalar. The following statements are <strong>incorrect</strong>. For each statement, write down the correct version of the right hand side.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent1" style="margin-top:12.0pt;">(<strong><em>A</em> </strong>+ <strong><em>B</em></strong>)<sup>2</sup> = <strong><em>A</em></strong><sup>2</sup> + 2<strong><em>AB</em> </strong>+ <strong><em>B</em></strong><sup>2</sup></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent1" style="margin-top:12.0pt;"> (<strong><em>A</em> </strong>– <em>k<strong>I</strong></em>)<sup>3</sup><strong> </strong>= <strong><em>A</em></strong><sup>3</sup> – 3<em>k<strong>A</strong></em><sup>2</sup><strong> </strong>+ 3<em>k</em><sup>2</sup><strong><em>A</em> </strong>– <em>k</em><sup>3 </sup></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent1" style="margin-top:12.0pt;"><strong><em>CA</em></strong> = <strong><em>B</em></strong> <strong><em>C</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{B}{A}">
<mfrac>
<mi>B</mi>
<mi>A</mi>
</mfrac>
</math></span></p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>(<strong><em>A</em></strong> + <strong><em>B</em></strong>)<sup>2</sup> = <strong><em>A</em></strong><sup>2</sup> + <strong><em>AB</em></strong> + <strong><em>BA</em></strong> + <strong><em>B</em></strong><sup>2</sup> <strong><em>A2</em></strong></p>
<p><em></em><em></em><strong>Note: </strong>Award <em><strong>A1</strong></em> in parts (a) to (c) if error is correctly identified, but not corrected.</p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>(<strong><em>A</em></strong> <em>–</em> <em>k<strong>I</strong></em>)<sup>3</sup> = <strong>A</strong><sup>3</sup> <em>–</em> 3<em>k<strong>A</strong></em><sup>2</sup> <em>+</em> 3<em>k</em><sup>2</sup><strong><em>A</em></strong> <em>–</em> <em>k</em><sup>3</sup><strong><em>I</em></strong> <strong><em>A2</em></strong></p>
<p><em></em><em></em><strong>Note: </strong>Award <em><strong>A1</strong></em> in parts (a) to (c) if error is correctly identified, but not corrected.</p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong><em>CA</em></strong> = <strong><em>B</em></strong> ⇒ <strong><em>C</em></strong> = <strong><em>BA</em></strong><sup>–1</sup> <strong><em>A2</em></strong></p>
<p><strong>Note: </strong>Award <em><strong>A1</strong></em> in parts (a) to (c) if error is correctly identified, but not corrected.</p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>+</mo><mn>5</mn><mtext>i</mtext></math> in exponential form.</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="text-align:center;"><img src="data:image/png;base64,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"></p>
<p style="text-align:left;">An equilateral triangle is to be drawn on the Argand plane with one of the vertices at the point corresponding to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>+</mo><mn>5</mn><mtext>i</mtext></math> and all the vertices equidistant from <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn></math>.</p>
<p style="text-align:left;">Find the points that correspond to the other two vertices. Give your answers in Cartesian form.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color:#999;font-size:90%;font-style:italic;">* This sample question was produced by experienced DP mathematics senior examiners to aid teachers in preparing for external assessment in the new MAA course. There may be minor differences in formatting compared to formal exam papers.</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mo>.</mo><mn>385</mn><mo>…</mo><msup><mtext>e</mtext><mrow><mn>1</mn><mo>.</mo><mn>1902</mn><mo>…</mo><mtext>i</mtext></mrow></msup><mo>≈</mo><mo> </mo><mn>5.39</mn><msup><mi mathvariant="normal">e</mi><mrow><mn>1</mn><mo>.</mo><mn>19</mn><mi mathvariant="normal">i</mi></mrow></msup></math> <strong>A1A1</strong></p>
<p> </p>
<p><strong>Note:</strong> Accept equivalent answers: <math xmlns="http://www.w3.org/1998/Math/MathML"><mo> </mo><mn>5.39</mn><msup><mi mathvariant="normal">e</mi><mrow><mi mathvariant="normal">-</mi><mn>5</mn><mo>.</mo><mn>09</mn><mi mathvariant="normal">i</mi></mrow></msup></math></p>
<p> </p>
<p><strong>[2 marks]</strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>multiply by <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>e</mtext><mrow><mfrac><mrow><mn>2</mn><mi mathvariant="normal">π</mi></mrow><mn>3</mn></mfrac><mtext>i</mtext></mrow></msup></math> <strong>(M1)</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>5</mn><mo>.</mo><mn>33</mn><mo>-</mo><mn>0</mn><mo>.</mo><mn>77</mn><mtext>i</mtext><mo>,</mo><mo> </mo><mn>3</mn><mo>.</mo><mn>33</mn><mo>-</mo><mn>4</mn><mo>.</mo><mn>23</mn><mtext>i</mtext></math> <strong>A1A1</strong></p>
<p> </p>
<p><strong>[3 marks]</strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>An infinite geometric sequence, with terms <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mi>n</mi></msub></math>, is such that <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>1</mn></msub><mo>=</mo><mn>2</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mtext>Σ</mtext><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mo>∞</mo></munderover><msub><mi>u</mi><mi>k</mi></msub><mo>=</mo><mn>10</mn></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the common ratio, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math>, for the sequence.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the least value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mi>n</mi></msub><mo><</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn><mo>=</mo><mfrac><mn>2</mn><mrow><mn>1</mn><mo>-</mo><mi>r</mi></mrow></mfrac></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>8</mn></math> <em><strong>A1</strong></em></p>
<p><em><strong><br>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>×</mo><msup><mfenced><mrow><mn>0</mn><mo>.</mo><mn>8</mn></mrow></mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><mo><</mo><mn>0</mn><mo>.</mo><mn>5</mn></math> <strong>OR </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>×</mo><msup><mfenced><mrow><mn>0</mn><mo>.</mo><mn>8</mn></mrow></mfenced><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup><mo>=</mo><mn>0</mn><mo>.</mo><mn>5</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>n</mi><mo>></mo></mrow></mfenced><mo> </mo><mn>7</mn><mo>.</mo><mn>212</mn><mo>…</mo></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>8</mn></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> If <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>7</mn></math> is seen, with or without seeing the value <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn><mo>.</mo><mn>212</mn><mo>…</mo></math> then award <em><strong>M1A1A0</strong></em>.</p>
<p><em><strong><br>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>A number of candidates did not attempt what should have been a straightforward question. Perhaps because it relied on a part of the syllabus that is restricted to HL and is not in common with SL. Some attempted it but were unaware of the formula for the sum of an infinite geometric sequence, although this is in the formula booklet. By far the biggest error was to fail to recognize that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> was the smallest integer value greater than that found from solving the equation. There were disappointingly few candidates who adopted a tabular or graphical approach to this question using technology. Some relied on trial-and-error.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>A number of candidates did not attempt what should have been a straightforward question. Perhaps because it relied on a part of the syllabus that is restricted to HL and is not in common with SL. Some attempted it but were unaware of the formula for the sum of an infinite geometric sequence, although this is in the formula booklet. By far the biggest error was to fail to recognize that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> was the smallest integer value greater than that found from solving the equation. There were disappointingly few candidates who adopted a tabular or graphical approach to this question using technology. Some relied on trial-and-error.</p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The 1st, 4th and 8th terms of an arithmetic sequence, with common difference <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d">
<mi>d</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d \ne 0">
<mi>d</mi>
<mo>≠<!-- ≠ --></mo>
<mn>0</mn>
</math></span>, are the first three terms of a geometric sequence, with common ratio <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r">
<mi>r</mi>
</math></span>. Given that the 1st term of both sequences is 9 find</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d">
<mi>d</mi>
</math></span>;</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r">
<mi>r</mi>
</math></span>;</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><strong>EITHER</strong></p>
<p>the first three terms of the geometric sequence are <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="9">
<mn>9</mn>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="9r">
<mn>9</mn>
<mi>r</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="9{r^2}">
<mn>9</mn>
<mrow>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
</math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="9 + 3d = 9r( \Rightarrow 3 + d = 3r)">
<mn>9</mn>
<mo>+</mo>
<mn>3</mn>
<mi>d</mi>
<mo>=</mo>
<mn>9</mn>
<mi>r</mi>
<mo stretchy="false">(</mo>
<mo stretchy="false">⇒</mo>
<mn>3</mn>
<mo>+</mo>
<mi>d</mi>
<mo>=</mo>
<mn>3</mn>
<mi>r</mi>
<mo stretchy="false">)</mo>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="9 + 7d = 9{r^2}">
<mn>9</mn>
<mo>+</mo>
<mn>7</mn>
<mi>d</mi>
<mo>=</mo>
<mn>9</mn>
<mrow>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
</math></span> <strong><em>(A1)</em></strong></p>
<p>attempt to solve simultaneously <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="9 + 7d = 9{\left( {\frac{{3 + d}}{3}} \right)^2}">
<mn>9</mn>
<mo>+</mo>
<mn>7</mn>
<mi>d</mi>
<mo>=</mo>
<mn>9</mn>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mn>3</mn>
<mo>+</mo>
<mi>d</mi>
</mrow>
<mn>3</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</math></span></p>
<p><strong>OR</strong></p>
<p>the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{{\text{1}}^{{\text{st}}}}">
<mrow>
<msup>
<mrow>
<mtext>1</mtext>
</mrow>
<mrow>
<mrow>
<mtext>st</mtext>
</mrow>
</mrow>
</msup>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{{\text{4}}^{{\text{th}}}}">
<mrow>
<msup>
<mrow>
<mtext>4</mtext>
</mrow>
<mrow>
<mrow>
<mtext>th</mtext>
</mrow>
</mrow>
</msup>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{{\text{8}}^{{\text{th}}}}">
<mrow>
<msup>
<mrow>
<mtext>8</mtext>
</mrow>
<mrow>
<mrow>
<mtext>th</mtext>
</mrow>
</mrow>
</msup>
</mrow>
</math></span> terms of the arithmetic sequence are</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="9,{\text{ }}9 + 3d,{\text{ }}9 + 7d">
<mn>9</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>9</mn>
<mo>+</mo>
<mn>3</mn>
<mi>d</mi>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>9</mn>
<mo>+</mo>
<mn>7</mn>
<mi>d</mi>
</math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{9 + 7d}}{{9 + 3d}} = \frac{{9 + 3d}}{9}">
<mfrac>
<mrow>
<mn>9</mn>
<mo>+</mo>
<mn>7</mn>
<mi>d</mi>
</mrow>
<mrow>
<mn>9</mn>
<mo>+</mo>
<mn>3</mn>
<mi>d</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mn>9</mn>
<mo>+</mo>
<mn>3</mn>
<mi>d</mi>
</mrow>
<mn>9</mn>
</mfrac>
</math></span> <strong><em>(A1)</em></strong></p>
<p>attempt to solve <strong><em>(M1)</em></strong></p>
<p><strong>THEN</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d = 1">
<mi>d</mi>
<mo>=</mo>
<mn>1</mn>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r = \frac{4}{3}">
<mi>r</mi>
<mo>=</mo>
<mfrac>
<mn>4</mn>
<mn>3</mn>
</mfrac>
</math></span> <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Accept answers where a candidate obtains <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d">
<mi>d</mi>
</math></span> by finding <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r">
<mi>r</mi>
</math></span> first. The first two marks in either method for part (a) are awarded for the same ideas and the third mark is awarded for attempting to solve an equation in <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r">
<mi>r</mi>
</math></span>.</p>
<p> </p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="w = 2\left( {{\text{cos}}\frac{\pi }{3} + {\text{i}}\,{\text{sin}}\frac{\pi }{3}} \right)">
<mi>w</mi>
<mo>=</mo>
<mn>2</mn>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>cos</mtext>
</mrow>
<mfrac>
<mi>π<!-- π --></mi>
<mn>3</mn>
</mfrac>
<mo>+</mo>
<mrow>
<mtext>i</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mfrac>
<mi>π<!-- π --></mi>
<mn>3</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
</div>
<div class="specification">
<p>These four points form the vertices of a quadrilateral, <em>Q</em>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Express <em>w</em><sup>2</sup> and <em>w</em><sup>3</sup> in modulus-argument form.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch on an Argand diagram the points represented by <em>w</em><sup>0</sup> , <em>w</em><sup>1</sup> , <em>w</em><sup>2</sup> and <em>w</em><sup>3</sup>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the area of the quadrilateral <em>Q</em> is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{21\sqrt 3 }}{2}"> <mfrac> <mrow> <mn>21</mn> <msqrt> <mn>3</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> </math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z = 2\left( {{\text{cos}}\frac{\pi }{n} + {\text{i}}\,{\text{sin}}\frac{\pi }{n}} \right),\,\,n \in {\mathbb{Z}^ + }"> <mi>z</mi> <mo>=</mo> <mn>2</mn> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>cos</mtext> </mrow> <mfrac> <mi>π</mi> <mi>n</mi> </mfrac> <mo>+</mo> <mrow> <mtext>i</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mi>π</mi> <mi>n</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mi>n</mi> <mo>∈</mo> <mrow> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>+</mo> </msup> </mrow> </math></span>. The points represented on an Argand diagram by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{z^0},\,\,{z^1},\,\,{z^2},\, \ldots \,,\,\,{z^n}"> <mrow> <msup> <mi>z</mi> <mn>0</mn> </msup> </mrow> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mrow> <msup> <mi>z</mi> <mn>1</mn> </msup> </mrow> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mrow> <msup> <mi>z</mi> <mn>2</mn> </msup> </mrow> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mo>…</mo> <mspace width="thinmathspace"></mspace> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mrow> <msup> <mi>z</mi> <mi>n</mi> </msup> </mrow> </math></span> form the vertices of a polygon <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{P_n}"> <mrow> <msub> <mi>P</mi> <mi>n</mi> </msub> </mrow> </math></span>.</p>
<p>Show that the area of the polygon <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{P_n}"> <mrow> <msub> <mi>P</mi> <mi>n</mi> </msub> </mrow> </math></span> can be expressed in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a\left( {{b^n} - 1} \right){\text{sin}}\frac{\pi }{n}"> <mi>a</mi> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mi>b</mi> <mi>n</mi> </msup> </mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mi>π</mi> <mi>n</mi> </mfrac> </math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a,\,\,b\, \in \mathbb{R}"> <mi>a</mi> <mo>,</mo> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mi>b</mi> <mspace width="thinmathspace"></mspace> <mo>∈</mo> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> </math></span>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{w^2} = 4\text{cis}\left( {\frac{{2\pi }}{3}} \right){\text{;}}\,\,{w^3} = 8{\text{cis}}\left( \pi \right)"> <mrow> <msup> <mi>w</mi> <mn>2</mn> </msup> </mrow> <mo>=</mo> <mn>4</mn> <mtext>cis</mtext> <mrow> <mo>(</mo> <mrow> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mn>3</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow> <mtext>;</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mrow> <msup> <mi>w</mi> <mn>3</mn> </msup> </mrow> <mo>=</mo> <mn>8</mn> <mrow> <mtext>cis</mtext> </mrow> <mrow> <mo>(</mo> <mi>π</mi> <mo>)</mo> </mrow> </math></span> <em><strong>(M1)A1A1</strong></em></p>
<p><strong>Note:</strong> Accept Euler form.</p>
<p><strong>Note:</strong> <em><strong>M1</strong></em> can be awarded for either both correct moduli or both correct arguments.</p>
<p><strong>Note:</strong> Allow multiplication of correct Cartesian form for <em><strong>M1</strong></em>, final answers must be in modulus-argument form.</p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img 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"> <em><strong>A1A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>use of area = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{2}ab\,\,{\text{sin}}\,C"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mi>a</mi> <mi>b</mi> <mspace width="thinmathspace"></mspace> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>C</mi> </math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{2} \times 1 \times 2 \times {\text{sin}}\frac{\pi }{3} + \frac{1}{2} \times 2 \times 4 \times {\text{sin}}\frac{\pi }{3} + \frac{1}{2} \times 4 \times 8 \times {\text{sin}}\frac{\pi }{3}"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>×</mo> <mn>1</mn> <mo>×</mo> <mn>2</mn> <mo>×</mo> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mi>π</mi> <mn>3</mn> </mfrac> <mo>+</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>×</mo> <mn>2</mn> <mo>×</mo> <mn>4</mn> <mo>×</mo> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mi>π</mi> <mn>3</mn> </mfrac> <mo>+</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>×</mo> <mn>4</mn> <mo>×</mo> <mn>8</mn> <mo>×</mo> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mi>π</mi> <mn>3</mn> </mfrac> </math></span> <em><strong>A1A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="C = \frac{\pi }{3}"> <mi>C</mi> <mo>=</mo> <mfrac> <mi>π</mi> <mn>3</mn> </mfrac> </math></span>, <em><strong>A1</strong> </em>for correct moduli.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{21\sqrt 3 }}{2}"> <mo>=</mo> <mfrac> <mrow> <mn>21</mn> <msqrt> <mn>3</mn> </msqrt> </mrow> <mn>2</mn> </mfrac> </math></span> <em><strong> AG</strong></em></p>
<p><strong>Note:</strong> Other methods of splitting the area may receive full marks.</p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{2} \times {2^0} \times {2^1} \times {\text{sin}}\frac{\pi }{n} + \frac{1}{2} \times {2^1} \times {2^2} \times {\text{sin}}\frac{\pi }{n} + \frac{1}{2} \times {2^2} \times {2^3} \times {\text{sin}}\frac{\pi }{n} + \, \ldots \, + \frac{1}{2} \times {2^{n - 1}} \times {2^n} \times {\text{sin}}\frac{\pi }{n}"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>×</mo> <mrow> <msup> <mn>2</mn> <mn>0</mn> </msup> </mrow> <mo>×</mo> <mrow> <msup> <mn>2</mn> <mn>1</mn> </msup> </mrow> <mo>×</mo> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mi>π</mi> <mi>n</mi> </mfrac> <mo>+</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>×</mo> <mrow> <msup> <mn>2</mn> <mn>1</mn> </msup> </mrow> <mo>×</mo> <mrow> <msup> <mn>2</mn> <mn>2</mn> </msup> </mrow> <mo>×</mo> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mi>π</mi> <mi>n</mi> </mfrac> <mo>+</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>×</mo> <mrow> <msup> <mn>2</mn> <mn>2</mn> </msup> </mrow> <mo>×</mo> <mrow> <msup> <mn>2</mn> <mn>3</mn> </msup> </mrow> <mo>×</mo> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mi>π</mi> <mi>n</mi> </mfrac> <mo>+</mo> <mspace width="thinmathspace"></mspace> <mo>…</mo> <mspace width="thinmathspace"></mspace> <mo>+</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo>×</mo> <mrow> <msup> <mn>2</mn> <mrow> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> <mo>×</mo> <mrow> <msup> <mn>2</mn> <mi>n</mi> </msup> </mrow> <mo>×</mo> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mi>π</mi> <mi>n</mi> </mfrac> </math></span> <em><strong>M1A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1</strong> </em>for powers of 2, <em><strong>A1</strong> </em>for any correct expression including both the first and last term.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {\text{sin}}\frac{\pi }{n} \times \left( {{2^0} + {2^2} + {2^4} + \, \ldots \, + {2^{n - 2}}} \right)"> <mo>=</mo> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mi>π</mi> <mi>n</mi> </mfrac> <mo>×</mo> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mn>2</mn> <mn>0</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mn>2</mn> <mn>2</mn> </msup> </mrow> <mo>+</mo> <mrow> <msup> <mn>2</mn> <mn>4</mn> </msup> </mrow> <mo>+</mo> <mspace width="thinmathspace"></mspace> <mo>…</mo> <mspace width="thinmathspace"></mspace> <mo>+</mo> <mrow> <msup> <mn>2</mn> <mrow> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> </mrow> </mrow> <mo>)</mo> </mrow> </math></span></p>
<p>identifying a geometric series with common ratio 2<sup>2</sup>(= 4) <em><strong>(</strong><strong>M1)A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{1 - {2^{2n}}}}{{1 - 4}} \times {\text{sin}}\frac{\pi }{n}"> <mo>=</mo> <mfrac> <mrow> <mn>1</mn> <mo>−</mo> <mrow> <msup> <mn>2</mn> <mrow> <mn>2</mn> <mi>n</mi> </mrow> </msup> </mrow> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mn>4</mn> </mrow> </mfrac> <mo>×</mo> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mi>π</mi> <mi>n</mi> </mfrac> </math></span> <em><strong>M1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1</strong></em> for use of formula for sum of geometric series.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{3}\left( {{4^n} - 1} \right){\text{sin}}\frac{\pi }{n}"> <mo>=</mo> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> <mrow> <mo>(</mo> <mrow> <mrow> <msup> <mn>4</mn> <mi>n</mi> </msup> </mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mrow> <mtext>sin</mtext> </mrow> <mfrac> <mi>π</mi> <mi>n</mi> </mfrac> </math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[6 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the matrices</p>
<p style="text-align: center;"><em><strong>A</strong></em> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 3&{ - 2} \\ 5&{ - 4} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>3</mn>
</mtd>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>2</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>5</mn>
</mtd>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>4</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, <em><strong>B</strong></em> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 1&3 \\ 2&{ - 2} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>2</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent1" style="margin-top:12.0pt;">Find <strong><em>BA</em></strong>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent1" style="margin-top:12.0pt;">Calculate det (<strong><em>BA</em></strong>)<em>.</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 class="indent1" style="margin-top:12.0pt;"> Find <strong><em>A</em></strong>(<strong><em>A</em></strong><sup>–1</sup><strong><em>B</em></strong> + 2<strong><em>A</em></strong><sup>–1</sup>)<strong><em>A</em></strong>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><strong><em>BA</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\left( {\begin{array}{*{20}{c}} 1&3 \\ 2&{ - 2} \end{array}} \right)\left( {\begin{array}{*{20}{c}} 3&{ - 2} \\ 5&{ - 4} \end{array}} \right)} \right) = \left( {\begin{array}{*{20}{c}} {18}&{ - 14} \\ { - 4}&4 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>3</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>5</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>4</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>18</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>14</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>4</mn>
</mrow>
</mtd>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A2</strong></em> </p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for one error, <em><strong>A0</strong></em> for two or more errors.</p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>det(<strong><em>BA</em></strong>) = (72 – 56) = 16 <em><strong>(M1)A1</strong></em> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong> </p>
<p><strong><em>A</em></strong>(<strong><em>A</em></strong><sup>–1</sup><strong><em>B</em></strong> + 2<strong><em>A</em></strong><sup>–1</sup>)<strong><em>A</em></strong> = <strong><em>BA</em></strong> <em>+</em> 2<strong><em>A (M1)A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left( {\begin{array}{*{20}{c}} {24}&{ - 18} \\ 6&{ - 4} \end{array}} \right)">
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>24</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>18</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>6</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>4</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>A1</em></strong></p>
<p><em><strong>OR</strong></em></p>
<p><strong><em>A</em></strong><sup>–1</sup> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = - \frac{1}{2}\left( {\begin{array}{*{20}{c}} { - 4}&2 \\ { - 5}&3 \end{array}} \right)">
<mo>=</mo>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>4</mn>
</mrow>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>5</mn>
</mrow>
</mtd>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span><strong><em> (A1)</em></strong></p>
<p>an attempt to evaluate <em><strong>(M1)</strong></em></p>
<p><strong><em>A</em></strong><sup>–1</sup><strong><em>B</em></strong> + 2<strong><em>A</em></strong><sup>–1 </sup><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = - \frac{1}{2}\left( {\begin{array}{*{20}{c}} 0&{ - 16} \\ 1&{ - 21} \end{array}} \right) - \left( {\begin{array}{*{20}{c}} { - 4}&2 \\ { - 5}&3 \end{array}} \right)">
<mo>=</mo>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>16</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>21</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>4</mn>
</mrow>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>5</mn>
</mrow>
</mtd>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p><strong><em>A</em></strong>(<strong><em>A</em></strong><sup>–1</sup><strong><em>B</em></strong> + 2<strong><em>A</em></strong><sup>–1</sup>)<strong><em>A</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 3&{ - 2} \\ 5&{ - 4} \end{array}} \right)\left( {\begin{array}{*{20}{c}} 4&6 \\ {4.5}&{7.5} \end{array}} \right)\left( {\begin{array}{*{20}{c}} 3&{ - 2} \\ 5&{ - 4} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>3</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>5</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>4</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>4</mn>
</mtd>
<mtd>
<mn>6</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>4.5</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>7.5</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>3</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>5</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>4</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left( {\begin{array}{*{20}{c}} 3&3 \\ 2&0 \end{array}} \right)\left( {\begin{array}{*{20}{c}} 3&{ - 2} \\ 5&{ - 4} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {24}&{ - 18} \\ 6&{ - 4} \end{array}} \right)">
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>3</mn>
</mtd>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>3</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>5</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>4</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>24</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>18</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>6</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>4</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1</strong></em> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>The strength of earthquakes is measured on the Richter magnitude scale, with values typically between <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn></math> is the most severe.</p>
<p>The Gutenberg–Richter equation gives the average number of earthquakes per year, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math>, which have a magnitude of at least <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>M</mi></math>. For a particular region the equation is</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>log</mi><mn>10</mn></msub><mo> </mo><mi>N</mi><mo>=</mo><mi>a</mi><mo>-</mo><mi>M</mi></math>, for some <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>∈</mo><mi mathvariant="normal">ℝ</mi></math>.</p>
<p>This region has an average of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>100</mn></math> earthquakes per year with a magnitude of at least <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math>.</p>
</div>
<div class="specification">
<p>The equation for this region can also be written as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mo>=</mo><mfrac><mi>b</mi><msup><mn>10</mn><mi>M</mi></msup></mfrac></math>.</p>
</div>
<div class="specification">
<p>Within this region the most severe earthquake recorded had a magnitude of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn><mo>.</mo><mn>2</mn></math>.</p>
</div>
<div class="specification">
