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<h2>HL Paper 1</h2><div class="specification">
<p>The following diagram shows a frame that is made from wire. The total length of wire is equal to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>15</mn><mo> </mo><mtext>cm</mtext></math>. The frame is made up of two identical sectors of a circle that are parallel to each other. The sectors have angle <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi></math> radians and radius <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo> </mo><mtext>cm</mtext></math>. They are connected by <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo> </mo><mtext>cm</mtext></math> lengths of wire perpendicular to the sectors. This is shown in the diagram below.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<div class="specification">
<p>The faces of the frame are covered by paper to enclose a volume, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mfrac><mn>6</mn><mrow><mn>2</mn><mo>+</mo><mi>θ</mi></mrow></mfrac></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 an expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi></math> in terms 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>Find the expression <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>V</mi></mrow><mrow><mo>d</mo><mi>θ</mi></mrow></mfrac></math>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Solve algebraically <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>V</mi></mrow><mrow><mo>d</mo><mi>θ</mi></mrow></mfrac><mo>=</mo><mn>0</mn></math> to find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi></math> that will maximize the volume, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.iii.</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>15</mn><mo>=</mo><mn>3</mn><mo>+</mo><mn>4</mn><mi>r</mi><mo>+</mo><mn>2</mn><mi>r</mi><mi>θ</mi></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>12</mn><mo>=</mo><mn>2</mn><mi>r</mi><mfenced><mrow><mn>2</mn><mo>+</mo><mi>θ</mi></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Award <em><strong>A1</strong></em> for any reasonable working leading to expected result e,g, factorizing <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math>.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mfrac><mn>6</mn><mrow><mn>2</mn><mo>+</mo><mi>θ</mi></mrow></mfrac></math> <em><strong>AG</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>attempt to use sector area to find volume <em><strong>(M1)</strong></em></p>
<p>volume <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi>r</mi><mn>2</mn></msup><mi>θ</mi><mo>×</mo><mn>1</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>×</mo><mfrac><mn>36</mn><msup><mfenced><mrow><mn>2</mn><mo>+</mo><mi>θ</mi></mrow></mfenced><mn>2</mn></msup></mfrac><mo>×</mo><mi>θ</mi><mo> </mo><mo> </mo><mo> </mo><mfenced><mrow><mo>=</mo><mfrac><mrow><mn>18</mn><mi>θ</mi></mrow><msup><mfenced><mrow><mn>2</mn><mo>+</mo><mi>θ</mi></mrow></mfenced><mn>2</mn></msup></mfrac></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.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>V</mi></mrow><mrow><mo>d</mo><mi>θ</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><msup><mfenced><mrow><mn>2</mn><mo>+</mo><mi>θ</mi></mrow></mfenced><mn>2</mn></msup><mo>×</mo><mn>18</mn><mo>-</mo><mn>36</mn><mi>θ</mi><mfenced><mrow><mn>2</mn><mo>+</mo><mi>θ</mi></mrow></mfenced></mrow><msup><mfenced><mrow><mn>2</mn><mo>+</mo><mi>θ</mi></mrow></mfenced><mn>4</mn></msup></mfrac></math> <em><strong>M1A1A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>V</mi></mrow><mrow><mo>d</mo><mi>θ</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>36</mn><mo>-</mo><mn>18</mn><mi>θ</mi></mrow><msup><mfenced><mrow><mn>2</mn><mo>+</mo><mi>θ</mi></mrow></mfenced><mn>3</mn></msup></mfrac></math></p>
<p> </p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>V</mi></mrow><mrow><mo>d</mo><mi>θ</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>36</mn><mo>-</mo><mn>18</mn><mi>θ</mi></mrow><msup><mfenced><mrow><mn>2</mn><mo>+</mo><mi>θ</mi></mrow></mfenced><mn>3</mn></msup></mfrac><mo>=</mo><mn>0</mn></math> <em><strong>M1</strong></em></p>
<p><br><strong>Note:</strong> Award this <em><strong>M1</strong></em> for simplified version equated to zero. The simplified version may have been seen in part (b)(ii).</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi><mo>=</mo><mn>2</mn></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.iii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>Several candidates missed that the angle <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi></math> was in radians and used arc and sector formulas with degrees instead. This aside, part (a) was often well done. Part (b)(i) was also correctly answered by many candidates, but their failure to make any attempt to simplify their answer often led to difficulties in part (b)(ii). Again, failing to simplify the result in part (b)(ii) led to yet more difficulties in part (b)(iii). Some candidates used the product rule to differentiate <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>18</mn><mi>θ</mi></mrow><msup><mfenced><mrow><mn>2</mn><mo>+</mo><mi>θ</mi></mrow></mfenced><mn>2</mn></msup></mfrac></math> as <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>18</mn><mi>θ</mi><msup><mfenced><mrow><mn>2</mn><mo>+</mo><mi>θ</mi></mrow></mfenced><mrow><mo>-</mo><mn>2</mn></mrow></msup></math> rather than the quotient rule. This was fine but made solving the equation in (b)(iii) less straightforward.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Several candidates missed that the angle <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi></math> was in radians and used arc and sector formulas with degrees instead. This aside, part (a) was often well done. Part (b)(i) was also correctly answered by many candidates, but their failure to make any attempt to simplify their answer often led to difficulties in part (b)(ii). Again, failing to simplify the result in part (b)(ii) led to yet more difficulties in part (b)(iii). Some candidates used the product rule to differentiate <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>18</mn><mi>θ</mi></mrow><msup><mfenced><mrow><mn>2</mn><mo>+</mo><mi>θ</mi></mrow></mfenced><mn>2</mn></msup></mfrac></math> as <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>18</mn><mi>θ</mi><msup><mfenced><mrow><mn>2</mn><mo>+</mo><mi>θ</mi></mrow></mfenced><mrow><mo>-</mo><mn>2</mn></mrow></msup></math> rather than the quotient rule. This was fine but made solving the equation in (b)(iii) less straightforward.</p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Several candidates missed that the angle <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi></math> was in radians and used arc and sector formulas with degrees instead. This aside, part (a) was often well done. Part (b)(i) was also correctly answered by many candidates, but their failure to make any attempt to simplify their answer often led to difficulties in part (b)(ii). Again, failing to simplify the result in part (b)(ii) led to yet more difficulties in part (b)(iii). Some candidates used the product rule to differentiate <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>18</mn><mi>θ</mi></mrow><msup><mfenced><mrow><mn>2</mn><mo>+</mo><mi>θ</mi></mrow></mfenced><mn>2</mn></msup></mfrac></math> as <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>18</mn><mi>θ</mi><msup><mfenced><mrow><mn>2</mn><mo>+</mo><mi>θ</mi></mrow></mfenced><mrow><mo>-</mo><mn>2</mn></mrow></msup></math> rather than the quotient rule. This was fine but made solving the equation in (b)(iii) less straightforward.</p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Several candidates missed that the angle <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi></math> was in radians and used arc and sector formulas with degrees instead. This aside, part (a) was often well done. Part (b)(i) was also correctly answered by many candidates, but their failure to make any attempt to simplify their answer often led to difficulties in part (b)(ii). Again, failing to simplify the result in part (b)(ii) led to yet more difficulties in part (b)(iii). Some candidates used the product rule to differentiate <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mn>18</mn><mi>θ</mi></mrow><msup><mfenced><mrow><mn>2</mn><mo>+</mo><mi>θ</mi></mrow></mfenced><mn>2</mn></msup></mfrac></math> as <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>18</mn><mi>θ</mi><msup><mfenced><mrow><mn>2</mn><mo>+</mo><mi>θ</mi></mrow></mfenced><mrow><mo>-</mo><mn>2</mn></mrow></msup></math> rather than the quotient rule. This was fine but made solving the equation in (b)(iii) less straightforward.</p>
<div class="question_part_label">b.iii.</div>
</div>
<br><hr><br><div class="specification">
<p>The position vector of a particle, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math>, relative to a fixed origin <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>O</mtext></math> at time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> is given by</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mtext>OP</mtext><mo>→</mo></mover><mo>=</mo><mfenced><mtable><mtr><mtd><mi>sin</mi><mo> </mo><mfenced><msup><mi>t</mi><mn>2</mn></msup></mfenced></mtd></mtr><mtr><mtd><mi>cos</mi><mo> </mo><mfenced><msup><mi>t</mi><mn>2</mn></msup></mfenced></mtd></mtr></mtable></mfenced></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the velocity vector of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the acceleration vector of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> is never parallel to the position vector of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math>.</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>attempt at chain rule <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi mathvariant="bold-italic">v</mi><mo>=</mo><mfrac><mrow><mo>d</mo><mi>O</mi><mi>P</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo></mrow></mfenced><mo> </mo><mfenced><mtable><mtr><mtd><mn>2</mn><mi>t</mi><mo> </mo><mi>cos</mi><mo> </mo><msup><mi>t</mi><mn>2</mn></msup></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn><mi>t</mi><mo> </mo><mi>sin</mi><mo> </mo><msup><mi>t</mi><mn>2</mn></msup></mtd></mtr></mtable></mfenced></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>attempt at product rule <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">a</mi><mo>=</mo><mfenced><mtable><mtr><mtd><mn>2</mn><mo> </mo><mi>cos</mi><mo> </mo><msup><mi>t</mi><mn>2</mn></msup><mo>-</mo><mn>4</mn><msup><mi>t</mi><mn>2</mn></msup><mo> </mo><mi>sin</mi><mo> </mo><msup><mi>t</mi><mn>2</mn></msup></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn><mo> </mo><mi>sin</mi><mo> </mo><msup><mi>t</mi><mn>2</mn></msup><mo>-</mo><mn>4</mn><msup><mi>t</mi><mn>2</mn></msup><mo> </mo><mi>cos</mi><mo> </mo><msup><mi>t</mi><mn>2</mn></msup></mtd></mtr></mtable></mfenced></math> <em><strong>A1</strong></em></p>
<p><br><strong>METHOD 1</strong></p>
<p>let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>S</mi><mo>=</mo><mi>sin</mi><mo> </mo><msup><mi>t</mi><mn>2</mn></msup></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>C</mi><mo>=</mo><mi>cos</mi><mo> </mo><msup><mi>t</mi><mn>2</mn></msup></math></p>
<p>finding <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mo> </mo><mi>θ</mi></math> using</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">a</mi><mo mathvariant="bold">·</mo><mover><mtext>OP</mtext><mo>→</mo></mover><mo>=</mo><mn>2</mn><mi>S</mi><mi>C</mi><mo>-</mo><mn>4</mn><msup><mi>t</mi><mn>2</mn></msup><msup><mi>S</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>S</mi><mi>C</mi><mo>-</mo><mn>4</mn><msup><mi>t</mi><mn>2</mn></msup><msup><mi>C</mi><mn>2</mn></msup><mo>=</mo><mo>-</mo><mn>4</mn><msup><mi>t</mi><mn>2</mn></msup></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mover><mtext>OP</mtext><mo>→</mo></mover></mfenced><mo>=</mo><mn>1</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="|" close="|"><mi mathvariant="bold-italic">a</mi></mfenced><mo>=</mo><msqrt><msup><mfenced><mrow><mn>2</mn><mi>C</mi><mo>-</mo><mn>4</mn><msup><mi>t</mi><mn>2</mn></msup><mi>S</mi></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mo>-</mo><mn>2</mn><mi>S</mi><mo>-</mo><mn>4</mn><msup><mi>t</mi><mn>2</mn></msup><mi>C</mi></mrow></mfenced><mn>2</mn></msup></msqrt></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msqrt><mn>4</mn><mo>+</mo><mn>16</mn><msup><mi>t</mi><mn>4</mn></msup></msqrt><mo>></mo><mn>4</mn><msup><mi>t</mi><mn>2</mn></msup></math></p>
<p>if <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>θ</mi></math> is the angle between them, then</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mo> </mo><mi>θ</mi><mo>=</mo><mo>-</mo><mfrac><mrow><mn>4</mn><msup><mi>t</mi><mn>2</mn></msup></mrow><msqrt><mn>4</mn><mo>+</mo><mn>16</mn><msup><mi>t</mi><mn>4</mn></msup></msqrt></mfrac></math> <em><strong>A1</strong></em></p>