<p>The number of earthquakes in a given year with a magnitude of at least <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn><mo>.</mo><mn>2</mn></math> can be modelled by a Poisson distribution, with mean <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi></math>. The number of earthquakes in one year is independent of the number of earthquakes in any other year.</p>
<p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi></math> be the number of years between the earthquake of magnitude <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn><mo>.</mo><mn>2</mn></math> and the next earthquake of at least this magnitude.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the average number of earthquakes in a year with a magnitude of at least <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn><mo>.</mo><mn>2</mn></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mo>(</mo><mi>Y</mi><mo>></mo><mn>100</mn><mo>)</mo></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>log</mi><mn>10</mn></msub><mo> </mo><mn>100</mn><mo>=</mo><mi>a</mi><mo>-</mo><mn>3</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>5</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mo>=</mo><msup><mn>10</mn><mrow><mn>5</mn><mo>-</mo><mi>M</mi></mrow></msup></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><msup><mn>10</mn><mn>5</mn></msup><msup><mn>10</mn><mi>M</mi></msup></mfrac><mfenced><mrow><mo>=</mo><mfrac><mn>100000</mn><msup><mn>10</mn><mi>M</mi></msup></mfrac></mrow></mfenced></math></p>
<p><br><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>100</mn><mo>=</mo><mfrac><mi>b</mi><msup><mn>10</mn><mn>3</mn></msup></mfrac></math> <em><strong>(M1)</strong></em></p>
<p><br><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>=</mo><mn>100000</mn><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><msup><mn>10</mn><mn>5</mn></msup></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mo>=</mo><mfrac><msup><mn>10</mn><mn>5</mn></msup><msup><mn>10</mn><mrow><mn>7</mn><mo>.</mo><mn>2</mn></mrow></msup></mfrac><mo>=</mo><mn>0</mn><mo>.</mo><mn>00631</mn><mo> </mo><mo> </mo><mo> </mo><mfenced><mrow><mn>0</mn><mo>.</mo><mn>0063095</mn><mo>…</mo></mrow></mfenced></math></strong> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Do not accept an answer of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mn>10</mn><mrow><mo>-</mo><mn>2</mn><mo>.</mo><mn>2</mn></mrow></msup></math>.</p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi><mo>></mo><mn>100</mn><mo>⇒</mo></math>no earthquakes in the first <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>100</mn></math> years <strong><em>(M1)</em></strong></p>
<p><br><strong>EITHER</strong></p>
<p>let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> be the number of earthquakes of at least magnitude <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn><mo>.</mo><mn>2</mn></math> in a year</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo>~</mo><mtext>Po</mtext><mfenced><mrow><mn>0</mn><mo>.</mo><mn>0063095</mn><mo>…</mo></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mtext>P</mtext><mfenced><mrow><mi>X</mi><mo>=</mo><mn>0</mn></mrow></mfenced></mrow></mfenced><mn>100</mn></msup></math> <strong><em>(M1)</em></strong></p>
<p><br><strong>OR</strong></p>
<p>let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> be the number of earthquakes in <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>100</mn></math> years</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo>~</mo><mtext>Po</mtext><mfenced><mrow><mn>0</mn><mo>.</mo><mn>0063095</mn><mo>…</mo><mo>×</mo><mn>100</mn></mrow></mfenced></math> <strong><em>(M1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mrow><mi>X</mi><mo>=</mo><mn>0</mn></mrow></mfenced></math></p>
<p><br><strong>THEN</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>532</mn><mo> </mo><mo> </mo><mfenced><mrow><mn>0</mn><mo>.</mo><mn>532082</mn><mo>…</mo></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>Y</mi><mo>></mo><mn>100</mn><mo>⇒</mo></math>no earthquakes in the first <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>100</mn></math> years <strong><em>(M1)</em></strong></p>
<p>let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi></math> be the number of earthquakes in <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>100</mn></math> years</p>
<p>since <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> is large and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math> is small</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>X</mi><mo>~</mo><mtext>B</mtext><mfenced><mrow><mn>100</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>.</mo><mn>0063095</mn><mo>…</mo></mrow></mfenced></math> <strong><em>(M1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>P</mtext><mfenced><mrow><mi>X</mi><mo>=</mo><mn>0</mn></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>531</mn><mo> </mo><mo> </mo><mfenced><mrow><mn>0</mn><mo>.</mo><mn>531019</mn><mo>…</mo></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">d.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>Parts (a), (b), and (c) were accessible to many candidates who earned full marks with the manipulation of logs and indices presenting no problems. Part (d), however, proved to be too difficult for most and very few correct attempts were seen. As in question 9, most candidates relied on calculator notation when using the Poisson distribution. The discipline of defining a random variable in terms of its distribution and parameters helps to conceptualize the problem in terms that aid a better understanding. Most candidates who attempted this question blindly entered values into the Poisson distribution calculator and were unable to earn any marks. There were a couple of correct solutions using a binomial distribution to approximate the given quantity.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<br><hr><br><div class="specification">
<p>Let <em><strong>A</strong></em><sup>2</sup> = 2<em><strong>A</strong></em> + <em><strong>I</strong></em> where <em><strong>A</strong></em> is a 2 × 2 matrix.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <em><strong>A</strong></em><sup>4</sup> = 12<em><strong>A</strong></em> + 5<em><strong>I</strong></em>.</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>Let <em><strong>B</strong></em> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left[ {\begin{array}{*{20}{c}} 4&2 \\ 1&{ - 3} \end{array}} \right]">
<mrow>
<mo>[</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>4</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>3</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>]</mo>
</mrow>
</math></span>.</p>
<p>Given that <em><strong>B</strong></em><sup>2</sup> – <em><strong>B</strong></em> – 4<em><strong>I</strong></em> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left[ {\begin{array}{*{20}{c}} k&0 \\ 0&k \end{array}} \right]">
<mrow>
<mo>[</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>k</mi>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mi>k</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>]</mo>
</mrow>
</math></span>, find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k">
<mi>k</mi>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><strong>METHOD 1</strong><br><em><strong>A</strong></em><sup>4</sup> = 4<em><strong>A</strong></em><sup>2</sup> + 4A<em><strong>I</strong></em> + <em><strong>I</strong></em><sup>2</sup> or equivalent <em><strong>M1A1</strong></em><br>= 4(2<em><strong>A</strong></em> + <em><strong>I</strong></em>) + 4<em><strong>A</strong></em> + <em><strong>I</strong></em> <em><strong>A1</strong></em><br>= 8<em><strong>A</strong></em> + 4I + 4<em><strong>A</strong></em> + <em><strong>I</strong></em><br>= 12<em><strong>A</strong></em> + 5<em><strong>I</strong></em> <em><strong>AG</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<p><strong>METHOD 2</strong><br><em><strong>A</strong></em><sup>3</sup> = <em><strong>A</strong></em>(2<em><strong>A</strong></em> + <em><strong>I</strong></em>) = 2<em><strong>A</strong></em><sup>2</sup> + <em><strong>A</strong><strong>I</strong></em> = 2(2<em><strong>A</strong></em> + <em><strong>I</strong></em>) + <em><strong>A</strong></em>(= 5<em><strong>A </strong></em>+ 2<em><strong>I</strong></em>) <em><strong>M1A1</strong></em><br><em><strong>A</strong></em><sup>4</sup> = <em><strong>A</strong></em>(5<em><strong>A </strong></em>+ 2<em><strong>I</strong></em>) <em><strong>A1</strong></em><br>= 5<em><strong>A</strong></em><sup>2</sup> + 2<em><strong>A</strong></em> = 5(2<em><strong>A </strong></em>+ <em><strong>I</strong></em>) + 2<em><strong>A</strong></em><br>= 12<em><strong>A</strong></em> + 5<em><strong>I</strong></em> <em><strong>AG</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><em><strong>B</strong></em><sup>2</sup> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left[ {\begin{array}{*{20}{c}} {18}&2 \\ 1&{11} \end{array}} \right]">
<mrow>
<mo>[</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>18</mn>
</mrow>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mrow>
<mn>11</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>]</mo>
</mrow>
</math></span> <em><strong> (A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left[ {\begin{array}{*{20}{c}} {18}&2 \\ 1&{11} \end{array}} \right] - \left[ {\begin{array}{*{20}{c}} 4&2 \\ 1&{ - 3} \end{array}} \right] - \left[ {\begin{array}{*{20}{c}} 4&0 \\ 0&4 \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {10}&0 \\ 0&{10} \end{array}} \right]">
<mrow>
<mo>[</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>18</mn>
</mrow>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mrow>
<mn>11</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>]</mo>
</mrow>
<mo>−</mo>
<mrow>
<mo>[</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>4</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>3</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>]</mo>
</mrow>
<mo>−</mo>
<mrow>
<mo>[</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>4</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>]</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>[</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>10</mn>
</mrow>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mrow>
<mn>10</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>]</mo>
</mrow>
</math></span> <em><strong>(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow k = 10">
<mo stretchy="false">⇒</mo>
<mi>k</mi>
<mo>=</mo>
<mn>10</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The matrix <strong><em>A </em></strong>= <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 1&2&0 \\ { - 3}&1&{ - 1} \\ 2&{ - 2}&1 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>3</mn>
</mrow>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>2</mn>
</mrow>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> has inverse <strong><em>A</em></strong><sup>−1</sup> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} { - 1}&{ - 2}&{ - 2} \\ 3&1&1 \\ a&6&b \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>2</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>2</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>3</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>a</mi>
</mtd>
<mtd>
<mn>6</mn>
</mtd>
<mtd>
<mi>b</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
</div>
<div class="specification">
<p>Consider the simultaneous equations</p>
<p><span class="mjpage mjpage__block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" alttext="x + 2y = 7">
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
<mi>y</mi>
<mo>=</mo>
<mn>7</mn>
</math></span></p>
<p><span class="mjpage mjpage__block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" alttext=" - 3x + y - z = 10">
<mo>−<!-- − --></mo>
<mn>3</mn>
<mi>x</mi>
<mo>+</mo>
<mi>y</mi>
<mo>−<!-- − --></mo>
<mi>z</mi>
<mo>=</mo>
<mn>10</mn>
</math></span></p>
<p><span class="mjpage mjpage__block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" alttext="2x - 2y + z = - 12">
<mn>2</mn>
<mi>x</mi>
<mo>−<!-- − --></mo>
<mn>2</mn>
<mi>y</mi>
<mo>+</mo>
<mi>z</mi>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mn>12</mn>
</math></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent2">Write down the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent2">Write down the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b">
<mi>b</mi>
</math></span>.</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 class="indent2">Write these equations as a matrix equation.</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="indent2">Solve the matrix equation.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span> = 4 <em><strong>A1 N1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b">
<mi>b</mi>
</math></span> = 7 <em><strong>A1 N1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong> EITHER</strong></p>
<p><em><strong>A</strong></em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} x \\ y \\ z \end{array}} \right) = \left( {\begin{array}{*{20}{c}} 7 \\ {10} \\ { - 12} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>x</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>y</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>z</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>7</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>10</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>12</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1 N1</strong></em></p>
<p><strong>OR</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 1&2&0 \\ { - 3}&1&{ - 1} \\ 2&{ - 2}&1 \end{array}} \right)\left( {\begin{array}{*{20}{c}} x \\ y \\ z \end{array}} \right) = \left( {\begin{array}{*{20}{c}} 7 \\ {10} \\ { - 12} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>3</mn>
</mrow>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>x</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>y</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>z</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>7</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>10</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>12</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1 N1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} x \\ y \\ z \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>x</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>y</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>z</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> = <strong><em>A</em></strong><sup>−1</sup> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 7 \\ {10} \\ { - 12} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>7</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>10</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>12</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> (accept algebraic method) <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} x \\ y \\ z \end{array}} \right) = \left( {\begin{array}{*{20}{c}} { - 3} \\ 5 \\ 4 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>x</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>y</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>z</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>3</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>5</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> (accept <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> = −3, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span> = 5, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z">
<mi>z</mi>
</math></span> = 4) <em><strong> A2 N3</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Let <strong><em>A</em></strong><em> = </em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 1&2&3 \\ 2&{ - 1}&2 \\ 3&{ - 3}&2 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>3</mn>
</mtd>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>3</mn>
</mrow>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span><em>, <strong>D</strong> =</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} { - 4}&{13}&{ - 7} \\ { - 2}&7&{ - 4} \\ 3&{ - 9}&5 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>4</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>13</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>7</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>2</mn>
</mrow>
</mtd>
<mtd>
<mn>7</mn>
</mtd>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>4</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>3</mn>
</mtd>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>9</mn>
</mrow>
</mtd>
<mtd>
<mn>5</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span><em>, </em>and<em> <strong>C</strong> =</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 5 \\ 7 \\ {10} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>5</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>7</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>10</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span><em>. </em></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent1" style="margin-top:12pt;text-align: left;">Given matrices <strong><em>A</em></strong><em>, <strong>B</strong>, <strong>C </strong></em>for which <strong><em>AB</em></strong><em> = <strong>C</strong> </em>and det <strong><em>A</em></strong> ≠ 0, express <strong><em>B</em></strong> in terms of <strong><em>A</em></strong> and <strong><em>C</em></strong><em>.</em></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="indent1" style="margin-top:12pt;text-align: left;">Find the matrix<em> <strong>DA</strong>.</em></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 class="indent1" style="margin-top:12pt;text-align: left;">Find <strong><em>B </em></strong>if <strong><em>AB</em></strong> = <strong><em>C</em></strong>.</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 class="indent1" style="margin-top:12pt;text-align: left;">Find the coordinates of the point of intersection of the planes <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x + 2y + 3z = 5">
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
<mi>y</mi>
<mo>+</mo>
<mn>3</mn>
<mi>z</mi>
<mo>=</mo>
<mn>5</mn>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2x - y + 2z = 7">
<mn>2</mn>
<mi>x</mi>
<mo>−</mo>
<mi>y</mi>
<mo>+</mo>
<mn>2</mn>
<mi>z</mi>
<mo>=</mo>
<mn>7</mn>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3x - 3y + 2z = 10">
<mn>3</mn>
<mi>x</mi>
<mo>−</mo>
<mn>3</mn>
<mi>y</mi>
<mo>+</mo>
<mn>2</mn>
<mi>z</mi>
<mo>=</mo>
<mn>10</mn>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>Since det <strong><em>A</em></strong> ≠ 0, <strong><em>A</em></strong><sup>–1</sup> exists. <em><strong> (M1) </strong></em></p>
<p>Hence <strong><em>AB</em></strong> = <strong><em>C</em></strong> ⇒ <strong><em>B</em></strong> = <strong><em>A</em></strong><sup>–1</sup><strong><em>C (C1)</em></strong></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><em><strong>DA</strong></em> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 1&0&0 \\ 0&1&0 \\ 0&0&1 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span><strong><em> (A1)</em></strong></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong><em>B </em></strong><em>=</em><strong><em> A</em></strong><sup><em>–1</em></sup><strong><em>C </em></strong><em>=</em><strong><em> DC (M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left( {\begin{array}{*{20}{c}} 1 \\ { - 1} \\ 2 \end{array}} \right)">
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span><strong><em> (A1)</em></strong></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>The system of equations is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\begin{array}{*{20}{c}} {x + 2y + 3z = 5} \\ {2x - y + 2z = 7} \\ {3x - 3y + 2z = 10} \end{array}">
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
<mi>y</mi>
<mo>+</mo>
<mn>3</mn>
<mi>z</mi>
<mo>=</mo>
<mn>5</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>2</mn>
<mi>x</mi>
<mo>−</mo>
<mi>y</mi>
<mo>+</mo>
<mn>2</mn>
<mi>z</mi>
<mo>=</mo>
<mn>7</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>3</mn>
<mi>x</mi>
<mo>−</mo>
<mn>3</mn>
<mi>y</mi>
<mo>+</mo>
<mn>2</mn>
<mi>z</mi>
<mo>=</mo>
<mn>10</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</math></span></p>
<p>or <em><strong>A</strong></em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} x \\ y \\ z \end{array}} \right) = ">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>x</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>y</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>z</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
</math></span> <em><strong>C (M1)</strong></em></p>
<p>The required point = (1, –1, 2).<strong><em> (A1)</em></strong></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p class="indent1" style="margin-top:12.0pt;">The square matrix <strong><em>X</em></strong> is such that <strong><em>X</em></strong><sup>3</sup> = 0. Show that the inverse of the matrix (<strong><em>I</em></strong> <em>– <strong>X</strong></em>) is <strong><em>I</em></strong> <em>+</em> <strong><em>X</em></strong> <em>+</em> <strong><em>X</em></strong><sup>2</sup><em>.</em></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p>For multiplying (<strong><em>I</em></strong> <em>–</em><strong> <em>X</em></strong>)(<strong><em>I</em></strong> <em>+</em> <strong><em>X</em></strong> <em>+</em> <strong><em>X</em></strong><sup>2</sup>) <em><strong>M1 </strong></em></p>
<p>= <strong><em>I</em></strong><sup>2</sup> + <strong><em>IX</em></strong> <em>+</em> <strong><em>IX</em></strong><sup>2</sup> <em>–</em> <strong><em>XI</em></strong><em> – <strong>X</strong></em><sup>2</sup> <em>–</em> <strong><em>X</em></strong><sup>3</sup> = <strong><em>I</em> </strong><em>+</em> <strong><em>X</em></strong> <em>+</em> <strong><em>X</em></strong><sup>2</sup><em> – <strong>X</strong> – <strong>X</strong></em><sup>2</sup><em> – <strong>X</strong></em><sup>3</sup> <em><strong>(A1)(A1)</strong></em></p>
<p>= <strong><em>I</em></strong> –<em> <strong>X</strong></em><sup>3</sup> <em><strong>A1 </strong></em></p>
<p>= <strong><em>I</em></strong> <em><strong>A1 </strong></em></p>
<p><strong><em>AB</em></strong> = <strong><em>I</em></strong> ⇒ <strong><em>A</em></strong><sup>–1</sup> = <strong><em>B</em></strong> <em><strong>(R1) </strong></em></p>
<p>(<strong><em>I</em></strong> <em>–</em> <strong><em>X</em></strong>) (<strong><em>I</em> </strong><em>+</em> <strong><em>X</em></strong> <em>+</em> <strong><em>X</em></strong><sup>2</sup>) = <strong><em>I</em></strong> ⇒ (<strong><em>I</em></strong> <em>–</em> <strong><em>X</em></strong>)<sup>–1</sup> = <strong><em>I</em></strong> <em>+</em> <strong><em>X</em></strong> <em>+</em> <strong><em>X</em></strong><sup>2</sup> <em><strong>AG N0</strong></em></p>
<p><em><strong>[5 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>Let <strong><em>A </em></strong>= <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 1&2 \\ 3&{ - 1} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>3</mn>
</mtd>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> and <strong><em>B</em></strong><strong><em> </em></strong>= <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 3&0 \\ { - 2}&1 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>3</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>2</mn>
</mrow>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <strong><em>A</em></strong> + <strong><em>B</em></strong>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find −3<strong><em>A</em></strong>. </p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <strong><em>AB</em></strong>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p>evidence of addition <em><strong>(M1)</strong></em></p>
<p><em>eg</em> at least two correct elements</p>
<p><strong><em>A</em></strong> + <strong><em>B</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 4&2 \\ 1&0 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>4</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong> A1 N2</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>evidence of multiplication <em><strong>(M1)</strong></em></p>
<p><em>eg</em> at least two correct elements</p>
<p>−3<strong><em>A</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} { - 3}&{ - 6} \\ { - 9}&3 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>3</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>6</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>9</mn>
</mrow>
</mtd>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong> A1 N2</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>evidence of matrix multiplication (in correct order) <em><strong>(M1)</strong></em></p>
<p><em>eg</em> <strong><em>AB</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} {1\left( 3 \right) + 2\left( { - 2} \right)}&{1\left( 0 \right) + 2\left( 1 \right)} \\ {3\left( 3 \right) + \left( { - 1} \right)\left( { - 2} \right)}&{3\left( 0 \right) + \left( { - 1} \right)\left( 1 \right)} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>1</mn>
<mrow>
<mo>(</mo>
<mn>3</mn>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mn>2</mn>
<mrow>
<mo>(</mo>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>1</mn>
<mrow>
<mo>(</mo>
<mn>0</mn>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mn>2</mn>
<mrow>
<mo>(</mo>
<mn>1</mn>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>3</mn>
<mrow>
<mo>(</mo>
<mn>3</mn>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>3</mn>
<mrow>
<mo>(</mo>
<mn>0</mn>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mn>1</mn>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p><strong><em>AB</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} { - 1}&2 \\ {11}&{ - 1} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>11</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong> A2 N3</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find a relationship between <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b">
<mi>b</mi>
</math></span> if the matrices <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="M = \left( {\begin{array}{*{20}{c}} 1&a \\ 2&3 \end{array}} \right)">
<mi>M</mi>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mi>a</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="N = \left( {\begin{array}{*{20}{c}} 1&b \\ 2&3 \end{array}} \right)">
<mi>N</mi>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mi>b</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> commute under matrix multiplication.</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>Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span> if the determinant of matrix <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="M">
<mi>M</mi>
</math></span> is −1.</p>
<p> </p>
<div class="marks">[2]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{M^{ - 1}}">
<mrow>
<msup>
<mi>M</mi>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</math></span> for this value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 1&a \\ 2&3 \end{array}} \right)\left( {\begin{array}{*{20}{c}} 1&b \\ 2&3 \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {1 + 2a}&{b + 3a} \\ 8&{2b + 9} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mi>a</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mi>b</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mn>2</mn>
<mi>a</mi>
</mrow>
</mtd>
<mtd>
<mrow>
<mi>b</mi>
<mo>+</mo>
<mn>3</mn>
<mi>a</mi>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>8</mn>
</mtd>
<mtd>
<mrow>
<mn>2</mn>
<mi>b</mi>
<mo>+</mo>
<mn>9</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 1&b \\ 2&3 \end{array}} \right)\left( {\begin{array}{*{20}{c}} 1&a \\ 2&3 \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {1 + 2b}&{a + 3b} \\ 8&{2a + 9} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mi>b</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mi>a</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mn>2</mn>
<mi>b</mi>
</mrow>
</mtd>
<mtd>
<mrow>
<mi>a</mi>
<mo>+</mo>
<mn>3</mn>
<mi>b</mi>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>8</mn>
</mtd>
<mtd>
<mrow>
<mn>2</mn>
<mi>a</mi>
<mo>+</mo>
<mn>9</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>M1A1</strong></em></p>
<p>So require <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a = b">
<mi>a</mi>
<mo>=</mo>
<mi>b</mi>
</math></span> <em><strong>M1A1</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left| {\begin{array}{*{20}{c}} 1&a \\ 2&3 \end{array}} \right| = 3 - 2a = - 1 \Rightarrow a = 2">
<mrow>
<mo>|</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mi>a</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>|</mo>
</mrow>
<mo>=</mo>
<mn>3</mn>
<mo>−</mo>
<mn>2</mn>
<mi>a</mi>
<mo>=</mo>
<mo>−</mo>
<mn>1</mn>
<mo stretchy="false">⇒</mo>
<mi>a</mi>
<mo>=</mo>
<mn>2</mn>
</math></span> <em><strong>M1A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {\begin{array}{*{20}{c}} 1&2 \\ 2&3 \end{array}} \right)^{ - 1}} = \left( {\begin{array}{*{20}{c}} { - 3}&2 \\ 2&{ - 1} \end{array}} \right)">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>3</mn>
</mrow>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<br><hr><br><div class="question">
<p>If <strong><em>A</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} x&4 \\ 4&2 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>x</mi>
</mtd>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>4</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> and <strong><em>B</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 2&y \\ 8&4 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mi>y</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>8</mn>