<p>so <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>1</mn><mo><</mo><mi>cos</mi><mo> </mo><mi>θ</mi><mo><</mo><mn>0</mn></math> therefore the vectors are never parallel <em><strong>R1</strong></em></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p>solve</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mn>2</mn><mo> </mo><mi>cos</mi><mo> </mo><msup><mi>t</mi><mn>2</mn></msup><mo>-</mo><mn>4</mn><msup><mi>t</mi><mn>2</mn></msup><mo> </mo><mi>sin</mi><mo> </mo><msup><mi>t</mi><mn>2</mn></msup></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn><mo> </mo><mi>sin</mi><mo> </mo><msup><mi>t</mi><mn>2</mn></msup><mo>-</mo><mn>4</mn><msup><mi>t</mi><mn>2</mn></msup><mo> </mo><mi>cos</mi><mo> </mo><msup><mi>t</mi><mn>2</mn></msup></mtd></mtr></mtable></mfenced><mo>=</mo><mi>k</mi><mfenced><mtable><mtr><mtd><mi>sin</mi><mo> </mo><msup><mi>t</mi><mn>2</mn></msup></mtd></mtr><mtr><mtd><mi>cos</mi><mo> </mo><msup><mi>t</mi><mn>2</mn></msup></mtd></mtr></mtable></mfenced></math> <em><strong>M1</strong></em></p>
<p>then</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>k</mi><mo>=</mo></mrow></mfenced><mfrac><mrow><mn>2</mn><mo> </mo><mi>cos</mi><mo> </mo><msup><mi>t</mi><mn>2</mn></msup><mo>-</mo><mn>4</mn><msup><mi>t</mi><mn>2</mn></msup><mo> </mo><mi>sin</mi><mo> </mo><msup><mi>t</mi><mn>2</mn></msup></mrow><mrow><mi>sin</mi><mo> </mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mfrac><mrow><mo>-</mo><mn>2</mn><mo> </mo><mi>sin</mi><mo> </mo><msup><mi>t</mi><mn>2</mn></msup><mo>-</mo><mn>4</mn><msup><mi>t</mi><mn>2</mn></msup><mo> </mo><mi>cos</mi><mo> </mo><msup><mi>t</mi><mn>2</mn></msup></mrow><mrow><mi>cos</mi><mo> </mo><msup><mi>t</mi><mn>2</mn></msup></mrow></mfrac></math></p>
<p><br><strong>Note:</strong> Condone candidates not excluding the division by zero case here. Some might go straight to the next line.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo> </mo><msup><mi>cos</mi><mn>2</mn></msup><mo> </mo><msup><mi>t</mi><mn>2</mn></msup><mo>-</mo><mn>4</mn><msup><mi>t</mi><mn>2</mn></msup><mi>cos</mi><mo> </mo><msup><mi>t</mi><mn>2</mn></msup><mo> </mo><mi>sin</mi><mo> </mo><msup><mi>t</mi><mn>2</mn></msup><mo>=</mo><mo>-</mo><mn>2</mn><mo> </mo><msup><mi>sin</mi><mn>2</mn></msup><mo> </mo><msup><mi>t</mi><mn>2</mn></msup><mo>-</mo><mn>4</mn><msup><mi>t</mi><mn>2</mn></msup><mo> </mo><mi>cos</mi><mo> </mo><msup><mi>t</mi><mn>2</mn></msup><mo> </mo><mi>sin</mi><mo> </mo><msup><mi>t</mi><mn>2</mn></msup></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo> </mo><msup><mi>cos</mi><mn>2</mn></msup><mo> </mo><msup><mi>t</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mo> </mo><msup><mi>sin</mi><mn>2</mn></msup><mo> </mo><msup><mi>t</mi><mn>2</mn></msup><mo>=</mo><mn>0</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>=</mo><mn>0</mn></math> <em><strong>A1</strong></em></p>
<p>this is never true so the two vectors are never parallel <em><strong>R1</strong></em></p>
<p> </p>
<p><strong>METHOD 3</strong></p>
<p>embedding vectors in a 3d space and taking the cross product: <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mtable><mtr><mtd><mi>sin</mi><mo> </mo><msup><mi>t</mi><mn>2</mn></msup></mtd></mtr><mtr><mtd><mi>cos</mi><mo> </mo><msup><mi>t</mi><mn>2</mn></msup></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr></mtable></mfenced><mo>×</mo><mfenced><mtable><mtr><mtd><mn>2</mn><mo> </mo><mi>cos</mi><mo> </mo><msup><mi>t</mi><mn>2</mn></msup><mo>-</mo><mn>4</mn><msup><mi>t</mi><mn>2</mn></msup><mo> </mo><mi>sin</mi><mo> </mo><msup><mi>t</mi><mn>2</mn></msup></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn><mo> </mo><mi>sin</mi><mo> </mo><msup><mi>t</mi><mn>2</mn></msup><mo>-</mo><mn>4</mn><msup><mi>t</mi><mn>2</mn></msup><mo> </mo><mi>cos</mi><mo> </mo><msup><mi>t</mi><mn>2</mn></msup><mo> </mo></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn><mo> </mo><msup><mi>sin</mi><mn>2</mn></msup><mo> </mo><msup><mi>t</mi><mn>2</mn></msup><mo>-</mo><mn>4</mn><msup><mi>t</mi><mn>2</mn></msup><mo> </mo><mi>cos</mi><mo> </mo><msup><mi>t</mi><mn>2</mn></msup><mo> </mo><mi>sin</mi><mo> </mo><msup><mi>t</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mo> </mo><msup><mi>cos</mi><mn>2</mn></msup><mo> </mo><msup><mi>t</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><msup><mi>t</mi><mn>2</mn></msup><mo> </mo><mi>cos</mi><mo> </mo><msup><mi>t</mi><mn>2</mn></msup><mo> </mo><mi>sin</mi><mo> </mo><msup><mi>t</mi><mn>2</mn></msup><mo> </mo></mtd></mtr></mtable></mfenced></math></p>
<p> <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfenced><mtable><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr></mtable></mfenced></math> <em><strong>A1</strong></em></p>
<p>since the cross product is never zero, the two vectors are never parallel <em><strong>R1</strong></em></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;">
<p>In part (a), many candidates found the velocity vector correctly. In part (b), however, many candidates failed to use the product rule correctly to find the acceleration vector. To show that the acceleration vector is never parallel to the position vector, a few candidates put <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi mathvariant="bold-italic">r</mi><mrow><mo mathvariant="bold">.</mo><mo mathvariant="bold">.</mo></mrow></mover><mo>=</mo><mi>k</mi><mi mathvariant="bold-italic">r</mi></math> presumably hoping to show that no value of the constant <em>k</em> existed for any <em>t</em> but this usually went nowhere.</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 graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = - {x^3}">
<mi>y</mi>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
</math></span> is transformed onto the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 33 - 0.08{x^3}">
<mi>y</mi>
<mo>=</mo>
<mn>33</mn>
<mo>−<!-- − --></mo>
<mn>0.08</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
</math></span> by a translation of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span> units vertically and a stretch parallel to the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>-axis of scale factor <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="b">
<mi>b</mi>
</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="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>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>The outer dome of a large cathedral has the shape of a hemisphere of diameter 32 m, supported by vertical walls of height 17 m. It is also supported by an inner dome which can be modelled by rotating the curve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 33 - 0.08{x^3}">
<mi>y</mi>
<mo>=</mo>
<mn>33</mn>
<mo>−</mo>
<mn>0.08</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
</math></span> through 360° about the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span>-axis between <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span> = 0 and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span> = 33, as indicated in the diagram.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<p>Find the volume of the space between the two domes.</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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="a">
<mi>a</mi>
</math></span> = 33 <em><strong> A1</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="\frac{1}{{\sqrt[3]{{0.08}}}} = 2.32">
<mfrac>
<mn>1</mn>
<mrow>
<mroot>
<mrow>
<mn>0.08</mn>
</mrow>
<mn>3</mn>
</mroot>
</mrow>
</mfrac>
<mo>=</mo>
<mn>2.32</mn>
</math></span> <em><strong> M1A1</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>volume within outer dome</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{2}{3}\pi + {16^3} + \pi \times {16^2} \times 17 = 22\,250.85">
<mfrac>
<mn>2</mn>
<mn>3</mn>
</mfrac>
<mi>π</mi>
<mo>+</mo>
<mrow>
<msup>
<mn>16</mn>
<mn>3</mn>
</msup>
</mrow>
<mo>+</mo>
<mi>π</mi>
<mo>×</mo>
<mrow>
<msup>
<mn>16</mn>
<mn>2</mn>
</msup>
</mrow>
<mo>×</mo>
<mn>17</mn>
<mo>=</mo>
<mn>22</mn>
<mspace width="thinmathspace"></mspace>
<mn>250.85</mn>
</math></span> <em><strong> M1A1</strong></em></p>
<p>volume within inner dome</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\pi {\int_0^{33} {\left( {\frac{{33 - y}}{{0.08}}} \right)} ^{\frac{2}{3}}}{\text{d}}y = 3446.92">
<mi>π</mi>
<mrow>
<msubsup>
<mo>∫</mo>
<mn>0</mn>
<mrow>
<mn>33</mn>
</mrow>
</msubsup>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mn>33</mn>
<mo>−</mo>
<mi>y</mi>
</mrow>
<mrow>
<mn>0.08</mn>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mfrac>
<mn>2</mn>
<mn>3</mn>
</mfrac>
</mrow>
</msup>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
<mo>=</mo>
<mn>3446.92</mn>
</math></span> <em><strong> M1A1</strong></em></p>
<p>volume between = 22 250.85 − 3446.92 = 18 803.93 m<sup>3</sup> <em><strong> A1</strong></em></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.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>A function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> is of the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>t</mi></mfenced><mo>=</mo><mi>p</mi><msup><mtext>e</mtext><mrow><mi>q</mi><mo> </mo><mi>cos</mi><mfenced><mrow><mi>r</mi><mi>t</mi></mrow></mfenced></mrow></msup><mo>,</mo><mo> </mo><mi>p</mi><mo>,</mo><mo> </mo><mi>q</mi><mo>,</mo><mo> </mo><mi>r</mi><mo>∈</mo><msup><mi mathvariant="normal">ℝ</mi><mo>+</mo></msup></math>. Part of the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> is shown.</p>
<p><img style="display:block;margin-left:auto;margin-right:auto;" src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAe4AAAFwCAYAAAB+e5hlAAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAADsMAAA7DAcdvqGQAAC/ZSURBVHhe7d0NnM113v/x99q2O62NrYiaKJMiN42y7YxViljUY2I3VsaqcCkl+ve/Cn+urtzVrjUtq7rokjUS7WIu4SJqZNFWDGm5lqloNoNN0jRur5r5n8/P7zDGGebm3Px+57ye+/A48/ueubMd5/373n2+3ysJEAAA8IUa7iMAAPABghsAAB8huAEA8BGCGwAAHyG4AQDwkSoGd7GK8lYos38bJSVdoesHZaug2H3qhELlvTVNmVlbVOS2nKJoi7Iyp+mtvEK3AQAAnE0VgjsQ2tte1dC7R2pL21f0t7fGKvmdXG0vKpXcxQVaNXqUFte+V8Mymukit/kUFzVTxmPd9P3XMjVzG+ENAEBFVH4fd/E2zUy/W881/w+9N+421XKbTzqmguwRGrSjp+YMu7lMaNtzv9Gfk4ZqSIr7TOEqjWy3UDcv/a3S6597vA0AAIRUyR53obbNytRzm27Sk/1uCRHaAYXrNHX4bnXv1vy00N7zwQyNfvqYWjS+0G0LqHWL+g3dp+FT1wW+OwAAOJOKB/eebPVPaqqOoxfroN7R6DueUPaeb90ng44ob8F0ZZ3XWi0ane+2BRTnK3tQW7XpMVZv7p+hvjf8XJm5wZnv89WoRWudlzVdC/KOuG0AACCUigd3vXS9vPMtPdOqpmpmzNbf8v+g9HrnuE8GfaEtaz6Sbrpa9Us/VSNJ6S9kBb62jlo985Z25i/XsOBQecA59a/WTfpIa7Z84bYAAIBQKjdUXrRbeXnnKe3ma0IPk3/7hXas3686zZJ0idsUVPzJu1qwqbm6p111+g+98GLVrblf63d8obJ9eAAAcFIlgrtYhblva8HB5mrb7FK3raKKVbTrU+XVKTOEHnRhLV1ynvsxAAAoVyWC+5A+3vieDta8Rg3rnnn199F9hYHPLm2/cle8LXVNUeOyo+vmUKH2HXU/BgAA5ap4cBfv0uacnarZ/Xal1Crny865VI1uqqODew+cGtzf5mvjEql7xxtCD7EfOqC9B+vopkaXKlSuAwCA4yoe3P/8H63ZdGn54eu4VM3aNpfWf6qCUpPVxTs2K+fo7ep4bYFmjp6rvDJV1r4t+FTrVZUheAAAEksFg7tYhX/P1dqagfBNqeO2hXK+krsPUMbRDdq8o+zWrm2a+2qumj3cXcmn/NQj2rF5g45mDFD35BDz3wAA4IQKVk7bp1Uj0/WQxpZTLa20M1VOC6HoA2X2nqdGL42nchoAAGdRsR534d+0YsFV5VdLO8W5qp8+Wr+tvVSTVhUE+upnckC5M5ar9m9HE9oAAFRA+T1u6wn/4hlpzETd+MZTerb2CP25Ij3oEwqVlz1Li9VRA9KbnP51RduUPX2F1K2v0pPPfjsAAADOFNxWpvThX2jIUunOJ57X2EdSVa/iS9kAAEAEVP50sCg4fPiwJk+erD59+qhBgwZuKwAA8GQfev78+Zo6dYrGjh3rtgAAAOO54M7Ly9OIEU85H2/dukUrV65wPgYAAB4M7pEjR2r8+GedjydPnqIxY8Zo//79zjUAAInOU8GdnZ3tPPbo0cN5bNmypdLS2mrixInONQAAic4zwb1r1y4NGfKIxo0bpwsuuMBtlZ544gmtXbtGH374odsCAEDi8kxwz549W4MHP6rk5GS35bg6depo1KhRgVB/1G0BACBxeWY7mPWor7322hO97aSkK5Sf/7nzsVm3bp1SU1PdKwAAEpMn93GbssENAAA8uKocAACUj+AGAMBHCG4AAHyE4AYAwEcIbgAAfITgBgDARwhuAAB8JAGDu1hFeSs0OfNP2lZU7Lb5QNEWZWVO01t5hW4DACARxUlwF6tw1Whdn3SPMnMPuG2hHNOeVb/R0MUX64Fhv1STi47/9Yv3rNPk/m2coi9JnUcqK3dP4DuGS6HyVv1JmcHv3z9be9xnQikuyNag6wOfZ5/r/LlPM/OOSBc1U8Zj3fT91zI1cxvhDQCJKj6Cu/hzvT13mQ7qA02bvykQlaEVFyzV05Mu1MMDWusit01FH+j3/QZpaf0Jen/np3r/qQv06n0TtKjgmPsJ1VC0Tdkj79MdfWdpX9unlf3+p8p/OV313KdPd0Cb5r2mnQMXamv+507luPz8V9Uv+fzjT9eor9uG3qq8nqOUHY7fDwDgO3ER3MWfvK0ZO5vqzqY1dXDB28otDNVf3qfVU6eooHsntXJ72tIR5f35eWV+lq6nnmivejXOVb12v1Sv5GUaPnVduTcAFVKcr+wnMjRkQQM9s+JVjevXRSn1znWfDK24YJWm/bGeBvVsefLGoqxat6jf0H3V//0AAL4UB8EdCOSZWVKvx/Vvgzqr5sFlmvv256cNdRfnLdakrHPUvkWDk3/p4s+0dsF6KS1F19VyW2tcpbTuN53hBqAijqlg0e80fKnUZcII9W1Sy20/E7uxmKil++dryB0PKXPBauWFnIM/X41atNZ5WdO1wIbQAQAJxf/BXfg3rVhygx648zpdefs96l6zQEtnvK1PTsm8b/XPLeu1SclqVN8ddjZFu5WXd1B1miXpErdJOkc/vLi2dPAT7dxbxeHo4h16c8YyHazZXNfk/buaOXPVbdR/8jrtKfde4BLdNm6N8neuV/ZvUrVv2gDdcfPDIeezz6l/tW7SR1qz5Qu3BQCQKHwe3EeUtyDQ8+x6j26vf65U6wZ17N5Q2rRCaz8p3Rs9ooIdeVKdq5V0yTluW8ChA9p70P34NPv01Tffuh9XTvEn72rBJqlp99t1R8aL+p9AGM9/ornWTrxfA2dtO/PCtxr1lHLXgxr35wV6Ji1Xox+bqdyyPe8LL1bdmvu1fscXgVsSAEAi8XdwF32kxXN/oCf73aLjg9F1dFOPe9VU7+i5mX+t5hzwJar9w1IhXwnF3+xXvi5V646djs9rB8L45gcGaWDTwD3FgnfLjAaU46Jm6vvkw2q1dYne2X7IbXRdWEuXnOd+DABIKD4O7mIVrn9D07au1Og7Grtbp5LUNP032hp4NuQc9dEDKjxUqu2y69W2VU0d3Veok9Ho9s5rXqOGdc+8mKxSLqqnRlcE0jZ/v76pSHAH1KjbUM1ruhelHSrUvqPuxwCAhOLf4C7O04JJa9V91iZ321Twz8d665lbA8n9tlbk7nc/+XzVb5QcaNuvA6WDO9RCtOJd2pyzUzW7366U4IK1SjqncYq61typBSv+dlqvv2bXFDWuYEe+eO9OfXRVV9167YVui8sZ4q+jmxpdqqqNCQAA/MqnwV2obbMy9Vx+V/Voe3JZ2XHnK7nzL3Wndirr2cnKdiqNnaPLmt2kVsrTjoLSc9/n65o7f6kuWqbZC/5HRYHvm7foVc3N66wJg1Pd4fcqqJWqwRN6SFm/08RVBSp2Cr/8US+tTdWTfVNCft/iPbl6I3uZcvfYgjir7patUY/+t9qP76uUE9vXjvu24FOtV3O1bXap2wIASBQ+DO4i5Wb+Uh1HLw50oCcp/epHlL2n1BKtPdnq3+YRvWkfb52hIXe0Vf/sf6hGcjc9nvGtcjbvOmVxWI36d2viohGqO7e7miY11R0vSYMWjVG6LXY7xT+U3b+FkpLu0mgnjM/kXNVPH6NFk2/U+33bqGHS1Wrz7GHd9+rTJ7eG2e95SiW1r/TBS48pvc3VgZ9xvX4x84A6TpmuYSkXO8+edEQ7Nm/Q0YwB6h4szAIASBjfKwlwP/YUm7O2oe9wsnKiDw/K18A5j5zWi62wwlUa+a8H9OhLZ6qAFkFFHyiz9zw1eml8iJsLAEC881yP+/Dhw3ph6lQdOXxIY8Y8o88/D194O73r3/5Y8yflnGE/9ZkcU8HbK1TYOUWXuS3RdUC5M5ar9m9HE9oAkKA81+Pu1+/XemNRtnslNWx4tdasXacLLrjAbakumz9eoumLpW4Duiq5oj3v4j3KffV1vfPDThqQ3qT8kqSRYnXPp68I/NJ9lZ5c5dl3AIDPeSq4P/74Y6Wl3qIa3z8Zpt99952ysxcpNTXNbQEAIHF5Yqg8Ly9PL774om6//Tb94Ac/cFsDdxWB/5177nl6+ul/07p169xWAAASV0yD2+azLbD7939Ql19+ubZty1OvX/XWsaNHnT9Hjx5RvXqX63e/y1RWVpZGjBih/fuDe7MBAEg8MQtu62V37txJhYWFWrZsudLT05157DFjxur530/R97///cDH4/XmmyvUvHlzJ+DbtGmje+5Jd74WAIBEFJM5bhv27tXrXs2d+7pSU1Pd1lOVtx3sww8/1JAhj2rChGfL/VoAAOJV1HvcFtrDhz+ld999r0rB27JlS82Z85rzPZj3BgAkmqgGdzC0LXgbNGjgtlaefW0wvBk2BwAkkqgFtwWsDY9XN7SD7HvYcLktbNu1a5fbCgBAfItKcFuwWsDanHY4QjvIhtpHjRqlYcOGOSvUAQCIdxEPbgvUsWPHasCAgRFZTNahQ0e1b99eY8aMcVsAAIhfEQ/u+fPnO499+vRxHiOhX79++uqrr5SdfbJUKgAA8Sii28FsXtuGyBcuzFadOnXc1oqp7OlgNhz/05/+RG+9laPk5GS3FQCA+BKxHrcNkY8cOdKZg65saFeFzZ3PmPGK8zOZ7wYAxKuIBbcNkTdu3NiZg44W+1n2M4PD8wAAxJuIDJUHh603bdpc5d52ZYfKg6yWuZVFffnl/2TIHAAQdyLS4546daomT/5DVIbIy7KfacPzDJkDAOJR2IPbaomvXbtGnTp1cluiLzhkvnz5crcFAID4EPah8nvvvVdDhw6t9p7tqg6VBwWH660mejiLvgAAEEth7XEHD/3wwqldFtbjxz/rDNsDABAvwhrczz//vNPb9ooePXo4w/Y2fA8AQDwIW3B7qbcddMEFFzgHkYwbN46FagCAuBC24PZabzvIbiRYqAYAiBdhCe7gmdhe6m2Xdv/99yszc5KzxxsAAD8LW4/bhqO9ygqxdOnSVfPmzXNbAADwp4geMlId1d0OVpb1tlu1alGtam4AAMRaWFeVe5mF9fDhIzVx4kS3BQAA/0mY4DZ2brdtDwvOyQMA4DcJFdy2PWzYsMf1yiuvuC0AAPhLQgW3sRrq9LoBAH6VcMFNrxsA4GcJF9yGXjcAwK8SMrjpdQMA/Cohg9vQ6wYA+FHCBje9bgCAHyVscBt63QAAv0no4KbXDQDwm4QObkOvGwDgJwkf3MFe98qVK90WAAC8K+GD27Rr104TJozjvG4AQKXEIjcI7oDgyWGc1w0AqKjDhw87x0VHO7wJblfPnj3pdQMAKmzjxo3q2vUup/MXTQS3K9jrXr16tdsCAED5/vKXvygjI8O9ih6Cu5QOHTooM3OSM/wBAEB5LCemTp2iG2+80W2JHoK7lOTkZKWltXW2hwEAUB7LicGDH3V2JkUbwV1Gt27dNG3adPcKAIDTzZ+/QD/72c/cq+giuMtITU11HtetW+c8AgBQmi1i3rp1S0yGyQ3BHcLQoUO1ePFi9woAgJNyczeoV69fxWSY3BDcIdhdlM1f7Nq1y20BAOA4m04Njs7GAsEdgt1FDRgwUIsWLXJbAACQ06Hbu3ePWrZs6bZEH8Fdji5dujgFWdgaBgAIysnJcYbJY4ngLocVZLGl/suXL3dbAACJzkZi7777bvcqNgjuM+jevbvmzJnjXgEAEtmHH37oPDZo0MB5jBWC+wysIIthaxgAwLIg1r1tQ3CfBVvDAABm7tzXnPVPsUZwnwVbwwAA1ttu2rRZ1E8CC4XgPgu2hgEA7CSwHj26u1exRXBXgA2N2BAJW8MAIPEETwJLSWnttsQWwV0BNjTCqWEAkJg2btzobA/2wjC5IbgrqGfPnpwaBgAJKCsrK2YngYVCcFdQsLxdXl6e8wgAiH92EtiSJW/E7CSwUAjuSujdu7dWrlzpXgEA4p2dBDZ8+MiYnQQWCsFdCe3ataN+OQAkkFifBBYKwV0JwfrlLFIDgPjnhZPAQiG4K6lz584sUgOABOCFk8BCIbgriUVqAJAYpk+f5ona5GUR3FXAIjUAiG92EljduvVifhJYKAR3FbBIDQDim1dOAguF4K4CFqkBQPyyTplXTgILheCuIhapAUB8shKnXjkJLBSCu4qCi9Q47hMA4ouXTgILheCuBlukxnGfABA/rMSpnQRmB0t5FcFdDbZIjeM+ASB+WIlTW8PkpRKnZRHc1RA87tPmQwAA/jd//gJPnQQWCsFdTd26ddPixYvdKwCAX9mapa1bt3iuNnlZBHc12X9g2xZm8yIAAP/yaonTsgjuMLD/0EuXLnWvAAB+ZIuNO3To4F55F8EdBlZdh9XlAOBfVuLUJCcnO49eRnCHQbCWbfA/PADAX7xc4rQsgjtMBg4c4PyHBwD4i9dLnJZFcIdJSkprDh4BAB/yeonTsgjuMAkePMKebgDwF9vS6+USp2UR3GFkm/azsrLcKwCA19lW3tmzZ3m6xGlZBHcY2Z5u27zPnm4A8Ac/lDgti+AOM9vTvXr1avcKAOBldjyzHdPsJwR3mNl2gjlz5rhXAACvshKne/fuOXFMs18Q3GEW3NOdl5fnPAIAvMkKZ/mhxGlZBHcE2DndK1eudK8AAF5ke7f9UnSlNII7AoLndAMAvMkqXdatW+/EKKmfENwRYHu6bTM/ldQAwJuWLVvmVLz0I4I7QjIyMvSXv/zFvQIAeIVVuJw6dYpT8dKPCO4IufHGG50XBiVQAcBb1q5d4+zd9kuJ07II7gixzfz2wrAXCADAO+bPX+BUuvQrgjuCbFO/vUAAAN5ge7etwqWNivoVwR1BtqmfEqgA4B05OTnO3m0/lTgti+COMHuBWC1cAEDsTZ8+TR06dHCv/IngjjB7gVgtXABAbAX3bicnJ7st/kRwR1jwBWLzKgCA2LG921bZ0u8I7iiwkno2rwIAiI3g3m2rbOl3BHcUtG/f3ilmDwCIDdua26dPX9/u3S6N4I4CTgwDgNiyrbndunVzr/yN4I4Sq4nLiWEAEH3xsHe7NII7SqwmLieGAUD0xcPe7dII7igJnhhm2xEAANFje7f9eO52eQjuKOrRoztHfQJAFNl7rl/P3S4PwR1FweFyTgwDgOiw45X9eu52eQjuKLLh8rS0ttq4caPbAgCIFDsnws/nbpeH4I4y245gd4AAgMiycyL8fO52eQjuKLPtCHYHyHA5AESWnRPh53O3y0NwR5ltR7A7QIbLASByrODV3r17lJqa6rbED4I7BuwOkOFyAIgcK3g1YMBA9yq+ENwxwHA5AESOvbfaDh47JyIeEdwxwHA5AESOvbdawat42rtdGsEdIwyXA0BkZGVlKSMjw72KP98rCXA/9pSkpCuUn/+5exV/bCinSZNkbduWFzf1cwEg1mzvdqtWLeL6vZUed4wwXA4A4bd06VINHz4yrjtEBHcMMVwOAOFlB4p06NDBvYpPBHcMsbocAMLHTl+0A0WSk5PdlvhEcMcQw+UAED7Lli2LuwNFQiG4Y8yGyxcvXuxeAQCqIl4PFAmF4I4xGy6fPXuW86IDAFTN6tWr4/JAkVAI7hgLDpfbKTYAgKqZM2eOOnfu7F7FN4LbA2y4fMOGXPcKAFAZtijNtGzZ0nmMdwS3B7C6HACqbt26derdu7d7Ff8Ibg9gdTkAVI11eCZMGKd27dq5LfGP4PYIirEAQOWtXbtGffr0TYhFaUEEt0cwXA4AlTdt2nR169bNvUoMBLdHMFwOAJWTl5envXv3KDU11W1JDAS3hzBcDgAVt3LlSg0YMNC9Shwc6+khHPUJABUTfL/ctGlzQs1vG3rcHsJwOQBUTCIuSgsiuD2G4XIAOLtEXJQWRHB7DKvLAeDMEnVRWhDB7TE2XG7DPwyXA0BoibooLYjg9qDbb2/PcDkAhBCslNalSxe3JfEQ3B5k58nacDkA4FSJvCgtiOD2IHtBdu1614kTbwAAxyXyorQggtujevTo7px4AwA4LtEXpQUR3B5lw+Vz577mXgEAEn1RWhDB7VE2XN60aTOGywEgYP/+/Qm/KC2I4PawtLQ0hssBIGD16tVOZclEXpQWRHB7WPv27RkuB4CAOXPmqHPnzu5VYiO4PaxBgwaqW7eesyADABJVcMqwZcuWzmOiI7g97u6779Z7773nXgFA4pk3b5569+7tXoHg9jgbLl+0aJF7BQCJxRalzZ49S506dXJbQHB7nA2Xm127djmPAJBIli5dquHDRzrnOOA4gtsHbLg8JyfHvQKAxDF9+jR16NDBvYIhuH2A4XIAici2w1o9i+TkZLcFhuD2AYbLASSirKwsZWRkuFcIIrh9guFyAInEtsFu3bpFN954o9uCIILbJxguB5BIgnXJWZR2OoLbJxguB5AoDh8+TF3yMyC4fYThcgCJYPny5erTpy91yctBcPuIDZfb1ggAiGdWl7xnz57uFcoiuH0kWLucoz4BxKvgiYjUJS8fwe0zNlzOUZ8A4tXixYs1cOAA9wqhENw+w1GfAOKVLb5du3aN0tLaui0IheD2GRsut0pCDJcDiDe25bVXr1+xBewsCG4f6tGju5YtW+ZeAYD/BbeAsSjt7AhuH0pJaa2pU6c4L3QAiAdsAas4gtuH7IVtL/CNGze6LQDgb2wBqziC26e6devmFOAHAL9jC1jlENw+ZYX3lyx5Q/v373dbAMCfrBMydOhQ9wpnQ3D7lK26HD58pFavXu22AID/cApY5RHcPpaamurMCwGAX3EKWOUR3D4WnA+yO1YA8Bub6uMUsMojuH3OSqDaHSsA+M3SpUudKT+2gFUOwe1zdqdqd6zs6QbgJ/aeZacddujQwW1BRRHcPseebgB+FKxJnpyc7LagogjuOMCebgB+M23adOe9C5VHcMcB20Zh2ynsZB0A8LpgwRXbGYPKI7jjgG2jsBN17GQdAPA6Cq5UD8EdJ2x1Oed0A/A6Cq5UH8EdJ4LndAeHoADAi1555RUNG/Y4BVeqgeCOIxkZGSxSA+BZVnBl9uxZ6tSpk9uCqiC44wiL1AB42bx585yCK/S2q+d7JQHux56SlHSF8vM/d69QUS+++KLz+NBDDzmPAOAF1ttu1aqFNm3aTKW0aqLHHWeCi9SopAbASyhvGj4Ed5yxRWpWjYhKagC8gvKm4UVwxyGrRvT888+7VwAQW8uXL6e8aRgR3HEoWI2I4z4BxJr1tjMzJ+n+++93W1BdBHec6t27txYsWOBeAUBs2LQdve3wIrjjlO2TnDp1irOSEwBixabtOEwkvAjuOGX7JG0Fp63kBIBY4DCRyCC445htDbOVnGwNAxAL1tvmMJHwI7jjWHBrmB1YDwDRRG87cgjuONezZ0/nwHoAiCZ625FDcMe5li1bOo+cGgYgWuhtRxbBnQDsrpdTwwBEy+LFi+ltRxDBnQDsrvfLL7+kIAuAiLP3mY8//pjedgQR3AnCCrLYAfYAEEn2PjNw4AD3CpFAcCcIK8hiq8vpdQOIFHt/sfcZ282CyCG4E4QVZBk27HF63QAixt5f7H3G3m8QOQR3Agn2uimDCiDcgr1te59BZBHcCcTuggcMGKh58+a5LQAQHtbbHjVqFL3tKCC4E0yXLl00YcI4et0Awia4krxDh45uCyKJ4E4wderUcQ4fodcNIFxGjhzJvu0oIrgTkJVBpdcNIByokhZ9BHcCotcNIFyoSR59BHeC6tevn+bOfY1eN4Aqo7cdGwR3ggru6544caLbAgCVQ287NgjuBEY1NQBVtXLlCjVu3JjedgwQ3AmMamoAquLw4cMaM2aM7r//frcF0URwJzh63QAqa/ny5U498uTkZLcF0URwJzjrdU+Y8KyzDxMAzsZ625mZk+htxxDBjRNzVMEVogBQnpkzZ6pXr1/R246h75UEuB97SlLSFcrP/9y9QqR9+OGHGjLkUS1btpxawwBCsu2jrVq10KZNm516EIgNetxwtGzZ0pmzsrkrAAhl+vTpGj/+WUI7xghunDB48GBn7oqiLADKsgWsS5cuUY8ePdwWxArBjRMaNGjgzF3ZXTUAlDZp0iSO7fQIghunsFKodlfN9jAAQbZw9csvv+TYTo8guHEKu5u2u2q7uwYA2/41fPhTbBn1EIIbpwneVWdnZzuPABJXsNiKLWCFN7AdDCHZUHn//g9q4cJsVpACCSq4/evdd99z1sDAG+hxIyQrrsBCNSCxBbd/EdreQnCjXMGFalacBUBiYfuXdxHcKJctVJs8eYrGjRvnLFABkDhsMZqdY8D2L+8huHFGtiDFztydP3++2wIg3tnCVM7a9i6CG2f1xBNPaPr0aeztBhKALUizCopWSRHeRHDjrGxVue3ttqEzhsyB+GYL0gYMGMiCNA8juFEhtrebIXMgvtlCVBakeR/BjQpjyByIXzaaZgtRbUEqC9K8jeBGhdmQua0ytcIsDJkD8cVG02xUjQpp3kdwo1JslWmXLl0Dd+WT3RYAfmejaDaaZqNq8D6CG5U2ZMgQbdiwQStXrnBbAPhZ8MhOyhv7A8GNSrP5L5sLGzNmjHbt2uW2AvAj27Ndu3Ztjuz0EYIbVWK1zIcNe1xjx45lvhvwKbvxtj3bDJH7C8GNKktPT3fu1GfOnOm2APATu/FmiNx/CG5Ui/2jnzv3Na1bt85tAeAHDJH7F+dxo9psuO2nP/0JZ/YCPsF5+/5GjxvVZmE9Y8YrGjZsGPPdgMfZv1ErX8wQuX8R3AgLG25r3769Hn/8cbcFgBfZmpTWrVszRO5jBDfC5qGHHnIeX3zxRecRgLdYLXJbk2K1GOBfBDfCygo52BsDxVkAb7HjOocMeVQvv/yf1CL3OYIbYWVvCHPmvOYUZ2GlOeAdNq9tx3VaDQb4G8GNsLPFanbC0PDhT1FZDfCA2bNnO499+vRxHuFvBDciwk4YspPEevf+FeENxJDNa9sBIlamGPGB4EbE2EliVhaVbWJAbATntW0EjK1f8YPgRkRZWdTgNjHCG4ie4H5tu3nmjO34QnAj4mybWIsWLQhvIIpsv7aVNLWbZ8QXghtRQXgD0WPbMXNycpzqaIg/BDeihvAGIs/qkNt2zMzMTPZrxymCG1FFeAORYzs47PAQW4zGgT/xi+BG1BHeQPjZvyXbwWHD4yxGi28EN2KC8AbCx/4N2b8l28HB4SHxj+BGzATD+9e//jVFWoBqmDx5sho2bOj8m0L8I7gRU/ZG07t3byqsAVVkp/Ht3LmTE78SCMGNmLN9psHyqBxMAlSchfbmzZudU/lYQZ44CG54gpVHDR5MwpGgwNllZ2c7e7UJ7cRDcMMzbCWsHQk6bdp0pycBIDQbmcrMnMRe7QRFcMNTbO/pH//4R2f4z+a/WXEOnMpC20am7CaXvdqJieCG51gPwnrcaWlp6ty5k1MJCgChjeMIbniWHfpvi9asEpTN5wGJzG5mCW0YghueZovW7I1q+fLlGjFiBEPnSEjB1eMLF2YT2iC44X32RmUrZ5s2beoMnX/44YfuM0B8sxvV0lu+6tSp4z6DRPa9kgD3Y09JSrpC+fmfu1fAcRbaQ4Y8ql69fqV+/fqxohZxK1jG1M7UtvrjvNYRRI8bvmJbxpYtW67CwkKnVCoL1xCPrIqghbaVBB4/fjyhjVPQ44ZvBXvfXbp0dco98uaGeBB8XVsvmwNDEAo9bvhWsPdtbO6bcqmFyntrmjKztqjIbYlPxSra9idlTl6hvKJity0+2O4JC22rIkhoozwEN3zNetlPPvmkXn75P/X88887K899PXy+J1v9k65wRpxO/9NG/TP/pFV5he4nl1JcoFWjR2lx7Xs1LKOZLjrRnK1B15f+HvdpZt4R99nTFe9Zp8n927if20SdR76q3D3H3GeroXiPPpj8oK53vu9dGpn1gfacMXOPac8HL6n/id+99NfU0EVNfqnH7q2h1ya8pm1xEN42n22vXds9YbsoOE8bZ2RD5V505ZUN3I+Ailu4cGFJu3Y/K3nhhRdKvvzyS7fVbw6XbH+ld+DfwOCShbv/1237umT7yt+XPHhdg5IrrxtSsnDXUbfdHC3ZtfD/lNw16f2Sb9yW474q2TDp3pJOp7WX45v3SyY9+HTJwu1fBy6Olux+/8XjP6/T70s2fPPd8c+pEvs90gPfZ0xJzu6jJd/tXlkyqlNayb8s/Kwk9Hf9ruSbDVNKHhyxsGS7/dzvdpe8//sHSq678toyf5fvSr7OGVXS8l8Wluyqzq8XY9u3bz/xmj106JDbCpSPHjfiip00Fhw+v+eedGfo0X97v8/RDy+u7X4cVEvJdwzUk0/eKh1cphlv7tCJfmbhOk0dvlvduzU/0dM2xQWrNO2P9TSoZ8tT2kMrVuH6Dfr+w48pPblW4Ppc1bu53/Gft3WJ3tl+6PinVUFxXraezvxCGU8N0m31zlWNemnK6HWVlg5/WasLQ/WW92v9Oxfo4eF3K/miwFtUjXq6+ZF/1ZOtAr/KH/+i7d+6nxboeddqd5+GFkzR1NX73DZ/mT17tlNgyIbGrcQv6zRQEQQ34o69+dmboBWreP/99535b38GeFnnqm7Da1TTvTruiPIWTFfWea3VotH5bpvZp9VTJ2rp/vkacsdDylyw+izzwYEQvG2QhqRc7F4b9wai5k90Y+ML3bbKOqJP1q7QJrXSzdcFv/f5uiato1odfFsrcve7baVdotuGPagUC+2gGhfq4svOU82uKWp8jttmajRQi/bnKGvSYuX5aMTcVo3ba3Tr1q3O65ShcVQGwY24ZcUqbCuNzX/7P8CLVZS3SM89O08Ha3bWA3c2cv/xfqEtaz6Sbrpa9UsHmoXfuDXK37le2b9J1b5pA3THzQ9r5rYQ8+PlOqC/f7BJNbvfrpRaVX2rKNKuvM8C/zGuVtIlJ3/BGj+8WJcFfvePdn55cuTgTAo/0Qdra6l7xxtk4wEnna/6jZKlTeu15Z8nuuKeZa8962Xb2fM9enR3Xp8UVUFlEdyIe8nJyT4N8GwNadPQXZyVpKZ3PKKsrZcq48VRSq9/7vFP+fYL7Vi/X3WaJQWiOoQa9ZRy14Ma9+cFeiYtV6Mfm6ncCi7mKi5Yo7lLWmvC4NQyYVkZh3Vgb3k3CweV/9XBCgT3MRW8vVBLbn1Cg9uV/VvW0IUX11FN5WlHQfmL7rzAtnlZ7YFgL5tV46gqghsJI1SAW+9n//5Qw7VekK7J7+906hnk53+q97P/oGF3HlNW3y7qP7OSW74uaqa+Tz6sVhWdry7O16JnstVkxr+fvEkIu5pKql3zrG9CxQVL9cyMazRj4t2qf9onB4K71sU6z73yIhsWtxXj48aN08iRI+llo9oIbiScYIBbr+ebb75Rq1Yt9Nxzz3l8G9m5qpeSrmGTfqOMmgV6c/R4/bnUtq6j+wp1tjiuUbehmp86QV6OQm2b9Yo29xqvx06Z866KS9WsbfPAL3hAhYdO9q2/LfhU6wPPNW/44zO/CRVt0aypn6rXtEGnznmfUKxDhQd01L3yErshtDrjNizepk0b55x55rIRDgQ3Epb1emyB0LZteWrSpInTG7r33nu1cuUK/8yDn3OpGt1URwf3HjhrcBfv3amPruqqW68900KzY9qz6mXN0i/0+G31w/AGEWoh2hHt2LxB+2vero4pZ+h52t70SX+W+vZ3VqOHFgjuA/t1UMlqVL/04rzYCQa23RAa2+Vgux1YMY5wIbiR8OwN1d5YX3/9dSe8N2zIDQR5stMLj81JZEUq2LHL/fik4j0fKOu53ynroFSzyy915zUWVG6Pdv2nKii1Nqt4T67eyF7mFk+xhW3ZGvXof6v9+L7l9FyNhfYfNGVnJw3vV6qIy5639Ezm6kA/vGpqXHO7HuhyTAtm/7e2FX0b+F2Waebcz9RlQn+1K2/Rm4X22Ona2XOY+jUJzrDb7zdZmatKb/06Evj/Kk9qdZOaXXbK6ryoKxvYmzZtZosXIsPdz+05FGBBLFnxlhUr3iwZNGjQieIYmzZtcp+NoN0LSx4MvPbt9R/yz3UPlEya/87xwiRBX+eUjLiud8kr2w+7DVazxIqcXOt+3bUlnUa8UpLjFFYpJfizHlxYstsKruSMKelU+med+JNWMiLnC/eLXMGvdYuqnNU3fy9ZOKKb+/26lYxY+PdShVT+N/DtBgfam5c8uDA/8MvvKskZFfzcMn+uG1WS83Wpv/t3fy955a7bT//9osgKqDz77LNxUPgHfsEhI8BZ2OKinJwcLVq0SHv37nGOFE1NTfXQfOUxFWSP0KAdPTVn2M0VKLYSLsUqXDVe/3rg13op/Uq3LZqKVZT7B/WelqSXXkgPsXAtcmwqZe3aNZo2bbpz3bt3b3Xq1IneNaKC4AYqIVSIW4DfeOONMX7TLtS2mZma13CA/l9Y5qYroDhf2Y/9hzT83yK48vwMij7Q5Anb1Wn4r9Sk3OH/8LKpk2XLlmnq1CkaPPhRde7cmQVniDqCG6gim9PMzd2gt9/O0ezZs9S1611KS0vTT37yE2flevQVKi97lharowakN4loz9vm21997T39sFtft0RqNNmc/RJNXyx1G9D1eFnUCLKwtpPn5s59TU2bNnMKp6SktGZLF2KG4AbCwIZOt2/fro8++khr167VkiVvqE+fvs42oGbNmumKK65gGNUngjdktkjRetZ2Q0ZYw0sIbiACQgW5BUCLFi0CvfHGuuqqhjHqlaMsm/747LPPnJ61TYPYFEiXLl31s5/9TNdddx1hDc8huIEosQIvW7Zs0bZt2wK9uQ3661/XnQjzyy+/3OmZ//jHPyYoIshCet++fdqxY4dTPc8WmBkL6tatU3T99U3VoEEDpw3wKoIbiBHrlX/++eeB3t7OQKh/rH/84x/OXLmxYfYf/ehHTmGYyy67TJdeemnMh9vXrVsb6I3+U3XrXqbU1DS31XuC/78eOnTICWi7Udq5c6cz6tGo0dVKS2vrTGE0atRIV155JTdK8B2CG/AYm2P98ssvnUAvKjro9AxNMNRvuSVVjRs3dkLHeurGeutBkRiCf/DBB5S9cL57JT3/+ynKyOjrXkVP6bK0Nnphdu/e7dz0fPXVV044GxvJqF27thPQF11U05maYJ0B4gXBDfhMMLyCwV5UVOScOGU+/vhjZwg+KBhgQU2bNg0E2enrza33eeGFoUuhfhborfa+r5d7ddLSpcudUYGK+uKLL/TPf/7TvTpdMIBLs6HsHTs+da9O3rSYUDcuhDMSAcENxKngkHFQcOi4rNLBH0p+/k6tyslxr066++50XVzqpqAiSodtKKVHDozdTDDnDJyK4AZwRjZ0n9y4kXt1XMOGV2tD7kb3CkA0RbFIIAA/ssVbbyxe6oS1SU37mf5r0fG5ZADRR48bAAAfoccNAEAYWJ0AO9M/0scBE9wAAISBLaS0k+Luuqurc56/LRCNBIIbAIAwSU9P16ZNm/X111+rc+dOWrlyhftM+DDHDQBABFhojxkzxqnW98QTT4StSl+Vg9uCFQAAnJ2VMR4/frx7VT30uAEAiADP9bgjjeAGAPiRrS4fO3astm7dosmTp6hly5buM+HB4jQAAMLAVpFnZ2frpz/9iXNc77Jly8Me2ib6wV20TQuH/1w/rvMj/bjDSC3cXug+AQCAf1l54Dlz5uitt3L00EMPhTjw5pgKssdqcm6Re1010Q3u4s+0cFhPPb5voD7a94X+NvI8Te0UCO+CY+4nAADgT7aP+/XXXy/naN1j2vPBDI1++phaNA59El9FRTG4i1W0caGmzm+i0f/356pf41xdfmtP3Xf9Ij0+Za3odwMA4lJxvrIHtVWbHmP15v4Z6nvDz5VZjV53FIP7kLatekMb696sFleff7ypRgO1uKOxCrM3KO/b400AAMSVGklKfyFLz7Sqo1bPvKWd+cs1LOX0c/ErKorB/ZXy//a51KiOap34qeer/tXXSns/Uf4XJDcAID4Vf/KuFmxqru5pV1U7eKMY3K66tXVR9H8qAAAxUqyiXZ8qr05rtWjkjjhXAxEKAEBE7VfuirelrilqfI7bVA1RDO7aSrrhCumvn6jgxKj4ERV8uj3QC79GSZeG4W8DAIDXfJuvjUuk7h1vUC23qTqiGNwXKjnlFtXa+4E2f3rkeFPxLm1+62PVSm+tZHIbABCHindsVs7R29Xx2gLNHD1XecXuE1UUxeCuoVq3DtSkHjv06qw12l1cqLz/mqVX/+duTXo0LSx3IQAAeNM2zX01V80e7q7kaiZv1GuVF+9eq8kjBmvMf+2QUh7Ryy88qXuuPT22qVUOAMDpOGQEAAAfYVU5AAA+QnADAOAjBDcAAD5CcAMA4CMENwAAPkJwAwDgIwQ3AAA+QnADAOAjni3AAgAATkePGwAAHyG4AQDwEYIbAAAfIbgBAPAN6f8DBkmUZEBz1vwAAAAASUVORK5CYII="></p>