</mtd>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, find 2 values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span>, given that <strong><em>AB</em></strong> = <strong><em>BA</em></strong>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><strong><em>AB</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} x&4 \\ 4&2 \end{array}} \right)\left( {\begin{array}{*{20}{c}} 2&y \\ 8&4 \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {2x + 32}&{xy + 16} \\ {24}&{4y + 8} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>x</mi>
</mtd>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>4</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mi>y</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>8</mn>
</mtd>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mn>32</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mi>x</mi>
<mi>y</mi>
<mo>+</mo>
<mn>16</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>24</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>4</mn>
<mi>y</mi>
<mo>+</mo>
<mn>8</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span><strong><em> (A1)</em></strong></p>
<p><strong><em>BA</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 2&y \\ 8&4 \end{array}} \right)\left( {\begin{array}{*{20}{c}} x&4 \\ 4&2 \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {2x + 4y}&{2y + 8} \\ {8x + 16}&{40} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mi>y</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>8</mn>
</mtd>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>x</mi>
</mtd>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>4</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mn>4</mn>
<mi>y</mi>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>2</mn>
<mi>y</mi>
<mo>+</mo>
<mn>8</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>8</mn>
<mi>x</mi>
<mo>+</mo>
<mn>16</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>40</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span><strong><em> (A1)</em></strong></p>
<p><strong><em>AB</em></strong> = <strong><em>BA</em></strong> ⇒ 8<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> + 16 = 24 and 4<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span> + 8 = 40</p>
<p>This gives <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 1">
<mi>x</mi>
<mo>=</mo>
<mn>1</mn>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 8">
<mi>y</mi>
<mo>=</mo>
<mn>8</mn>
</math></span>.<strong><em> (A1)</em></strong><strong><em> (C3)</em></strong></p>
<p><em><strong>[3 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>The equation of the line <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>m</mi><mi>x</mi><mo>+</mo><mi>c</mi></math> can be expressed in vector form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">r</mi><mo>=</mo><mi mathvariant="bold-italic">a</mi><mo>+</mo><mi>λ</mi><mi mathvariant="bold-italic">b</mi></math>.</p>
</div>
<div class="specification">
<p>The matrix <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">M</mi></math> is defined by <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>6</mn><mo> </mo><mo> </mo></mtd><mtd><mn>3</mn></mtd></mtr><mtr><mtd><mn>4</mn><mo> </mo><mo> </mo></mtd><mtd><mn>2</mn></mtd></mtr></mtable></mfenced></math>.</p>
</div>
<div class="specification">
<p>The line <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>m</mi><mi>x</mi><mo>+</mo><mi>c</mi></math> (where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>≠</mo><mo>−</mo><mn>2</mn></math>) is transformed into a new line using the transformation described by matrix <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">M</mi></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the vectors <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">a</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">b</mi></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math> and/or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>det </mtext><mi mathvariant="bold-italic">M</mi></math>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the equation of the resulting line does not depend on <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>(one vector to the line is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mi>c</mi></mtd></mtr></mtable></mfenced></math> therefore) <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">a</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mi>c</mi></mtd></mtr></mtable></mfenced></math> <strong><em>A1</em></strong></p>
<p>the line goes <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math> up for every <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math> across</p>
<p>(so the direction vector is) <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">b</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mi>m</mi></mtd></mtr></mtable></mfenced></math> <strong><em>A1</em></strong></p>
<p><br><strong>Note:</strong> Although these are the most likely answers, many others are possible.</p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>(from GDC <strong>OR </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mo>×</mo><mn>2</mn><mo>-</mo><mn>4</mn><mo>×</mo><mn>3</mn></math>) <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mrow><mo> </mo><mi mathvariant="bold-italic">M</mi><mo> </mo></mrow></mfenced><mo>=</mo><mn>0</mn></math> <strong><em>A1</em></strong></p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mi>X</mi></mtd></mtr><mtr><mtd><mi>Y</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mtable><mtr><mtd><mn>6</mn><mo> </mo><mo> </mo></mtd><mtd><mn>3</mn></mtd></mtr><mtr><mtd><mn>4</mn><mo> </mo><mo> </mo></mtd><mtd><mn>2</mn></mtd></mtr></mtable></mfenced><mfenced><mtable><mtr><mtd><mi>x</mi></mtd></mtr><mtr><mtd><mi>m</mi><mi>x</mi><mo>+</mo><mi>c</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mtable><mtr><mtd><mn>6</mn><mi>x</mi><mo>+</mo><mn>3</mn><mi>m</mi><mi>x</mi><mo>+</mo><mn>3</mn><mi>c</mi></mtd></mtr><mtr><mtd><mn>4</mn><mi>x</mi><mo>+</mo><mn>2</mn><mi>m</mi><mi>x</mi><mo>+</mo><mn>2</mn><mi>c</mi></mtd></mtr></mtable></mfenced></math> <em><strong>M1A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced><mtable><mtr><mtd><mn>3</mn><mfenced><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mi>m</mi><mi>x</mi><mo>+</mo><mi>c</mi></mrow></mfenced></mtd></mtr><mtr><mtd><mn>2</mn><mfenced><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mi>m</mi><mi>x</mi><mo>+</mo><mi>c</mi></mrow></mfenced></mtd></mtr></mtable></mfenced></math> <em><strong>A1</strong></em></p>
<p>therefore the new line has equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mi>Y</mi><mo>=</mo><mn>2</mn><mi>X</mi></math> <em><strong>A1</strong></em></p>
<p>which is independent of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math> <em><strong>AG</strong></em></p>
<p><br><strong>Note:</strong> The <em><strong>AG</strong> </em>line (or equivalent) must be seen for the final <em><strong>A1</strong> </em>line to be awarded.</p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>take two points on the line, e.g <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>0</mn><mo>,</mo><mo> </mo><mi>c</mi></mrow></mfenced></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>1</mn><mo>,</mo><mo> </mo><mi>m</mi><mo>+</mo><mi>c</mi></mrow></mfenced></math> <em><strong>M1</strong></em></p>
<p>these map to <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>6</mn><mo> </mo><mo> </mo></mtd><mtd><mn>3</mn></mtd></mtr><mtr><mtd><mn>4</mn><mo> </mo><mo> </mo></mtd><mtd><mn>2</mn></mtd></mtr></mtable></mfenced><mfenced><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mi>c</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mtable><mtr><mtd><mn>3</mn><mi>c</mi></mtd></mtr><mtr><mtd><mn>2</mn><mi>c</mi></mtd></mtr></mtable></mfenced></math></p>
<p>and <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>6</mn><mo> </mo><mo> </mo></mtd><mtd><mn>3</mn></mtd></mtr><mtr><mtd><mn>4</mn><mo> </mo><mo> </mo></mtd><mtd><mn>2</mn></mtd></mtr></mtable></mfenced><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mi>m</mi><mo>+</mo><mi>c</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mtable><mtr><mtd><mn>6</mn><mo>+</mo><mn>3</mn><mi>m</mi><mo>+</mo><mn>3</mn><mi>c</mi></mtd></mtr><mtr><mtd><mn>4</mn><mo>+</mo><mn>2</mn><mi>m</mi><mo>+</mo><mn>2</mn><mi>c</mi></mtd></mtr></mtable></mfenced></math> <em><strong>A1</strong></em></p>
<p>therefore a direction vector is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>6</mn><mo>+</mo><mn>3</mn><mi>m</mi></mtd></mtr><mtr><mtd><mn>4</mn><mo>+</mo><mn>2</mn><mi>m</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mrow><mn>2</mn><mo>+</mo><mi>m</mi></mrow></mfenced><mfenced><mtable><mtr><mtd><mn>3</mn></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable></mfenced></math></p>
<p>(since <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>≠</mo><mo>−</mo><mn>2</mn></math>) a direction vector is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>3</mn></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable></mfenced></math></p>
<p>the line passes through <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>3</mn><mi>c</mi></mtd></mtr><mtr><mtd><mn>2</mn><mi>c</mi></mtd></mtr></mtable></mfenced><mo>-</mo><mi>c</mi><mfenced><mtable><mtr><mtd><mn>3</mn></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr></mtable></mfenced></math> therefore it always has the origin as a jump-on vector <em><strong>A1</strong></em></p>
<p>the vector equation is therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">r</mi><mo>=</mo><mi>μ</mi><mfenced><mtable><mtr><mtd><mn>3</mn></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable></mfenced></math> <em><strong>A1</strong></em></p>
<p>which is independent of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math> <em><strong>AG</strong></em></p>
<p><br><strong>Note:</strong> The <em><strong>AG</strong> </em>line (or equivalent) must be seen for the final <em><strong>A1</strong> </em>line to be awarded.</p>
<p> </p>
<p><strong>METHOD 3</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">r</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mn>6</mn><mo> </mo><mo> </mo></mtd><mtd><mn>3</mn></mtd></mtr><mtr><mtd><mn>4</mn><mo> </mo><mo> </mo></mtd><mtd><mn>2</mn></mtd></mtr></mtable></mfenced><mfenced><mrow><mfenced><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mi>c</mi></mtd></mtr></mtable></mfenced><mo>+</mo><mi>λ</mi><mfenced><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mi>m</mi></mtd></mtr></mtable></mfenced></mrow></mfenced><mo>=</mo><mfenced><mtable><mtr><mtd><mn>3</mn><mi>c</mi></mtd></mtr><mtr><mtd><mn>2</mn><mi>c</mi></mtd></mtr></mtable></mfenced><mo>+</mo><mi>λ</mi><mfenced><mtable><mtr><mtd><mn>6</mn><mo>+</mo><mn>3</mn><mi>m</mi></mtd></mtr><mtr><mtd><mn>4</mn><mo>+</mo><mn>2</mn><mi>m</mi></mtd></mtr></mtable></mfenced></math> <em><strong>M1A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi>c</mi><mfenced><mtable><mtr><mtd><mn>3</mn></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mfenced><mrow><mn>2</mn><mo>+</mo><mi>m</mi></mrow></mfenced><mi>λ</mi><mfenced><mtable><mtr><mtd><mn>3</mn></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi>μ</mi><mfenced><mtable><mtr><mtd><mn>3</mn></mtd></mtr><mtr><mtd><mn>2</mn></mtd></mtr></mtable></mfenced></math></p>
<p>where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>μ</mi><mo>=</mo><mi>c</mi><mo>+</mo><mfenced><mrow><mn>2</mn><mo>+</mo><mi>m</mi></mrow></mfenced><mi>λ</mi></math> is an arbitrary parameter. <em><strong>A1</strong></em></p>
<p>which is independent of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math> (as <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>μ</mi></math> can take any value) <em><strong>AG</strong></em></p>
<p><br><strong>Note:</strong> The <em><strong>AG</strong> </em>line (or equivalent) must be seen for the final <em><strong>A1</strong> </em>line to be awarded.</p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>In part (a), most candidates were unable to convert the Cartesian equation of a line into its vector form. In part (b), almost every candidate showed that the value of the determinant was zero. In part (c), the great majority of candidates failed to come up with any sort of strategy to solve the problem.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A">
<mi>A</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="B">
<mi>B</mi>
</math></span> are 2 × 2 matrices, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A = \left[ {\begin{array}{*{20}{c}} 5&2 \\ 2&0 \end{array}} \right]">
<mi>A</mi>
<mo>=</mo>
<mrow>
<mo>[</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>5</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>]</mo>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="BA = \left[ {\begin{array}{*{20}{c}} {11}&2 \\ {44}&8 \end{array}} \right]">
<mi>B</mi>
<mi>A</mi>
<mo>=</mo>
<mrow>
<mo>[</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>11</mn>
</mrow>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>44</mn>
</mrow>
</mtd>
<mtd>
<mn>8</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>]</mo>
</mrow>
</math></span>. Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="B">
<mi>B</mi>
</math></span></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="B = \left( {BA} \right){A^{ - 1}}">
<mi>B</mi>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>B</mi>
<mi>A</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<msup>
<mi>A</mi>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = - \frac{1}{4}\left( {\begin{array}{*{20}{c}} {11}&2 \\ {44}&8 \end{array}} \right)\left( {\begin{array}{*{20}{c}} 0&{ - 2} \\ { - 2}&5 \end{array}} \right)">
<mo>=</mo>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mn>4</mn>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>11</mn>
</mrow>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>44</mn>
</mrow>
</mtd>
<mtd>
<mn>8</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
</mtd>
<mtd>
<mn>5</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = - \frac{1}{4}\left( {\begin{array}{*{20}{c}} { - 4}&{ - 12} \\ { - 16}&{ - 48} \end{array}} \right)">
<mo>=</mo>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mn>4</mn>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>4</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>12</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>16</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>48</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left( {\begin{array}{*{20}{c}} 1&3 \\ 4&{12} \end{array}} \right)">
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>4</mn>
</mtd>
<mtd>
<mrow>
<mn>12</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>(A1)</strong></em></p>
<p><strong>OR</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} a&b \\ c&d \end{array}} \right)\left( {\begin{array}{*{20}{c}} 5&2 \\ 2&0 \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {11}&2 \\ {44}&8 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>a</mi>
</mtd>
<mtd>
<mi>b</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>c</mi>
</mtd>
<mtd>
<mi>d</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>5</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>11</mn>
</mrow>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>44</mn>
</mrow>
</mtd>
<mtd>
<mn>8</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left. { \Rightarrow \begin{array}{*{20}{c}} {5a + 2b = 11} \\ {2a\,\,\,\,\,\,\,\,\,\,\,\,\, = 2} \end{array}} \right\}">
<mrow>
<mo fence="true" stretchy="true" symmetric="true"></mo>
<mrow>
<mo stretchy="false">⇒</mo>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>5</mn>
<mi>a</mi>
<mo>+</mo>
<mn>2</mn>
<mi>b</mi>
<mo>=</mo>
<mn>11</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>2</mn>
<mi>a</mi>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mo>=</mo>
<mn>2</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>}</mo>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow a = 1">
<mo stretchy="false">⇒</mo>
<mi>a</mi>
<mo>=</mo>
<mn>1</mn>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b = 3">
<mi>b</mi>
<mo>=</mo>
<mn>3</mn>
</math></span> <em><strong>(A1)</strong></em></p>
<p><span style="background-color: #ffffff;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left. {\begin{array}{*{20}{c}} {5c + 2d = 44} \\ {2c{\text{ }} = 8} \end{array}} \right\}">
<mrow>
<mo fence="true" stretchy="true" symmetric="true"></mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>5</mn>
<mi>c</mi>
<mo>+</mo>
<mn>2</mn>
<mi>d</mi>
<mo>=</mo>
<mn>44</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>2</mn>
<mi>c</mi>
<mrow>
<mtext> </mtext>
</mrow>
<mo>=</mo>
<mn>8</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>}</mo>
</mrow>
</math></span></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow c = 4">
<mo stretchy="false">⇒</mo>
<mi>c</mi>
<mo>=</mo>
<mn>4</mn>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d = 12">
<mi>d</mi>
<mo>=</mo>
<mn>12</mn>
</math></span> <em><strong>(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="B = \left( {\begin{array}{*{20}{c}} 1&3 \\ 4&{12} \end{array}} \right)">
<mi>B</mi>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>4</mn>
</mtd>
<mtd>
<mrow>
<mn>12</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>(A1)</strong></em> <em><strong>(C4)</strong></em></p>
<p class="accept"><strong>Note:</strong> Correct solution with inversion (ie <em>AB</em> instead of <em>BA</em>) earns FT marks, (maximum <em><strong>[3 marks]</strong></em>).</p>
<p><em><strong>[4 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>Let <strong><em>A</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 1&2&3 \\ 3&1&2 \\ 2&0&1 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>3</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, <strong><em>B</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} {18} \\ {23} \\ {13} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>18</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>23</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>13</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, and <strong><em>X</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} x \\ y \\ z \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>x</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>y</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>z</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
</div>
<div class="specification">
<p>Consider the equation <strong><em>AX</em></strong> = <strong><em>B</em></strong>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent2">Write down the inverse matrix <strong><em>A</em></strong><sup>−1</sup>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent2">Express <strong><em>X</em></strong> in terms of <strong><em>A</em></strong><sup>−1</sup> and <strong><em>B</em></strong>.</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 class="indent2"><strong>Hence</strong>, solve for <strong><em>X</em></strong>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p><strong><em>A</em></strong><sup>−1</sup> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} { - \frac{1}{3}}&{\frac{2}{3}}&{ - \frac{1}{3}} \\ { - \frac{1}{3}}&{\frac{5}{3}}&{ - \frac{7}{3}} \\ {\frac{2}{3}}&{ - \frac{4}{3}}&{\frac{5}{3}} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
</mrow>
</mtd>
<mtd>
<mrow>
<mfrac>
<mn>2</mn>
<mn>3</mn>
</mfrac>
</mrow>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
</mrow>
</mtd>
<mtd>
<mrow>
<mfrac>
<mn>5</mn>
<mn>3</mn>
</mfrac>
</mrow>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mfrac>
<mn>7</mn>
<mn>3</mn>
</mfrac>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mfrac>
<mn>2</mn>
<mn>3</mn>
</mfrac>
</mrow>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mfrac>
<mn>4</mn>
<mn>3</mn>
</mfrac>
</mrow>
</mtd>
<mtd>
<mrow>
<mfrac>
<mn>5</mn>
<mn>3</mn>
</mfrac>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} { - 0.333}&{0.667}&{ - 0.333} \\ { - 0.333}&{1.67}&{ - 2.33} \\ {0.667}&{ - 1.33}&{1.67} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>0.333</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>0.667</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>0.333</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>0.333</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>1.67</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>2.33</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>0.667</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>1.33</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>1.67</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A2 N2</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong><em>X</em></strong> = <strong><em>A</em></strong><sup>−1</sup><strong><em>B</em></strong> <em><strong>A1 N1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong><em>X</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 5 \\ 2 \\ 3 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>5</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A3 N3</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<br><hr><br><div class="question">
<p>The sum of an infinite geometric sequence is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9</mn></math>.</p>
<p>The first term is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn></math> more than the second term.</p>
<p>Find the third term. Justify your answer.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><strong>METHOD 1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msub><mi>u</mi><mn>1</mn></msub><mrow><mn>1</mn><mo>-</mo><mi>r</mi></mrow></mfrac><mo>=</mo><mn>9</mn></math> <strong><em>A1</em></strong></p>
<p>therefore <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>1</mn></msub><mo>=</mo><mn>9</mn><mo>-</mo><mn>9</mn><mi>r</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>1</mn></msub><mo>=</mo><mn>4</mn><mo>+</mo><msub><mi>u</mi><mn>1</mn></msub><mi>r</mi></math> <strong><em>A1</em></strong></p>
<p>substitute or solve graphically: <strong><em>M1</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9</mn><mo>-</mo><mn>9</mn><mi>r</mi><mo>=</mo><mn>4</mn><mo>+</mo><mfenced><mrow><mn>9</mn><mo>-</mo><mn>9</mn><mi>r</mi></mrow></mfenced><mi>r</mi></math> <strong>OR</strong> <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>4</mn><msup><mfenced><mrow><mn>1</mn><mo>-</mo><mi>r</mi></mrow></mfenced><mn>2</mn></msup></mfrac><mo>=</mo><mn>9</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9</mn><msup><mi>r</mi><mn>2</mn></msup><mo>-</mo><mn>18</mn><mi>r</mi><mo>+</mo><mn>5</mn><mo>=</mo><mn>0</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mfrac><mn>5</mn><mn>3</mn></mfrac></math></p>
<p>only <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></math> is possible as the sum to infinity exists <strong><em>R1</em></strong></p>
<p>then <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>1</mn></msub><mo>=</mo><mn>9</mn><mo>-</mo><mfenced><mrow><mn>9</mn><mo>×</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></mrow></mfenced><mo>=</mo><mn>6</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>3</mn></msub><mo>=</mo><mn>6</mn><mo>×</mo><msup><mfrac><mn>1</mn><mn>3</mn></mfrac><mn>2</mn></msup><mo>=</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></math> <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msub><mi>u</mi><mn>1</mn></msub><mrow><mn>1</mn><mo>-</mo><mi>r</mi></mrow></mfrac><mo>=</mo><mn>9</mn></math> <strong><em>A1</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mfrac><mrow><msub><mi>u</mi><mn>1</mn></msub><mo>-</mo><mn>4</mn></mrow><msub><mi>u</mi><mn>1</mn></msub></mfrac></math> <strong><em>A1</em></strong></p>
<p>attempt to solve <strong><em>M1</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msub><mi>u</mi><mn>1</mn></msub><mrow><mn>1</mn><mo>-</mo><mfenced><mstyle displaystyle="true"><mfrac><mrow><msub><mi>u</mi><mn>1</mn></msub><mo>-</mo><mn>4</mn></mrow><msub><mi>u</mi><mn>1</mn></msub></mfrac></mstyle></mfenced></mrow></mfrac><mo>=</mo><mn>9</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><msub><mi>u</mi><mn>1</mn></msub><mfenced><mstyle displaystyle="true"><mfrac><mn>4</mn><msub><mi>u</mi><mn>1</mn></msub></mfrac></mstyle></mfenced></mfrac><mo>=</mo><mn>9</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><msub><mi>u</mi><mn>1</mn></msub></mfenced><mn>2</mn></msup><mo>=</mo><mn>36</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>1</mn></msub><mo>=</mo><mo>±</mo><mn>6</mn></math></p>
<p>attempting to solve both possible sequences</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mo>,</mo><mo> </mo><mn>2</mn><mo>,</mo><mo> </mo><mo>…</mo></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>6</mn><mo>,</mo><mo> </mo><mo>-</mo><mn>10</mn><mo> </mo><mo>…</mo></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mfrac><mn>5</mn><mn>3</mn></mfrac></math></p>
<p>only <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></math> is possible as the sum to infinity exists <strong><em>R1</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>3</mn></msub><mo>=</mo><mn>6</mn><mo>×</mo><msup><mfenced><mfrac><mn>1</mn><mn>3</mn></mfrac></mfenced><mn>2</mn></msup><mo>=</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></math> <strong><em>A1</em></strong></p>
<p> </p>
<p><em><strong>[5 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p>Many candidates submitted quite poor attempts at this question. Many managed to state the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>1</mn></msub><mo>=</mo><mn>9</mn><mo>(</mo><mn>1</mn><mo>-</mo><mi>r</mi><mo>)</mo></math> obtained by considering the sum to infinity but few managed to find the second equation <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>u</mi><mn>1</mn></msub><mo>(</mo><mn>1</mn><mo>-</mo><mi>r</mi><mo>)</mo><mo>=</mo><mn>4</mn></math>. Common errors in failing to obtain this equation were that “four more” meant multiplied by four or thinking that the second term was four more than the first term. Even those candidates who obtained both equations were often unable to solve them. Attempted solutions often filled the page with algebra going nowhere. Most of those candidates who actually found the third term correctly then failed to realize that there were two solutions to the equations, one of which had to be rejected. Consequently, the final “reasoning” mark was seldom awarded.</p>
</div>
<br><hr><br><div class="specification">
<p>Consider the following system of equations where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a \in \mathbb{R}">
<mi>a</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
</math></span>.</p>
<p style="text-align: center;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2x + 4y - z = 10">
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mn>4</mn>
<mi>y</mi>
<mo>−<!-- − --></mo>
<mi>z</mi>
<mo>=</mo>
<mn>10</mn>
</math></span></p>
<p style="text-align: center;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x + 2y + az = 5">
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
<mi>y</mi>
<mo>+</mo>
<mi>a</mi>
<mi>z</mi>
<mo>=</mo>
<mn>5</mn>
</math></span></p>
<p style="text-align: center;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="5x + 12y = 2a">
<mn>5</mn>
<mi>x</mi>
<mo>+</mo>
<mn>12</mn>
<mi>y</mi>
<mo>=</mo>
<mn>2</mn>
<mi>a</mi>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span> for which the system of equations does not have a unique solution.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the solution of the system of equations when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a = 2">
<mi>a</mi>
<mo>=</mo>
<mn>2</mn>
</math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p>an attempt at a valid method<em> eg</em> by inspection or row reduction <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2 \times {R_2} = {R_1} \Rightarrow 2a = - 1">
<mn>2</mn>
<mo>×</mo>
<mrow>
<msub>
<mi>R</mi>
<mn>2</mn>
</msub>
</mrow>
<mo>=</mo>
<mrow>
<msub>
<mi>R</mi>
<mn>1</mn>
</msub>
</mrow>
<mo stretchy="false">⇒</mo>
<mn>2</mn>
<mi>a</mi>
<mo>=</mo>
<mo>−</mo>
<mn>1</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow a = - \frac{1}{2}">
<mo stretchy="false">⇒</mo>
<mi>a</mi>
<mo>=</mo>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</math></span> <em><strong> A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>using elimination or row reduction to eliminate one variable <em><strong>(M1)</strong></em></p>
<p>correct pair of equations in 2 variables, such as</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left. {\begin{array}{*{20}{c}} {5x + 10y = 25} \\ {5x + 12y = 4} \end{array}} \right\}">
<mrow>
<mo fence="true" stretchy="true" symmetric="true"></mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>5</mn>
<mi>x</mi>
<mo>+</mo>
<mn>10</mn>
<mi>y</mi>
<mo>=</mo>
<mn>25</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>5</mn>
<mi>x</mi>
<mo>+</mo>
<mn>12</mn>
<mi>y</mi>
<mo>=</mo>
<mn>4</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>}</mo>
</mrow>
</math></span> <em><strong> A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z">
<mi>z</mi>
</math></span> = 0 and one other equation in two variables.</p>
<p> </p>
<p>attempting to solve for these two variables <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 26">
<mi>x</mi>
<mo>=</mo>
<mn>26</mn>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = - 10.5">
<mi>y</mi>
<mo>=</mo>
<mo>−</mo>
<mn>10.5</mn>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z = 0">
<mi>z</mi>
<mo>=</mo>
<mn>0</mn>
</math></span> <em><strong>A1A1</strong></em></p>
<p><strong>Note:</strong> Award<em><strong> A1A0</strong></em> for only two correct values, and <em><strong>A0A0</strong></em> for only one.</p>
<p><strong>Note:</strong> Award marks in part (b) for equivalent steps seen in part (a).</p>
<p> </p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>An arithmetic sequence <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_1}{\text{, }}{u_2}{\text{, }}{u_3} \ldots ">
<mrow>
<msub>
<mi>u</mi>
<mn>1</mn>
</msub>
</mrow>
<mrow>
<mtext>, </mtext>
</mrow>
<mrow>
<msub>
<mi>u</mi>
<mn>2</mn>
</msub>
</mrow>
<mrow>
<mtext>, </mtext>
</mrow>
<mrow>
<msub>
<mi>u</mi>
<mn>3</mn>
</msub>
</mrow>
<mo>…<!-- … --></mo>
</math></span> has <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_1} = 1">
<mrow>
<msub>
<mi>u</mi>
<mn>1</mn>
</msub>
</mrow>
<mo>=</mo>
<mn>1</mn>
</math></span> and common difference <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d \ne 0">
<mi>d</mi>
<mo>≠<!-- ≠ --></mo>
<mn>0</mn>
</math></span>. Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_2}{\text{, }}{u_3}">
<mrow>
<msub>
<mi>u</mi>
<mn>2</mn>
</msub>
</mrow>
<mrow>
<mtext>, </mtext>
</mrow>
<mrow>
<msub>
<mi>u</mi>
<mn>3</mn>
</msub>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_6}">
<mrow>
<msub>
<mi>u</mi>
<mn>6</mn>
</msub>
</mrow>
</math></span> are the first three terms of a geometric sequence</p>
</div>
<div class="specification">
<p>Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_N} = - 15">
<mrow>
<msub>
<mi>u</mi>
<mi>N</mi>
</msub>
</mrow>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mn>15</mn>
</math></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d">
<mi>d</mi>
</math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>determine the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sum\limits_{r = 1}^N {{u_r}} ">
<munderover>
<mo movablelimits="false">∑</mo>
<mrow>
<mi>r</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>N</mi>
</munderover>
<mrow>
<mrow>
<msub>
<mi>u</mi>
<mi>r</mi>
</msub>
</mrow>
</mrow>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p>use of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{u_n} = {u_1} + (n - 1)d">
<mrow>
<msub>
<mi>u</mi>
<mi>n</mi>
</msub>
</mrow>
<mo>=</mo>
<mrow>
<msub>
<mi>u</mi>
<mn>1</mn>
</msub>
</mrow>
<mo>+</mo>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo>−</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mi>d</mi>
</math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{(1 + 2d)^2} = (1 + d)(1 + 5d)">
<mrow>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>+</mo>
<mn>2</mn>
<mi>d</mi>
<msup>
<mo stretchy="false">)</mo>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>+</mo>
<mi>d</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>+</mo>
<mn>5</mn>
<mi>d</mi>
<mo stretchy="false">)</mo>
</math></span> (or equivalent) <strong><em>M1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d = - 2">
<mi>d</mi>
<mo>=</mo>
<mo>−</mo>
<mn>2</mn>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1 + (N - 1) \times - 2 = - 15">
<mn>1</mn>
<mo>+</mo>
<mo stretchy="false">(</mo>
<mi>N</mi>
<mo>−</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>×</mo>
<mo>−</mo>
<mn>2</mn>
<mo>=</mo>
<mo>−</mo>
<mn>15</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="N = 9">
<mi>N</mi>
<mo>=</mo>
<mn>9</mn>
</math></span> <strong><em>(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sum\limits_{r = 1}^9 {{u_r}} = \frac{9}{2}(2 + 8 \times - 2)">
<munderover>
<mo movablelimits="false">∑</mo>
<mrow>
<mi>r</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mn>9</mn>
</munderover>
<mrow>
<mrow>
<msub>
<mi>u</mi>
<mi>r</mi>
</msub>
</mrow>
</mrow>
<mo>=</mo>
<mfrac>
<mn>9</mn>
<mn>2</mn>
</mfrac>
<mo stretchy="false">(</mo>
<mn>2</mn>
<mo>+</mo>
<mn>8</mn>
<mo>×</mo>
<mo>−</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
</math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = - 63">
<mo>=</mo>
<mo>−</mo>
<mn>63</mn>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p class="indent1" style="margin-top:12.0pt;">Find the determinant of <strong><em>A</em></strong>, where <strong><em>A</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 3&1&2 \\ 9&5&8 \\ 7&4&6 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>3</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>9</mn>
</mtd>
<mtd>
<mn>5</mn>
</mtd>
<mtd>
<mn>8</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>7</mn>
</mtd>
<mtd>
<mn>4</mn>
</mtd>
<mtd>
<mn>6</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p>det <strong><em>A</em></strong> = −2 <em><strong>A2</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>It is given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{lo}}{{\text{g}}_2}\,y + {\text{lo}}{{\text{g}}_4}\,x + {\text{lo}}{{\text{g}}_4}\,2x = 0">
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>2</mn>
</msub>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>y</mi>
<mo>+</mo>
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>4</mn>
</msub>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>+</mo>
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>4</mn>
</msub>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mi>x</mi>
<mo>=</mo>
<mn>0</mn>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{lo}}{{\text{g}}_{{r^2}}}x = \frac{1}{2}{\text{lo}}{{\text{g}}_r}\,x">
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mrow>
<mrow>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</msub>
</mrow>
<mi>x</mi>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mi>r</mi>
</msub>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r,\,x \in {\mathbb{R}^ + }">
<mi>r</mi>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>∈</mo>
<mrow>
<msup>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
<mo>+</mo>
</msup>
</mrow>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Express <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span> in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>. Give your answer in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = p{x^q}">
<mi>y</mi>
<mo>=</mo>
<mi>p</mi>
<mrow>
<msup>
<mi>x</mi>
<mi>q</mi>
</msup>
</mrow>
</math></span>, where <em>p</em> , <em>q</em> are constants.</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>The region <em>R</em>, is bounded by the graph of the function found in part (b), the <em>x</em>-axis, and the lines <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 1">
<mi>x</mi>
<mo>=</mo>
<mn>1</mn>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \alpha ">
<mi>x</mi>
<mo>=</mo>
<mi>α</mi>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\alpha > 1">
<mi>α</mi>
<mo>></mo>
<mn>1</mn>
</math></span>. The area of <em>R</em> is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sqrt 2 ">
<msqrt>
<mn>2</mn>
</msqrt>
</math></span>.</p>
<p>Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\alpha ">
<mi>α</mi>
</math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><strong>METHOD 1</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{lo}}{{\text{g}}_{{r^2}}}x = \frac{{{\text{lo}}{{\text{g}}_r}\,x}}{{{\text{lo}}{{\text{g}}_r}\,{r^2}}}\left( { = \frac{{{\text{lo}}{{\text{g}}_r}\,x}}{{{\text{2}}\,{\text{lo}}{{\text{g}}_r}\,r}}} \right)">
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mrow>
<mrow>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</msub>
</mrow>
<mi>x</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mi>r</mi>
</msub>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mrow>
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mi>r</mi>
</msub>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mi>r</mi>
</msub>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mrow>
<mrow>
<mtext>2</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mi>r</mi>
</msub>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>r</mi>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong> M1A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{{\text{lo}}{{\text{g}}_r}\,x}}{2}">
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mi>r</mi>
</msub>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mn>2</mn>
</mfrac>
</math></span> <em><strong>AG</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{lo}}{{\text{g}}_{{r^2}}}x = \frac{1}{{{\text{lo}}{{\text{g}}_x}\,{r^2}}}">
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mrow>
<mrow>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</msub>
</mrow>
<mi>x</mi>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mi>x</mi>
</msub>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{{2\,{\text{lo}}{{\text{g}}_x}\,r}}">
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mi>x</mi>
</msub>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>r</mi>
</mrow>
</mfrac>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{{\text{lo}}{{\text{g}}_r}\,x}}{2}">
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mi>r</mi>
</msub>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mn>2</mn>
</mfrac>
</math></span> <em><strong>AG</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<p> </p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{lo}}{{\text{g}}_2}\,y + {\text{lo}}{{\text{g}}_4}\,x + {\text{lo}}{{\text{g}}_4}\,2x = 0">
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>2</mn>
</msub>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>y</mi>
<mo>+</mo>
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>4</mn>
</msub>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>+</mo>
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>4</mn>
</msub>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mi>x</mi>
<mo>=</mo>
<mn>0</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{lo}}{{\text{g}}_2}\,y + {\text{lo}}{{\text{g}}_4}\,2{x^2} = 0">
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>2</mn>
</msub>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>y</mi>
<mo>+</mo>
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>4</mn>
</msub>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{lo}}{{\text{g}}_2}\,y + \frac{1}{2}{\text{lo}}{{\text{g}}_2}\,2{x^2} = 0">
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>2</mn>
</msub>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>y</mi>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>2</mn>
</msub>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{lo}}{{\text{g}}_2}\,y = - \frac{1}{2}{\text{lo}}{{\text{g}}_2}\,2{x^2}">
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>2</mn>
</msub>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>y</mi>
<mo>=</mo>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>2</mn>
</msub>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{lo}}{{\text{g}}_2}\,y = {\text{lo}}{{\text{g}}_2}\left( {\frac{1}{{\sqrt {2x} }}} \right)">
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>2</mn>
</msub>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>y</mi>
<mo>=</mo>
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>2</mn>
</msub>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
<mi>x</mi>
</msqrt>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>M1A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = \frac{1}{{\sqrt 2 }}{x^{ - 1}}">
<mi>y</mi>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
<mrow>
<msup>
<mi>x</mi>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p><strong>Note</strong>: For the final <em><strong>A</strong></em> mark, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span> must be expressed in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p{x^q}">
<mi>p</mi>
<mrow>
<msup>
<mi>x</mi>
<mi>q</mi>
</msup>
</mrow>
</math></span>.</p>
<p><em><strong>[5 marks]</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{lo}}{{\text{g}}_2}\,y + {\text{lo}}{{\text{g}}_4}\,x + {\text{lo}}{{\text{g}}_4}\,2x = 0">
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>2</mn>
</msub>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>y</mi>
<mo>+</mo>
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>4</mn>
</msub>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>+</mo>
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>4</mn>
</msub>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mi>x</mi>
<mo>=</mo>
<mn>0</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{lo}}{{\text{g}}_2}\,y + \frac{1}{2}{\text{lo}}{{\text{g}}_2}\,x + \frac{1}{2}{\text{lo}}{{\text{g}}_2}\,2x = 0">
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>2</mn>
</msub>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>y</mi>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>2</mn>
</msub>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>2</mn>
</msub>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mi>x</mi>
<mo>=</mo>
<mn>0</mn>
</math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{lo}}{{\text{g}}_2}\,y + {\text{lo}}{{\text{g}}_2}\,{x^{\frac{1}{2}}} + {\text{lo}}{{\text{g}}_2}\,{\left( {2x} \right)^{\frac{1}{2}}} = 0">
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>2</mn>
</msub>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>y</mi>
<mo>+</mo>
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>2</mn>
</msub>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<msup>
<mi>x</mi>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>2</mn>
</msub>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
</msup>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{lo}}{{\text{g}}_2}\,\left( {\sqrt 2 xy} \right) = 0">
<mrow>
<mtext>lo</mtext>
</mrow>
<mrow>
<msub>
<mrow>
<mtext>g</mtext>
</mrow>
<mn>2</mn>
</msub>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mo>(</mo>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
<mi>x</mi>
<mi>y</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sqrt 2 xy = 1">
<msqrt>
<mn>2</mn>
</msqrt>
<mi>x</mi>
<mi>y</mi>
<mo>=</mo>
<mn>1</mn>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = \frac{1}{{\sqrt 2 }}{x^{ - 1}}">
<mi>y</mi>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
<mrow>
<msup>
<mi>x</mi>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p><strong>Note</strong>: For the final <em><strong>A</strong></em> mark, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span> must be expressed in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p{x^q}">
<mi>p</mi>
<mrow>
<msup>
<mi>x</mi>
<mi>q</mi>
</msup>
</mrow>
</math></span>.</p>
<p><em><strong>[5 marks]</strong></em></p>
<p> </p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>the area of <em>R</em> is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int\limits_1^\alpha {\frac{1}{{\sqrt 2 }}} {x^{ - 1}}{\text{d}}x">
<munderover>
<mo>∫</mo>
<mn>1</mn>
<mi>α</mi>
</munderover>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
</mrow>
<mrow>
<msup>
<mi>x</mi>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left[ {\frac{1}{{\sqrt 2 }}{\text{ln}}\,x} \right]_1^\alpha ">
<mo>=</mo>
<msubsup>
<mrow>
<mo>[</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mo>]</mo>
</mrow>
<mn>1</mn>
<mi>α</mi>
</msubsup>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{{\sqrt 2 }}{\text{ln}}\,\alpha ">
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>α</mi>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{{\sqrt 2 }}{\text{ln}}\,\alpha = \sqrt 2 ">
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>α</mi>
<mo>=</mo>
<msqrt>
<mn>2</mn>
</msqrt>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\alpha = {{\text{e}}^2}">
<mi>α</mi>
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Only follow through from part (b) if <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span> is in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = p{x^q}">
<mi>y</mi>
<mo>=</mo>
<mi>p</mi>
<mrow>
<msup>
<mi>x</mi>
<mi>q</mi>
</msup>
</mrow>
</math></span></p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the system of equations <strong><em>A</em></strong><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} x \\ y \end{array}} \right) = \left( {\begin{array}{*{20}{c}} 2 \\ { - 3} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>x</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>y</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>3</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> where <strong><em>A</em></strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left( {\begin{array}{*{20}{c}} {k + 1}&{ - k} \\ 2&{k - 1} \end{array}} \right)">
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mi>k</mi>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mrow>
<mi>k</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k \in \mathbb{R}">
<mi>k</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find det <strong><em>A</em></strong>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the set of values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k">
<mi>k</mi>
</math></span> for which the system has a unique solution.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="questionCharChar">Attempting to find det <strong><em>A</em></strong> <em><strong>(M1) </strong></em></p>
<p class="questionCharChar">det <strong><em>A </em></strong><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {k^2} + 2k - 1">
<mo>=</mo>
<mrow>
<msup>
<mi>k</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>2</mn>
<mi>k</mi>
<mo>−</mo>
<mn>1</mn>
</math></span> <em><strong>A1 N2</strong></em></p>
<p class="questionCharChar"><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="questionCharChar">System has a unique solution provided det <strong><em>A</em></strong> ≠ 0 <em><strong> (R1)</strong></em></p>
<p class="questionCharChar"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{k^2} + 2k - 1 \ne 0">
<mrow>
<msup>
<mi>k</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>2</mn>
<mi>k</mi>
<mo>−</mo>
<mn>1</mn>
<mo>≠</mo>
<mn>0</mn>
</math></span> <em><strong> (A1)</strong></em></p>
<p class="questionCharChar">Solving <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{k^2} + 2k - 1 \ne 0">
<mrow>
<msup>
<mi>k</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>2</mn>
<mi>k</mi>
<mo>−</mo>
<mn>1</mn>
<mo>≠</mo>
<mn>0</mn>
</math></span> or equivalent for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k">
<mi>k</mi>
</math></span> <em><strong>M1</strong></em></p>
<p class="questionCharChar"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k \in \mathbb{R}\,{\text{\ }}\left\{ { - 1 \pm \,\sqrt 2 } \right\}\,\,\left( {{\text{accept}}\,\,k \ne - 1 \pm \,\sqrt 2 {\text{,}}\,\,\,k \ne - 2.41{\text{,}}\,\,0.414} \right)">
<mi>k</mi>
<mo>∈</mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>\ </mtext>
</mrow>
<mrow>
<mo>{</mo>
<mrow>
<mo>−</mo>
<mn>1</mn>
<mo>±</mo>
<mspace width="thinmathspace"></mspace>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
<mo>}</mo>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>accept</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>k</mi>
<mo>≠</mo>
<mo>−</mo>
<mn>1</mn>
<mo>±</mo>
<mspace width="thinmathspace"></mspace>
<msqrt>
<mn>2</mn>
</msqrt>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>k</mi>
<mo>≠</mo>
<mo>−</mo>
<mn>2.41</mn>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mn>0.414</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1 N3</strong></em></p>
<p class="questionCharChar"><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The matrix <em><strong>A</strong></em> is given by <em><strong>A </strong></em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left[ {\begin{array}{*{20}{c}} a&b \\ c&d \end{array}} \right]">
<mo>=</mo>
<mrow>
<mo>[</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>a</mi>
</mtd>
<mtd>
<mi>b</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>c</mi>
</mtd>
<mtd>
<mi>d</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>]</mo>
</mrow>
</math></span>.</p>
</div>
<div class="specification">
<p>The matrix <em><strong>B</strong></em> is given by <em><strong>B</strong></em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left[ {\begin{array}{*{20}{c}} 3&2 \\ 2&3 \end{array}} \right]">
<mo>=</mo>
<mrow>
<mo>[</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>3</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>]</mo>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the eigenvalues of <em><strong>A</strong></em> are real if <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {a - d} \right)^2} + 4bc \geqslant 0">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mi>a</mi>
<mo>−</mo>
<mi>d</mi>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>4</mn>
<mi>b</mi>
<mi>c</mi>
<mo>⩾</mo>
<mn>0</mn>
</math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Deduce that the eigenvalues are real if <em><strong>A</strong></em> is symmetric.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the eigenvalues of <em><strong>B</strong></em>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the corresponding eigenvectors.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>the eigenvalues satisfy</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left| {\begin{array}{*{20}{c}} {a - \lambda }&b \\ c&{d - \lambda } \end{array}} \right| = 0">
<mrow>
<mo>|</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mi>a</mi>
<mo>−</mo>
<mi>λ</mi>
</mrow>
</mtd>
<mtd>
<mi>b</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>c</mi>
</mtd>
<mtd>
<mrow>
<mi>d</mi>
<mo>−</mo>
<mi>λ</mi>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>|</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {a - \lambda } \right)\left( {d - \lambda } \right) - bc = 0">
<mrow>
<mo>(</mo>
<mrow>
<mi>a</mi>
<mo>−</mo>
<mi>λ</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>d</mi>
<mo>−</mo>
<mi>λ</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mi>b</mi>
<mi>c</mi>
<mo>=</mo>
<mn>0</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\lambda ^2} - \left( {a + d} \right)\lambda + ad - bc = 0">
<mrow>
<msup>
<mi>λ</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>a</mi>
<mo>+</mo>
<mi>d</mi>
</mrow>
<mo>)</mo>
</mrow>
<mi>λ</mi>
<mo>+</mo>
<mi>a</mi>
<mi>d</mi>
<mo>−</mo>
<mi>b</mi>
<mi>c</mi>
<mo>=</mo>
<mn>0</mn>
</math></span> <em><strong>A1</strong></em></p>
<p>the condition for real roots is </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {a + d} \right)^2} - 4\left( {ad - bc} \right) \geqslant 0">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mi>a</mi>
<mo>+</mo>
<mi>d</mi>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>4</mn>
<mrow>
<mo>(</mo>
<mrow>
<mi>a</mi>
<mi>d</mi>
<mo>−</mo>
<mi>b</mi>
<mi>c</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>⩾</mo>
<mn>0</mn>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {a - d} \right)^2} + 4bc \geqslant 0">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mi>a</mi>
<mo>−</mo>
<mi>d</mi>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>4</mn>
<mi>b</mi>
<mi>c</mi>
<mo>⩾</mo>
<mn>0</mn>
</math></span> <em><strong>AG</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>if the matrix is symmetric, <em>b</em> = <em>c</em>. In this case, <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {a - d} \right)^2} + 4bc = {\left( {a - d} \right)^2} + 4{b^2} \geqslant 0">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mi>a</mi>
<mo>−</mo>
<mi>d</mi>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>4</mn>
<mi>b</mi>
<mi>c</mi>
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mi>a</mi>
<mo>−</mo>
<mi>d</mi>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>4</mn>
<mrow>
<msup>
<mi>b</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>⩾</mo>
<mn>0</mn>
</math></span></p>
<p>because each square term is non-negative <em><strong>R1</strong><strong>AG</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>the characteristic equation is</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\lambda ^2} - 6\lambda + 5 = 0">
<mrow>
<msup>
<mi>λ</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>6</mn>
<mi>λ</mi>
<mo>+</mo>
<mn>5</mn>
<mo>=</mo>
<mn>0</mn>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\lambda = 1,5">
<mi>λ</mi>
<mo>=</mo>
<mn>1</mn>
<mo>,</mo>
<mn>5</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>taking <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\lambda = 1">
<mi>λ</mi>
<mo>=</mo>
<mn>1</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left[ {\begin{array}{*{20}{c}} 2&2 \\ 2&2 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} x \\ y \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 0 \\ 0 \end{array}} \right]">
<mrow>
<mo>[</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>]</mo>
</mrow>
<mrow>
<mo>[</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>x</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>y</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>]</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>[</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>]</mo>
</mrow>
</math></span> <em><strong>M1</strong></em></p>
<p>giving eigenvector <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left[ {\begin{array}{*{20}{c}} 1 \\ { - 1} \end{array}} \right]">
<mo>=</mo>
<mrow>
<mo>[</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>]</mo>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p> </p>
<p>taking <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\lambda = 5">
<mi>λ</mi>
<mo>=</mo>
<mn>5</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left[ {\begin{array}{*{20}{c}} { - 2}&2 \\ 2&{ - 2} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} x \\ y \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 0 \\ 0 \end{array}} \right]">
<mrow>
<mo>[</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>]</mo>
</mrow>
<mrow>
<mo>[</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>x</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>y</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>]</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>[</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>]</mo>
</mrow>
</math></span> <em><strong>M1</strong></em></p>
<p>giving eigenvector <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left[ {\begin{array}{*{20}{c}} 1 \\ 1 \end{array}} \right]">
<mo>=</mo>
<mrow>
<mo>[</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>]</mo>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<br><hr><br><div class="question">
<p>Solve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {{\text{ln}}\,x} \right)^2} - \left( {{\text{ln}}\,2} \right)\left( {{\text{ln}}\,x} \right) < 2{\left( {{\text{ln}}\,2} \right)^2}">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo><</mo>
<mn>2</mn>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</math></span>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {{\text{ln}}\,x} \right)^2} - \left( {{\text{ln}}\,2} \right)\left( {{\text{ln}}\,x} \right) - 2{\left( {{\text{ln}}\,2} \right)^2}\left( { = 0} \right)">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mn>2</mn>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mn>0</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p><strong>EITHER</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ln}}\,x = \frac{{{\text{ln}}\,2 \pm \sqrt {{{\left( {{\text{ln}}\,2} \right)}^2} + 8{{\left( {{\text{ln}}\,2} \right)}^2}} }}{2}">
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mo>±</mo>
<msqrt>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>8</mn>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
</mrow>
<mn>2</mn>
</mfrac>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{{\text{ln}}\,2 \pm 3\,{\text{ln}}\,2}}{2}">
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
<mo>±</mo>
<mn>3</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
</mrow>
<mn>2</mn>
</mfrac>
</math></span> <em><strong>A1</strong></em></p>
<p><strong>OR</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {{\text{ln}}\,x - 2\,{\text{ln}}\,2} \right)\left( {{\text{ln}}\,x + 2\,{\text{ln}}\,2} \right)\left( { = 0} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>−</mo>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mn>0</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong> M1A1</strong></em></p>
<p><strong>THEN</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{ln}}\,x = 2\,{\text{ln}}\,2">
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>=</mo>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
</math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - {\text{ln}}\,2">
<mo>−</mo>
<mrow>
<mtext>ln</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow x = 4">
<mo stretchy="false">⇒</mo>
<mi>x</mi>
<mo>=</mo>
<mn>4</mn>
</math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \frac{1}{2}">
<mi>x</mi>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</math></span> <em><strong> (M1)A1</strong></em> </p>
<p><strong>Note:</strong> <em><strong>(M1)</strong></em> is for an appropriate use of a log law in either case, dependent on the previous <em><strong>M1</strong></em> being awarded, <strong>A1</strong> for both correct answers.</p>
<p>solution is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{2} < x < 4">
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mo><</mo>
<mi>x</mi>
<mo><</mo>
<mn>4</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[6 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>Matrices <strong><em>A</em></strong>, <strong><em>B</em> </strong>and <strong><em>C</em> </strong>are defined as</p>
<p style="text-align: center;"><strong><em>A</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 1&5&1 \\ 3&{ - 1}&3 \\ { - 9}&3&7 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>5</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>3</mn>
</mtd>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</mtd>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>9</mn>
</mrow>
</mtd>
<mtd>
<mn>3</mn>
</mtd>
<mtd>
<mn>7</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, <strong><em>B</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 1&2&{ - 1} \\ 3&{ - 1}&0 \\ 0&3&1 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>3</mn>
</mtd>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>3</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, <strong><em>C</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 8 \\ 0 \\ { - 4} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>8</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>4</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent1" style="margin-top:12.0pt;">Given that <strong><em>AB</em> </strong>= <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} a&0&0 \\ 0&a&0 \\ 0&0&a \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>a</mi>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mi>a</mi>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mi>a</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent1" style="margin-top:12.0pt;">Hence, or otherwise, find <strong><em>A</em></strong><sup>–1</sup>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent1" style="margin-top:12.0pt;">Find the matrix <strong><em>X</em></strong>, such that <strong><em>AX</em> </strong>= <strong><em>C</em></strong>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a = 16">
<mi>a</mi>
<mo>=</mo>
<mn>16</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong><em>A</em></strong><sup>–1</sup> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{{16}}\left( {\begin{array}{*{20}{c}} 1&2&{ - 1} \\ 3&{ - 1}&0 \\ 0&3&1 \end{array}} \right)">
<mfrac>
<mn>1</mn>
<mrow>
<mn>16</mn>
</mrow>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>3</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>3</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>(M1)A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong><em>AX</em></strong> = <strong><em>C</em></strong> ⇒ <strong><em>X</em></strong> = <strong><em>A</em></strong><sup>–1</sup><strong><em>C (M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{{16}}\left( {\begin{array}{*{20}{c}} 1&2&{ - 1} \\ 3&{ - 1}&0 \\ 0&3&1 \end{array}} \right)\left( {\begin{array}{*{20}{c}} 8 \\ 0 \\ { - 4} \end{array}} \right)">
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mn>16</mn>
</mrow>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>3</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>3</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>8</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>4</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{{16}}\left( {\begin{array}{*{20}{c}} {12} \\ {24} \\ { - 4} \end{array}} \right)\,\,\,\left( { = \left( {\begin{array}{*{20}{c}} {0.75} \\ {1.5} \\ { - 0.25} \end{array}} \right)} \right)">
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mn>16</mn>
</mrow>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>12</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>24</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>4</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>0.75</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>1.5</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>0.25</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Let <strong><em>M</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} a&b \\ { - b}&a \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>a</mi>
</mtd>
<mtd>
<mi>b</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mi>b</mi>
</mrow>
</mtd>
<mtd>
<mi>a</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b">
<mi>b</mi>
</math></span> are non-zero real numbers.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent1" style="margin-top:12.0pt;">Show that <strong><em>M</em> </strong>is non-singular.</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="indent1" style="margin-top:12.0pt;"> Calculate <strong><em>M</em></strong><sup>2</sup>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent1" style="margin-top:12.0pt;"> Show that det(<strong><em>M</em></strong><sup>2</sup>) is positive.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>finding det<strong><em> M </em></strong><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {a^2} + {b^2}">
<mo>=</mo>
<mrow>
<msup>
<mi>a</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mi>b</mi>
<mn>2</mn>
</msup>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{a^2} + {b^2} > 0">
<mrow>
<msup>
<mi>a</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mi>b</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>></mo>
<mn>0</mn>
</math></span>, therefore <strong><em>M</em></strong> is non-singular or equivalent statement <em><strong>R1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong><em>M</em></strong><sup>2</sup> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} a&b \\ { - b}&a \end{array}} \right)\left( {\begin{array}{*{20}{c}} a&b \\ { - b}&a \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {{a^2} - {b^2}}&{2ab} \\ { - 2ab}&{{a^2} - {b^2}} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>a</mi>
</mtd>
<mtd>
<mi>b</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mi>b</mi>
</mrow>
</mtd>
<mtd>
<mi>a</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>a</mi>
</mtd>
<mtd>
<mi>b</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mi>b</mi>
</mrow>
</mtd>
<mtd>
<mi>a</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mrow>
<msup>
<mi>a</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mrow>
<msup>
<mi>b</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>2</mn>
<mi>a</mi>
<mi>b</mi>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>2</mn>
<mi>a</mi>
<mi>b</mi>
</mrow>
</mtd>
<mtd>
<mrow>
<mrow>
<msup>
<mi>a</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mrow>
<msup>
<mi>b</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>M1A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong> </p>
<p>det(<strong><em>M</em></strong><sup>2</sup>) <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left( {{a^2} - {b^2}} \right)\left( {{a^2} - {b^2}} \right) + \left( {2ab} \right)\left( {2ab} \right)">
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msup>
<mi>a</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mrow>
<msup>
<mi>b</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msup>
<mi>a</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mrow>
<msup>
<mi>b</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mi>a</mi>
<mi>b</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mi>a</mi>
<mi>b</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p>det(<strong><em>M</em></strong><sup>2</sup>) <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = {\left( {{a^2} - {b^2}} \right)^2} + {\left( {2ab} \right)^2}">
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msup>
<mi>a</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mrow>
<msup>
<mi>b</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mi>a</mi>
<mi>b</mi>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</math></span> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( { = {{\left( {{a^2} + {b^2}} \right)}^2}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msup>
<mi>a</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mi>b</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p>since the first term is non-negative and the second is positive <em><strong>R1 </strong></em></p>
<p>therefore det(<strong><em>M</em></strong>2) > 0 </p>
<p><strong>Note:</strong> Do not penalise first term stated as positive. </p>
<p><strong>OR</strong> </p>
<p>det(<strong><em>M</em></strong><sup>2</sup>) = (det <strong><em>M</em></strong>)<sup>2</sup> <em><strong>A1</strong></em></p>
<p>since det <strong><em>M</em></strong> is positive so too is det (<strong><em>M</em></strong><sup>2</sup>) <em><strong>R1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="question">
<p class="indent1" style="margin-top:12pt;text-align: left;">Given the matrix <strong><em>A</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 3&2 \\ { - 1}&0 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>3</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> find the values of the real number <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k">
<mi>k</mi>
</math></span> for which <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{det}}\left( {A - kI} \right) = 0">
<mrow>
<mtext>det</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>A</mi>
<mo>−</mo>
<mi>k</mi>
<mi>I</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="I = \left( {\begin{array}{*{20}{c}} 1&0 \\ 0&1 \end{array}} \right)">
<mi>I</mi>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p class="indent1" style="margin-top:12pt;text-align: left;"> </p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{det}}\left( {A - kI} \right) = 0">
<mrow>
<mtext>det</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>A</mi>
<mo>−</mo>
<mi>k</mi>
<mi>I</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow \left| {\begin{array}{*{20}{c}} {3 - k}&2 \\ { - 1}&{ - k} \end{array}} \right| = 0">
<mo stretchy="false">⇒</mo>
<mrow>
<mo>|</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>3</mn>
<mo>−</mo>
<mi>k</mi>
</mrow>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mi>k</mi>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>|</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow {k^2} - 3k + 2 = 0">
<mo stretchy="false">⇒</mo>
<mrow>
<msup>
<mi>k</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>3</mn>
<mi>k</mi>
<mo>+</mo>
<mn>2</mn>
<mo>=</mo>
<mn>0</mn>
</math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow \left( {k - 2} \right)\left( {k - 1} \right) = 0">
<mo stretchy="false">⇒</mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>k</mi>
<mo>−</mo>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>k</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow k = 1{\text{,}}\,\,2">
<mo stretchy="false">⇒</mo>
<mi>k</mi>
<mo>=</mo>
<mn>1</mn>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mn>2</mn>
</math></span> <em><strong>(A2)</strong></em> <em><strong>(C4)</strong></em></p>
<p><em><strong>[4 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>The matrix <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">M</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mn>0</mn><mo>.</mo><mn>2</mn><mo> </mo><mo> </mo></mtd><mtd><mn>0</mn><mo>.</mo><mn>7</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo>.</mo><mn>8</mn><mo> </mo><mo> </mo></mtd><mtd><mn>0</mn><mo>.</mo><mn>3</mn></mtd></mtr></mtable></mfenced></math> has eigenvalues <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>5</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math>.</p>
</div>
<div class="specification">
<p>A switch has two states, <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math>. Each second it either remains in the same state or moves according to the following rule: If it is in state <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> it will move to state <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math> with a probability of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>8</mn></math> and if it is in state <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math> it will move to state <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> with a probability of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>7</mn></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an eigenvector corresponding to the eigenvalue of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math>. Give your answer in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mi>a</mi></mtd></mtr><mtr><mtd><mi>b</mi></mtd></mtr></mtable></mfenced></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>,</mo><mo> </mo><mi>b</mi><mo>∈</mo><mi mathvariant="normal">ℤ</mi></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Using your answer to (a), or otherwise, find the long-term probability of the switch being in state <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math>. Give your answer in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>c</mi><mi>d</mi></mfrac></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>,</mo><mo> </mo><mi>d</mi><mo>∈</mo><msup><mi mathvariant="normal">ℤ</mi><mo>+</mo></msup></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi><mo>=</mo><mn>1</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mo>-</mo><mn>0</mn><mo>.</mo><mn>8</mn></mtd><mtd><mn>0</mn><mo>.</mo><mn>7</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo>.</mo><mn>8</mn></mtd><mtd><mo>-</mo><mn>0</mn><mo>.</mo><mn>7</mn></mtd></mtr></mtable></mfenced><mfenced><mtable><mtr><mtd><mi>x</mi></mtd></mtr><mtr><mtd><mi>y</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr></mtable></mfenced></math> <strong>OR </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>0</mn><mo>.</mo><mn>2</mn></mtd><mtd><mn>0</mn><mo>.</mo><mn>7</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo>.</mo><mn>8</mn></mtd><mtd><mn>0</mn><mo>.</mo><mn>3</mn></mtd></mtr></mtable></mfenced><mfenced><mtable><mtr><mtd><mi>x</mi></mtd></mtr><mtr><mtd><mi>y</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mtable><mtr><mtd><mi>x</mi></mtd></mtr><mtr><mtd><mi>y</mi></mtd></mtr></mtable></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>8</mn><mi>x</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>7</mn><mi>y</mi></math> <em><strong>(A1)</strong></em></p>
<p>an eigenvector is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>7</mn></mtd></mtr><mtr><mtd><mn>8</mn></mtd></mtr></mtable></mfenced></math> (or equivalent with integer values) <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p>(the long-term probability matrix is given by the eigenvector corresponding to the eigenvalue equal to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math>, scaled so that the sum of the entries is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math>)</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8</mn><mo>+</mo><mn>7</mn><mo>=</mo><mn>15</mn></math> <em><strong>(M1)</strong></em></p>
<p><br><strong>OR</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>0</mn><mo>.</mo><mn>2</mn></mtd><mtd><mn>0</mn><mo>.</mo><mn>7</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo>.</mo><mn>8</mn></mtd><mtd><mn>0</mn><mo>.</mo><mn>3</mn></mtd></mtr></mtable></mfenced><mfenced><mtable><mtr><mtd><mi>p</mi></mtd></mtr><mtr><mtd><mn>1</mn><mo>-</mo><mi>p</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mtable><mtr><mtd><mi>p</mi></mtd></mtr><mtr><mtd><mn>1</mn><mo>-</mo><mi>p</mi></mtd></mtr></mtable></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><br><strong>OR</strong></p>
<p>considering high powers of the matrix <em>e.g.</em> <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mtable><mtr><mtd><mn>0</mn><mo>.</mo><mn>2</mn></mtd><mtd><mn>0</mn><mo>.</mo><mn>7</mn></mtd></mtr><mtr><mtd><mn>0</mn><mo>.</mo><mn>8</mn></mtd><mtd><mn>0</mn><mo>.</mo><mn>3</mn></mtd></mtr></mtable></mfenced><mn>50</mn></msup></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mfrac><mn>7</mn><mn>15</mn></mfrac></mtd><mtd><mfrac><mn>7</mn><mn>15</mn></mfrac></mtd></mtr><mtr><mtd><mfrac><mn>8</mn><mn>15</mn></mfrac></mtd><mtd><mfrac><mn>8</mn><mn>15</mn></mfrac></mtd></mtr></mtable></mfenced></math></p>
<p><br><strong>THEN</strong></p>
<p>probability of being in state <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>7</mn><mn>15</mn></mfrac></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>In part (a), some candidates could correctly use either <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi mathvariant="bold-italic">A</mi><mo>-</mo><mi>λ</mi><mi mathvariant="bold-italic">I</mi><mo>)</mo><mi mathvariant="bold-italic">x</mi><mo>=</mo><mn>0</mn></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">A</mi><mi mathvariant="bold-italic">x</mi><mo>=</mo><mi>λ</mi><mi mathvariant="bold-italic">x</mi></math>to find an eigenvector but many did not pay attention to the fact that integer values of the eigenvector were required. Some candidates used the method of finding the steady state by finding <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi mathvariant="bold-italic">A</mi><mi>n</mi></msup></math> for some high value of n in part (b) but ignored the fact that they needed to express their answer in rational form. Some did try to convert their calculated answer of 0.467 to <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>467</mn><mn>1000</mn></mfrac></math> but this could only receive partial credit as an exact answer was required.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the complex numbers <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{z_1} = 1 + \sqrt 3 {\text{i, }}{z_2} = 1 + {\text{i}}">
<mrow>
<msub>
<mi>z</mi>
<mn>1</mn>
</msub>
</mrow>
<mo>=</mo>
<mn>1</mn>
<mo>+</mo>
<msqrt>
<mn>3</mn>
</msqrt>
<mrow>
<mtext>i, </mtext>
</mrow>
<mrow>
<msub>
<mi>z</mi>
<mn>2</mn>
</msub>
</mrow>
<mo>=</mo>
<mn>1</mn>
<mo>+</mo>
<mrow>
<mtext>i</mtext>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="w = \frac{{{z_1}}}{{{z_2}}}">
<mi>w</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<msub>
<mi>z</mi>
<mn>1</mn>
</msub>
</mrow>
</mrow>
<mrow>
<mrow>
<msub>
<mi>z</mi>
<mn>2</mn>
</msub>
</mrow>
</mrow>
</mfrac>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By expressing <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{z_1}">
<mrow>
<msub>
<mi>z</mi>
<mn>1</mn>
</msub>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{z_2}">
<mrow>
<msub>
<mi>z</mi>
<mn>2</mn>
</msub>
</mrow>
</math></span> in modulus-argument form write down the modulus of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="w">
<mi>w</mi>
</math></span>;</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By expressing <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{z_1}">
<mrow>
<msub>
<mi>z</mi>
<mn>1</mn>
</msub>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{z_2}">
<mrow>
<msub>
<mi>z</mi>
<mn>2</mn>
</msub>
</mrow>
</math></span> in modulus-argument form write down the argument of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="w">
<mi>w</mi>
</math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the smallest positive integer value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n">
<mi>n</mi>
</math></span>, such that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{w^n}">
<mrow>
<msup>
<mi>w</mi>
<mi>n</mi>
</msup>
</mrow>
</math></span> is a real number.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{z_1} = 2{\text{cis}}\left( {\frac{\pi }{3}} \right)">
<mrow>
<msub>
<mi>z</mi>
<mn>1</mn>
</msub>
</mrow>
<mo>=</mo>
<mn>2</mn>
<mrow>
<mtext>cis</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mi>π</mi>
<mn>3</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{z_2} = \sqrt 2 {\text{cis}}\left( {\frac{\pi }{4}} \right)">
<mrow>
<msub>
<mi>z</mi>
<mn>2</mn>
</msub>
</mrow>
<mo>=</mo>
<msqrt>
<mn>2</mn>
</msqrt>
<mrow>
<mtext>cis</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mi>π</mi>
<mn>4</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A1A0 </strong></em>for correct moduli and arguments found, but not written in mod-arg form.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left| w \right| = \sqrt 2 ">
<mrow>
<mo>|</mo>
<mi>w</mi>
<mo>|</mo>
</mrow>
<mo>=</mo>
<msqrt>
<mn>2</mn>
</msqrt>
</math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{z_1} = 2{\text{cis}}\left( {\frac{\pi }{3}} \right)">
<mrow>
<msub>
<mi>z</mi>
<mn>1</mn>
</msub>
</mrow>
<mo>=</mo>
<mn>2</mn>
<mrow>
<mtext>cis</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mi>π</mi>
<mn>3</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{z_2} = \sqrt 2 {\text{cis}}\left( {\frac{\pi }{4}} \right)">
<mrow>
<msub>
<mi>z</mi>
<mn>2</mn>
</msub>
</mrow>
<mo>=</mo>
<msqrt>
<mn>2</mn>
</msqrt>
<mrow>
<mtext>cis</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mi>π</mi>
<mn>4</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1A1</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A1A0 </strong></em>for correct moduli and arguments found, but not written in mod-arg form.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\arg w = \frac{\pi }{{12}}">
<mi>arg</mi>
<mo></mo>
<mi>w</mi>
<mo>=</mo>
<mfrac>
<mi>π</mi>
<mrow>
<mn>12</mn>
</mrow>
</mfrac>
</math></span> <em><strong>A1</strong></em></p>
<p> </p>
<p><strong>Notes:</strong> Allow <em><strong>FT </strong></em>from incorrect answers for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{z_1}">
<mrow>
<msub>
<mi>z</mi>
<mn>1</mn>
</msub>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{z_2}">
<mrow>
<msub>
<mi>z</mi>
<mn>2</mn>
</msub>
</mrow>
</math></span> in modulus-argument form.</p>
<p> </p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sin \left( {\frac{{\pi n}}{{12}}} \right) = 0">
<mi>sin</mi>
<mo></mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mi>π</mi>
<mi>n</mi>
</mrow>
<mrow>
<mn>12</mn>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span> <em><strong>(M1)</strong></em></p>
<p><strong>OR</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\arg ({w^n}) = \pi ">
<mi>arg</mi>
<mo></mo>
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mi>w</mi>
<mi>n</mi>
</msup>
</mrow>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>π</mi>
</math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{n\pi }}{{12}} = \pi ">
<mfrac>
<mrow>
<mi>n</mi>
<mi>π</mi>
</mrow>
<mrow>
<mn>12</mn>
</mrow>
</mfrac>
<mo>=</mo>
<mi>π</mi>
</math></span></p>
<p><strong>THEN</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\therefore n = 12">
<mo>∴</mo>
<mi>n</mi>
<mo>=</mo>
<mn>12</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the matrix <strong><em>A </em></strong><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left( {\begin{array}{*{20}{c}} 5&{ - 2} \\ 7&1 \end{array}} \right)">
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>5</mn>
</mtd>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>2</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>7</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
</div>
<div class="specification">
<p><strong><em>B</em></strong>, <strong><em>C</em></strong> and <strong><em>X</em></strong> are also 2 × 2 matrices.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the inverse, <strong><em>A</em></strong><sup>–1</sup>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that <strong><em>XA</em></strong> + <strong><em>B</em></strong> = <strong><em>C</em></strong>, express <strong><em>X</em></strong> in terms of <strong><em>A</em></strong><sup>–1</sup>, <strong><em>B</em></strong> and <strong><em>C</em></strong><em>.</em></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that <em><strong>B</strong></em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left( {\begin{array}{*{20}{c}} 6&7 \\ 5&{ - 2} \end{array}} \right)">
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>6</mn>
</mtd>
<mtd>
<mn>7</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>5</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, and <em><strong>C</strong></em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left( {\begin{array}{*{20}{c}} { - 5}&0 \\ { - 8}&7 \end{array}} \right)">
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>5</mn>
</mrow>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>8</mn>
</mrow>
</mtd>
<mtd>
<mn>7</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, find <em><strong>X</strong></em>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p>det <em><strong>A</strong></em> = 5(1) − 7(−2) = 19</p>
<p><strong><em>A</em></strong><sup>–1</sup> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{{19}}\left( {\begin{array}{*{20}{c}} 1&2 \\ { - 7}&5 \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {\frac{1}{{19}}}&{\frac{2}{{19}}} \\ {\frac{{ - 7}}{{19}}}&{\frac{5}{{19}}} \end{array}} \right)">
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mn>19</mn>
</mrow>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>7</mn>
</mrow>
</mtd>
<mtd>
<mn>5</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<mn>19</mn>
</mrow>
</mfrac>
</mrow>
</mtd>
<mtd>
<mrow>
<mfrac>
<mn>2</mn>
<mrow>
<mn>19</mn>
</mrow>
</mfrac>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mfrac>
<mrow>
<mo>−</mo>
<mn>7</mn>
</mrow>
<mrow>
<mn>19</mn>
</mrow>
</mfrac>
</mrow>
</mtd>
<mtd>
<mrow>
<mfrac>
<mn>5</mn>
<mrow>
<mn>19</mn>
</mrow>
</mfrac>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>(A2)</strong></em></p>
<p><strong>Note: </strong>Award <em><strong>(A1)</strong></em> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 1&2 \\ { - 7}&5 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>7</mn>
</mrow>
</mtd>
<mtd>
<mn>5</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, <em><strong>(A1)</strong></em> for dividing by 19.</p>
<p><strong>OR</strong></p>
<p><strong><em>A</em></strong><sup>–1</sup> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left( {\begin{array}{*{20}{c}} {0.0526}&{0.105} \\ { - 0.368}&{0.263} \end{array}} \right)">
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>0.0526</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>0.105</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>0.368</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>0.263</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong> (G2)</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong><em>XA</em> </strong>+ <strong><em>B</em></strong> = <strong><em>C</em> ⇒</strong> <strong><em>XA</em> </strong>= <strong><em>C</em></strong> – <strong><em>Β</em></strong> <em><strong>(M1) </strong></em></p>
<p><strong><em>X</em> </strong>= (<strong><em>C</em> </strong>– <strong><em>Β</em></strong>)<strong><em>Α</em></strong><sup>–1</sup> <em><strong>(A1)</strong></em></p>
<p><strong>OR</strong></p>
<p><strong><em>X</em> </strong>= (<strong><em>C</em></strong> – <strong><em>B</em></strong>)<strong><em>A</em></strong><sup>–1</sup> <em><strong> (A2)</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>(<strong><em>C</em> </strong>– <strong><em>Β</em></strong>)<strong><em>Α</em></strong><sup>–1</sup> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} { - 11}&{ - 7} \\ { - 13}&9 \end{array}} \right)\left( {\begin{array}{*{20}{c}} {\frac{1}{{19}}}&{\frac{2}{{19}}} \\ {\frac{{ - 7}}{{19}}}&{\frac{5}{{19}}} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>11</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>7</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>13</mn>
</mrow>
</mtd>
<mtd>
<mn>9</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<mn>19</mn>
</mrow>
</mfrac>
</mrow>
</mtd>
<mtd>
<mrow>
<mfrac>
<mn>2</mn>
<mrow>
<mn>19</mn>
</mrow>
</mfrac>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mfrac>
<mrow>
<mo>−</mo>
<mn>7</mn>
</mrow>
<mrow>
<mn>19</mn>
</mrow>
</mfrac>
</mrow>
</mtd>
<mtd>
<mrow>
<mfrac>
<mn>5</mn>
<mrow>
<mn>19</mn>
</mrow>
</mfrac>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>(A1) </strong></em></p>
<p>⇒<strong><em> X</em> </strong>= <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} {\frac{{38}}{{19}}}&{\frac{{ - 57}}{{19}}} \\ {\frac{{ - 76}}{{19}}}&{\frac{{19}}{{19}}} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} 2&{ - 3} \\ { - 4}&1 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mfrac>
<mrow>
<mn>38</mn>
</mrow>
<mrow>
<mn>19</mn>
</mrow>
</mfrac>
</mrow>
</mtd>
<mtd>
<mrow>
<mfrac>
<mrow>
<mo>−</mo>
<mn>57</mn>
</mrow>
<mrow>
<mn>19</mn>
</mrow>
</mfrac>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mfrac>
<mrow>
<mo>−</mo>
<mn>76</mn>
</mrow>
<mrow>
<mn>19</mn>
</mrow>
</mfrac>
</mrow>
</mtd>
<mtd>
<mrow>
<mfrac>
<mrow>
<mn>19</mn>
</mrow>
<mrow>
<mn>19</mn>
</mrow>
</mfrac>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>3</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>4</mn>
</mrow>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>(A1)</strong></em></p>
<p><strong>OR</strong></p>
<p><strong><em>X</em> </strong>= <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 2&{ - 3} \\ { - 4}&1 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>3</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>4</mn>
</mrow>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong> (G2)</strong></em></p>
<p><strong>Note: </strong>If premultiplication by <strong>A</strong><sup>–1</sup><em> </em>is used, award <em><strong>(M1)(M0)</strong></em> in part (i) but award <em><strong>(A2)</strong></em> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} {\frac{{ - 37}}{{19}}}&{\frac{{11}}{{19}}} \\ {\frac{{12}}{{19}}}&{\frac{{94}}{{19}}} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mfrac>
<mrow>
<mo>−</mo>
<mn>37</mn>
</mrow>
<mrow>
<mn>19</mn>
</mrow>
</mfrac>
</mrow>
</mtd>
<mtd>
<mrow>
<mfrac>
<mrow>
<mn>11</mn>
</mrow>
<mrow>
<mn>19</mn>
</mrow>
</mfrac>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mfrac>
<mrow>
<mn>12</mn>
</mrow>
<mrow>
<mn>19</mn>
</mrow>
</mfrac>
</mrow>
</mtd>
<mtd>
<mrow>
<mfrac>
<mrow>
<mn>94</mn>
</mrow>
<mrow>
<mn>19</mn>
</mrow>
</mfrac>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> in part (ii).</p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent1" style="margin-top:12pt;text-align: left;">Find the values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b">
<mi>b</mi>
</math></span> given that the matrix <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A = \left( {\begin{array}{*{20}{c}} a&{ - 4}&{ - 6} \\ { - 8}&5&7 \\ { - 5}&3&4 \end{array}} \right)">