<p>The points <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> have coordinates <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext><mo>(</mo><mn>0</mn><mo>,</mo><mo> </mo><mn>6</mn><mo>.</mo><mn>5</mn><mo>)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext><mo>(</mo><mn>5</mn><mo>.</mo><mn>2</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>.</mo><mn>2</mn><mo>)</mo></math>, and lie on <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math>.</p>
<p>The point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math> is a local maximum and the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math> is a local minimum.</p>
<p>Find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math>, of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi></math> and of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi></math>.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p>substitute coordinates of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>A</mtext></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mn>0</mn></mfenced><mo>=</mo><mi>p</mi><msup><mtext>e</mtext><mrow><mi>q</mi><mo> </mo><mi>cos</mi><mfenced><mn>0</mn></mfenced></mrow></msup><mo>=</mo><mn>6</mn><mo>.</mo><mn>5</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>6</mn><mo>.</mo><mn>5</mn><mo>=</mo><mi>p</mi><msup><mtext>e</mtext><mi>q</mi></msup></math> <em><strong>(A1)</strong></em></p>
<p><br>substitute coordinates of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>B</mtext></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mrow><mn>5</mn><mo>.</mo><mn>2</mn></mrow></mfenced><mo>=</mo><mi>p</mi><msup><mtext>e</mtext><mrow><mi>q</mi><mo> </mo><mi>cos</mi><mfenced><mrow><mn>5</mn><mo>.</mo><mn>2</mn><mi>r</mi></mrow></mfenced></mrow></msup><mo>=</mo><mn>0</mn><mo>.</mo><mn>2</mn></math></p>
<p><br><strong>EITHER</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>t</mi></mfenced><mo>=</mo><mo>-</mo><mi>p</mi><mi>q</mi><mi>r</mi><mo> </mo><mi>sin</mi><mfenced><mrow><mi>r</mi><mi>t</mi></mrow></mfenced><msup><mtext>e</mtext><mrow><mi>q</mi><mo> </mo><mi>cos</mi><mfenced><mrow><mi>r</mi><mi>t</mi></mrow></mfenced></mrow></msup></math> <em><strong>(M1)</strong></em></p>
<p>minimum occurs when <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mi>p</mi><mi>q</mi><mi>r</mi><mo> </mo><mi>sin</mi><mfenced><mrow><mn>5</mn><mo>.</mo><mn>2</mn><mi>r</mi></mrow></mfenced><msup><mtext>e</mtext><mrow><mi>q</mi><mo> </mo><mi>cos</mi><mfenced><mrow><mn>5</mn><mo>.</mo><mn>2</mn><mi>r</mi></mrow></mfenced></mrow></msup><mo>=</mo><mn>0</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sin</mi><mfenced><mrow><mi>r</mi><mi>t</mi></mrow></mfenced><mo>=</mo><mn>0</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>×</mo><mn>5</mn><mo>.</mo><mn>2</mn><mo>=</mo><mi mathvariant="normal">π</mi></math> <em><strong>(A1)</strong></em></p>
<p><br><strong>OR</strong></p>
<p>minimum value occurs when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>cos</mi><mfenced><mrow><mi>r</mi><mi>t</mi></mrow></mfenced><mo>=</mo><mo>-</mo><mn>1</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>×</mo><mn>5</mn><mo>.</mo><mn>2</mn><mo>=</mo><mi mathvariant="normal">π</mi></math> <em><strong>(A1)</strong></em></p>
<p><strong><br>OR</strong></p>
<p>period <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>2</mn><mo>×</mo><mn>5</mn><mo>.</mo><mn>2</mn><mo>=</mo><mn>10</mn><mo>.</mo><mn>4</mn></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r</mi><mo>=</mo><mfrac><mrow><mn>2</mn><mi mathvariant="normal">π</mi></mrow><mrow><mn>10</mn><mo>.</mo><mn>4</mn></mrow></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>r</mi><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mrow><mn>5</mn><mo>.</mo><mn>2</mn></mrow></mfrac><mo>=</mo><mn>0</mn><mo>.</mo><mn>604152</mn><mo>…</mo><mo> </mo><mfenced><mrow><mn>0</mn><mo>.</mo><mn>604</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>2</mn><mo>=</mo><mi>p</mi><mo> </mo><msup><mtext>e</mtext><mrow><mo>-</mo><mi>q</mi></mrow></msup></math> <em><strong>(A1)</strong></em></p>
<p>eliminate <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi></math> or <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>e</mtext><mrow><mn>2</mn><mi>q</mi></mrow></msup><mo>=</mo><mfrac><mrow><mn>6</mn><mo>.</mo><mn>5</mn></mrow><mrow><mn>0</mn><mo>.</mo><mn>2</mn></mrow></mfrac></math> <strong>OR </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>2</mn><mo>=</mo><mfrac><msup><mi>p</mi><mn>2</mn></msup><mrow><mn>6</mn><mo>.</mo><mn>5</mn></mrow></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>q</mi><mo>=</mo><mn>1</mn><mo>.</mo><mn>74</mn><mo> </mo><mfenced><mrow><mn>1</mn><mo>.</mo><mn>74062</mn><mo>…</mo></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mn>1</mn><mo>.</mo><mn>14017</mn><mo>…</mo><mo> </mo><mo> </mo><mfenced><mrow><mn>1</mn><mo>.</mo><mn>14</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[8 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p>This was a challenging question and suitably positioned at the end of the examination. Candidates who attempted it were normally able to substitute points A and B into the given equation. Some were able to determine the first derivative. Only a few candidates were able to earn significant marks for this question.</p>
</div>
<br><hr><br><div class="specification">
<p>The production of oil <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mi>P</mi></mfenced></math>, in barrels per day, from an oil field satisfies the differential equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>P</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mfrac><mn>1000</mn><mrow><mn>2</mn><mo>+</mo><mi>t</mi></mrow></mfrac></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> is measured in days from the start of production.</p>
</div>
<div class="specification">
<p>The production of oil at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>20</mn><mo>,</mo><mn>000</mn></math> barrels per day.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mn>0</mn><mn>5</mn></msubsup><mfrac><mn>1000</mn><mrow><mn>2</mn><mo>+</mo><mi>t</mi></mrow></mfrac><mtext>d</mtext><mi>t</mi></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>State in context what this value represents.</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 an expression for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mn>0</mn><mn>365</mn></msubsup><mi>P</mi><mfenced><mi>t</mi></mfenced><mo> </mo><mtext>d</mtext><mi>t</mi></math> and state what it represents.</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 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>1252</mn><mo>.</mo><mn>7</mn><mo>…</mo><mo>≈</mo><mn>1250</mn></math> (barrels per day) <strong>A1</strong></p>
<p> </p>
<p><strong>[1 mark]</strong></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>This is the increase (change) in <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi></math> (production per day) between <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>5</mn></math> (or during the first <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn></math> days) <strong>A1</strong></p>
<p> </p>
<p><strong>[1 mark]</strong></p>
<div class="question_part_label">a.ii.</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>P</mi><mo>=</mo><mn>1000</mn><mo> </mo><mi>ln</mi><mfenced><mrow><mn>2</mn><mo>+</mo><mi>t</mi></mrow></mfenced><mo>+</mo><mi>c</mi></math> <strong>(M1)A1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>=</mo><mn>20000</mn><mo>-</mo><mn>1000</mn><mo> </mo><mi>ln</mi><mo> </mo><mn>2</mn><mo>≈</mo><mn>19306</mn><mo>.</mo><mn>8</mn><mo>…</mo></math> <strong>(M1)A1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>=</mo><mn>1000</mn><mo> </mo><mi>ln</mi><mfenced><mrow><mn>2</mn><mo>+</mo><mi>t</mi></mrow></mfenced><mo>+</mo><mn>19300</mn></math></p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∫</mo><mn>20000</mn><mi>P</mi></munderover><mtext>d</mtext><mi>P</mi><mo>=</mo><munderover><mo>∫</mo><mn>0</mn><mi>t</mi></munderover><mfrac><mn>1000</mn><mrow><mn>2</mn><mo>+</mo><mi>x</mi></mrow></mfrac><mtext>d</mtext><mi>x</mi></math> <strong>(M1)</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mfenced open="[" close="]"><mi>P</mi></mfenced><mn>20000</mn><mi>P</mi></msubsup><mo>=</mo><mn>1000</mn><msubsup><mfenced open="[" close="]"><mrow><mi>ln</mi><mfenced><mrow><mn>2</mn><mo>+</mo><mi>x</mi></mrow></mfenced></mrow></mfenced><mn>0</mn><mi>t</mi></msubsup></math> <strong>A1</strong></p>
<p> </p>
<p><strong>Note: A1</strong> is for the correct integral, with the correct limits.</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>-</mo><mn>20000</mn><mo>=</mo><mn>1000</mn><mfenced><mrow><mi>ln</mi><mfenced><mrow><mn>2</mn><mo>+</mo><mi>t</mi></mrow></mfenced><mo>-</mo><mi>ln</mi><mo> </mo><mn>2</mn></mrow></mfenced></math> <strong>(M1)A1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mo>=</mo><mn>1000</mn><mo> </mo><mi>ln</mi><mfenced><mfrac><mrow><mn>2</mn><mo>+</mo><mi>t</mi></mrow><mn>2</mn></mfrac></mfenced><mo>+</mo><mn>20000</mn></math></p>
<p> </p>
<p><strong>[4 marks]</strong></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"><mn>8847883</mn><mo>≈</mo><mn>8850000</mn></math> (barrels) <strong>A1</strong></p>
<p>Total production of oil in barrels in the first year (or first <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>365</mn></math> days) <strong>A1</strong></p>
<p> </p>
<p><strong>Note:</strong> For the final <strong>A1</strong> “barrels”’ must be present either in the statement or as the units.</p>
<p> </p>
<p>Accept any value which rounds correctly to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>8850000</mn></math></p>
<p> </p>
<p><strong>[2 marks]</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.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>A tank of water initially contains <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>400</mn></math> litres. Water is leaking from the tank such that after <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn></math> minutes there are <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>324</mn></math> litres remaining in the tank.</p>
<p>The volume of water, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi></math> litres, remaining in the tank after <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> minutes, can be modelled by the differential equation</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>V</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mo>-</mo><mi>k</mi><msqrt><mi>V</mi></msqrt></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>k</mi></math> is a constant.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mo>=</mo><msup><mfenced><mrow><mn>20</mn><mo>-</mo><mfrac><mi>t</mi><mn>5</mn></mfrac></mrow></mfenced><mn>2</mn></msup></math>.</p>
<div class="marks">[6]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the time taken for the tank to empty.</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"><mfrac><mrow><mo>d</mo><mi>V</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mo>-</mo><mi>k</mi><msup><mi>V</mi><mfrac><mn>1</mn><mn>2</mn></mfrac></msup></math></p>
<p>use of separation of variables <em><strong> (M1)</strong></em> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo>∫</mo><msup><mi>V</mi><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup><mo> </mo><mtext>d</mtext><mi>V</mi><mo>=</mo><mo>∫</mo><mo>-</mo><mi>k</mi><mo> </mo><mtext>d</mtext><mi>t</mi></math> <em><strong> A1</strong></em> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><msup><mi>V</mi><mfrac><mn>1</mn><mn>2</mn></mfrac></msup><mo>=</mo><mo>-</mo><mi>k</mi><mi>t</mi><mo> </mo><mfenced><mrow><mo>+</mo><mi>c</mi></mrow></mfenced></math> <em><strong> A1</strong></em> </p>
<p>considering initial conditions <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>40</mn><mo>=</mo><mi>c</mi></math> <em><strong> A1</strong></em> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><msqrt><mn>324</mn></msqrt><mo>=</mo><mo>-</mo><mn>10</mn><mi>k</mi><mo>+</mo><mn>40</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>k</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>4</mn></math> <em><strong> A1</strong></em> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><msqrt><mi>V</mi></msqrt><mo>=</mo><mo>-</mo><mn>0</mn><mo>.</mo><mn>4</mn><mi>t</mi><mo>+</mo><mn>40</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msqrt><mi>V</mi></msqrt><mo>=</mo><mn>20</mn><mo>-</mo><mn>0</mn><mo>.</mo><mn>2</mn><mi>t</mi></math> <em><strong> A1</strong></em> </p>
<p> </p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for any correct intermediate step that leads to the <em><strong>AG</strong></em>.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>V</mi><mo>=</mo><msup><mfenced><mrow><mn>20</mn><mo>-</mo><mfrac><mi>t</mi><mn>5</mn></mfrac></mrow></mfenced><mn>2</mn></msup></math> <em><strong>AG</strong></em></p>
<p><br><strong>Note:</strong> Do not award the final <strong><em>A1</em></strong> if the <em><strong>AG</strong></em> line is not stated.</p>