<mi>A</mi>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>a</mi>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>4</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>6</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>8</mn>
</mrow>
</mtd>
<mtd>
<mn>5</mn>
</mtd>
<mtd>
<mn>7</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>5</mn>
</mrow>
</mtd>
<mtd>
<mn>3</mn>
</mtd>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> is the inverse of the matrix <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="B = \left( {\begin{array}{*{20}{c}} 1&2&{ - 2} \\ 3&b&1 \\ { - 1}&1&{ - 3} \end{array}} \right)">
<mi>B</mi>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>3</mn>
</mtd>
<mtd>
<mi>b</mi>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>3</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent1" style="margin-top:12pt;text-align: left;">For the values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b">
<mi>b</mi>
</math></span> found in part (a), solve the system of linear equations</p>
<p class="indent1" style="margin-top:12pt;text-align: left;"><span class="mjpage mjpage__block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" alttext="\begin{array}{*{20}{c}} {x + 2y - 2z = 5} \\ {3x + by + z = 0} \\ { - x + y - 3z = a - 1.} \end{array}">
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
<mi>y</mi>
<mo>−</mo>
<mn>2</mn>
<mi>z</mi>
<mo>=</mo>
<mn>5</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>3</mn>
<mi>x</mi>
<mo>+</mo>
<mi>b</mi>
<mi>y</mi>
<mo>+</mo>
<mi>z</mi>
<mo>=</mo>
<mn>0</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mi>x</mi>
<mo>+</mo>
<mi>y</mi>
<mo>−</mo>
<mn>3</mn>
<mi>z</mi>
<mo>=</mo>
<mi>a</mi>
<mo>−</mo>
<mn>1.</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</math></span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><em>AB</em> = <em>I</em></p>
<p>(<em>AB</em>)<sub>11</sub> = 1 ⇒ <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span> – 12 + 6 = 1, giving <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span> = 7 <em><strong>(A1) (C1) </strong></em></p>
<p>(<em>AB</em>)<sub>22</sub> = 1 ⇒ –16 + 5<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b">
<mi>b</mi>
</math></span> + 7 = 1, giving <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b">
<mi>b</mi>
</math></span> = 2 <em><strong>(A1) (C1)</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>the system is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="BX = \left( {\begin{array}{*{20}{c}} 5 \\ 0 \\ 6 \end{array}} \right)">
<mi>B</mi>
<mi>X</mi>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>5</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>6</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="X = \left( {\begin{array}{*{20}{c}} x \\ y \\ z \end{array}} \right)">
<mi>X</mi>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>x</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>y</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>z</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
<p>Then, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="X = A\left( {\begin{array}{*{20}{c}} 5 \\ 0 \\ 6 \end{array}} \right) = \left( {\begin{array}{*{20}{c}} 7&{ - 4}&{ - 6} \\ { - 8}&5&7 \\ { - 5}&3&4 \end{array}} \right)\left( {\begin{array}{*{20}{c}} 5 \\ 0 \\ 6 \end{array}} \right).">
<mi>X</mi>
<mo>=</mo>
<mi>A</mi>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>5</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>6</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>7</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>4</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>6</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>8</mn>
</mrow>
</mtd>
<mtd>
<mn>5</mn>
</mtd>
<mtd>
<mn>7</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>5</mn>
</mrow>
</mtd>
<mtd>
<mn>3</mn>
</mtd>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>5</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>6</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>.</mo>
</math></span> <em><strong>(M1)</strong></em></p>
<p>Thus <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = - 1">
<mi>x</mi>
<mo>=</mo>
<mo>−</mo>
<mn>1</mn>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 2">
<mi>y</mi>
<mo>=</mo>
<mn>2</mn>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z = - 1">
<mi>z</mi>
<mo>=</mo>
<mo>−</mo>
<mn>1</mn>
</math></span> <em><strong>(A1) (C2)</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The function <em>f</em> : <em><strong>M</strong></em> → <em><strong>M</strong></em> where <em><strong>M</strong></em> is the set of 2 × 2 matrices, is given by <em>f</em>(<em><strong>X</strong></em>) = <em><strong>AX</strong></em> where <em><strong>A</strong></em> is a 2 × 2 matrix.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Given that <em><strong>A</strong></em> is non-singular, prove that<em> f</em> is a bijection.</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 now given that <em><strong>A</strong></em> is singular.</p>
<p>By considering appropriate determinants, prove that <em>f</em> is not a bijection.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p>suppose <em>f</em>(<em><strong>X</strong></em>) = <em>f</em>(<em><strong>Y</strong></em>) , ie <em><strong>AX</strong></em> = <em><strong>AY</strong></em> <em><strong>(M1)</strong></em></p>
<p>then <em><strong>A</strong></em><sup>−1</sup><strong><em>AX</em></strong> = <em><strong>A</strong></em><sup>−1</sup><strong><em>AY A1</em></strong></p>
<p><em><strong>X</strong></em> = <em><strong>Y A1</strong></em></p>
<p>since <em>f</em>(<em><strong>X</strong></em>) = <em>f</em>(<em><strong>Y</strong></em>) ⇒ <em><strong>X</strong></em> = <em><strong>Y</strong></em>, <em>f</em> is an injection <em><strong>R1</strong></em></p>
<p>now suppose <strong>C</strong> ∈ <em><strong>M</strong></em> and consider <em>f</em>(<em><strong>D</strong></em>) = <em><strong>C</strong></em> , ie <em><strong>AD</strong></em> = <em><strong>C M1</strong></em></p>
<p>then <em><strong>D</strong></em> = <em><strong>A</strong></em><sup>−1</sup> <em><strong>C</strong></em> (<em><strong>A</strong></em><sup>−1</sup> exists since <em><strong>A</strong></em> is non- singular) <em><strong>A1</strong></em></p>
<p>since given <strong>C</strong> ∈ <em><strong>M</strong></em>, there exists <em><strong>D</strong></em> ∈ <em><strong>M</strong></em> such that <em>f</em>(<em><strong>D</strong></em>) = <em><strong>C</strong></em> , <em>f</em> is a surjection <em><strong>R1</strong></em></p>
<p>therefore<em> f</em> is a bijection <em><strong>AG</strong></em></p>
<p><em><strong>[7</strong></em><strong style="font-style: italic;"> marks]</strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>suppose <em>f</em>(<em><strong>X</strong></em>) = <em><strong>Y</strong></em>, ie <em><strong>AX</strong></em> = <em><strong>Y</strong></em> <em><strong>(M1)</strong></em></p>
<p>then det(<em><strong>A</strong></em>)det(<em><strong>X</strong></em>) = det(<em><strong>Y</strong></em>) <em><strong>A1</strong></em></p>
<p>since det(<em><strong>A</strong></em>) = 0, it follows that det(<em><strong>Y</strong></em>) = 0 <em><strong>A1</strong></em></p>
<p>it follows that <em>f</em> is not surjective since the function cannot reach non-singular matrices <em><strong>R1</strong></em></p>
<p>therefore <em>f</em> is not a bijection <em><strong>AG</strong></em></p>
<p><em><strong>[4</strong></em><strong style="font-style: italic;"> marks]</strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z = 1 - \cos 2\theta - {\text{i}}\sin 2\theta ,{\text{ }}z \in \mathbb{C},{\text{ }}0 \leqslant \theta \leqslant \pi ">
<mi>z</mi>
<mo>=</mo>
<mn>1</mn>
<mo>−<!-- − --></mo>
<mi>cos</mi>
<mo><!-- --></mo>
<mn>2</mn>
<mi>θ<!-- θ --></mi>
<mo>−<!-- − --></mo>
<mrow>
<mtext>i</mtext>
</mrow>
<mi>sin</mi>
<mo><!-- --></mo>
<mn>2</mn>
<mi>θ<!-- θ --></mi>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>z</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">C</mi>
</mrow>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>0</mn>
<mo>⩽<!-- ⩽ --></mo>
<mi>θ<!-- θ --></mi>
<mo>⩽<!-- ⩽ --></mo>
<mi>π<!-- π --></mi>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Solve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2\sin (x + 60^\circ ) = \cos (x + 30^\circ ),{\text{ }}0^\circ \leqslant x \leqslant 180^\circ ">
<mn>2</mn>
<mi>sin</mi>
<mo></mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>+</mo>
<msup>
<mn>60</mn>
<mo>∘</mo>
</msup>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>cos</mi>
<mo></mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>+</mo>
<msup>
<mn>30</mn>
<mo>∘</mo>
</msup>
<mo stretchy="false">)</mo>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<msup>
<mn>0</mn>
<mo>∘</mo>
</msup>
<mo>⩽</mo>
<mi>x</mi>
<mo>⩽</mo>
<msup>
<mn>180</mn>
<mo>∘</mo>
</msup>
</math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sin 105^\circ + \cos 105^\circ = \frac{1}{{\sqrt 2 }}">
<mi>sin</mi>
<mo></mo>
<msup>
<mn>105</mn>
<mo>∘</mo>
</msup>
<mo>+</mo>
<mi>cos</mi>
<mo></mo>
<msup>
<mn>105</mn>
<mo>∘</mo>
</msup>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the modulus and argument of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z">
<mi>z</mi>
</math></span> in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\theta ">
<mi>θ</mi>
</math></span>. Express each answer in its simplest form.</p>
<div class="marks">[9]</div>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find the cube roots of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z">
<mi>z</mi>
</math></span> in modulus-argument form.</p>
<div class="marks">[5]</div>
<div class="question_part_label">c.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2\sin (x + 60^\circ ) = \cos (x + 30^\circ )">
<mn>2</mn>
<mi>sin</mi>
<mo></mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>+</mo>
<msup>
<mn>60</mn>
<mo>∘</mo>
</msup>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>cos</mi>
<mo></mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>+</mo>
<msup>
<mn>30</mn>
<mo>∘</mo>
</msup>
<mo stretchy="false">)</mo>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2(\sin x\cos 60^\circ + \cos x\sin 60^\circ ) = \cos x\cos 30^\circ - \sin x\sin 30^\circ ">
<mn>2</mn>
<mo stretchy="false">(</mo>
<mi>sin</mi>
<mo></mo>
<mi>x</mi>
<mi>cos</mi>
<mo></mo>
<msup>
<mn>60</mn>
<mo>∘</mo>
</msup>
<mo>+</mo>
<mi>cos</mi>
<mo></mo>
<mi>x</mi>
<mi>sin</mi>
<mo></mo>
<msup>
<mn>60</mn>
<mo>∘</mo>
</msup>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>cos</mi>
<mo></mo>
<mi>x</mi>
<mi>cos</mi>
<mo></mo>
<msup>
<mn>30</mn>
<mo>∘</mo>
</msup>
<mo>−</mo>
<mi>sin</mi>
<mo></mo>
<mi>x</mi>
<mi>sin</mi>
<mo></mo>
<msup>
<mn>30</mn>
<mo>∘</mo>
</msup>
</math></span> <strong><em>(M1)(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2\sin x \times \frac{1}{2} + 2\cos x \times \frac{{\sqrt 3 }}{2} = \cos x \times \frac{{\sqrt 3 }}{2} - \sin x \times \frac{1}{2}">
<mn>2</mn>
<mi>sin</mi>
<mo></mo>
<mi>x</mi>
<mo>×</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mo>+</mo>
<mn>2</mn>
<mi>cos</mi>
<mo></mo>
<mi>x</mi>
<mo>×</mo>
<mfrac>
<mrow>
<msqrt>
<mn>3</mn>
</msqrt>
</mrow>
<mn>2</mn>
</mfrac>
<mo>=</mo>
<mi>cos</mi>
<mo></mo>
<mi>x</mi>
<mo>×</mo>
<mfrac>
<mrow>
<msqrt>
<mn>3</mn>
</msqrt>
</mrow>
<mn>2</mn>
</mfrac>
<mo>−</mo>
<mi>sin</mi>
<mo></mo>
<mi>x</mi>
<mo>×</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow \frac{3}{2}\sin x = - \frac{{\sqrt 3 }}{2}\cos x">
<mo stretchy="false">⇒</mo>
<mfrac>
<mn>3</mn>
<mn>2</mn>
</mfrac>
<mi>sin</mi>
<mo></mo>
<mi>x</mi>
<mo>=</mo>
<mo>−</mo>
<mfrac>
<mrow>
<msqrt>
<mn>3</mn>
</msqrt>
</mrow>
<mn>2</mn>
</mfrac>
<mi>cos</mi>
<mo></mo>
<mi>x</mi>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow \tan x = - \frac{1}{{\sqrt 3 }}">
<mo stretchy="false">⇒</mo>
<mi>tan</mi>
<mo></mo>
<mi>x</mi>
<mo>=</mo>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>3</mn>
</msqrt>
</mrow>
</mfrac>
</math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow x = 150^\circ ">
<mo stretchy="false">⇒</mo>
<mi>x</mi>
<mo>=</mo>
<msup>
<mn>150</mn>
<mo>∘</mo>
</msup>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[5 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p>choosing two appropriate angles, for example 60° and 45° <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sin 105^\circ = \sin 60^\circ \cos 45^\circ + \cos 60^\circ \sin 45^\circ ">
<mi>sin</mi>
<mo></mo>
<msup>
<mn>105</mn>
<mo>∘</mo>
</msup>
<mo>=</mo>
<mi>sin</mi>
<mo></mo>
<msup>
<mn>60</mn>
<mo>∘</mo>
</msup>
<mi>cos</mi>
<mo></mo>
<msup>
<mn>45</mn>
<mo>∘</mo>
</msup>
<mo>+</mo>
<mi>cos</mi>
<mo></mo>
<msup>
<mn>60</mn>
<mo>∘</mo>
</msup>
<mi>sin</mi>
<mo></mo>
<msup>
<mn>45</mn>
<mo>∘</mo>
</msup>
</math></span> and</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\cos 105^\circ = \cos 60^\circ \cos 45^\circ - \sin 60^\circ \sin 45^\circ ">
<mi>cos</mi>
<mo></mo>
<msup>
<mn>105</mn>
<mo>∘</mo>
</msup>
<mo>=</mo>
<mi>cos</mi>
<mo></mo>
<msup>
<mn>60</mn>
<mo>∘</mo>
</msup>
<mi>cos</mi>
<mo></mo>
<msup>
<mn>45</mn>
<mo>∘</mo>
</msup>
<mo>−</mo>
<mi>sin</mi>
<mo></mo>
<msup>
<mn>60</mn>
<mo>∘</mo>
</msup>
<mi>sin</mi>
<mo></mo>
<msup>
<mn>45</mn>
<mo>∘</mo>
</msup>
</math></span> <strong><em>(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sin 105^\circ + \cos 105^\circ = \frac{{\sqrt 3 }}{2} \times \frac{1}{{\sqrt 2 }} + \frac{1}{2} \times \frac{1}{{\sqrt 2 }} + \frac{1}{2} \times \frac{1}{{\sqrt 2 }} - \frac{{\sqrt 3 }}{2} \times \frac{1}{{\sqrt 2 }}">
<mi>sin</mi>
<mo></mo>
<msup>
<mn>105</mn>
<mo>∘</mo>
</msup>
<mo>+</mo>
<mi>cos</mi>
<mo></mo>
<msup>
<mn>105</mn>
<mo>∘</mo>
</msup>
<mo>=</mo>
<mfrac>
<mrow>
<msqrt>
<mn>3</mn>
</msqrt>
</mrow>
<mn>2</mn>
</mfrac>
<mo>×</mo>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mo>×</mo>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mo>×</mo>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
<mo>−</mo>
<mfrac>
<mrow>
<msqrt>
<mn>3</mn>
</msqrt>
</mrow>
<mn>2</mn>
</mfrac>
<mo>×</mo>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{{\sqrt 2 }}">
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
</math></span> <strong><em>AG</em></strong></p>
<p><strong>OR</strong></p>
<p>attempt to square the expression <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{(\sin 105^\circ + \cos 105^\circ )^2} = {\sin ^2}105^\circ + 2\sin 105^\circ \cos 105^\circ + {\cos ^2}105^\circ ">
<mrow>
<mo stretchy="false">(</mo>
<mi>sin</mi>
<mo></mo>
<msup>
<mn>105</mn>
<mo>∘</mo>
</msup>
<mo>+</mo>
<mi>cos</mi>
<mo></mo>
<msup>
<mn>105</mn>
<mo>∘</mo>
</msup>
<msup>
<mo stretchy="false">)</mo>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mi>sin</mi>
<mn>2</mn>
</msup>
</mrow>
<msup>
<mn>105</mn>
<mo>∘</mo>
</msup>
<mo>+</mo>
<mn>2</mn>
<mi>sin</mi>
<mo></mo>
<msup>
<mn>105</mn>
<mo>∘</mo>
</msup>
<mi>cos</mi>
<mo></mo>
<msup>
<mn>105</mn>
<mo>∘</mo>
</msup>
<mo>+</mo>
<mrow>
<msup>
<mi>cos</mi>
<mn>2</mn>
</msup>
</mrow>
<msup>
<mn>105</mn>
<mo>∘</mo>
</msup>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{(\sin 105^\circ + \cos 105^\circ )^2} = 1 + \sin 210^\circ ">
<mrow>
<mo stretchy="false">(</mo>
<mi>sin</mi>
<mo></mo>
<msup>
<mn>105</mn>
<mo>∘</mo>
</msup>
<mo>+</mo>
<mi>cos</mi>
<mo></mo>
<msup>
<mn>105</mn>
<mo>∘</mo>
</msup>
<msup>
<mo stretchy="false">)</mo>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>1</mn>
<mo>+</mo>
<mi>sin</mi>
<mo></mo>
<msup>
<mn>210</mn>
<mo>∘</mo>
</msup>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{2}">
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sin 105^\circ + \cos 105^\circ = \frac{1}{{\sqrt 2 }}">
<mi>sin</mi>
<mo></mo>
<msup>
<mn>105</mn>
<mo>∘</mo>
</msup>
<mo>+</mo>
<mi>cos</mi>
<mo></mo>
<msup>
<mn>105</mn>
<mo>∘</mo>
</msup>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
</mfrac>
</math></span> <strong><em>AG</em></strong></p>
<p> </p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z = (1 - \cos 2\theta ) - {\text{i}}\sin 2\theta ">
<mi>z</mi>
<mo>=</mo>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>−</mo>
<mi>cos</mi>
<mo></mo>
<mn>2</mn>
<mi>θ</mi>
<mo stretchy="false">)</mo>
<mo>−</mo>
<mrow>
<mtext>i</mtext>
</mrow>
<mi>sin</mi>
<mo></mo>
<mn>2</mn>
<mi>θ</mi>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left| z \right| = \sqrt {{{(1 - \cos 2\theta )}^2} + {{(\sin 2\theta )}^2}} ">
<mrow>
<mo>|</mo>
<mi>z</mi>
<mo>|</mo>
</mrow>
<mo>=</mo>
<msqrt>
<mrow>
<msup>
<mrow>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>−</mo>
<mi>cos</mi>
<mo></mo>
<mn>2</mn>
<mi>θ</mi>
<mo stretchy="false">)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mrow>
<mo stretchy="false">(</mo>
<mi>sin</mi>
<mo></mo>
<mn>2</mn>
<mi>θ</mi>
<mo stretchy="false">)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
</math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left| z \right| = \sqrt {1 - 2\cos 2\theta + {{\cos }^2}2\theta + {{\sin }^2}2\theta } ">
<mrow>
<mo>|</mo>
<mi>z</mi>
<mo>|</mo>
</mrow>
<mo>=</mo>
<msqrt>
<mn>1</mn>
<mo>−</mo>
<mn>2</mn>
<mi>cos</mi>
<mo></mo>
<mn>2</mn>
<mi>θ</mi>
<mo>+</mo>
<mrow>
<msup>
<mrow>
<mi>cos</mi>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mn>2</mn>
<mi>θ</mi>
<mo>+</mo>
<mrow>
<msup>
<mrow>
<mi>sin</mi>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mn>2</mn>
<mi>θ</mi>
</msqrt>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \sqrt 2 \sqrt {(1 - \cos 2\theta )} ">
<mo>=</mo>
<msqrt>
<mn>2</mn>
</msqrt>
<msqrt>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>−</mo>
<mi>cos</mi>
<mo></mo>
<mn>2</mn>
<mi>θ</mi>
<mo stretchy="false">)</mo>
</msqrt>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \sqrt {2(2{{\sin }^2}\theta )} ">
<mo>=</mo>
<msqrt>
<mn>2</mn>
<mo stretchy="false">(</mo>
<mn>2</mn>
<mrow>
<msup>
<mrow>
<mi>sin</mi>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mi>θ</mi>
<mo stretchy="false">)</mo>
</msqrt>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 2\sin \theta ">
<mo>=</mo>
<mn>2</mn>
<mi>sin</mi>
<mo></mo>
<mi>θ</mi>
</math></span> <strong><em>A1</em></strong></p>
<p>let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\arg (z) = \alpha ">
<mi>arg</mi>
<mo></mo>
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>α</mi>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\tan \alpha = - \frac{{\sin 2\theta }}{{1 - \cos 2\theta }}">
<mi>tan</mi>
<mo></mo>
<mi>α</mi>
<mo>=</mo>
<mo>−</mo>
<mfrac>
<mrow>
<mi>sin</mi>
<mo></mo>
<mn>2</mn>
<mi>θ</mi>
</mrow>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>cos</mi>
<mo></mo>
<mn>2</mn>
<mi>θ</mi>
</mrow>
</mfrac>
</math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{ - 2\sin \theta \cos \theta }}{{2{{\sin }^2}\theta }}">
<mo>=</mo>
<mfrac>
<mrow>
<mo>−</mo>
<mn>2</mn>
<mi>sin</mi>
<mo></mo>
<mi>θ</mi>
<mi>cos</mi>
<mo></mo>
<mi>θ</mi>
</mrow>
<mrow>
<mn>2</mn>
<mrow>
<msup>
<mrow>
<mi>sin</mi>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mi>θ</mi>
</mrow>
</mfrac>
</math></span> <strong><em>(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = - \cot \theta ">
<mo>=</mo>
<mo>−</mo>
<mi>cot</mi>
<mo></mo>
<mi>θ</mi>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\arg (z) = \alpha = - \arctan \left( {\tan \left( {\frac{\pi }{2} - \theta } \right)} \right)">
<mi>arg</mi>
<mo></mo>
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>α</mi>
<mo>=</mo>
<mo>−</mo>
<mi>arctan</mi>
<mo></mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>tan</mi>
<mo></mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
<mo>−</mo>
<mi>θ</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \theta - \frac{\pi }{2}">
<mo>=</mo>
<mi>θ</mi>
<mo>−</mo>
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
</math></span> <strong><em>A1</em></strong></p>
<p><strong>OR</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z = (1 - \cos 2\theta ) - {\text{i}}\sin 2\theta ">
<mi>z</mi>
<mo>=</mo>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>−</mo>
<mi>cos</mi>
<mo></mo>
<mn>2</mn>
<mi>θ</mi>
<mo stretchy="false">)</mo>
<mo>−</mo>
<mrow>
<mtext>i</mtext>
</mrow>
<mi>sin</mi>
<mo></mo>
<mn>2</mn>
<mi>θ</mi>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 2{\sin ^2}\theta - 2{\text{i}}\sin \theta \cos \theta ">
<mo>=</mo>
<mn>2</mn>
<mrow>
<msup>
<mi>sin</mi>
<mn>2</mn>
</msup>
</mrow>
<mi>θ</mi>
<mo>−</mo>
<mn>2</mn>
<mrow>
<mtext>i</mtext>
</mrow>
<mi>sin</mi>
<mo></mo>
<mi>θ</mi>
<mi>cos</mi>
<mo></mo>
<mi>θ</mi>
</math></span> <strong><em>M1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 2\sin \theta (\sin \theta - {\text{i}}\cos \theta )">
<mo>=</mo>
<mn>2</mn>
<mi>sin</mi>
<mo></mo>
<mi>θ</mi>
<mo stretchy="false">(</mo>
<mi>sin</mi>
<mo></mo>
<mi>θ</mi>
<mo>−</mo>
<mrow>
<mtext>i</mtext>
</mrow>
<mi>cos</mi>
<mo></mo>
<mi>θ</mi>
<mo stretchy="false">)</mo>
</math></span> <strong><em>(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = - 2{\text{i}}\sin \theta (\cos \theta + {\text{i}}\sin \theta )">
<mo>=</mo>
<mo>−</mo>
<mn>2</mn>
<mrow>
<mtext>i</mtext>
</mrow>
<mi>sin</mi>
<mo></mo>
<mi>θ</mi>
<mo stretchy="false">(</mo>
<mi>cos</mi>
<mo></mo>
<mi>θ</mi>
<mo>+</mo>
<mrow>
<mtext>i</mtext>
</mrow>
<mi>sin</mi>
<mo></mo>
<mi>θ</mi>
<mo stretchy="false">)</mo>
</math></span> <strong><em>M1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 2\sin \theta \left( {\cos \left( {\theta - \frac{\pi }{2}} \right) + {\text{i}}\sin \left( {\theta - \frac{\pi }{2}} \right)} \right)">
<mo>=</mo>
<mn>2</mn>
<mi>sin</mi>
<mo></mo>
<mi>θ</mi>
<mrow>
<mo>(</mo>
<mrow>
<mi>cos</mi>
<mo></mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>θ</mi>
<mo>−</mo>
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mrow>
<mtext>i</mtext>
</mrow>
<mi>sin</mi>
<mo></mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>θ</mi>
<mo>−</mo>
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>M1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left| z \right| = 2\sin \theta ">
<mrow>
<mo>|</mo>
<mi>z</mi>
<mo>|</mo>
</mrow>
<mo>=</mo>
<mn>2</mn>
<mi>sin</mi>
<mo></mo>
<mi>θ</mi>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\arg (z) = \theta - \frac{\pi }{2}">
<mi>arg</mi>
<mo></mo>
<mo stretchy="false">(</mo>
<mi>z</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>θ</mi>
<mo>−</mo>
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[9 marks]</em></strong></p>
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>attempt to apply De Moivre’s theorem <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{(1 - \cos 2\theta - {\text{i}}\sin 2\theta )^{\frac{1}{3}}} = {2^{\frac{1}{3}}}{(\sin \theta )^{\frac{1}{3}}}\left[ {\cos \left( {\frac{{\theta - \frac{\pi }{2} + 2n\pi }}{3}} \right) + {\text{i}}\sin \left( {\frac{{\theta - \frac{\pi }{2} + 2n\pi }}{3}} \right)} \right]">
<mrow>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>−</mo>
<mi>cos</mi>
<mo></mo>
<mn>2</mn>
<mi>θ</mi>
<mo>−</mo>
<mrow>
<mtext>i</mtext>
</mrow>
<mi>sin</mi>
<mo></mo>
<mn>2</mn>
<mi>θ</mi>
<msup>
<mo stretchy="false">)</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
</mrow>
</msup>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mn>2</mn>
<mrow>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
</mrow>
</msup>
</mrow>
<mrow>
<mo stretchy="false">(</mo>
<mi>sin</mi>
<mo></mo>
<mi>θ</mi>
<msup>
<mo stretchy="false">)</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
</mrow>
</msup>
</mrow>
<mrow>
<mo>[</mo>
<mrow>
<mi>cos</mi>
<mo></mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mi>θ</mi>
<mo>−</mo>
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
<mo>+</mo>
<mn>2</mn>
<mi>n</mi>
<mi>π</mi>
</mrow>
<mn>3</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mrow>
<mtext>i</mtext>
</mrow>
<mi>sin</mi>
<mo></mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mi>θ</mi>
<mo>−</mo>
<mfrac>
<mi>π</mi>
<mn>2</mn>
</mfrac>
<mo>+</mo>
<mn>2</mn>
<mi>n</mi>
<mi>π</mi>
</mrow>
<mn>3</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>]</mo>
</mrow>
</math></span> <strong><em>A1A1A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> <strong><em>A1 </em></strong>for modulus, <strong><em>A1 </em></strong>for dividing argument of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z">
<mi>z</mi>
</math></span> by 3 and <strong><em>A1 </em></strong>for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2n\pi ">
<mn>2</mn>
<mi>n</mi>
<mi>π</mi>
</math></span>.</p>
<p> </p>
<p>Hence cube roots are the above expression when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="n = - 1,{\text{ }}0,{\text{ }}1">
<mi>n</mi>
<mo>=</mo>
<mo>−</mo>
<mn>1</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>0</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>1</mn>
</math></span>. Equivalent forms are acceptable. <strong><em>A1</em></strong></p>
<p><strong><em>[5 marks]</em></strong></p>
<div class="question_part_label">c.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.ii.</div>
</div>
<br><hr><br><div class="question">
<p>Find the solution of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\log _2}x - {\log _2}5 = 2 + {\log _2}3">
<mrow>
<msub>
<mi>log</mi>
<mn>2</mn>
</msub>
</mrow>
<mi>x</mi>
<mo>−</mo>
<mrow>
<msub>
<mi>log</mi>
<mn>2</mn>
</msub>
</mrow>
<mn>5</mn>
<mo>=</mo>
<mn>2</mn>
<mo>+</mo>
<mrow>
<msub>
<mi>log</mi>
<mn>2</mn>
</msub>
</mrow>
<mn>3</mn>
</math></span>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\log _2}x - {\log _2}5 = 2 + {\log _2}3">
<mrow>
<msub>
<mi>log</mi>
<mn>2</mn>
</msub>
</mrow>
<mi>x</mi>
<mo>−</mo>
<mrow>
<msub>
<mi>log</mi>
<mn>2</mn>
</msub>
</mrow>
<mn>5</mn>
<mo>=</mo>
<mn>2</mn>
<mo>+</mo>
<mrow>
<msub>
<mi>log</mi>
<mn>2</mn>
</msub>
</mrow>
<mn>3</mn>
</math></span></p>
<p>collecting at least two log terms <strong><em>(M1)</em></strong></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\log _2}\frac{x}{5} = 2 + {\log _2}3{\text{ or }}{\log _2}\frac{x}{{15}} = 2">
<mrow>
<msub>
<mi>log</mi>
<mn>2</mn>
</msub>
</mrow>
<mfrac>
<mi>x</mi>
<mn>5</mn>
</mfrac>
<mo>=</mo>
<mn>2</mn>
<mo>+</mo>
<mrow>
<msub>
<mi>log</mi>
<mn>2</mn>
</msub>
</mrow>
<mn>3</mn>
<mrow>
<mtext> or </mtext>
</mrow>
<mrow>
<msub>
<mi>log</mi>
<mn>2</mn>
</msub>
</mrow>
<mfrac>
<mi>x</mi>
<mrow>
<mn>15</mn>
</mrow>
</mfrac>
<mo>=</mo>
<mn>2</mn>
</math></span></p>
<p>obtaining a correct equation without logs <strong><em>(M1)</em></strong></p>
<p><em>eg</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{x}{5} = 12">
<mfrac>
<mi>x</mi>
<mn>5</mn>
</mfrac>
<mo>=</mo>
<mn>12</mn>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><strong>OR</strong><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,\,\,">
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
</math></span><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{x}{{15}} = {2^2}">
<mfrac>
<mi>x</mi>
<mrow>
<mn>15</mn>
</mrow>
</mfrac>
<mo>=</mo>
<mrow>
<msup>
<mn>2</mn>
<mn>2</mn>
</msup>
</mrow>
</math></span> <strong><em>(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 60">
<mi>x</mi>
<mo>=</mo>
<mn>60</mn>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[4 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent1" style="margin-top:12.0pt;">Write down the inverse of the matrix</p>
<p class="indent1" style="margin-top:12pt;text-align: center;"><strong><em>A</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 1&{ - 3}&1 \\ 2&2&{ - 1} \\ 1&{ - 5}&3 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>3</mn>
</mrow>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>5</mn>
</mrow>
</mtd>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent1" style="margin-top:12.0pt;"><strong>Hence</strong>, find the point of intersection of the three planes.</p>
<p class="indent1" style="margin-top:12pt;text-align: center;"><span class="mjpage mjpage__block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" alttext="\begin{array}{*{20}{c}} {x - 3y + z = 1} \\ {2x + 2y - z = 2} \\ {x - 5y + 3z = 3} \end{array}">
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mn>3</mn>
<mi>y</mi>
<mo>+</mo>
<mi>z</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
<mi>y</mi>
<mo>−</mo>
<mi>z</mi>
<mo>=</mo>
<mn>2</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mn>5</mn>
<mi>y</mi>
<mo>+</mo>
<mn>3</mn>
<mi>z</mi>
<mo>=</mo>
<mn>3</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</math></span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent1" style="margin-top:12.0pt;">A fourth plane with equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x + y + z = d">
<mi>x</mi>
<mo>+</mo>
<mi>y</mi>
<mo>+</mo>
<mi>z</mi>
<mo>=</mo>
<mi>d</mi>
</math></span> passes through the point of intersection. Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d">
<mi>d</mi>
</math></span><em>.</em></p>
<div class="marks">[1]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><strong><em>A</em></strong><sup>–1</sup> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} {0.1}&{0.4}&{0.1} \\ { - 0.7}&{0.2}&{0.3} \\ { - 1.2}&{0.2}&{0.8} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>0.1</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>0.4</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>0.1</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>0.7</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>0.2</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>0.3</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>1.2</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>0.2</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>0.8</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A2 N2</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>For attempting to calculate <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} x \\ y \\ z \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mi>x</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>y</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>z</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> = <em><strong>A</strong></em><sup>−1</sup><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 1 \\ 2 \\ 3 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 1.2{\text{,}}\,\,y = 0.6{\text{,}}\,\,z = 1.6">
<mi>x</mi>
<mo>=</mo>
<mn>1.2</mn>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>y</mi>
<mo>=</mo>
<mn>0.6</mn>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>z</mi>
<mo>=</mo>
<mn>1.6</mn>
</math></span> (so the point is (1.2, 0.6, 1.6)) <em><strong> A2 N2</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>(1.2, 0.6, 1.6) lies on <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x + y + z = d">
<mi>x</mi>
<mo>+</mo>
<mi>y</mi>
<mo>+</mo>
<mi>z</mi>
<mo>=</mo>
<mi>d</mi>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\therefore d = 3.4">
<mo>∴</mo>
<mi>d</mi>
<mo>=</mo>
<mn>3.4</mn>
</math></span> <em><strong> A1 N1</strong></em></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Let <strong><em>A</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 3&2 \\ k&4 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>3</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>k</mi>
</mtd>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> and <strong><em>B</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 2&2 \\ 1&3 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>. Find, in terms of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k">
<mi>k</mi>
</math></span>,</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent2">2<strong><em>A </em></strong>− <strong><em>B</em></strong>.</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="indent2">det (2<strong><em>A </em></strong>− <strong><em>B</em></strong>).</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p> </p>
<p><em>2<strong>A </strong></em>= <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 6&4 \\ {2k}&8 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>6</mn>
</mtd>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>2</mn>
<mi>k</mi>
</mrow>
</mtd>
<mtd>
<mn>8</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong> (A1)</strong></em></p>
<p>2<strong><em>A </em></strong>− <strong><em>B</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 4&2 \\ {2k - 1}&5 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>4</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>2</mn>
<mi>k</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
<mtd>
<mn>5</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A2 N3</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Evidence of using the definition of determinant <em><strong> (M1)</strong></em></p>
<p>Correct substitution <em><strong>(A1) </strong></em></p>
<p><em>eg</em> 4(5) − 2(2<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k">
<mi>k</mi>
</math></span> − 1), 20 − 2(2<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k">
<mi>k</mi>
</math></span> <span style="display: inline !important;float: none;background-color: #ffffff;color: #000000;font-family: Verdana,Arial,Helvetica,sans-serif;font-size: 14px;font-style: normal;font-variant: normal;font-weight: 400;letter-spacing: normal;text-align: left;text-decoration: none;text-indent: 0px;white-space: normal;">−</span> 1), 20 − 4<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k">
<mi>k</mi>
</math></span> + 2</p>
<p>det (2<strong><em>A </em></strong>− <strong><em>B</em></strong>) = 22 − 4<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k">
<mi>k</mi>
</math></span> <em><strong>A1 N3</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>The square matrix <strong><em>X</em></strong> is such that <strong><em>X</em></strong><sup>3</sup> = 0. Show that the inverse of the matrix (<strong><em>I</em> </strong>–<strong> <em>X</em></strong>) is <strong><em>I</em></strong> + <strong><em>X</em></strong> + <strong><em>X</em></strong><sup>2</sup>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p>For multiplying (<strong><em>I</em></strong> – <strong><em>X</em></strong>)(<strong><em>I</em></strong> + <strong><em>X</em></strong> + <strong><em>X</em></strong><sup>2</sup>) <em><strong>M1</strong></em></p>
<p>= <strong><em>I</em></strong><sup>2</sup> + <strong><em>IX</em></strong> + <strong><em>IX</em></strong><sup>2</sup> – <strong><em>XI </em></strong>– <strong><em>X</em></strong><sup>2</sup> – <strong><em>X</em></strong><sup>3</sup> = <strong><em>I</em> </strong>+ <strong><em>X</em></strong> + <strong><em>X</em></strong><sup>2</sup> – <strong><em>X</em></strong> – <strong><em>X</em></strong><sup>2</sup> – <strong><em>X</em></strong><sup>3</sup> <em><strong>(A1)(A1) </strong></em></p>
<p>= <strong><em>I</em></strong> – <strong><em>X</em></strong><sup>3</sup> <em><strong>A1 </strong></em></p>
<p>= <strong><em>I</em></strong> <em><strong>A1 </strong></em></p>
<p><strong><em>AB</em></strong> = <strong><em>I</em></strong> ⇒ <strong><em>A</em></strong><sup>–1</sup> = <strong><em>B</em></strong> <em><strong> (R1) </strong></em></p>
<p>(<strong><em>I</em></strong> – <strong><em>X</em></strong>)(<strong><em>I</em></strong> + <strong><em>X</em></strong> + <strong><em>X</em></strong><sup>2</sup>) = <strong><em>I</em></strong> ⇒ (<strong><em>I</em></strong> – <strong><em>X</em></strong>)<sup>–1</sup> = <strong><em>I</em></strong> + <strong><em>X</em></strong> + <strong><em>X</em></strong><sup>2</sup> <em><strong>AG N0</strong></em> </p>
<p><em><strong>[6 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>It is given that <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>1</mn></msub><mo>=</mo><mn>3</mn><mtext> cis</mtext><mfenced><mfrac><mrow><mn>3</mn><mi mathvariant="normal">π</mi></mrow><mn>4</mn></mfrac></mfenced></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>2</mn></msub><mo>=</mo><mn>2</mn><mtext> cis</mtext><mfenced><mfrac><mrow><mi>n</mi><mi mathvariant="normal">π</mi></mrow><mn>16</mn></mfrac></mfenced><mo>,</mo><mo> </mo><mi>n</mi><mo>∈</mo><msup><mi mathvariant="normal">ℤ</mi><mo>+</mo></msup></math>.</p>
</div>
<div class="specification">
<p>In parts (a)(i) and (a)(ii), give your answers in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><msup><mtext>e</mtext><mrow><mtext>i</mtext><mi>θ</mi></mrow></msup><mo>,</mo><mo> </mo><mi>r</mi><mo>≥</mo><mn>0</mn><mo>,</mo><mo> </mo><mo>−</mo><mi>π</mi><mo><</mo><mi>θ</mi><mo>≤</mo><mi>π</mi></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><msub><mi>z</mi><mn>1</mn></msub><mn>3</mn></msup></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mfrac><msub><mi>z</mi><mn>1</mn></msub><msub><mi>z</mi><mn>2</mn></msub></mfrac></mfenced><mn>4</mn></msup></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>2</mn></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the least value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>1</mn></msub><msub><mi>z</mi><mn>2</mn></msub><mo>∈</mo><msup><mi mathvariant="normal">ℝ</mi><mo>+</mo></msup></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><msub><mi>z</mi><mn>1</mn></msub><mn>3</mn></msup><mo>=</mo><mn>27</mn><msup><mtext>e</mtext><mfrac><mi>iπ</mi><mn>4</mn></mfrac></msup><mo> </mo><mfenced><mrow><mo>=</mo><mn>27</mn><msup><mtext>e</mtext><mrow><mn>0</mn><mo>.</mo><mn>785398</mn><mo>…</mo><mo> </mo><mtext>i</mtext></mrow></msup></mrow></mfenced></math> <em><strong>A1A1</strong></em><br><br><br><strong>Note:</strong> Award <em><strong>A1</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>27</mn></math> and <em><strong>A1</strong></em> for the angle in the correct form.</p>
<p><em><strong><br>[2 marks]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mfrac><msub><mi>z</mi><mn>1</mn></msub><msub><mi>z</mi><mn>2</mn></msub></mfrac></mfenced><mn>4</mn></msup><mo>=</mo><mfenced><mfrac><mn>81</mn><mn>16</mn></mfrac></mfenced><msup><mtext>e</mtext><mfrac><mi>iπ</mi><mn>2</mn></mfrac></msup><mo> </mo><mfenced><mrow><mo>=</mo><mn>5</mn><mo>.</mo><mn>0625</mn><msup><mtext>e</mtext><mrow><mn>1</mn><mo>.</mo><mn>57079</mn><mo>…</mo><mo> </mo><mtext>i</mtext></mrow></msup></mrow></mfenced></math> <em><strong>A1A2</strong></em><br><br><br><strong>Note:</strong> Award <em><strong>A1</strong></em> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>81</mn><mn>16</mn></mfrac></math>, <em><strong>A2</strong></em> for the angle in the correct form and <em><strong>A1</strong></em> for the angle in incorrect form e.g. <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>cis</mtext><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></math> and/or <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>5</mn><mi mathvariant="normal">π</mi></mrow><mn>2</mn></mfrac></math>. Award <em><strong>A1</strong></em> if <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>i</mtext></math> is given in place of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>cis</mtext><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></math>.</p>
<p><em><strong><br>[3 marks]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>z</mi><mn>1</mn></msub><msub><mi>z</mi><mn>2</mn></msub><mo>=</mo><mn>6</mn><mo> </mo><mtext>cis</mtext><mo> </mo><mfenced><mrow><mfrac><mrow><mn>3</mn><mi mathvariant="normal">π</mi></mrow><mn>4</mn></mfrac><mo>+</mo><mfrac><mrow><mi>n</mi><mi mathvariant="normal">π</mi></mrow><mn>16</mn></mfrac></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>6</mn><mo> </mo><mtext>cis</mtext><mo> </mo><mfenced><mfrac><mrow><mn>12</mn><mi mathvariant="normal">π</mi><mo>+</mo><mi>n</mi><mi mathvariant="normal">π</mi></mrow><mn>16</mn></mfrac></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>12</mn><mi mathvariant="normal">π</mi><mo>+</mo><mi>n</mi><mi mathvariant="normal">π</mi><mo>=</mo><mn>32</mn><mi mathvariant="normal">π</mi></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><mn>20</mn></math> <em><strong>A1</strong></em><br><br><em><strong><br>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>Given that<strong><em> A</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 3&{ - 2} \\ { - 3}&4 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>3</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>3</mn>
</mrow>
</mtd>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> and <strong><em>I</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 1&0 \\ 0&1 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, find the values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\lambda ">
<mi>λ</mi>
</math></span> for which (<strong><em>A</em> </strong>– <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\lambda ">
<mi>λ</mi>
</math></span><em><strong>I</strong></em>) is a singular matrix.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p>singular matrix ⇒ det = 0<strong><em> (R1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left| {\begin{array}{*{20}{c}} {3 - \lambda }&{ - 2} \\ { - 3}&{4 - \lambda } \end{array}} \right|">
<mrow>
<mo>|</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>3</mn>
<mo>−</mo>
<mi>λ</mi>
</mrow>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>3</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>4</mn>
<mo>−</mo>
<mi>λ</mi>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>|</mo>
</mrow>
</math></span><strong><em> (A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {3 - \lambda } \right)\left( {4 - \lambda } \right) - 6 = 0">
<mrow>
<mo>(</mo>
<mrow>
<mn>3</mn>
<mo>−</mo>
<mi>λ</mi>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>4</mn>
<mo>−</mo>
<mi>λ</mi>
</mrow>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mn>6</mn>
<mo>=</mo>
<mn>0</mn>
</math></span><strong><em> (M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow {\lambda ^2} - 7\lambda + 6 = 0">
<mo stretchy="false">⇒</mo>
<mrow>
<msup>
<mi>λ</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>7</mn>
<mi>λ</mi>
<mo>+</mo>
<mn>6</mn>
<mo>=</mo>
<mn>0</mn>
</math></span><strong><em> (A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\lambda = 1">
<mi>λ</mi>
<mo>=</mo>
<mn>1</mn>
</math></span> or 6 <strong><em>(A1)</em></strong><strong><em>(A1) (C6)</em></strong></p>
<p><strong>Note:</strong><em> </em>Award <em><strong>(C2)</strong></em> for one correct answer with no working.</p>
<p><em><strong>[6 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the inverse of the matrix <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 1&2&1 \\ 1&1&2 \\ 2&1&4 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><strong>Hence</strong> solve the system of equations</p>
<p><span class="mjpage mjpage__block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" alttext="x + 2y + z = 0">
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
<mi>y</mi>
<mo>+</mo>
<mi>z</mi>
<mo>=</mo>
<mn>0</mn>
</math></span></p>
<p><span class="mjpage mjpage__block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" alttext="x + y + 2z = 7">
<mi>x</mi>
<mo>+</mo>
<mi>y</mi>
<mo>+</mo>
<mn>2</mn>
<mi>z</mi>
<mo>=</mo>
<mn>7</mn>
</math></span></p>
<p><span class="mjpage mjpage__block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" alttext="2x + y + z = 17">
<mn>2</mn>
<mi>x</mi>
<mo>+</mo>
<mi>y</mi>
<mo>+</mo>
<mi>z</mi>
<mo>=</mo>
<mn>17</mn>
</math></span></p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p class="questionCharChar"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\left( {\begin{array}{*{20}{c}} 1&2&1 \\ 1&1&2 \\ 2&1&4 \end{array}} \right)^{ - 1}} = \left( {\begin{array}{*{20}{c}} 2&{ - 7}&3 \\ 0&2&{ - 1} \\ { - 1}&3&{ - 1} \end{array}} \right)">
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>7</mn>
</mrow>
</mtd>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
<mtd>
<mn>3</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A2 N2</strong></em></p>
<p class="indent1"><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p class="questionCharChar">In matrix form <em>A</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> = <em>B</em> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> = <em>A<sup>−</sup></em><sup>1</sup> <em>B</em> <em><strong>M1</strong></em></p>
<p class="questionCharChar"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 2">
<mi>x</mi>
<mo>=</mo>
<mn>2</mn>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = -3">
<mi>y</mi>
<mo>=</mo>
<mo>−</mo>
<mn>3</mn>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="z = 4">
<mi>z</mi>
<mo>=</mo>
<mn>4</mn>
</math></span> <em><strong> A1A1A1 N0</strong></em></p>
<p class="indent1"><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>Given that <strong><em>A</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 2&3 \\ 1&{ - 2} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> and <strong><em>B</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 2&0 \\ 0&{ - 3} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>3</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, find <strong><em>X</em></strong> if <strong><em>BX</em></strong> = <strong><em>A</em></strong> <em>–</em> <strong><em>AB</em></strong>.</p>
<p> </p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p class="questionCharChar"><strong>METHOD 1</strong></p>
<p><strong><em>A</em></strong> <em>–</em> <strong><em>AB</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 2&3 \\ 1&{ - 2} \end{array}} \right) - \left( {\begin{array}{*{20}{c}} 4&{ - 9} \\ 2&6 \end{array}} \right) = \left( {\begin{array}{*{20}{c}} { - 2}&{12} \\ { - 1}&{ - 8} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>4</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>9</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mn>6</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>12</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>8</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>(M1)(A1)</strong></em></p>
<p><strong><em>X</em></strong> = <strong><em>B</em></strong><sup>–1</sup>(<strong><em>A</em></strong> <em>–</em> <strong><em>AB</em></strong>) = <strong><em>B</em></strong><sup>–1 </sup><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} { - 2}&{12} \\ { - 1}&{ - 8} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>12</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>8</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = - \frac{1}{6}\left( {\begin{array}{*{20}{c}} { - 3}&0 \\ 0&2 \end{array}} \right)\left( {\begin{array}{*{20}{c}} { - 2}&{12} \\ { - 1}&{ - 8} \end{array}} \right)">
<mo>=</mo>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mn>6</mn>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>3</mn>
</mrow>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>12</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>8</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left( {\begin{array}{*{20}{c}} { - 1}&6 \\ {\frac{1}{3}}&{\frac{8}{3}} \end{array}} \right)">
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
<mtd>
<mn>6</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
</mrow>
</mtd>
<mtd>
<mrow>
<mfrac>
<mn>8</mn>
<mn>3</mn>
</mfrac>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>(A2) (C6)</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>Attempting to set up a matrix equation <em><strong>(M2)</strong></em></p>
<p><strong><em>X</em></strong> = <strong><em>B</em></strong><sup>–1</sup>(<strong><em>A</em></strong> <em>–</em> <strong><em>AB</em></strong>) <em><strong>(A2)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left( {\begin{array}{*{20}{c}} { - 1}&6 \\ {\frac{1}{3}}&{\frac{8}{3}} \end{array}} \right)">
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
<mtd>
<mn>6</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
</mrow>
</mtd>
<mtd>
<mrow>
<mfrac>
<mn>8</mn>
<mn>3</mn>
</mfrac>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> (from GDC) <em><strong>(A2) (C6)</strong></em></p>
<p> </p>
<p><em><strong>[6 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>The matrix <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">A</mi></math> is given by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">A</mi><mo>=</mo><mfenced open="[" close="]"><mtable><mtr><mtd><mi>a</mi></mtd><mtd><mi>b</mi></mtd></mtr><mtr><mtd><mi>c</mi></mtd><mtd><mi>d</mi></mtd></mtr></mtable></mfenced></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>By considering the determinant of a relevant matrix, show that the eigenvalues, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi></math>, of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">A</mi></math> satisfy the equation</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>λ</mi><mn>2</mn></msup><mo>-</mo><mi>α</mi><mi>λ</mi><mo>+</mo><mi>β</mi><mo>=</mo><mn>0</mn></math>,</p>
<p style="text-align: left;">where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>β</mi></math> are functions of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>,</mo><mo> </mo><mi>b</mi><mo>,</mo><mo> </mo><mi>c</mi><mo>,</mo><mo> </mo><mi>d</mi></math> to be determined.</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>Verify that</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi mathvariant="bold-italic">A</mi><mn>2</mn></msup><mo>-</mo><mi>α</mi><mi mathvariant="bold-italic">A</mi><mo>+</mo><mi>β</mi><mi mathvariant="bold-italic">I</mi><mo>=</mo></math> <em><strong>0</strong></em>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Assuming that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">A</mi></math> is non-singular, use the result in part (b)(i) to show that</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi mathvariant="bold-italic">A</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo>=</mo><mfrac><mn>1</mn><mi>β</mi></mfrac><mfenced><mrow><mi>α</mi><mi mathvariant="bold-italic">I</mi><mo>-</mo><mi mathvariant="bold-italic">A</mi></mrow></mfenced></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mtable><mtr><mtd><mi>a</mi><mo>-</mo><mi>λ</mi></mtd><mtd><mi>b</mi></mtd></mtr><mtr><mtd><mi>c</mi></mtd><mtd><mi>d</mi><mo>-</mo><mi>λ</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mn>0</mn></math> <em><strong>M</strong><strong>1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>a</mi><mo>-</mo><mi>λ</mi></mrow></mfenced><mfenced><mrow><mi>d</mi><mo>-</mo><mi>λ</mi></mrow></mfenced><mo>-</mo><mi>b</mi><mi>c</mi><mo>=</mo><mn>0</mn></math> <em><strong>M</strong><strong>1A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>λ</mi><mn>2</mn></msup><mo>-</mo><mfenced><mrow><mi>a</mi><mo>+</mo><mi>d</mi></mrow></mfenced><mi>λ</mi><mo>+</mo><mi>a</mi><mi>d</mi><mo>-</mo><mi>b</mi><mi>c</mi><mo>=</mo><mn>0</mn></math> <em><strong>A</strong><strong>1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>α</mi><mo>=</mo><mfenced><mrow><mi>a</mi><mo>+</mo><mi>d</mi></mrow></mfenced><mo>;</mo><mo> </mo><mi>β</mi><mo>=</mo><mi>a</mi><mi>d</mi><mo>-</mo><mi>b</mi><mi>c</mi></math></p>
<p><em><strong><br>[4 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi mathvariant="bold-italic">A</mi><mn>2</mn></msup><mo>=</mo><mfenced open="[" close="]"><mtable><mtr><mtd><mi>a</mi></mtd><mtd><mi>b</mi></mtd></mtr><mtr><mtd><mi>c</mi></mtd><mtd><mi>d</mi></mtd></mtr></mtable></mfenced><mfenced open="[" close="]"><mtable><mtr><mtd><mi>a</mi></mtd><mtd><mi>b</mi></mtd></mtr><mtr><mtd><mi>c</mi></mtd><mtd><mi>d</mi></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced open="[" close="]"><mtable><mtr><mtd><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><mi>b</mi><mi>c</mi></mtd><mtd><mi>a</mi><mi>b</mi><mo>+</mo><mi>b</mi><mi>d</mi></mtd></mtr><mtr><mtd><mi>a</mi><mi>c</mi><mo>+</mo><mi>c</mi><mi>d</mi></mtd><mtd><mi>b</mi><mi>c</mi><mo>+</mo><msup><mi>d</mi><mn>2</mn></msup></mtd></mtr></mtable></mfenced></math> <em><strong>(M</strong><strong>1)A1</strong></em></p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi mathvariant="bold-italic">A</mi><mn>2</mn></msup><mo>-</mo><mfenced><mrow><mi>a</mi><mo>+</mo><mi>d</mi></mrow></mfenced><mi mathvariant="bold-italic">A</mi><mo>+</mo><mfenced><mrow><mi>a</mi><mi>d</mi><mo>-</mo><mi>b</mi><mi>c</mi></mrow></mfenced><mi mathvariant="bold-italic">I</mi><mo>=</mo></math></p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="[" close="]"><mtable><mtr><mtd><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><mi>b</mi><mi>c</mi></mtd><mtd><mi>a</mi><mi>b</mi><mo>+</mo><mi>b</mi><mi>d</mi></mtd></mtr><mtr><mtd><mi>a</mi><mi>c</mi><mo>+</mo><mi>c</mi><mi>d</mi></mtd><mtd><mi>b</mi><mi>c</mi><mo>+</mo><msup><mi>d</mi><mn>2</mn></msup></mtd></mtr></mtable></mfenced><mo>-</mo><mfenced><mrow><mi>a</mi><mo>+</mo><mi>d</mi></mrow></mfenced><mfenced open="[" close="]"><mtable><mtr><mtd><mi>a</mi></mtd><mtd><mi>b</mi></mtd></mtr><mtr><mtd><mi>c</mi></mtd><mtd><mi>d</mi></mtd></mtr></mtable></mfenced><mo>+</mo><mfenced><mrow><mi>a</mi><mi>d</mi><mo>-</mo><mi>b</mi><mi>c</mi></mrow></mfenced><mfenced open="[" close="]"><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd></mtr></mtable></mfenced></math> <em><strong>M</strong><strong>1</strong></em></p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced open="[" close="]"><mtable><mtr><mtd><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><mi>b</mi><mi>c</mi><mo>-</mo><mi>a</mi><mfenced><mrow><mi>a</mi><mo>+</mo><mi>d</mi></mrow></mfenced><mo>+</mo><mi>a</mi><mi>d</mi><mo>-</mo><mi>b</mi><mi>c</mi></mtd><mtd><mi>a</mi><mi>b</mi><mo>+</mo><mi>b</mi><mi>d</mi><mo>-</mo><mi>b</mi><mfenced><mrow><mi>a</mi><mo>+</mo><mi>d</mi></mrow></mfenced></mtd></mtr><mtr><mtd><mi>a</mi><mi>c</mi><mo>+</mo><mi>c</mi><mi>d</mi><mo>-</mo><mi>c</mi><mfenced><mrow><mi>a</mi><mo>+</mo><mi>d</mi></mrow></mfenced></mtd><mtd><mi>b</mi><mi>c</mi><mo>+</mo><msup><mi>d</mi><mn>2</mn></msup><mo>-</mo><mi>d</mi><mfenced><mrow><mi>a</mi><mo>+</mo><mi>d</mi></mrow></mfenced><mo>+</mo><mi>a</mi><mi>d</mi><mo>-</mo><mi>b</mi><mi>c</mi></mtd></mtr></mtable></mfenced></math> <em><strong>A2</strong></em></p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo></math> <em><strong>0</strong></em> <em><strong>AG</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>A1A0</strong></em> for a single error.</p>
<p><em><strong><br>[5 marks]</strong></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>multiply throughout by <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi mathvariant="bold-italic">A</mi><mrow><mo>–</mo><mn>1</mn></mrow></msup></math> giving <em><strong>M</strong><strong>1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">A</mi><mo>-</mo><mi>α</mi><mi mathvariant="bold-italic">I</mi><mo>+</mo><mi>β</mi><msup><mi mathvariant="bold-italic">A</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo>=</mo></math> <em><strong>0 A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi mathvariant="bold-italic">A</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo>=</mo><mfrac><mn>1</mn><mi>β</mi></mfrac><mfenced><mrow><mi>α</mi><mi mathvariant="bold-italic">I</mi><mo>-</mo><mi mathvariant="bold-italic">A</mi></mrow></mfenced></math><em><strong> AG</strong></em></p>
<p><em><strong><br>[2 marks]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>Roger buys a new laptop for himself at a cost of <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>£</mo><mn>495</mn></math>. At the same time, he buys his daughter Chloe a higher specification laptop at a cost of <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>£</mo><mn>2200</mn></math>.</p>
<p>It is anticipated that Roger’s laptop will depreciate at a rate of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn><mo>%</mo></math> per year, whereas Chloe’s laptop will depreciate at a rate of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>15</mn><mo>%</mo></math> per year.</p>
</div>
<div class="specification">
<p>Roger and Chloe’s laptops will have the same value <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> years after they were purchased.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Estimate the value of Roger’s laptop after <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn></math> years.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Comment on the validity of your answer to part (b).</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>£</mo><mn>495</mn><mo>×</mo><mn>0</mn><mo>.</mo><msup><mn>9</mn><mn>5</mn></msup><mo>=</mo><mo>£</mo><mn>292</mn><mo> </mo><mo> </mo><mo>(</mo><mo>£</mo><mn>292</mn><mo>.</mo><mn>292</mn><mo>…</mo><mo>)</mo></math> <em><strong>(M1)A1</strong></em> </p>
<p><em><strong><br>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>£</mo><mn>495</mn><mo>×</mo><mn>0</mn><mo>.</mo><msup><mn>9</mn><mi>k</mi></msup><mo>=</mo><mn>2200</mn><mo>×</mo><mn>0</mn><mo>.</mo><msup><mn>85</mn><mi>k</mi></msup></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>=</mo><mn>26</mn><mo>.</mo><mn>1</mn><mo> </mo><mo> </mo><mfenced><mrow><mn>26</mn><mo>.</mo><mn>0968</mn><mo>…</mo></mrow></mfenced></math> <em><strong>A1</strong></em> </p>
<p><em><strong><br></strong></em><strong>Note:</strong> Award<em> <strong>M1A0 </strong></em>for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi><mo>-</mo><mn>1</mn></math> in place of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math>.<em><strong><br><br></strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>depreciation rates unlikely to be constant (especially over a long time period) <em><strong>R1</strong></em></p>
<p><br><strong>Note:</strong> Accept reasonable answers based on the magnitude of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> or the fact that “value” depends on factors other than time.</p>
<p><em><strong><br></strong></em><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Let <strong><em>A</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 1&x&{ - 1} \\ 3&1&4 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mi>x</mi>
</mtd>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>3</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> and <strong><em>B</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 3 \\ x \\ 2 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>x</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent2">Find <strong><em>AB</em></strong>.</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="indent2">The matrix <strong><em>C</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} {20} \\ {28} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>20</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>28</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> and 2<strong><em>AB</em></strong> = <strong><em>C</em></strong>. Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p>Attempting to multiply matrices <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 1&x&{ - 1} \\ 3&1&4 \end{array}} \right)\left( {\begin{array}{*{20}{c}} 3 \\ x \\ 2 \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {3 + {x^2} - 2} \\ {9 + x + 8} \end{array}} \right)\left( { = \left( {\begin{array}{*{20}{c}} {1 + {x^2}} \\ {17 + x} \end{array}} \right)} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mi>x</mi>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>3</mn>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>x</mi>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>3</mn>
<mo>+</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>9</mn>
<mo>+</mo>
<mi>x</mi>
<mo>+</mo>
<mn>8</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>17</mn>
<mo>+</mo>
<mi>x</mi>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong> A1A1 N3</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Setting up equation <em><strong>M1</strong></em></p>