<p><em><strong><br>[6 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>0</mn><mo>=</mo><msup><mfenced><mrow><mn>20</mn><mo>-</mo><mfrac><mi>t</mi><mn>5</mn></mfrac></mrow></mfenced><mn>2</mn></msup><mo>⇒</mo><mi>t</mi><mo>=</mo><mn>100</mn></math> minutes <em><strong> (M1)A1</strong></em></p>
<p><em><strong><br>[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>Consider the function <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><msqrt><mo>-</mo><mi>a</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>x</mi><mo>+</mo><mi>a</mi></msqrt><mo>,</mo><mo> </mo><mi>a</mi><mo>∈</mo><msup><mi mathvariant="normal">ℝ</mi><mo>+</mo></msup></math>.</p>
</div>
<div class="specification">
<p>For <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>></mo><mn>0</mn></math> the curve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mi>f</mi><mfenced><mi>x</mi></mfenced></math> has a single local maximum.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced></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 in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> at which the maximum occurs.</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>Hence find the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math> for which <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> has the smallest possible maximum value.</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 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"><mi>f</mi><mo>'</mo><mfenced><mi>x</mi></mfenced><mo>=</mo><mfenced><mrow><mo>-</mo><mn>2</mn><mi>a</mi><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>×</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>×</mo><msup><mfenced><mrow><mo>-</mo><mi>a</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>x</mi><mo>+</mo><mi>a</mi></mrow></mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup></math></p>
<p> </p>
<p><strong>Note: M1</strong> is for use of the chain rule.</p>
<p> </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mrow><mo>-</mo><mn>2</mn><mi>a</mi><mi>x</mi><mo>+</mo><mn>1</mn></mrow><mrow><mn>2</mn><msqrt><mo>-</mo><mi>a</mi><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>x</mi><mo>+</mo><mi>a</mi></msqrt></mrow></mfrac></math> <strong>M1A1</strong></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><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>2</mn><mi>a</mi><mi>x</mi><mo>+</mo><mn>1</mn><mo>=</mo><mn>0</mn></math> <strong>(M1)</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>a</mi></mrow></mfrac></math> <strong>A1</strong></p>
<p> </p>
<p><strong>[2 marks]</strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Value of local maximum <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msqrt><mo>-</mo><mi>a</mi><mo>×</mo><mfrac><mn>1</mn><mrow><mn>4</mn><msup><mi>a</mi><mn>2</mn></msup></mrow></mfrac><mo>+</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi>a</mi></mrow></mfrac><mo>+</mo><mi>a</mi></msqrt></math> <strong>M1A1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msqrt><mfrac><mn>1</mn><mrow><mn>4</mn><mi>a</mi></mrow></mfrac><mo>+</mo><mi>a</mi></msqrt></math></p>
<p>This has a minimum value when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>5</mn></math> <strong>(M1)A1</strong></p>
<p> </p>
<p><strong>[4 marks]</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 <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = \frac{{2 - 3{x^5}}}{{2{x^3}}},\,\,x \in \mathbb{R},\,\,x \ne 0">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mn>2</mn>
<mo>−<!-- − --></mo>
<mn>3</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>5</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mn>2</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
</mrow>
</mfrac>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>≠<!-- ≠ --></mo>
<mn>0</mn>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f\left( x \right)">
<mi>y</mi>
<mo>=</mo>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span> has a local maximum at A. Find the coordinates of A.</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 there is exactly one point of inflexion, B, on the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f\left( x \right)">
<mi>y</mi>
<mo>=</mo>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The coordinates of B can be expressed in the form B<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {{2^a},\,b \times {2^{ - 3a}}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msup>
<mn>2</mn>
<mi>a</mi>
</msup>
</mrow>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mi>b</mi>
<mo>×</mo>
<mrow>
<msup>
<mn>2</mn>
<mrow>
<mo>−</mo>
<mn>3</mn>
<mi>a</mi>
</mrow>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span> where <em>a</em>, <em>b</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \in \mathbb{Q}">
<mo>∈</mo>
<mrow>
<mi mathvariant="double-struck">Q</mi>
</mrow>
</math></span>. Find the value of <em>a</em> and the value of <em>b</em>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f\left( x \right)">
<mi>y</mi>
<mo>=</mo>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span> showing clearly the position of the points A and B.</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>attempt to differentiate <em><strong> (M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f'\left( x \right) = - 3{x^{ - 4}} - 3x">
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mo>−</mo>
<mn>3</mn>
<mrow>
<msup>
<mi>x</mi>
<mrow>
<mo>−</mo>
<mn>4</mn>
</mrow>
</msup>
</mrow>
<mo>−</mo>
<mn>3</mn>
<mi>x</mi>
</math></span> <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1</strong> </em>for using quotient or product rule award <em><strong>A1</strong> </em>if correct derivative seen even in unsimplified form, for example <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f'\left( x \right) = \frac{{ - 15{x^4} \times 2{x^3} - 6{x^2}\left( {2 - 3{x^5}} \right)}}{{{{\left( {2{x^3}} \right)}^2}}}">
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mo>−</mo>
<mn>15</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>4</mn>
</msup>
</mrow>
<mo>×</mo>
<mn>2</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>6</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mo>−</mo>
<mn>3</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>5</mn>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</math></span>.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" - \frac{3}{{{x^4}}} - 3x = 0">
<mo>−</mo>
<mfrac>
<mn>3</mn>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>4</mn>
</msup>
</mrow>
</mrow>
</mfrac>
<mo>−</mo>
<mn>3</mn>
<mi>x</mi>
<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 {x^5} = - 1 \Rightarrow x = - 1">
<mo stretchy="false">⇒</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>5</mn>
</msup>
</mrow>
<mo>=</mo>
<mo>−</mo>
<mn>1</mn>
<mo stretchy="false">⇒</mo>
<mi>x</mi>
<mo>=</mo>
<mo>−</mo>
<mn>1</mn>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{A}}\left( { - 1,\, - \frac{5}{2}} \right)">
<mrow>
<mtext>A</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mo>−</mo>
<mn>1</mn>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mo>−</mo>
<mfrac>
<mn>5</mn>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p><em><strong>[5 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="f''\left( x \right) = 0">
<msup>
<mi>f</mi>
<mo>″</mo>
</msup>
<mrow>
<mo>(</mo>
<mi>x</mi>
<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="f''\left( x \right) = 12{x^{ - 5}} - 3\left( { = 0} \right)">
<msup>
<mi>f</mi>
<mo>″</mo>
</msup>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>12</mn>
<mrow>
<msup>
<mi>x</mi>
<mrow>
<mo>−</mo>
<mn>5</mn>
</mrow>
</msup>
</mrow>
<mo>−</mo>
<mn>3</mn>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mn>0</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for correct derivative seen even if not simplified.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow x = \sqrt[5]{4}\left( { = {2^{\frac{2}{5}}}} \right)">
<mo stretchy="false">⇒</mo>
<mi>x</mi>
<mo>=</mo>
<mroot>
<mn>4</mn>
<mn>5</mn>
</mroot>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mrow>
<msup>
<mn>2</mn>
<mrow>
<mfrac>
<mn>2</mn>
<mn>5</mn>
</mfrac>
</mrow>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p>hence (at most) one point of inflexion <em><strong>R1</strong></em></p>
<p><strong>Note:</strong> This mark is independent of the two <em><strong>A1</strong> </em>marks above. If they have shown or stated their equation has only one solution this mark can be awarded.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f''\left( x \right)">
<msup>
<mi>f</mi>
<mo>″</mo>
</msup>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span> changes sign at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \sqrt[5]{4}\left( { = {2^{\frac{2}{5}}}} \right)">
<mi>x</mi>
<mo>=</mo>
<mroot>
<mn>4</mn>
<mn>5</mn>
</mroot>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mrow>
<msup>
<mn>2</mn>
<mrow>
<mfrac>
<mn>2</mn>
<mn>5</mn>
</mfrac>
</mrow>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>R1</strong></em></p>
<p>so exactly one point of inflexion</p>
<p><em><strong>[5 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="x = \sqrt[5]{4} = {2^{\frac{2}{5}}}\left( { \Rightarrow a = \frac{2}{5}} \right)">
<mi>x</mi>
<mo>=</mo>
<mroot>
<mn>4</mn>
<mn>5</mn>
</mroot>
<mo>=</mo>
<mrow>
<msup>
<mn>2</mn>
<mrow>
<mfrac>
<mn>2</mn>
<mn>5</mn>
</mfrac>
</mrow>
</msup>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mo stretchy="false">⇒</mo>
<mi>a</mi>
<mo>=</mo>
<mfrac>
<mn>2</mn>
<mn>5</mn>
</mfrac>
</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="f\left( {{2^{\frac{2}{5}}}} \right) = \frac{{2 - 3 \times {2^2}}}{{2 \times {2^{\frac{6}{5}}}}} = - 5 \times {2^{ - \frac{6}{5}}}\left( { \Rightarrow b = - 5} \right)">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msup>
<mn>2</mn>
<mrow>
<mfrac>
<mn>2</mn>
<mn>5</mn>
</mfrac>
</mrow>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mn>2</mn>
<mo>−</mo>
<mn>3</mn>
<mo>×</mo>
<mrow>
<msup>
<mn>2</mn>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mn>2</mn>
<mo>×</mo>
<mrow>
<msup>
<mn>2</mn>
<mrow>
<mfrac>
<mn>6</mn>
<mn>5</mn>
</mfrac>
</mrow>
</msup>
</mrow>
</mrow>
</mfrac>
<mo>=</mo>
<mo>−</mo>
<mn>5</mn>
<mo>×</mo>
<mrow>
<msup>
<mn>2</mn>
<mrow>
<mo>−</mo>
<mfrac>
<mn>6</mn>
<mn>5</mn>
</mfrac>
</mrow>
</msup>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mo stretchy="false">⇒</mo>
<mi>b</mi>
<mo>=</mo>
<mo>−</mo>
<mn>5</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>(M1)A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1</strong> </em>for the substitution of their value for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> into <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right)">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span>.</p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img src="data:image/png;base64,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"><em><strong>A1A1A1A1</strong></em></p>
<p><em><strong>A1</strong></em> for shape for <em>x</em> < 0<br><em><strong>A1 </strong></em>for shape for <em>x</em> > 0<br><em><strong>A1 </strong></em>for maximum at A<br><em><strong>A1 </strong></em>for POI at B.</p>
<p><strong>Note:</strong> Only award last two <em><strong>A1</strong></em>s if A and B are placed in the correct quadrants, allowing for follow through.</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;">
[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="specification">
<p>Juri skis from the top of a hill to a finishing point at the bottom of the hill. She takes the shortest route, heading directly to the finishing point <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mtext>F</mtext><mo>)</mo></math>.</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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"></p>
<p>Let <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> define the height of the hill above <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>F</mtext></math> at a horizontal distance <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> from the starting point at the top of the hill.</p>
<p>The graph of the <strong>derivative</strong> of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> is shown below. The graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>′</mo><mo>(</mo><mi>x</mi><mo>)</mo></math> has local minima and maxima when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> is equal to <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>,</mo><mo> </mo><mi>c</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi></math>. The graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>′</mo><mo>(</mo><mi>x</mi><mo>)</mo></math> intersects the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> is equal to <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>,</mo><mo> </mo><mi>d</mi></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math>.</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" 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"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Identify the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> value of the point where <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>|</mo><mi>h</mi><mo>′</mo><mo>(</mo><mi>x</mi><mo>)</mo><mo>|</mo></math> has its maximum value.</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>Interpret this point in the given context.</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>Juri starts at a height of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>60</mn></math> metres and finishes at <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>F</mtext></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mi>f</mi></math>.</p>