<p><em>eg</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2\left( {\begin{array}{*{20}{c}} {1 + {x^2}} \\ {17 + x} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {20} \\ {28} \end{array}} \right)">
<mn>2</mn>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>17</mn>
<mo>+</mo>
<mi>x</mi>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>20</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>28</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} {2 + 2{x^2}} \\ {34 + 2x} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {20} \\ {28} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>2</mn>
<mo>+</mo>
<mn>2</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>34</mn>
<mo>+</mo>
<mn>2</mn>
<mi>x</mi>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>20</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>28</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} {1 + {x^2}} \\ {17 + x} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} {10} \\ {14} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>17</mn>
<mo>+</mo>
<mi>x</mi>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>10</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>14</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\begin{array}{*{20}{c}} {2 + 2{x^2} = 20} \\ {34 + 2x = 28} \end{array}\,\,\,\,\,\,\left( {\begin{array}{*{20}{c}} {1 + {x^2} = 10} \\ {17 + x = 14} \end{array}} \right)">
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>2</mn>
<mo>+</mo>
<mn>2</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>20</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>34</mn>
<mo>+</mo>
<mn>2</mn>
<mi>x</mi>
<mo>=</mo>
<mn>28</mn>
</mrow>
</mtd>
</mtr>
</mtable>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>10</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>17</mn>
<mo>+</mo>
<mi>x</mi>
<mo>=</mo>
<mn>14</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = - 3">
<mi>x</mi>
<mo>=</mo>
<mo>−</mo>
<mn>3</mn>
</math></span> <em><strong> A1 N2</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>A meteorologist models the height of a hot air balloon launched from the ground. The model assumes the balloon travels vertically upwards and travels <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>450</mn></math> metres in the first minute.</p>
<p>Due to the decrease in temperature as the balloon rises, the balloon will continually slow down. The model suggests that each minute the balloon will travel only <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>82</mn><mo>%</mo></math> of the distance travelled in the previous minute.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find how high the balloon will travel in the first <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn></math> minutes after it is launched.</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>The balloon is required to reach a height of at least <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2520</mn></math> metres.<br><br>Determine whether it will reach this height.</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>Suggest a limitation of the given model.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>recognition of geometric sequence <em>eg</em> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>82</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mn>10</mn></msub><mo>=</mo><mfrac><mrow><mn>450</mn><mfenced><mrow><mn>1</mn><mo>-</mo><mn>0</mn><mo>.</mo><msup><mn>82</mn><mn>10</mn></msup></mrow></mfenced></mrow><mrow><mn>1</mn><mo>-</mo><mn>0</mn><mo>.</mo><mn>82</mn></mrow></mfrac></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>2160</mn><mo> </mo><mtext>m</mtext><mo> </mo><mo> </mo><mfenced><mrow><mn>2156</mn><mo>.</mo><mn>37</mn><mo>…</mo></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><em><strong><br></strong></em><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>S</mi><mo>∞</mo></msub><mo>=</mo><mfrac><mn>450</mn><mrow><mn>1</mn><mo>-</mo><mn>0</mn><mo>.</mo><mn>82</mn></mrow></mfrac></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>2500</mn><mo><</mo><mn>2520</mn></math> so the balloon will not reach the required height. <em><strong>A1</strong></em></p>
<p><em><strong><br></strong></em><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>horizontal motion not taken into account,</p>
<p>rate of cooling will not likely be linear,</p>
<p>balloon is considered a point mass / size of balloon not considered,</p>
<p>effects of wind/weather unlikely to be consistent,</p>
<p>a discrete model has been used, whereas a continuous one may offer greater accuracy <em><strong>R1</strong></em></p>
<p> <br><strong>Note:</strong> Accept any other sensible answer.</p>
<p><em><strong><br></strong></em><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the matrix <strong><em>A</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 0&2 \\ a&{ - 1} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>a</mi>
</mtd>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent1" style="margin-top:12.0pt;">Find the matrix <strong><em>A</em></strong><sup>2</sup>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="indent1" style="margin-top:12.0pt;">If det <strong><em>A</em></strong><sup>2</sup> = 16, determine the possible values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span><em>.</em></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><strong><em>A</em></strong><sup>2</sup> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} {2a}&{ - 2} \\ { - a}&{2a + 1} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>2</mn>
<mi>a</mi>
</mrow>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mi>a</mi>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>2</mn>
<mi>a</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>(M1)A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>METHOD 1</strong> </p>
<p>det <strong><em>A</em></strong><sup>2</sup> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="4{a^2} + 2a - 2a = 4{a^2}">
<mn>4</mn>
<mrow>
<msup>
<mi>a</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>2</mn>
<mi>a</mi>
<mo>−</mo>
<mn>2</mn>
<mi>a</mi>
<mo>=</mo>
<mn>4</mn>
<mrow>
<msup>
<mi>a</mi>
<mn>2</mn>
</msup>
</mrow>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span> = ±2 <em><strong>A1A1 N2</strong></em> </p>
<p> </p>
<p><strong>METHOD 2</strong> </p>
<p>det <strong><em>A</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="- 2a">
<mo>−</mo>
<mn>2</mn>
<mi>a</mi>
</math></span> <em><strong>M1 </strong></em></p>
<p>det <strong><em>A</em></strong> = ±4</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span> = ±2 <em><strong>A1A1 N2</strong></em> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p class="indent1" style="margin-top:12.0pt;">Consider the matrix <strong><em>A </em></strong><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left( {\begin{array}{*{20}{c}} {{{\text{e}}^x}}&{{{\text{e}}^{ - x}}} \\ {2 + {{\text{e}}^x}}&1 \end{array}} \right)">
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mi>x</mi>
</msup>
</mrow>
</mrow>
</mtd>
<mtd>
<mrow>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mi>x</mi>
</mrow>
</msup>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>2</mn>
<mo>+</mo>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mi>x</mi>
</msup>
</mrow>
</mrow>
</mtd>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x \in \mathbb{R}">
<mi>x</mi>
<mo>∈</mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
</math></span>.</p>
<p class="question" style="margin-top:12.0pt;">Find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> for which <strong><em>A</em> </strong>is singular.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p>finding det<strong> <em>A </em></strong>= <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{{\text{e}}^x} - {{\text{e}}^{ - x}}\left( {2 + {{\text{e}}^x}} \right)">
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mi>x</mi>
</msup>
</mrow>
<mo>−</mo>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mi>x</mi>
</mrow>
</msup>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mo>+</mo>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mi>x</mi>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span> or equivalent <em><strong>A1</strong></em></p>
<p><strong><em>A</em></strong> is singular ⇒ det <strong><em>A</em></strong> = 0 <em><strong>(R1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{{\text{e}}^x} - {{\text{e}}^{ - x}}\left( {2 + {{\text{e}}^x}} \right) = 0">
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mi>x</mi>
</msup>
</mrow>
<mo>−</mo>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mi>x</mi>
</mrow>
</msup>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mo>+</mo>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mi>x</mi>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{{\text{e}}^{2x}} - {{\text{e}}^x} - 2 = 0">
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mn>2</mn>
<mi>x</mi>
</mrow>
</msup>
</mrow>
<mo>−</mo>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mi>x</mi>
</msup>
</mrow>
<mo>−</mo>
<mn>2</mn>
<mo>=</mo>
<mn>0</mn>
</math></span> <em><strong>A1</strong></em></p>
<p>solving for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{{\text{e}}^x}">
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mi>x</mi>
</msup>
</mrow>
</math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{{\text{e}}^x}">
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mi>x</mi>
</msup>
</mrow>
</math></span> > 0 (or equivalent explanation) <em><strong>(R1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{{\text{e}}^x} = 2">
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mi>x</mi>
</msup>
</mrow>
<mo>=</mo>
<mn>2</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = ">
<mi>x</mi>
<mo>=</mo>
</math></span> ln 2 (only) <em><strong>A1 N0</strong></em></p>
<p><em><strong>[6 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="question">
<p>The matrices <strong><em>A</em></strong>, <strong><em>B</em></strong>, <strong><em>C</em></strong> and <strong><em>X</em></strong> are all non-singular 3 × 3 matrices.</p>
<p>Given that <strong><em>A</em></strong><sup><em>–</em>1</sup><strong><em>XB</em></strong> = <strong><em>C</em></strong>, express <strong><em>X</em></strong> in terms of the other matrices.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><strong><em>AA</em></strong><sup>–1</sup><strong><em>XB</em> </strong>= <strong><em>ΑC</em></strong> <strong><em>(M1)(A1) </em></strong></p>
<p><strong><em>IXBB</em></strong><sup>–1</sup><strong> </strong>= <strong><em>ACB</em></strong><sup>–1</sup> <strong><em>(M1)(A1) </em></strong></p>
<p><strong><em>X</em> <em>=</em> <em>ACB</em></strong><sup>–1</sup> <em><strong>(M1)(A1) (C6)</strong></em></p>
<p><em><strong>[6 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="question">
<p class="indent1" style="margin-top:12.0pt;">If <strong><em>A</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 1&2 \\ k&{ - 1} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>k</mi>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> and <strong><em>A</em></strong><sup>2</sup> is a matrix whose entries are all 0, find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k">
<mi>k</mi>
</math></span>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><strong><em>A</em></strong><sup>2</sup> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 1&2 \\ k&{ - 1} \end{array}} \right)\left( {\begin{array}{*{20}{c}} 1&2 \\ k&{ - 1} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>k</mi>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>1</mn>
</mtd>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mi>k</mi>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left( {\begin{array}{*{20}{c}} {1 + 2k}&0 \\ 0&{2k + 1} \end{array}} \right)">
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mn>2</mn>
<mi>k</mi>
</mrow>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mrow>
<mn>2</mn>
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A2</strong></em></p>
<p class="question"><strong>Note:</strong> Award <em><strong>A2</strong> </em>for 4 correct, <em><strong>A1</strong> </em>for 2 or 3 correct.</p>
<p class="question"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1 + 2k = 0">
<mn>1</mn>
<mo>+</mo>
<mn>2</mn>
<mi>k</mi>
<mo>=</mo>
<mn>0</mn>
</math></span> <em><strong>M1</strong></em></p>
<p class="question"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k = - \frac{1}{2}">
<mi>k</mi>
<mo>=</mo>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[5 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="question">
<p class="indent1" style="margin-top:12.0pt;">Given that <strong><em>M</em></strong> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 2&{ - 1} \\ { - 3}&4 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>3</mn>
</mrow>
</mtd>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> and that <strong><em>M</em></strong><sup>2</sup> <em>–</em> 6<strong><em>M</em></strong> <em>+</em> <em>k<strong>I</strong></em> = 0 find <em>k</em>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><strong><em>M</em></strong><sup>2</sup> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 2&{ - 1} \\ { - 3}&4 \end{array}} \right)\left( {\begin{array}{*{20}{c}} 2&{ - 1} \\ { - 3}&4 \end{array}} \right) = \left( {\begin{array}{*{20}{c}} 7&{ - 6} \\ { - 18}&{19} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>3</mn>
</mrow>
</mtd>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>2</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>3</mn>
</mrow>
</mtd>
<mtd>
<mn>4</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>7</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>6</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>18</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>19</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>M1A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow \left( {\begin{array}{*{20}{c}} 7&{ - 6} \\ { - 18}&{19} \end{array}} \right) - \left( {\begin{array}{*{20}{c}} {12}&{ - 6} \\ { - 18}&{24} \end{array}} \right)">
<mo stretchy="false">⇒</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>7</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>6</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>18</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>19</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>12</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>6</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>18</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>24</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em>+</em> <em>k<strong>I</strong></em> = 0 <em><strong>(M1)</strong></em></p>
<p class="question"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow \left( {\begin{array}{*{20}{c}} { - 5}&0 \\ 0&{ - 5} \end{array}} \right)">
<mo stretchy="false">⇒</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>5</mn>
</mrow>
</mtd>
<mtd>
<mn>0</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>0</mn>
</mtd>
<mtd>
<mrow>
<mo>−</mo>
<mn>5</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em>+</em> <em>k<strong>I</strong></em> = 0 <em><strong>(A1)</strong></em></p>
<p class="question">⇒ <em>k</em> = 5 <em><strong>A1</strong></em></p>
<p><em><strong>[5 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="question">
<p>In the following Argand diagram the point A represents the complex number <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 1 + 4{\text{i}}">
<mo>−</mo>
<mn>1</mn>
<mo>+</mo>
<mn>4</mn>
<mrow>
<mtext>i</mtext>
</mrow>
</math></span> and the point B represents the complex number <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - 3 + 0{\text{i}}">
<mo>−</mo>
<mn>3</mn>
<mo>+</mo>
<mn>0</mn>
<mrow>
<mtext>i</mtext>
</mrow>
</math></span>. The shape of ABCD is a square. Determine the complex numbers represented by the points C and D.</p>
<p><img src="images/Schermafbeelding_2017-08-09_om_06.11.20.png" alt="M17/5/MATHL/HP1/ENG/TZ2/05"></p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p>C represents the complex number <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="1 - 2{\text{i}}">
<mn>1</mn>
<mo>−</mo>
<mn>2</mn>
<mrow>
<mtext>i</mtext>
</mrow>
</math></span> <strong><em>A2</em></strong></p>
<p>D represents the complex number <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3 + 2{\text{i}}">
<mn>3</mn>
<mo>+</mo>
<mn>2</mn>
<mrow>
<mtext>i</mtext>
</mrow>
</math></span> <strong><em>A2</em></strong></p>
<p><strong><em>[4 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="question">
<p class="indent1" style="margin-top:12pt;text-align: left;">Find the values of the real number <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k">
<mi>k</mi>
</math></span> for which the determinant of the matrix <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} {k - 4}&3 \\ { - 2}&{k + 1} \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mi>k</mi>
<mo>−</mo>
<mn>4</mn>
</mrow>
</mtd>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> is equal to zero.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left| {\begin{array}{*{20}{c}} {k - 4}&3 \\ { - 2}&{k + 1} \end{array}} \right| = 0">
<mrow>
<mo>|</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mi>k</mi>
<mo>−</mo>
<mn>4</mn>
</mrow>
</mtd>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>2</mn>
</mrow>
</mtd>
<mtd>
<mrow>
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>|</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow \left( {k - 4} \right)\left( {k + 1} \right) + 6 = 0">
<mo stretchy="false">⇒</mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>k</mi>
<mo>−</mo>
<mn>4</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>k</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mn>6</mn>
<mo>=</mo>
<mn>0</mn>
</math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow {k^2} - 3k + 2 = 0">
<mo stretchy="false">⇒</mo>
<mrow>
<msup>
<mi>k</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>3</mn>
<mi>k</mi>
<mo>+</mo>
<mn>2</mn>
<mo>=</mo>
<mn>0</mn>
</math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow \left( {k - 2} \right)\left( {k - 1} \right) = 0">
<mo stretchy="false">⇒</mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>k</mi>
<mo>−</mo>
<mn>2</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>k</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow k = 2">
<mo stretchy="false">⇒</mo>
<mi>k</mi>
<mo>=</mo>
<mn>2</mn>
</math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k = 1">
<mi>k</mi>
<mo>=</mo>
<mn>1</mn>
</math></span> <em><strong>(A1) (C3)</strong></em></p>
<p><em><strong>[3 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="question">
<p>If <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A = \left( {\begin{array}{*{20}{c}} {2p}&3 \\ { - 4p}&p \end{array}} \right)">
<mi>A</mi>
<mo>=</mo>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mrow>
<mn>2</mn>
<mi>p</mi>
</mrow>
</mtd>
<mtd>
<mn>3</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>−</mo>
<mn>4</mn>
<mi>p</mi>
</mrow>
</mtd>
<mtd>
<mi>p</mi>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> and det <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A = 14">
<mi>A</mi>
<mo>=</mo>
<mn>14</mn>
</math></span>, find the possible values of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
<mi>p</mi>
</math></span>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2{p^2} + 12p = 14">
<mn>2</mn>
<mrow>
<msup>
<mi>p</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>12</mn>
<mi>p</mi>
<mo>=</mo>
<mn>14</mn>
</math></span> <em><strong>(M1)</strong></em><em><strong>(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{p^2} + 6p - 7 = 0">
<mrow>
<msup>
<mi>p</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>6</mn>
<mi>p</mi>
<mo>−</mo>
<mn>7</mn>
<mo>=</mo>
<mn>0</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {p + 7} \right)\left( {p - 1} \right) = 0">
<mrow>
<mo>(</mo>
<mrow>
<mi>p</mi>
<mo>+</mo>
<mn>7</mn>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>p</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span> <em><strong>(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p = - 7">
<mi>p</mi>
<mo>=</mo>
<mo>−</mo>
<mn>7</mn>
</math></span> or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p = 1">
<mi>p</mi>
<mo>=</mo>
<mn>1</mn>
</math></span> <em><strong>(A1) (C4)</strong></em></p>
<p><strong>Note: </strong>Both answers are required for the final <em><strong>(A1)</strong></em>.</p>
<p><em><strong>[4 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="question">
<p>Solve the equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{4^x} + {2^{x + 2}} = 3">
<mrow>
<msup>
<mn>4</mn>
<mi>x</mi>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mn>2</mn>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
</mrow>
</msup>
</mrow>
<mo>=</mo>
<mn>3</mn>
</math></span>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p>attempt to form a quadratic in <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{2^x}">
<mrow>
<msup>
<mn>2</mn>
<mi>x</mi>
</msup>
</mrow>
</math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{({2^x})^2} + 4 \bullet {2^x} - 3 = 0">
<mrow>
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mn>2</mn>
<mi>x</mi>
</msup>
</mrow>
<msup>
<mo stretchy="false">)</mo>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>4</mn>
<mo>∙</mo>
<mrow>
<msup>
<mn>2</mn>
<mi>x</mi>
</msup>
</mrow>
<mo>−</mo>
<mn>3</mn>
<mo>=</mo>
<mn>0</mn>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{2^x} = \frac{{ - 4 \pm \sqrt {16 + 12} }}{2}{\text{ }}\left( { = - 2 \pm \sqrt 7 } \right)">
<mrow>
<msup>
<mn>2</mn>
<mi>x</mi>
</msup>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mo>−</mo>
<mn>4</mn>
<mo>±</mo>
<msqrt>
<mn>16</mn>
<mo>+</mo>
<mn>12</mn>
</msqrt>
</mrow>
<mn>2</mn>
</mfrac>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mo>−</mo>
<mn>2</mn>
<mo>±</mo>
<msqrt>
<mn>7</mn>
</msqrt>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{2^x} = - 2 + \sqrt 7 {\text{ }}\left( {{\text{as }} - 2 - \sqrt 7 < 0} \right)">
<mrow>
<msup>
<mn>2</mn>
<mi>x</mi>
</msup>
</mrow>
<mo>=</mo>
<mo>−</mo>
<mn>2</mn>
<mo>+</mo>
<msqrt>
<mn>7</mn>
</msqrt>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mtext>as </mtext>
</mrow>
<mo>−</mo>
<mn>2</mn>
<mo>−</mo>
<msqrt>
<mn>7</mn>
</msqrt>
<mo><</mo>
<mn>0</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>R1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = {\log _2}\left( { - 2 + \sqrt 7 } \right){\text{ }}\left( {x = \frac{{\ln \left( { - 2 + \sqrt 7 } \right)}}{{\ln 2}}} \right)">
<mi>x</mi>
<mo>=</mo>
<mrow>
<msub>
<mi>log</mi>
<mn>2</mn>
</msub>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mo>−</mo>
<mn>2</mn>
<mo>+</mo>
<msqrt>
<mn>7</mn>
</msqrt>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mi>ln</mi>
<mo></mo>
<mrow>
<mo>(</mo>
<mrow>
<mo>−</mo>
<mn>2</mn>
<mo>+</mo>
<msqrt>
<mn>7</mn>
</msqrt>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mi>ln</mi>
<mo></mo>
<mn>2</mn>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>R0 A1 </em></strong>if final answer is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = {\log _2}\left( { - 2 + \sqrt 7 } \right)">
<mi>x</mi>
<mo>=</mo>
<mrow>
<msub>
<mi>log</mi>
<mn>2</mn>
</msub>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mo>−</mo>
<mn>2</mn>
<mo>+</mo>
<msqrt>
<mn>7</mn>
</msqrt>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
<p> </p>
<p><strong><em>[5 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="question">
<p>Solve the equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\log _2}(x + 3) + {\log _2}(x - 3) = 4">
<mrow>
<msub>
<mi>log</mi>
<mn>2</mn>
</msub>
</mrow>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>+</mo>
<mn>3</mn>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mrow>
<msub>
<mi>log</mi>
<mn>2</mn>
</msub>
</mrow>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>−</mo>
<mn>3</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>4</mn>
</math></span>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p style="color: #999;font-size: 90%;font-style: italic;">* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.</p><p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\log _2}(x + 3) + {\log _2}(x - 3) = 4">
<mrow>
<msub>
<mi>log</mi>
<mn>2</mn>
</msub>
</mrow>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>+</mo>
<mn>3</mn>
<mo stretchy="false">)</mo>
<mo>+</mo>
<mrow>
<msub>
<mi>log</mi>
<mn>2</mn>
</msub>
</mrow>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>−</mo>
<mn>3</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>4</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\log _2}({x^2} - 9) = 4">
<mrow>
<msub>
<mi>log</mi>
<mn>2</mn>
</msub>
</mrow>
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>9</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>4</mn>
</math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^2} - 9 = {2^4}{\text{ }}( = 16)">
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>9</mn>
<mo>=</mo>
<mrow>
<msup>
<mn>2</mn>
<mn>4</mn>
</msup>
</mrow>
<mrow>
<mtext> </mtext>
</mrow>
<mo stretchy="false">(</mo>
<mo>=</mo>
<mn>16</mn>
<mo stretchy="false">)</mo>
</math></span> <strong><em>M1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^2} = 25">
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>25</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \pm 5">
<mi>x</mi>
<mo>=</mo>
<mo>±</mo>
<mn>5</mn>
</math></span> <strong><em>(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 5">
<mi>x</mi>
<mo>=</mo>
<mn>5</mn>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[5 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="question">
<p>The rate, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A">
<mi>A</mi>
</math></span>, of a chemical reaction at a fixed temperature is related to the concentration of two compounds, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="B">
<mi>B</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="C">
<mi>C</mi>
</math></span>, by the equation</p>
<p style="text-align: center;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A = k{B^x}{C^y}">
<mi>A</mi>
<mo>=</mo>
<mi>k</mi>
<mrow>
<msup>
<mi>B</mi>
<mi>x</mi>
</msup>
</mrow>
<mrow>
<msup>
<mi>C</mi>
<mi>y</mi>
</msup>
</mrow>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k \in \mathbb{R}">
<mi>k</mi>
<mo>∈</mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
</math></span>.</p>
<p>A scientist measures the three variables three times during the reaction and obtains the following values.</p>
<p><img style="display: block;margin-left:auto;margin-right:auto;" src="data:image/png;base64,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"></p>
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k">
<mi>k</mi>
</math></span>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{log}}\,A = x\,{\text{log}}\,B + y\,{\text{log}}\,C + {\text{log}}\,k">
<mrow>
<mtext>log</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>A</mi>
<mo>=</mo>
<mi>x</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>log</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>B</mi>
<mo>+</mo>
<mi>y</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>log</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>C</mi>
<mo>+</mo>
<mrow>
<mtext>log</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>k</mi>
</math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{log}}\,5.74 = x\,{\text{log}}\,2.1 + y\,{\text{log}}\,3.4 + {\text{log}}\,k">
<mrow>
<mtext>log</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>5.74</mn>
<mo>=</mo>
<mi>x</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>log</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2.1</mn>
<mo>+</mo>
<mi>y</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>log</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>3.4</mn>
<mo>+</mo>
<mrow>
<mtext>log</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>k</mi>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{log}}\,2.88 = x\,{\text{log}}\,1.5 + y\,{\text{log}}\,2.4 + {\text{log}}\,k">
<mrow>
<mtext>log</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2.88</mn>
<mo>=</mo>
<mi>x</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>log</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>1.5</mn>
<mo>+</mo>
<mi>y</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>log</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>2.4</mn>
<mo>+</mo>
<mrow>
<mtext>log</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>k</mi>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{log}}\,0.980 = x\,{\text{log}}\,0.8 + y\,{\text{log}}\,1.9 + {\text{log}}\,k">
<mrow>
<mtext>log</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>0.980</mn>
<mo>=</mo>
<mi>x</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>log</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>0.8</mn>
<mo>+</mo>
<mi>y</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>log</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mn>1.9</mn>
<mo>+</mo>
<mrow>
<mtext>log</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>k</mi>
</math></span> <em><strong>M1A1</strong></em></p>
<p><strong>Note:</strong> Allow any consistent base, allow numerical equivalents.</p>
<p>attempting to solve their system of equations <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> = 1.53, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span> = 0.505 <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="k">
<mi>k</mi>
</math></span> = 0.997 <em><strong>A1</strong></em></p>
<p><em><strong>[6 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br>