<p>Sketch a possible diagram of the hill on the following pair of coordinate axes.</p>
<p><img src="data:image/png;base64,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"></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"><mi>a</mi></math> <em><strong>A1</strong></em></p>
<p><em><strong><br>[1 mark]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>the hill is at its steepest / largest slope of hill <em><strong>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><img 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"> <em><strong>A1A1A1</strong></em></p>
<p><strong><br>Note: </strong>Award <em><strong>(A1)</strong></em> for decreasing function from <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> and increasing from <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math> to <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math>; <em><strong>(A1)</strong></em> for minimum at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi></math> and max at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math>; <em><strong>(A1)</strong></em> for starting at height of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>60</mn></math> and finishing at a height of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn></math> at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math>. If reasonable curvature not evident on graph (i.e. only straight lines used) award <em><strong>A1A0A1</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>This was one of the weakest questions on the paper. Many candidates failed to appreciate the significance of the absolute value and gave <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math> as the maximum value rather than <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math>. Another common error was to interpret the maximum value as greatest velocity or highest point rather than the point where the hill was steepest. A few candidates drew a graph that went from the starting point to the finishing point. What happened in between, often, showed little understanding of the relationship between the graphs of a function and its derivative. The section of the syllabus that mentions understanding derivatives through graphical methods needs more support from teachers.</p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>This was one of the weakest questions on the paper. Many candidates failed to appreciate the significance of the absolute value and gave <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math> as the maximum value rather than <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi></math>. Another common error was to interpret the maximum value as greatest velocity or highest point rather than the point where the hill was steepest. A few candidates drew a graph that went from the starting point to the finishing point. What happened in between, often, showed little understanding of the relationship between the graphs of a function and its derivative. The section of the syllabus that mentions understanding derivatives through graphical methods needs more support from teachers.</p>
<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 slope field for the differential equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><msup><mtext>e</mtext><mrow><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></msup><mo>-</mo><mi>y</mi></math> is shown in the following two graphs.</p>
</div>
<div class="specification">
<p>On the second graph,</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate the value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac></math> at the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>0</mn><mo>,</mo><mo> </mo><mn>1</mn><mo>)</mo></math>.</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>Sketch, on the first graph, a curve that represents the points where <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mn>0</mn></math>.</p>
<p><img 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"></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>(i) sketch the solution curve that passes through the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>0</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math>.</p>
<p>(ii) sketch the solution curve that passes through the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>0</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>.</mo><mn>75</mn><mo>)</mo></math>.</p>
<p><img 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"></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><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><msup><mtext>e</mtext><mn>0</mn></msup><mo>-</mo><mn>1</mn></mrow></mfenced><mo>=</mo><mn>0</mn></math> <em><strong>A1</strong></em></p>
<p> </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 style="padding-left:30px;"><img 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"></p>
<p>gradient <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>0</mn></math> at <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>0</mn><mo>,</mo><mo> </mo><mn>1</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p>correct shape <em><strong>A1</strong></em></p>
<p> <br><strong>Note:</strong> Award second <em><strong>A1</strong></em> for horizontal asymptote of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>0</mn></math>, and general symmetry about the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-axis.</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 style="padding-left:30px;"><img 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EpyQA7YrNmyS83dief74UcfS18yRVJ82de8ePGi9HcKCn1TWPrAGoPqv/9eI0pygD7Kmct1knhZGDaa2PfwxagD+zS7d+8WpagxusykcezYMVqyVGlRMpfnz545VWZ4rni+MpcRQWDgYTp//nxRCgMd00qFi3vFPu7t27dFjed4I+P9+/f5LAzLxLLImzcvH2xi1imz7wQHB9N69errGvyZNUC8ceOGZSsmT130CwyCGjRoKErywGpJ5AEY2qOVRg0Da2+W3x3xVEa0WUwiZOv99u07cN3kSwwbzaBz5/gIa9IkuUuXAMZEz4s2c/MYI8nt27eLknlEdgRyBCNmbITLbqAYre/cuVOUrHEEigzeZ40aNb02YN7KiEFJrVq1DO9/60Xb08T+puwlYm2Q8ODBA1HjHCOOQJHB/cFYm92Go5pNYp+xTZu2UvuNMwc+qxyBHGnUqJGhpWFvZISxlr3KeGD/ft52ZetCR0wxmqBZs+bS1/Mxap4w4TdRco3ZHlfYpzF7HzcqownWrl1Hx48fL0pygAHLlTsPf85AhtEEMGAYxXrzjI3IiI4IhTdv3jxRYx2OjkC4TzjNPLVYsTqCWSTacVSrMGYaTXDy5L98Zm0m7pZgx4wZSxcsWChKcsCsz1EvyTCaGAi5cuDTg7cyYh/XCudAV0CX+8JYO2LYaF4X3kwYtaLjyyYgoKVbZWP27AEddfTo0aJkDljScbcchucre5kWyzaY0QPIJ6uDQGEvXrxYlPRjhoxmLedHhUzvWVdgkBDVSo0VzwFel2a24cjes85Av5G59I7nCg9/bbBplfdsZDBAcFwZ8gRvZcTS++DBv4iS9Wi6HMZaxjN1hqmxZxHJBgfzZcJG6SRatGj8cL5MmJEjH3zwgSjJAa8KP7LvVWE+Kvbsmw0znPwnevToosZ6oBvYQEe6XvKF3mcGkz9bX+hCU68o+8GB9957zycPTnbDBIigowymQmF/0E9lGkwA/eALveQLvR8jRgyf6ULDV2XTc/HJvpgZo9EqMELErNnOQD6M8OyMP8joLzyxQZxPdyB5u91Be7R73/YHGYEddLlho/n40SPxyb74QyZ/xH38z+aBxiHfq5cvRcme+IOM/oI/KNGnoaHik31Be7R73/YHGYEddLla61MoFAqFQieGjWYcH6yhe0rs998Xn+zLe1ijjxZNlOwJ5IsmeZ/GU/xBRn8B/gJ2J3asWOKTfUF7tHvf9gcZgR10uaneswqFQh/Ke1ah8E/U8qxCoVAoFDpR3rM2QXnPmoM/yOgvKO9Zc1Des+ahvGclobxnzeFN9Z49fPiw+KSPPXv2kIULF4b/nD17VvzmzcIflKjynjUH5T2rH6+M5pkzZ8QneTx79ow8fPhQlORx7lyQ+CSPp0+fik/yQPSSt42FCxeQHDlykEqVKoka91y+fJkULVqUVKtWjf/UrFmDH7RW+B5fuGeEMqPti+sGBcnXS0ePHpU+kILeR/Q1O+GV0Vy9ejU5ffo0/5wqdWr+r9W8ZCOhrl27iZJnJE+RQnzynICWLfnSqdUkSJiQxBSegOvXr/d49mOUtWvXkiNHjoiScyCfmd5rQUHnTVc4nshYtWo1UrlyZVHSx++//062bNnC5cbPixcvSQoD7cvOxI8fX3zSDwZfMpXcJ598wv9FKLfevfvwzzKBHlyyZIkoOQftUevbZtEioKWpy+d6ZIShHjp0mCjJAUa6Z89eomRMl5uFV0YzICCAtGzVWursJE6cOOTLLzOQixcviho5zJ83l3Tr1l2U5FC+fHnSsVNnqaO6UqVKkWHDhklVeFu3biEtWgRIG6njOgcOHBSlMOJ4MAjAqPfq1askZ86cosa+YJB57NgxUZIHdELTps3IXcl7jQjllihRIrJLskdylixZyMqVq6Svgs2dM1u6XqpYsSK5cuUyH6DIAmEBP//8c7Jp02ZRYwOYIvEKJGRG+qb796POyWcmrEPSIt9+Jz3TB1L97N9/QJTkgCwfkfPyWQ2eq1XJgl2hJUWWdU0kLq9duzY9ffoMz5KAbDWpU6cWv42ayZMnw7rTXLly0Xbt2rlNS6XRu3dvOnLkqAip86zOcoK+gvRJaEMyM3wAvEskgL5w4YKokQPuGfdrJIm5NyBDCNoUri+ThQsX0UOHDomSHNBnkFJOJniuefPlD88a42u8Nppg6tSprJE2FyU5QFFlzpLVo1yL169dE5+8B50CncMqoNgi//01a9bQ1atXi5IckCwYaXecKQA8cysUMK6J9GNmDIb0yKgZywYNfvLIaG7evJmOGTOGt4UECRLQuHHj0n82bhS/jZpbt27RVq1a05YtW9GgoCCaJ08e8RtrQco5GBIYUE8HJkaMHtoP7nfLli2ixhrOnj3L09dp4B7RlqxIaxYVmEQgNZcz0B490Vd6wTOGATNDL3kiIyYQS5cuFSU5aCkKL5qcG9kbov3MCJtzek62bNnIggULyXvvRSdp0qThUeexzn7t6lXy4P59/qPtizjWs5dD3hfLYq7q77PyzRs3XqvHEtmdO3fI3LnzSPHixcKjlsAV+Tb7cbwmwHLapZAQwjoTn+prkfGdfT8q2fH/DRgwkOTJnZswhRle7+09QW5N9hvXr5MQJiOWPbCvEDNmTF6P6zDFQzJk+JIkT/4ZrwOu7jUkOJjXOdZHdU/O6llHJNeuXSdTpkwlRYt+Fy4j61T879+8eZPEih2bxBL7H67uyVFGV/XaNfG3PogTh9Sr34DkyJ6dJEuWjNcDT+/p4oULXEZkfNCiVTl+H3Lgevny5SPp06cn586d5d6wbdq04d8Fru4J7/EzJluRwoVJm7Zt+VLg2LFjSZMmTcgtdk1HWerWrct/N336dP6zYMECcuPGdXLq1Cm+9wYX/+zZshNmsPn3HWV01ZaA9twj35MmY+R6bGuUK1eOLFmylDRvEYClJb7NobUxx+fr+HdQf+nSJR4lxjGLhbP3AbR6TXY8/0yZMpFevXqTffv2kYxfZSTx4of9Hb2ya/Wu2hKcstBvcM3YrE2inr/3OO+z6/YhuXLmJB9++CH/LtArO3CURU89dMvw4SPY832HpE6TOlxGLFOjTaL/YN9Qq3eUxfGe9NRrsuNeXzx/QQYN/oUUKVI4XBbgyT2hvV84fz5siZn9Ta3e1XOPEeM90rZte/LVl1/yJXFPZHdW7ygLcFaPz/Dw3b59Jylc+Bte5zOE8fQapjj4Mq2zmYmVTJw4iY/u9IBs32awdu06vtRmBUim6yxzOmssfMnUcTQtg+XLl/MfRzCiZQMWUTIf3CtGzkaShnsqozbTXLFiBT1+/ISo1QeW0OPHj+c0izzkwHKsq5+MGTPSyZOn8FkrZp6yQBLomjVrUTaYFDWu0du/3HHixElaq1YtS7ZVsDyJdhMZpvClb29ABzpbkUJ7jFxnJliNgm4ygqcyYrWmRo2a0vV++XIVfL5Ma9hoQgGgc40bN07UyAEvC9N1Zx0mMnjBZvHzz33pqlWrRMk8YBRdNUA24vd4SdoMunbtyvfxNCCf1cYb12jcuAndvXuPqPEMT2XUjCYM9aRJk2hAQEvKRt7it+5hszg6f/58UdIPm+nyf/FONeNpZLDgCbjm0KHD6PDhw6NsU+jbZoHlPwz+8K+ZPGUDTVf9Zvv27XxAL2u/HDx48IDv5zoSVd82i0aNGhl6tt7IGBh4mO+ryuTy5ctc71v9PKPCsNHUaN++vW7HCLO4du0a3zeRDZS6J4rVDOC40qJFC1GSAxomZtZwhJIJrtuxYyc6ZMhQyztH5D1NKL2uXbvpnq1WqlSJ78l6SmRHIBgv2Q4s6K/t23cQJevBTBOzXJn3iVkuZpwyleymTZv43rXMa2LvGg51smndug1fcZEJZppwrPMVhiMCaQwYMIDUqFlLlOSQJEkSEjfuB+TSpcuiRg4jR46QfiYsXbq05PPPU/I9MVlgz6Rv37783CjrGKLWenDdYcOGko8//ojvq8gkbty4ZPDgQSRhwoSi5v+wzio+hXHhwgWSOXOm8D1uIyDLP/aHZIIzjiNGDBcl68G+4/Tp08jYseNEjfVg/zZXrpxkzZo1osZ6ihQpQu7cvSP1mtre9aRJv4saOQwaNJB0795DlOSQPHlyfvTQZ0EPhPH0muvXr4tPlC9bYrlEJvCSW7duvSg5x4plL2+8EaMCI0V3y8iYjZw44dm+mxlALizV4V8r92bMwFMZMXtnCofeuHFD1ISBv9G0aVPWtsL2ivD7mDFj8pklvMZnzZpFf/311yiXN6PC6iMnZiDbA9Ub4AvgDsz4vFkNMAJ0g6aX0JbM3CJyBe6ze/fuXuklIzIuW7Zcyv0BTZdjpc9qz2xXGE4Ndj4oSFpUIG+BN5YdIklEBQJPYyRuduQQM3kWGsq9a/VG3PEFnsi4ceNG7omt8d1334V7WzIlQFq3bk2qVqlCihYrxuuuXLlCduzYwb1JCxQowGel3uIPqcHgVenoZWpH4B2vRQWyK4iXCu97O/dtf5AR2EGXK6NpE5TRNAd/kBEoo2kOymiagzKa+lFJqBUKH6CSUCsU/olpjkAKhUKhULzpKKOpUCgUCoVODBtN7GnaHayD2x3saWI/zs5APjskgY0Kf5DRX8Cept3BnqbdQXu0e9/2BxmBHXS59Jmm2kJVKBQKhb8i1WjigDxyo8k8KK9QKBQKhVlI9Z7FUYCSJUuSgwcPksDAQB7ZQaF4G1HeswqFfyJ1polzQMuXL+cHyFOlSkW2b98ufqNQKBQKhf3x2Ghitnj8+HFRCsvR5gk4wI+ZZrp06UjRokUj/K2oOHfunNf7oZ7KqPHs2TOe41IGL54/5zk/AXLw2RG8e01Gu+IPMvoLL168EJ/sS+jTp+KTb9DTV9Ee0S7NAttbZusldzK+fPmSbNiwUZTkceTI0Qh631tdbiYeG03MFn///Q+eoBZcv3aN/+sJSNh64sQJUrFiRZI3b16eNNgdCGjdqVNnUfIMJFL1BiRLrVK1GnkqoWM+fvyYvBRKqlu37rqeiZmUr1DRrbckDPtzNpAwAyhkK56rmTK+7XijoHgiY4ncFf4R8+bN4yEOZTNt2nSyc+dOUXIO2iPapVlAf1auUtXUQY07GaF/8XyXLl0qauQQLdq7Ycm9Bd7qcjPxanl29OhRpHPnLuTGjRuixnOQyWL+/Pk83meaNGncOgfBgQjZA1auXCVqrAcDhL8WLyJNmzaT6vX7228T+ABBpsv/sqVLSEBAQIRYrFYSGhpKfqhUmd3rRGnPFkpG5nv0JTB4vrjXa2wQXadOHWkrNBo//vgjWb58BVm3br2okUO7dm3JzJmzSEhIiKixHhgw6KWffmok9R336dObnDlzVvfqoBlkzJiR24eBAweJGhvAHrpXICtHxoyZ6JIlS0WNd7x69YqWLl2aJkqUSFdeNuTtXL9+gyjpw2iWk4sXL9K2bduKkjVEznKC54IEukz5iRrr+e+//3gCXVeZ0SGfJxlE9HD8+AmenBiJts0gKhmRPxLJegcNGszzLFqdUBsg68/06dPp2bNnI2REsTrLCa5Xtlx5nncQeWe9wdssJ3gHyB6DBMVoU1YSOcvJrFmzeQYimeAekcvSVSYVq7KcXLhwgXbo0FGUjKFXRtxrnTp1ed5ZmYwZM5aOG/ertETtUeG10QRQAiVLljHcIPD/p0uXjiZLlsxt9nG8NCR43bBho6iRw9q166QnY4YhRZZyGFBZ4PlC4e3bt0/UWA/ub/jw4exnhJR7RZozvE8kE+/ffwAfFFkJ3iOeJ4x1pcqV+TW//PJLKYnModzmzp1LS5cpS+fMmSs1UTwy+2NAdOrUKVEjh6VLl3LjKRO0W9wr3rVMVq9eze9XJtDXsvUSgNGUrYOdYchogl27dvMcbkbBCCJWrFg0Q4YMosY1UOx58+X3OpehtzRp0pTu3btXlORw+vQZ2rVrN1GSg/Z8Q0LMmf3pZfPmLTRrtuxSZoAaMCoVv/+er5rgWcsARvvTT5PRHDlz0TRp0vIZmQxle/Lkv7TQN4Vpga8L0uDgEFFrLVgpwfXM0BGeMHjwYJ73VCbILVy4yLeiJA8Ya8e8xjLASs2AAQNFSQ7QS1myZJOqH5xh2GhiCWfo0GH02LFjosZ7/vnnH/ruu+/SDh06iBrXYCkEMwU9mJVMF4oNyvXWrVuixjyeP3vmsjHMnDmTKwE0GllA2WE06WhIMLK0usFiJoRlaW/bk7cyYtY3e/YcWrJUaTYzm2e5EtKWZ7VZ6MCBA2mVqlUtWcaLDJ7xiBEjaf36DejOnTtdtiszZdm9ezdfwjR7dv3UxfYF7ql16zZ8WU9mv1mzZg2dP3++KIWB9mjlrAxtqGbNWobu0xsZsVVm9SpNZCZPnkIDAlqKkm8wbDSDzp3jL8usdW4YTmy15mNKxV2n3b59O5+yuyPYxBeLxoUGanZWe+zNhD59Kkqv88cff9DOnbuIkhzQiQICAujixYt5GfI9kZDNH+1p4sRJ3Hh6upxohox32LvYsWOHKFmD1XuaesBzPnPmjMt+q8fHwBOwb4alf+w5mmXI3LUPzDabNWtu2vX0gEEXZmHaNdEeo+rbZgCfgFatWouS53gjI+6vdu3aUpekocuxtSJ739oRU4wmgIHDKN2MEdWyZcu44YSDkLu/h72LrVu3iZJzzDSaAI2kRo2apnZEd0YTOBvFymD58uVcEcgymhoYoGzatEmU9CFbRm+xg9F0h9lGUwN7nWY5lOkZVN24cYPPcmUaThhrzMSADKMJNmzYwAe53tyntzJi8mC2LowKTZdjAoElYl9g2Gg6LmPdvn2bbtmyRZSMcfr0ae5Rq8dw9u7dW3xyjhUeV/ib+/fvFyXjwBDrWQ6zegbkCsxGIJ9Zys4q/EFG4A9G0+zVFCuI7D3rCuwjyzSa4NChwPD2qKdvm0FgYCA9ePCQKOnHiIyXL1+Wppc0XY532bdvP/5ZNlJjz3oKM8L8fOY333xDVq5cyc9NKhRvAir2rELhn9jaCiFG7fnz58nWrVv54WUb23eFQqFQvAUYNppGogLp4eOPP+aGc/Xq1aR69epeGc5bNgi95I7Hjx/bPtYn5HtuYjgwK/AHGf0Fb8LoyQbJ2+0O2qPd+7Y/yAjsoMsNG83Hjx6JT9YBw4mMKAsXLiQ9e/YUtfrxh0z+iPv4n92DoTP5Xr18KUr2xB9k9Bf8QYk+DQ0Vn+wL2qPd+7Y/yAjsoMv9ZpMwW7ZsZObMmWTQoEGkR48eaqlWoVAoFNIxbDTjfPCB+GQ9tWvXDjecnizVxn7/ffHJvrwXIwZ5N1o0UbInkC9a9OiiZE/8QUZ/AVl+7E7sWLHEJ/uC9mj3vu0PMgI76HJbe8+6IjAwkBQsWJCUKVOGpwRSXrUKf0N5zyoU/olfWhss1WrOQeXKlTM1watCoVAoFK4wbDSt9p51heZVi+Mo7gyn8p41B3/wTPUHGf0F5T1rDsp71jyU96xBYDgvXLhA9uzZww3n06dPxW8iorxnzUF5z/6fS5cuSU0S7gv8QYkq71lzUN6z+vH7zcCPPvqInDt3jhvOrFmzujScCoUZLFy4gJQtW5YkT56cnD17VtQqFIq3BY+N5v79+yNMkRMmSiQ+WcdLNgpavHixKL1OggQJyO7du0lQUBApVKgQeRVpxBQvfnzxyTOWLFkqPllPrNixuQcblmnt6psV/b33+I+dsVrGqlWreXVW2B+JFTOm+GRfPpDovR8Z6JnIusYZZrZJbEP16fOzKJmHOxm3bdsmfbkeE6AVK1aKUhje6nIz8dhoZsiQgVT/sUb4jA4Gy2qiM2Oyd+8+snnzZlHzOunSpSPr168nhw4dIpkyZSKPHJaN43v5oJMkSUxat24jxYjFihWL3+edO3dIo0aNpDo3zZ8/nz3fvaLkmmjRotn+GIIMGWW0eTsQ0w+Oc8SNG5c8ePCA/PbbROmDTewD1q1bz+3qFtoj2qUZ4KQA9FmnTp1NvV93MmbOnJnky1+A3Lx5U9RYD3TimjV/84mahre63FTYg/eYhw8f8lyHVmQPiQqkg3GXKoqNiCh72DxDChJVGwUJgpEDEFH1ZYF8jkgALTNPHRvR8aTeTBGIGmtBwmc2wKEvXrwQNf7DqX//hbYylOXGH7KceArazpQpUy1P4O2MCxcu0GLFS9CrV6+KGjloutCqNGqu8IVeQqq+Bg0a8lRrssD94T6Ras0ueGU0ARpLjpy56KRJf4gaOcBwIut8VI0FxrxYsWKUjUL5C75+7Zr4jXecPPkvbdiwoaUNFM/TMaUVDCYMJwyoLJBVH0llYcycAQMHOc0Af+vw4cM8QXC/fv1oSEiI+I0xopIRyhxtA8/WyLs0w2jmyZNHfLIGpGtCqj4j6b1giDwFKeTGjx9Py5WvQHft2sUVrZWcPXs2/Bp49x06dOQJzGUaE6QuRL5OV88L7dGKwSEMp1kJtvXKiGv5wnBC50P3X5M8KHKGoeAG+F8rVa5KavxYnVSpUpkvHWCp4pbDFD55ihT8X8f69+PE4RlMgKt6LLncv3ePf3asP3PmDBnyy1Dy7MUz8tv4CeSDuGF7Gkgj9uTxY/4Z10T5hx9+4MuOvXr1Ig0aNCBJkyYND4QQ+fsgKtnnzp1H1qxZQ/r360fSZ0gfXu/tPX308cckduzY/DNkYY2BxIgZkyRLloy8L6JenDx5ktSv35AM6N+PFC9RnNcBZ7KDkOBg8UnfPTmrZ8qWdO3ajcSIEZP88ssg7qEMsJ/BRvEklP1/SNf2Qdy4vD6qe9JkdFWvXfPUqVOkf/8B5Oy5c6RN69akbJkyJF78ePx3ntxTaGgolx8yJv3kk9fex/z5C8i9+w/IA/Zz8/ZNcvP6DYJF8Lx585A0qdOwe/2IJE/+GWvY75B33oGT2YevyY5rwGM8fYYMfNkoZ86cTmWcO3cOuXbtOj+StXff/5eXNAIPHSL58uUn70YLa4/0Pwxg/yN169YlxYoVddqW4idIQOLFC3sujvWOMmr1U6dNY7I+J3fv3iVssMD/jREzBsmdKydJnTotSZz4Y+7M9OjRQ/6OEVnH8e9cuXKFBLP7SpkyJe83Gs7uFWj1WnvH9RYtWszlYAMEUqF8eVKoUEG+DOhOduCqzTjWX7t2jQSxNvMJe9effPopX84DPbr3ICf/PUW6dO5E8ubLy+uAXtmBoyx66tHu6tSpR35kurBOndrhMsLD+lJICPdZQN/WZHSUxfGe9NQ7ys5m9lwvjR49inzKnoGGJ/f06OFDcvHiRS7jR6wO7Qy4eu7or1iWHjpkCMnD+o4nsjurd5QFOKtnEwrSo0dPvu02ceJvvM5nwGga4XBgIB/tbNu+XdTI4cCBA7RJ02ZRjmQxAqxVqxZlhpLOmzdP1HoPRleZMmcxbVakgZEUUwB8lB551Ih7aNeuPR0wYKAlo1VX3Lx1i5YoUYouXLiIP2PIhRkaZmuRZTQL/N1z587x9tSlazf+jj25Z29l1P6/mzdvUTZQoStXrqSzZs12mcxaz0yTKRs+enf2g/ecK1cut8nVrQDXfPgQz+gGl3/x4r/4jD8y+N4RVm+GjHiOO3fuouXKVaCjx4wxbVsHsqGNsAHda+8bS8XNm7eg/foNkJZMGzLMnjOHTp48Jbzdog7t0ejqRlRAd9SsVdtr/eCNjNAJ9es3MF0XRgXed9u2bfmM05cYNppaRnSsO2O5QCZ61vXxu1atWnElt2DBAlHrPWgsWJ5go3BRYw6aYXIFFBuWTmXtOQLIs3r1alqzZi2+Z4NyVIMUM4GixbXRMT1ZCpIhoxnLs/n8YE8TfdtsoJyHDh1Gx44dJ2qMwWZPUfab8+cv8PYbGPj6wMAq1q1bTwsX+ZYPjoC7vm0G8N/ABMFbw+mNjPg+9C+bpYoa64H+27t3L+3WrbuokY9ho6mhPUBfGM7WrdtE+cLxu2nTpnFFV6RIET4yNYKv7hWdEE4HWmeUBRpqq1at6b/MWCiUI5C/gf6KFZO1a9eJGuvBbBqDXJnOkhjY5s2XX6pe8qXe95XhNM1oAjxAOAfJJjAwkK5fv0GUXNO1a1eu7DJkyGB42Qn3WqlyZVGSB5ZQZDdQRUSU0VToAbM3PXrJTHBN2XoJuhCDednAo/bKlSuiJA/DWU7g6JA4cWJRsicIxoDIQbjVsWPHks6dO/OoLosWLbJNhhQENYgRI4atz0G+ePGCP0PIaVdkyDht2jTuWMY6Lalfv76o9Qx/yHIC5y/NKc2uIPZsgoQJRcmePH/+nLzzzju27tv+ICPQdLkvMWwxfBl7Vi9avEI0ijZt2nDvr3/++YcHamAzN/47X6Niz5qD1TJu3LiRxIkThyxYsID/Gxh4SPzmzQMDELujYs+ag4o9qx97TLMkAzd6uFjj+ACOT/hDFhSFPShatCipWrVq+E+2bNnFbxQKxduAXyahNgucW6pSpQrZsGEDWbduHSlcuLD4jUJhLSoJtULhn7yVM00NHLJduXIlqVevHilSpAiZNWuW+I1CoVAoFK/zVs80HZk7dy6pWbMmyZgxI4/on9DmzgUK/8bXM010ezifYU8/JCSERw3Syg8fPuS/R+YgOPoBBPOG8xxAVCI4CCHDCBIlJEqUiEe6QdkujnUKhVUYNprng4JIqtSpRcmeICyTY+gpV8BYIpn1q1eveGiqggULit9YD7wAMfO1c2aJZ6GhPPtKbBt7VPqDjECW0URbRuacffv28TCImzZtIjt27OAh8vCcNODUhOwt+NFCJ2K/HzlqNZYvX87/RXg1hIe7efMm3+JwJAXrZ19//TUpWbIk+eqrr8gXX3zBjSyc8KwA94QwenYGzisYTNi5b/uDjECvLrcSZTQjASWQPn16PvqGEZVlOJXRNAdlNMNi8MI4Dh48mOzcufO1nI/ZsmXjcZlLly7N45XiyJizowYwjO5SMSEmKGapMMyrV68mf/31FwkMDIxwTaS8g7/AkCFDSJYsWUw91qCMpjkoo6kfZTSdgGWqypUrc8WD0TVGzVajjKY5vI1GEwO9Y8eOkYMHD5KJEyeSI0eO8GeAWSMMY/ny5UmuXLn4DBKzPr1LqHqMpjNwbfy/ONoFuXCu9ejRozy4P87P5s6dm7Ru3Zp7IjsG6vYGZTTNQRlN/ag9TRfgsaBj//rrr6Rt27Zk5MiRli0xKd4+jBrNS5cu8aTsv/zyCzl+/Divw1EqOLVhkIe9eS0jil24d+8en/kiqMjixYu5EcVB9Vq1avFBKgy7lgFDobArymi6YdWqVaRSpUp8eQlLT3B2UCiM4qnRRKABzNyWLl1KxowZw88WwwEnICCAVKhQge892j2aiyNQO0inBcM/evRocvjwYfLy5Us+Cx0+fDjJly+frSNPKd5elNHUAXILIicgPAtnzJjBl7vUrFNhBD1GE0YEDjzjxo0jCxcu5GXMJvv27Uu+++478tlnn4lv+j+4t0OHDvEBwfz58/kSb/XqyE1Zh99rzJgxxTcVCt/isX/48uUr+ChRA3uaVoPE00bC3WEd3FMQixEehgDOEufPnydlypQhFStW5K73Zoffw54m9uOwn2pXIB/2PuyMP8gYFTAWcKRBbFt4tMIRDY49CPuI5Uy0QySqlmEwsS8pCzgLYZaJs9KIebtlyxZy9uxZ3udwvAXHwTAbjTzGx56mr9DbV9Ee0S7NAA6KOBJkNlHJiDYJj2vZ/Pvvv6/pWW90udl4bDThht6gQUOpyh0Zz1u3bsNHorLAUteSJUv4yBcdFeU5c+ZwhQZHCxjSAwcOiG+bR79+/cnYseN4Q5UBvB///vtvPkhQ+Aa0L7QrGAbs6WXPnp3PMDHjgqHEtkChQoVI3Lhxxf/xZoO+hsHC3r17uQHdunUr37eF1y+cfvBcHNsrFLp2nlQms2fPJh07dpIaoxfOXd279wjXSzKAg1Bo6DOSL38BvpcuCxxfatmyFRssHRE1NoE9eI9B9nnkikOS4HNnz4la65k7dx4dMmSIx0mGjeS0YyN7Wqx4CZ6rToN1WFqvXj20WNq6dWtTstuzEVV4gmkkuMY1WWPhZSthxpmeOnWK5yTFT1SpdiAfkkMbBddEG8K/ZmOWjNq7sAqkBkPi4OHDh9NEiRLxtsRmW3TEiBG8XhZ4B67aLxsYi0/6YMbD4//HE9A2+/fvT9nAgj+rZs2a0SV//cX797hx43jC8kOHAsW35YCk2nXq1KW7du0SNa+D9mh2ezpx4uRreskIemSE3mWDBJ6bVBZon+PHj6fMeHKdITM/qSu83tPE/4YAAKNHj2Uj4UV8KUkGWJ6oXKUqGTrkFx76TgYYScKDtnTpMqRcubCoKAAzNJx3w0wYo2H8axZ4vphhrF+/gY0qR0vZ02GdhkydOo0wBUBatWpFcuTIbsneLZ4n63jcmWXb9m18P6tM6dIkadKk3IMSzlb4sXLfGEueOKoBGU6e/JfNVK6T/fsPkJu3bvLflyhegjRq9BOPhGMmeMb79+8n33zzDT/LiNkjUtVhlolVHLOvh3aE2RqWua5du8aXd3Guct369XwGkSF9BlK7di1+NtkoWCpds2Yt2bNnD3+OmTNlJl999SX5+OPEJGPGr/g7NSPVGNoLwl/Cux36IEmSJIQpVn68BjP2Pj/3JS0DWvBtFCz7ygBewRMnTiLjxo316piON6AftWrVmlSsWIHfuyzQTyZMmMDu9zdpzlpYSejcuQvT/VVIeQcd7BNgNI3AFA7NkOFLPlOSBUYfGHlMnjxZ1Mjh55/70gJfF4ww0z179iyNFSsWZcqOzp49W9SaR0jIJZomTVq6fv16UWM9mHkMGDCQfvFFenrs2DFRay2YXeFagwcPprnz5KXx4ifgCc3r1a9PmbIX3zLGH3/8wTPbx/kgLn+m5StU5KsXGLVbPbs7d+4ca7MtKRsI8Fkl2gtWTcxYpXAG+geeH+4V/zZs+BPduXMXX1V48OCB+Ja1YOYZFHSe7t69h79H6IkPP/qYlilbji5evFh8y3ugBxYvXkRTfv45f6YJEiSg27Zt479bu3YdTcHq8c7NWHnQA2ZCuEc2aBA1cujduzefgcskODiEZsyYibIBmaixHrzvwoW/5e/Wlxg2mliOwRIBlgouXbokaq0HD3DMmLF09OjR/HNUmLlkpC2LON4rG2FTNlLnHbd69epeDSCes47tatkZnR5GrHHjJrxjygLKFUalRMlSdMuWLfT69RseL417C94pniOWv4KCgkSte2CEXMkIw4h7sspQOePkyZOUzaB528BPxowZ6YIFC/jyrJXcvn2bPz93fSMqrFiixru5c+cOPX36tKgxxlOmtNE/0D5TpkzJn/EPP/zA2wyM9vr1G/iyLZZvQ0NDxf9lHbgmBkPt2rXjbRfgnq1uc8uXL6ctWrTwevLijYx4j9CFUW3pmM3Fixfp0KHDuF7yFYaNZhAbQQN0zvbt29MlS5bwsiyOHz/B91ejMibB7EGbCRpXp06d6fz588OVEv5dsWIF/eijj2j8+PHZqH4nr9fLXdYAQ58+FSXnoEM0bdqMja7/EjXywEwJ+xlPLNyzMgM8Q1/LiPaxf/9+rryhxOPFi0dHjhxJ79+/L74Rtqdpd8zaL7OSyAp73759NFOmTPy5Z8mSha5Zs4YbMhgw7B/LGvShr0Ifwlhfv3bNbd82A23ysmDBQo8HS+gz3siItt62bVtpel/T5QcPHqQNGjSUtorgiGlGU2PDho107969oiQHzCT79u0nSq9jttHUCAw8zJ12HMHovEGDBrzTYtapd2aox2hqYInNyAzCW+xgkNzhSxkxCm7SpAmNESMGf/+5cuWiu3fvdvqulNE0B1ezHGyblC1blr8HLNv++eef3HjKBvJhJUqG0dQ4dOiQx6t+3hpNDej92bPniJJ1OOpyrBphBU42ho2mtgRhZ3zhcRUYGEgTJ07MPf2WLl3q1shhZGrFcpiZQD5fjOw8QbaMUMRYGtSW55MnT06nT5/udsnKH4ymlZ6wZoHBZlRgmRr7ffC29ZXxRHu0e9/2BxmBHbxnDRtNhWvQOdFhoUwzZ84szalGYT2YPQwYMIDGiROHvvvuu7RWrVp8dqMXfzCabxJP2SwqsvGUtVSreLNQGWMtBO7uPXv25G7xrIOSTJkykVKlSvGYmwr/g43Guat92rRp+SF7xExF+i1EzkEkmzRp0ohvKuwGkmSjLz58+JB06NCBNGnShB9/YQMf/l4VCr0YNpq+iMThKTiL50sQ8gwRTRDRCKH5UMYZs+cOUU0eP34sNbKIN0A+R5ntiBUyIpwbQtdByXbs2JFnEUHEHrR9nGd9U4P443yn3UH4SU/QjCfOqvbq1YuMGDGCn6vEv1b1P7RHu/dtf5AR+FqXA8NG87HJMVitwA6xSHFQH2HAYDzbtWvHg3DjUDbiaYIXrNH+FylZsN2AfK/YjNnOmCkjwiR+8cUXJF26dGTmzJmkZcuWvNPiID2CMLzpQfv9QYk+9TKmK1KSwXhi1QeJsTEYQjJuK0Jjoj3avW/7g4zADrpcLc9KBooWuTnRORENJkeOHKRfv37knITA9wr3INYrYg7nzJmTG8bbt2/zVFX37t3j8T7NiGijsA94n4hxi/i16I/IZtS8eXPTEzIo3hwMG804frA0Zccs/jCWFy9e5Al5R40axRV0/QYNeKgzu/JutGgkmqSwZN7ijYzYb0YYtO+//54v1SF/KvYnEe4OM0vsgckKjWYn/CE/Z+xYscQnYxQoUID3xw0bNpAFCxaQhAkT8i0UMwKUoz2iXdoZf5AR2EGXGzaaWNKwO1iKsSOYdSJ2LfbGJk2axBU3HEzq169vS+MJJWr3xMB6ZaSUkiA2u8e+Vrx48cjXX3/NY7Hu3r2bO4ZAcWK2+aYvwUaFP8yqEzDjZhZ4199++y3f78T2CZbkkydPzvujkb00tEe7D0D8QUZgB12ulmdtABorvPlOnjxJtm/fznMnasZTZiqeN51Xr15xZyw8VxgEzCbh9Tpt2jTuVYkA+Xnz5rX9wEBhLfB6b9asGblz5w7vj0hMAf8DJP9WnrYKw0ZTec+aA7xnYTwx48EyETor9lkw0sWy4caNG/kyoi+xwjPVbCLLqBlKDEq0PWSUJ0+ezAckyPpRrVq1N9YD1ghvovesJ2Dmif6I7C1YecARIyzT41/0V70o71nzUN6zkrCDx5U7HL1ntc565swZ7jCEc57FihXjCWh/+eUXqRn1HfEX71ksuyJtW6NGjcINJVK3TZ06lTv0wGMZqbiQyg2fV69ezVOF6QWprxYuXBj+gyMpbyJvsvesJ6A/Vq5cmU8Q0P+6du3KB1l6HYaU96x52EKXh8U48J7IsWftiFWxZ83EXexZxAFF8GctYTGCgSPGr8wYtHaNPYtngGTh06ZNo6lTp+bPBz8I2L1o0SKnMVQRNgyB/hFsmhlTmjNnTp6RxB2I6YkoQNo1okePxmPOeoqKPWsOMjNsaCBbysyZM3lfRIQhZkijDN1oNK6rDPxBRmAHXW7YaMrMMu8t/tD5EeaLjexFyTXIKoDUQ8jHCKX94Ycf0lWrVnmc1scbEHbsuY3iU8JYDRw4kIdF04xYsk8/pXPnzImQUcQZHTp04EZVY8yYMbRIkSKi5BrkLkQGE6P4g9H0ByUqKzeoM9BnCxUqxNsd2qCrRBXoM3YP2ecPMgI76HKPjKavHqoMg+BvYGSLWVTatGl5p0Ui7Hr16vGOq8f4+iMYoG3fvp127949fMaNHxi/GTNm8IwzUGTuOHPmDP//unbtKmooDWH/L+qQcsgVmGE0btxYlIzhzGi+qe/tTQeZbBCoHzGImzVr5lNDbgW+0r927Q8e7Wlib61c+Qrk8OEjosZ64PzStGlTHvNT5h4LwmoNHTqUe1XKAs4+u3bt0uXwAw9P7LNg3xMOLfAAPXHiBD+cHSdOHL5nx2agUuU3E+wV/fvvv2Tx4sU8hB3OzeGHzQbJ+vXreaAB3C8cGLAvWadOHe40hTBp7pg/fz7/N2PGjPxfkOyzz/jfX7p0CT/svmzZMvGb/zN79mzyxx9/kNy5c5P27dtzBxEzgDMS5N+2bZuokQOcWRYuXOSzPXJ/5sCBg2TlylVcJ8HjGs576L/wwGYDOtKiRQuuL80ETm1lypYjc+fOk9qvDx0K5O0T/YzZDFFrLdCBTZrI1/u6CLOd+sH+0Y4dO2jZcuXp2rXr6NGjR8VvrIU1QJ50dPz48R5nJ/c2nQxGjFgKRUb0y5cvi1prwD1h6QGzIFyvVavWPPGzp+DvIIt71apVw/M6YumINXrKOjS9ceNGhJHjP//8QydM+I2nL0PmjidPnojfvA5WGjx99u7A37x16xYfrU+ZMoUvkTout2LPKE2atDzhN/Yt8f2o0CNjlSpV+N/WltPQpq9cvkK//PJLWqFCBXrp0mWno9zNmzfzZVzshULGuHHj0n82bhS/9QymaGlQUBD/W5MnT5E2qsa97tu3nzZs2JC37aj2BK1o85itY8UAKQXN2I8/ffo0bdOmLW8f+CxrlgfZL1y4QPv160cbN27Cc1iiDu8RGVSwbYI2hn64detWU1fp0F8GD/6FBgS0NG1PF30mKhlxX9D39erVZzpjk7T2qun9nj170WvXrtkiNdg7+A97uR6D/w3nCtu160jKlClN6tWrw707nz59Sm7dvCm+RUjyFCn4v47177OZEGtU/LOreoQzu3/vHv/sWD91ylQye85c8tHHH5Pu3brwuJEA4c6eCDdw7ZoAHm8Y8eHcY9KkSfkBduDs+65kP3jwIBkyZCi/5x9//JGUKFGce88ZuSfIHzt2bP4Zslxjs5YYMWNyj06cIcQMcvq0GWT9hvXk++9/IBUrViDp0qXlx1Jc3WtIcLD4FFYPr07MXDdu2ECOHT/G3te//BgBzqExA0FYhyZfffUV99BNnTo1k+8hH02ePXeOn2Es8k1hkiJFcjYLS0YSxE9Anjx9QkLZvX3++efkg7hx+XWiuqeH7HeYCcZk9xX67Bk/NH716hVy9OgxsnbtWi4fGyjw74NYsWLzZ5UhfXryKXsOiRJ+SB48vE/Sf/EFmznXcPue4AGLWKKQMSl7387eR0I2C0BACUR+wX2vXbuezy4H9B9AOnRsR5IkScqjND0Qsy/He2Idlty9c4d7U+K91qpViz8vNrgJ9yLXZJk7dw65du06//wa7P/v2aMnqfFjDdKVtWFkTdHTluKz/oVADMCx3lFGZ/U4W7ht23Y2M1pJdu/eQ2rXrsVkr8ln1o5txvHvMGXMkwukTJmS9xuNyG1MA/X32DPD+3z27Dl/12gbB1jfCToXxPrOf/y4BmZhH8T5gESL9i6pW7cOycDaYVSyA8f27liPACBB7NlHixaN3GTfeXD/Adm7dz/Zt28Pjxzzw/ffk2SfJiMpU6Xk/f/5s/972kaWHejpw87q0Y5Xr/6b6aU5XBf+wPrrRx99yPvdz337cm9urJb079ePpGB9B7h67nrqNdnZIIEwY8I+vUMCAlqQQoUKhus3T+7pEZu1YqYci13rI1aHdgZcPfexY8eRGTNm8jZUj91Xog8T8Xpv78lRFuCs/u81a8iE8b+xuvfJ2DGj+blZX+G10dTAEhkMSJ8+P5MsWbOQZk2bhj8UK4EyZiM6pgT2sobTnQfWdgZuD402VapU4Q3KW7A8AoU/dtw4bmyaN2vKl0KNAhlh3GEs8SyhlB2B8kHAgz8nTyGFChYk1apV5ffjKbgOluIQ9QYh4hBjFQHkHQ9sI+JGOqbEU7K/f+b0GRJy+RJ5FvqMGdvHTMkm4goXy6BsVC3+j7DzfGyUSk6dOsUPhENhQulGPsOL548ML1DEOBuJaFL4nDpVatZR43NjbgTcH2TBsuPHrHNGfo4AcuGoAJuN8whMWKLVDBEGCgiqjyVhPeBZfv55CjJt2nR+ltYRuMa/cnhGUFhjx44lS5ctI23btOERoPAejLbJqEC7nzd/Pjl75hxp1qwJvze9gRvwfo8fO0YyZsqkW0YMKrQsPlgmhx6wMlAEZAwMDOQDQFwv8vtGu3346CEf1Jw4cZINjpaR/v37hg+0rQDtDwZ00u9/kPz58rIB2vd80N27d29u5Nuwd4+EDehDZoG+t33HDjLklyF8W6ZSpR880kvoN3h3+H+gg5z1G2dgW2HQ4MEke/YcpEH9epbrfbzvXazP7tm7jw8Y+vb92SfG07DRRCPRwm1hb6137z58jb9p0ya8zmrQYDDq2bTpH/Lrr786jRH6jM1AYpoUo1IDhmHQoMFcQXTq1FHUeg/uA8rJnYLCXgb2Kr/88iuSlQ1SzADvEEodnQCzchg+KFwYFVwPMzjIByOBUT3AqoIGPmMGXKFCBa680PlwPvLTTz8NV6D48aRDeguaMwY3zgww0nkdPnKEXGfKCxkusP+EWSfA/WHAgrB6PXr04HV6KF++PN/vwSDAFZjt92aDyuHDhoW/s/z58/N6q9iwYSM/CF6uXFl+X97g2LftyiM2w0d7s7pdeQMGo6tWreYz0ZYtA/heefXq1fl7+eabb3iQhKxZs4pvm8OZM2f5fj8GgO3atRW17kH7R9/25jlqeh+rZL169bR0oKTpcm3ShEE/+jJWEqQBo2k27tz9rQBr7HCblo2vPMsU3oPzmGj62I/SYIqN1x07dkzU6KNSpUpuj105+70/HDlRmA/2PeFHkDJlSt7eMmXKRNlsWfzWPHyhlx48eCD9utD5ZvtZuMOStSFtuUsmmFn4IuCwlctrCmvIkCED98h29JDF5++++y6CR21ksPToyIULF0jmzJkizLqd4e73ircHzOQQFB6rOlgxwjIjls2LFi3KV3vMwhd6CatLsq8LnW/GFpknGL7DyPtWdsQO8QrdgX04NlsWJXsC+bAsYmf0yoicpkmTJOEh0bC/OH/ePDJx4kTx27A9cwTtxvEWgD0f7JvjmA+O9+D4CZRemCPGmwmWZ+2OlbFnzQJtyVnfLlOmDN/y6N+/P48zjX14bDF5EtfWLFzJaDfsoMsNG00Ve9YcHGPP2hV/iT2rR0bs1a1kRg8OPNj/2bFzJ0mXLp347f95V+zxQKFhnxfe09g/wf5tQECAYeclO+MPSlRG7FmjRBXXFXv92JODsx+yqHTs2JHvQWOPHCsZslCxZ/Vj2BHoPDxTU6cWJXsCF2ZHF3M7ghEzvM/MdlgyE2zCw4PNDolgXeEPMoIC+fOTnRY6ApkBZkF2SL6NWRCWLuFYg8846obZGOq041KOHuDwjHYEAxvM6gCWENHPYJjwo3neYhCFf60Aih7Llnr6NgYqSDaAFRDcJ5INwOEle/bs4hvW4ImMvsQOutyw0VQoFJ7jD0ZTFlgGhhHEPh/OJyMSFJbF4ZWJc7fOwD4WjF7KlJ+T0NBn3I9C8+zOly9fhOT42ELC8R6AZXbM6uAhHnkZFMYUR6Dw/2N/GysKmTNn5n/bKoMaFTgq06FDB77fDrmmT5/Osx9p96nwDcpoKhQ+4G00mjgKBGOFoz84JwxjgNmUowMMzt3BQJQoUYLvIcOJCsvnOOSOIBnajxlHTKD6cG0cqcJeGWTBWcp58+bxM8eOYRIhFwwpjhjlypWLB3zArFUGGDj06dOHp7aDAUeIPuy3m3nWU6EfZTQVCh/wNhhNGEg4uCDazO+//87P1GHpHMulWBatVKkS9xzFXjJmdVgKttMsCrLinDLOWSKICxy/EMlKi2KFmSniECPBOWanKVKksNSDH2dS4bQ2YMAALgNmwXiuOBdvx3OqbypqT9MmqD1Nc1B7mubh6Z4mlj0RdB6JuRFtCmUcdIdRRMhGKHckFED4PrOUPGaDUg+2M2C8cG8ITrFixQr+gwECSJ8+PQ+ZlzNnTm5QETbQiv1CLN0iEMfcuXP5bBwz0Z9++snr2a/a0/QAGE0jqCTU5uAuCbUdgHxPbJiE2hF/kBG8KUmon7LnPW/ePMpmPRh8h/+wmRddt26d5QFHzApYbhQEb+/bty/9+OOPIzwHlFu1amVKDlZnIB1exowZw69XqlQpXucp6DN21z/ADrpcncxXKBQeAQ9WzCgR5xQzGxzDQfhFzHz27NnDHXswCytevLhPAo74AiQxQHxZOB3dvXuXJwQYPHgwnwWOGzeO74MiXjSWVhGLFrNVM8C+5rFjx7gDFcI5woEKy8Q4sgLPYqbjxTcVZqH2NBUKH+Bve5pwljl06BAZNWpUeED7ihUrkvr165NChQrxJVeFc7CEjGw4CC6PfKw47wuQHAH7oXAuwv6kGZFtMKDBdQYNGsQdreA41K1bN75kjFjQCuMoo6lQ+AB/MJoIXBLIlD2ycsDjFSDAPcrYn7QyMPebDI66IHABPIgRmerIkbCk/jhv2rhxY1KqVCmeLs7bQPsa2HedMWMGN6Aw3Pib3bt3517AmBkrvEMZTYXCB9jZaELBYkY0YsQIvowIJ5569erxzByy43y+DSCgwdGjR8nq1au5NyyWugGWWREjuUqVKtygGok+BSON8I8I04c8mTjH2rlzZ760juwkCg+A0TSCcgQyB+UIZA5myMhG6JZna4jsCIRs+FY7zThDy8CC+927dy/NkSMHdyj55JNP6NChQ6VnkPAUuzgCRYWnTjZPnjyhp06dogMHDgzPhvLuu+/S3Llz0zlz5vB7ZoZWfNtzLl26RNkMlyZJkoT/bfzLjDM9cOAAz8JiZ5QjkELhBuT5kxUDFddBBJb+/Qfwg/gywDVxVADu/jKdZjCDxFJgp06decxTzCBxHATOKti7RLJu7LdZNbP0RVByfwFHzxDYAUupiJKEs6JbtmwhmTJl4kENsDeJAA94XzhygihHTJeL/9s9mFliiR2BHDCr7dKlCw/dh2MyiHyE87PYD8XyrsIJYbZTH127dqVLliyht27dCh+RXL9+nf9rFYcPH6EzZ87k+Q69HV2xRiU+6YMparp48V80KCjI0IjOEzCixwjQzmAm9OzZM1GyjtDQULp8+XJapmw5GhDQUtfRBw1vZcQsq3bt2pa3Zw3MNHE0oFjxElLzz6LfLl26lKZMlZrncoQKSJw4MZ02bdpr984Mm/hkDnfu3KFjxoylZcuVp7t37xa1xsAKjS/wZFUA7dGMVQRm4Oi8efPpqlWr6J9//kkbNGhA48WLx99h3Lhxaf369elff/1Fb9y44fGMETKePn2at406depQZpT5302RIgXt2bMnXbFiJf+70I0y2L//AJ08eQqXyfHZearLrcCjPU18FYd44W6+Zs1acvPWTfJFuvSkYcP6PPSVVSNljLQQkWP9+g1kz969pEzp0jzmJLLfW5WrEBE3EFZry5atZNfu3SRnjhwke/Zs/MAyDnxjNGg2M2fNIju27yB37t7hM51iRYtxDzvcK1zLMQqEi7+ZUVNwHYz68S9rnHzmc+LESfLkyWNW/4Q8exbKnv8jHg/08uUrZPr0qZZ4SuK6cIxAcOp06b4gtWrVJJ999pnlkU7Qttq0aUuqVq1CSrN2JYsE7Bk2btSYDB06RFo0l5Mn/yU1atYgp0+d4uHj4BiCeKboR1bJgByk//zzD5k3fwFp2KA+d3KBR6eZYI8O+31ovwjcfurU6fD2i39xr4hGdIt9r3OnTvz4h1Hmzp3H+slx8i97lqBI4cI81N/HHycmGTN+xfUD+itmhGYB/YsjLegnR44cJXv37eWBHa5dvcb08m0eChD9FH0ZegJ7oQ0bNuRRlzx1KsLfwD7ookWL+D6r5vGL54wg8s2bNycFCxbkDkVW6X3oJejgdevWMR28h+vgr776irWhkqa3IY+A0TTC3r376ZAhQ2nSTz6l3xUtxrOQswcufmsNTNFRZkBp4SLf8uvi+iEh1s7SMNo5fPgwbdKkKRulp6IZMnxJR48ezWYLIeIb5oPR37lz59iofA9t0SKAFi9RkiZOkpT/NG3aTHzLGIMHD6bfFC5Cy1eoSBs3bkI3b95CDx06xEZ4Z/jeCfa8rNznwOF4vL9knyXnqxhWtx1H8Gy/+CK91Jke6N27N82aNZsoWQ/eX7t27fi+GLo8fn755RdL3+v27dtpmjRpaaXKlenFi54fttfL1SvXaOYsWXnfaNjwJz4T27dvH2UDP9430X5lrBZBP+B6e/bspQMHDuK6EG36w48+pkWKFBXfMh/oCOilqtWq8Wtly56DP3c24Ax/1/hB8Ik1a9Z4PePFKtikSZPC91i1HzbYooULF+bPHHrZSrAC1bNnb5ojZy5+jwsXLqJswCR+Kw/DRtMRLL9gWQ0KHZ1GBmgEZ86coWzETAcNGiRqrQfLqdisx9KxbKWLezZ76Uw2aB8w1C1atOBGWtayD0Dn7t69O23btq3XSsQboMAx0EMfkRUR6OrVq7RkyZLhSq5evXq8zgow4Jk1azYtWao0396wWon6A2jXFyU5r2AQhK0M6KUZM2by6+7atYu39USJEoW3gSxZsrDfz6Dnz5/3apAKfYf+i7+bLFmy8L+Ln2+//ZbrRMgAhzqrgP47ePAgG3AP4QMzKweAkTFsNJX3rDm8bd6z2JuwoqG7kxHKHIM6zWtUFmvXrqPNmjUPH+xYbTTxbEeMGMFnAtifGj58uMeesJ7sJQMoYE//H6O8id6zVoFZGfZDsUepGVG0jRo1atAJEybQE8ePi296BtoVDNjs2bP5oEzzysXPjh07xLfMwQ66XBlNm6COnJiDOxm9ictpFKzAwJHCESuN5tGjR2m6dOn4cmxAQABf1vIG2QbQG5TR9B4YUTgOVa9encaIEYMbuVixYtHKbOaGeiNLnzg2A+dNb9ueK96IIydxDEatkIHdM16A92LEIO/aPLks5Itm4IC1DNzJ6IschHCcKleurChZB1JwIR4sQrLheAKSOOMwu7fOKP4QNza2zbNyALRHO/ZtZGBBhCfkD7179y7ZuXMnjyCESE9s9skdm9B2GzVqxFO84TvMZoj/O2rgCIU8qGY6QgE76HIVEUih8AFmRgSCBzDOXM6fP5+fwVu+fDn3cFQovAVmAZ7P27dvJ6NHj+ZB5v/77z9+bhdeuWw2ymMOW3V6wc6o4AYKhZ8CxYbjAB9//DFXbkiSjMPqymAqjIIjSDhOUqtWLR53GEd5EF5xypQpPCk3DCdmoUmSJCE9evQgu3fv5sf03oY5mOGZJs4NIY6hncH5JZx3tDM4k4RlETsvieEsJZqLnQN1+4OMwOhM8/Lly6Ro0aI8JdTw4cNJmzZteFQhM0GKr/dtvrWB5O0482pnnj9/zo2Qnfu2pzK+evWKz0S3bt3KZ6LIrAJwfhJLvpiJIlaxJ0nM9WAHXW64lyETgt1BVnK784I12v9YQ7QzkO/Vy5eiZE/8QUajbNq0ic8CEGINI3yERDPbYAIMQOzO09BQ8cm+oD3avW97KiMCrCDkIgL5I+wiwjIihyqCDyBgRoUKFXigf7Oxgy5Xy7MKhZ+A0T1mlcWKFeM/169f5xGqFApfg73OfPnyccOJFYq9e/eSCRMmiN++WRg2msp71hyU96w5+IOM3rBixQq+1IX9I3g7IsC21U4YynvWHOzqPeuImTLCcxaDOYRpNBs76HLDRtPu+5nA7vuZACM1uyspyGf3vUJ/kNETMGqvVq0aX+6CUwZml1j2wv6T1dh9PxPYfT8T2N1XAfiDjMAOulwtzyoUNgXOFThCgrRQx44dI5MmTXorXfwVCjth2GjCe9buwOPK7sB71u6OF5APXnZ2xh9kdAe8fzt27EiyZ89Oypcvz4MUZMyYUfxWHpjl2h14z9odtEe7921/kBHYQZcbNprKe9YclPesOfi79+y5c+fIl19+ScaNG8fPXSJCi6+WzfxBiSrvWXPwBxmB8p5VKBQczC5x1hLOE/CSxd5lmTJlxG8VCoVdMGw0EyZKJD7Zl3gmH7C1glixY9ve6zM6m/Hgx84YlfElG3Ej8a1MMKNDMuGxY8fyBMm4voy9S4Tfi2opO5bJcUOtwNPkyr7gbeg3svBGl2MQisTdZmHYaPqDY4LZUSmsAFnekRXdzuBAs8ylQjT2NWvWiJI+jMiIDP+ly5Rl7yK2qLEeGK2SJUuSmzdvkjlz5vCQZTLawf3790mlylVEyTkxLTrOceTIUf6szSBu3Ljik31Be0S7tDP+ICPwRpfjvjAgrVSpEh8oGkUtzyp0gdnQ9u07eNBmq8E1kJ2jVOky0hxgYEQqVPyeLF3yF0mVKqWotZbbt2+TlClT8ogqR48e5ZklZIBsKK1atSZr1/wt9XgOFFaTJk3JmTOn+SBRJlj+DgoK8nsnMYV3wDcAZ5yhTxYvXszbg7cYNppXrlwRn+zLm+I9i+9gRoKlhs2bN4ta64BS/+uvv8hPP/1EfvnlFx602awZgisOHDhISpYqTSpWrEjWr1vrcSovzE7xnDwBRqRNm7bciOC8rAzOnD1LPv30U96J4R2L8GMy2LJlKxk5chSZPn2a29B7ZvVtvJNJk34nXbt2JePGjeVxSc06Z4p352ogh9BuGzZsJK1btyFNmzYj165dk7JSAoWMNoj+c+DAAXbvf/BnYGcgr91lBEZ0ec6cOfnACckNkD7P2/atO2A7HmidunVfa6DHjhwn0aK/3vmQeaF+/Xqi5D3I43b6zFn+GTLcvXOHf3YGOkTbtm1EKQzcHs67Zc2aVXd8TlznwYOH/F8Yqd179kS49r379/iIFVH9nz97TmrXrkWKFy/Of+cNlP5Hzp49x5eakiZNyq/15MlTfo2FixZxxXDl8mU+Oo/zQRx+wDddunSkapWol9c8BXoMreEMe97TmFJ9yJ7BF+m/INWrVWPv8yNy48ZN9lwe8Dx5VhyuX/zXErJn927SqVNH3n48Be8a78sTGdFxhgwZSrp378YzNsgAZy6HDBnCI6f07duX72daDZ7Fzl27ycKFC0hPNuJGLsWoQD9HfsUCBQp4HdcWjz8o6AIZPmIESZs2DWnXtq34jTlAxm3btjEFmIkkTJiI3yNkxfufyIz06dOn+KCrQvnyvP+bifZMXrx4SZ6Gsr7K9MBfS5aQ4OCL7CeExGQzeASHSJQoLBPI999/z9u0kX4TGhrK9QLuO7JeQt2t22EGRdNLderU5uEW3YF+A69tBFuPLCP2+GfOmi1KhA+cH7Ln6wosnzZp0liUvGft2nXkyJEjovR/Hj18RD6I+/o+dtWqVTzqR/BORz9s3bo1G0SOFLX6MJzlJE+ePDykl5HGYCW4PciIWIieyIjGKGuNHzLWqVOHfJw4ManxYw2uBHBt7G15q7D0gnQ/mE3COGO2U7tWbfa8cjvdV5s1axY33FWqVGYlc943luzq1W/Akyb36d3L8DPXZNQzm+ncpQszCl+T7ytWEDXWAiOEqD64Zwx6UJbRb46fOEECAlqSWTNnkM8++0zURg3aJAymtzKOHz+eLF78F5k3b65lUcMgIwaQvXv3Jus3bCDPQp/xAVfhwoVNT37sCK6LGTuWtrNnz+amr1IyevQYUrp0aZI+fXpR5x24Ln7M1wmOMn7Byta3SW84wdoxjmG1b99e1HgHBso//vgjH1ChnaIvegR7CYbIly+f+GRfMmbMKD7Zl/r169M1a9aIkjzu3LlD2YiLslmtqHEN5GMje1EyBhuU0OXLl9NixUvQ4OBgUWscPTI+evSIFi7yLWWDPVFjPVu3bqVMkVM2+6CBgYHS+g3eb/ESJenjx49FjX7y5s0rPumHzfJo48ZN6KJFiyib/Yha64AKmzFjBr+uXZk2bRq9ffu2KNkTf5Dxxo0b/F0bgU2eaLJkyeiSJUu8bp+Ghyz2HJMo9IJEsthX07vXY9bMCMmSkVpo3do1Hu9buiMqGeHw80OlymTB/HmkVKlSotZaFixYwHML4nq4PpYKZfQbPON27dqTNX+v9iqObDQvZjS7d+8xfd/SHZV++MHWXrSSHoMh/EFG8K6XgmJ/u2bNmmTYsGHk33//5cvl3rZPa9f+FAoXIB8klmRlKVYN7BOvXrVSWqKBiRMnkurVq3MvWXjtyTxWhGW8qVOnWL7E70jx4sUsXRpVKDyFTQ55DGdsP2EAa/RsrzKaircKODpYabj27d1LWrZsSRo0aEB69OhOAgIC+D4mHC1k7ZFrQFHIHpQonAPFbXf8QcaXr7zz8MWRrp49e4qSMZTRVChMAnku27ZrR7p06UISxI9PBg0aTEqWLEGmTXN/vEPxZqOMpjnAS9hTMHBMkSKFKBlH9WSFwgSwZ9K8eXO+FIvII6PHjCGNGjXiASFwBEKhULwZKKOpUJgAZpk4H4clIJz/QvCJ3377je+hzpgxQ3xL8bbiDysN/iCjTJ8AVxg+p4lRNA7F2hmcy0H0FTsDGXEwWFZEGm9ANCB42dqh4brCVzLWr1+fTJ8+nWcmwb9ahvl06dKSvHnz8fOjjhTIn5/s3LVLlOwJzpPaPbYrBiroN3beu0VAAJzpVDIaA6YKg1BfO5oZNpoKxdsOotKUKFGCR385d/Ycz1ijkSVLZpI06Sdk3bp1oiYMfzCaCoXidQzPx88HBYlP9iUkOFh8si/IQP/M5gl1IZ/dE3rLlnH9+vWkSJEiPG4wQrk5Gkzw/PkL6V6zZoEzpXYHEa3sDtqj3fv2Ez+QEdhBl3s800R4OcSDvX37FilYsBCfMus584bp//Lly0UpjAoVKkiZauNBJxfeU1Buu9gIH/IXK1bc58tPWE5EwO73mbLFkp6edEybNm2KELhY1nNEp4L3WmyHg/JYnkfCZI9DUZnMnTt3yMaNG3nIRCzF6wnWgODNCKitkT17dp4EWi9ozwgsjyMlSJGHfc3IAaUR47Vhw4Zk+PDhoiYMO8w0Xzx/TjZv2Uzix0/An1tkYDQjp2KK3I8//PBD8t1334mSfO6ywSaeua/bHzh8+LDTGLcwmtgvjNy3jbY/szh79izPtJMtW1b2HBFGzzc42hZXutlRl2vACQ+hXDUsb5MwmnpB6DOmJGBkw3/69Okjfhs1TKFE+P8KFMgvfmM9wRcvhv+bKlWqcBnixYtHd+7cyX8nm9DQUDps2DD60Ycf0i5dutCgc+do6NOn4reuQQi4OHHihN/DF198IX5jPZDviQjHxkb4tHbt2lwGyO9LDh48QJnSpD169ODPgw0qxG+ipnLlyuHPET9M6YnfuIcpGf7/fP311/xdzpw5k5cfPnwovkEpMzq8bvLkyaLm/+T3cfhJNlijhQoVoo0aNaLFixenxYoViyA7uHfvnvj0fyL3YzxzX2Cn9rdhwwbKDB5NnTq1qIkI+oyzvm2k/ZkFwtI5ylCuXDneNmTz8uXLCLbFlW7WdLkjvXr1inAPVrdJj2aaf/zxO9m8eQspX748d1wZPnwYuXr1Gvnnn43k22+jtuwYbSMgsEa2bNmkjw4R2os1bJ4iBvL379+fOzEhrBIyTsgEaYMePXpIChcuwoMHI/WWHhBgGK9My8iBSDO5c+fmn2USzEZ88BItW7YsP5eoV36zQUojhAHEWUgsk2LEzDo+D9CPw/2ugOyjRo0KD6UXPXo08v33P+hyhEDWnIIFC/LQeCtXruSzCMy4U6RITpYsWcr6wrf8e1u3biUVKpTnWS8iO8v5eqbZrFkzfq/w8MWM87uiRUnqVKnIdDeevpH7MYKjy4qu5Ihd2h/499+TZMyYsXzfGkEs9GCk/ZnF5cuXSdWqVfnRKLRhtAX0m2bNmrLPE8W35ID2CMczzbbo1c24h1o1a5KAli1FjYQ2CaOpF/aSxacwEOA3adKktF27dqLGOZv++YdOnTpVlHzD2bNn6erVq0QpDATsxiPw1WwTMOWpe6SMGU3jxo1Fyfdcv3aNPz9fjvRZ56Jp0qQRpTDy5M5N8+TJ4zIgM55jgQIF6DUmv6cMHTqU3zNGxVh5cWTy5D/5Csrdu3f5D2ahCITtDF/ONM+cOcPvAYHjNfAZdQcPHhQ1r2OHfuyIHdqfxqiRI13ONCNjpP2ZyTT2LrEaooFVrLRp09LPPvtM1MgBunn06Ii2Ra9u7tChA2UDFVGSg0dG8/r16+LT/4HSHzBggCg5h42maIIECfhyxObNm0WtXNBAIytR1OHF4KX5CihWvZ0ey3yQF8tqEyZMoM+fPRO/8Q1oD75WWlmzZqUVKlQQpTAaN2rE5Tpx4rioiYj2HKHkMOBjI1vxm7BlV1egg+L/6969u0uDvH79evrTTz/xwQ0yKrjC0WgeOhTIFaks0F9xH//+e0rUhG29JEyYkPbu3UvUvI4d+rEjdmh/GqNHj9ZtNKNqfzJxdt3q1arxPiUT6OHIA1A9uvnmzZuUzULpJ58kldomPfKejTzlhVMAuzkePd4VSGKKXJHt2rUlJ0+e5EtonTp1Er+VAzaPsZwZeenj1KlTJF++fDxhsa+BF7I77zXIOWbMGJ6ZpEWLFiR/gQJRJuU2GyyJyfJMhYNUq1at+TtyBdrf8ePHIizzw1HpE3Em98CBg/zfyOA5/vrrr6RMmdKEzZx4nkM2iyJ//72GbNmyRXwrIh07diQjRowgy5YtIwMHDnS5jIakv3/++Sf5/fffdS2bw4Fp0qRJUs+eYXkZYcWQWFwDy3NIgB4YeFjURPSexfswox/D6a1Ll66iJBc4AfqCyN6zrtqfbByTNkNGJLYOPHyY1K1bV9TKAbo5cmAFV7rZ0XsWy7ljx44lDRv+JNe2COPpFdhw9WSUh9EEltNw2X79+ola67nhZIYMqlWrGuVsQAbaTFOvIxDAkgWWLxIlSkSrV68uaq2DGSc6ZcpU/uM4I7JqpI/rdevWg27cuFHUOAcj5bC21FfUhLUxlFHPjJGodQ3yTRYuXJjPoI4cOSpqI6LNMJnBFDXGwUwTOUyRTzTyKNtq4PSTOXMmUfo/qMPvNJw5AgHHfrx48WJR657Nm7dQNtgzLc+m3vaH9jR48C904kT37cFbXM004VyFduyqbzu2v8iOWDJhAwo6Z84cygyP1FUPV7jSzc4cgYC3bdIbuCPQkCFDIrg+OwObxHDl1VizZg2ZPXs2d0zRnBwQnQPHOeAcUbt2bZI/f35eHxk45GzYsIE7wxiN3IKZbFQzXQA37kGDBolSGJAfm/GYQXgKe0Hk/PnzZOHCRWTX7t1k0cIFPJoGwqXh3qOiTp3apHz5CqJEuEMJftq3a0cSJkrk8qgERvpwchkwcBApVbIkz6KxevUq/i+cUMyIgONpO7hx4wYfJTKlZZojBkaM+LtoV1GB5/jdd0V5mh+EroPjAGDNmXTr1o3fy9y5c7mTlTvQblOm/JxMmzad59lzBO0DDhvIGK85bZgBZqFffvkVu89fpR57QlvBjBEjeThZOIJRPRz0kMIM4OxeVHk40Y9jxYrF9YA75s2bz/rqC64XzMJd+8NKxPDhI0hQ0Dn+e2S4sQqsAGHWA0cgXPcfNnNctGgRn0XCMSVHjhwu+zZm9J9/nsJp+/MEOMWdOXOGr5hALy1etFDX0SsAGbB69ccff3icexXHfrBCA8c3Ngng7cIIUelmHC/C0TysGly4cIFs27adrGJ6cNHChXz1x5M26TUwmp7CFBtlRogeP36CshdN+/btRytVrkznzp1HL7KRgLuR8/Tp02mMGO/x0Y2nYLP66NGj/G+UK1+Brl+/QfxGP5r8esHIODg4mB8rKFO2HA0IaEn37NnDZwpGiWpPE27YmFXWr9+A76PBgcNxlI4RYfTo0bx6jlFx5MgRPlpjDVTUOMfsmSYzTLRjp06ipA/sv9SoUUOUwmCGksulZ68a7alPn77c1X7+/PmiNgxthrlu3TpRYx7vvReDMsUgSnJgRpK2a9+e72nGjBkzQj9FW0KdO/8ER9AHq1SpIkrOgaNJga8L0g0bol418AZX7Q/9Ydy4X3k/DQm5JGqtRZtpLl++nOmWwZQNSjyaUTtrf+5w1Etly5WnI0aMZDr5uMd6Cc+rXdu2lA2oRE3U4J2uXr2aNmvWnB9Z0nSFGSsIrnQz+unp06fZu+7G9f7AgQPpvn37eL0jetqkUTw2mqGhz2jTpk3ppk1baNFixenoUWPomdNnxG/1gRurWlX/jaERsNETzZEzF82Q4UvKRkP08uXL4reewUZjXH53hh3gvNKYMWNpss+S80HB+fMXxG/Mw5nRROPDUiCuO2HCby5lhaLLZ7IXJox0mjRpufenO8w0mmvXrmNGqqMo6adNmzaveaKijPO4ejpx4SLf0u3bt3Ol5dimNGcNDFbMJijoPM2WLbsoySE4OIS3J7SZHTt28Hu74nC/x5kBR53eM64A/XjEiBGi9DpQxuivOFNpBc7aHxQ6rvnPP/rvwyi7du32yBHIGZHbX1TAAQZ6Keknn9Kq1arxiYoRMFCCkY8KDD6wvA3dkDtPXj5ZMWPS4AiW0R1185MnT3hd+/Yd+XWHDBnK3mvUzj7u2qQZeGQ0ly9fSTt16siMx/nwB4YbXLBggctZCTpO5LX6tmxUc+mSvhEgRhPYB4FiQ4cwAhpXzZo1ufwaMKJTpkQ8fA5li2v26fMzu/5+0xuHI45GE3t048aN46PGv/5a8pqyiWzIFi5cSFeuWCFKxoAyxXtB59c7czXLaMJgtmczIG9Gqlh1QLAHKBKAPaK4cePSNWvW8LIztOeI70yZMoW33Y4dO/A6AIMZI0YMNrCrasro2RE82++KFqP588sL7gHlg71TrR+iz2bJkpkPPjXgxZkpUyaXAzQoL2f92NW+J2YtuKar35uBY/tDP/799z/4zMeontADdMKCBQtp8RIlmV5c4ZHRjNyPI7c/V0A/lC5Tlt3n76bpJexjLly4QJTCwAoNfCY0atepw40RZrGRZ3Zmgfal2RYQfDGY7tq5m+vm26xPOwPtEatxjkTVJs1Ct9FEZ4dbLxpp5B+cidP47bff6M8//yxKlL9gKLHOnTvTuXPn0vG//ir9XM1V1tgwtU+ZMqVT+RGZxxegAWJGhCMTeNE7tu8I71BHjhymdevWDTcGaMhYPqtUqRKdOWMGd3JxbNhGQFSnpk2buV2ecYwIBLZs2cKfH+TXa2gjgxEzlI8R0KaKFv2Ot6+SJUvyNueIY5u8ceMGf444Jzd+/Hg6a9Ys+iv7/zX5R44cye/JCoOJ9505S1b+rmWd08Q91KpV6zWDd+LECe74g0EDDGbu3Ll5H9GI3P6aNWumux+HhITwaD0w1lbi2P4OHDhI9x84IH5jPRjQOh7ZwCAbgze0L0e02ZO2xO+u/ckE7x7bO5oedPzBO5QF2l22bNmcyuGomyPbFkSyQgQwfEembdEdEQhOO9jcdsaXX35J2CiVf544cSJ3Xe7Vqxcvg927dxP2ErjzAyLYyAayh4QEkzNnzoqaiDBFKz29GZvZkRUrVohSGIiXqDnZHD16hIwcOYqwBhGeZop1Uh6bMXny5CRXrlympb9iI1buyOQOx9izrKHzWJsaTBHwGLieULRYcdKv78/cecAo2rNBlB7Ee3V0gIjcJnfv3sPbAyKNwElDc8RB+i44yXTv3p0MGDDA1Ogs6Ga1a9cho0eP4g4pMiIC4ZrNm7cgAwcOiODEp4E2CIc8NnDjUZUccdb+9PRjZpxJz569+H1aGd3GjPZnFoh5jKNDGo79GH2rdevWpGqVKqy9F+N1WluN3P5kgxjCzKiL0v9BekKkuJNFYOAhXbo5cj+GcySePf6VaVsMpwaDB5svwmh5wi3hcWVn4PkGw6XX280XsNEwV8R6DKyv0CPj8ePHXzMSCxcuJNWqVSMzZ8401cNTgwflZ4MNGHQgw2jiWVy8GEzSpvXuHLLewZQvQXagBAkTipI9wXvAOUQ7Z7vBu8Ygx876B0BP+jrnsGGjqVD4O1q2EqsMpjN8HXtWoVB4h+F8mgqFP4MtBxjMDh06SDOYCoXCfzFsNHEA2u74g4zINPHq1StRsifYz3yTZNy5cycpXrw4ad68+Wv5LhVhy4p2J/TpU/HJvqA9ol3aGX+QEdhBlxs2mtevXROf7MvtSImB7QjW6l/aXEnBsD934jhgJ/TKePToUVK0aFHyxRdfuI0+9LbiD4PNu/fuiU/2Be0R7dLO+IOMwA66XC3PKt464IGH0IVwzIFHqJVengqF4s3CsNGM88EH4pN9wREJu/NejBjkXRt71wHIF82kYy5W4U7Ge2xmgqwo8GbEsYX48eOL3ygiY3dPShA7Vizxyb6gPdq9b/uDjMAOulx5zyreGh49esSD9+Pc14kTJ0w75+oNyntWofBP1PKs4q0AB6CzZs3KDSUOxfvSYCoUCv/FsNFEcAO7g+AGdgeOQHb3VoR8OARtZ5zJCK/AH374gS/NIvUYIrEo3OMPjkAIbmB30B7t3rf9QUZgB11u2Gg+fvRIfLIvyEpud+C59p/dj3Mw+V69fClK9iSyjNh9QE7N7du38z1MX4Us80f8QYk+DQ0Vn+wL2qPd+7Y/yAjsoMvV8qzCtpihtJFYd9myZWTv3r2vxV9F7N9y5cqS6tWr8+8hDmdwcLD4rXPwna+++op73Go/iH9phLVr14pP8sBy9dWrV0VJHhjEKDcKhT9j2GgmFLE07Uw8P/CQjBU7tq09UxGIG6P66JI8KhHAOVfuPKKkH8inyQhjhCDPgwYN4obOEQTMRgSgmTNnkfnz5/ND8n37/ux26Xbp0qUkW7ZspE+fPvxn4MCBPEC3t2D2+++//4qSPPr0+dmt8YoVM6b4ZB7Dhg03dUvnA53e+zi876vAHI5t0q74g4zAFrqcdRzTiZyGSAbPnz+3NO+lM/77Lyxzukxwj8htJ5PAwMO0YcOGPCm3DNB+kIuRzTRFjecwQ8RTNSGdHd6TI/j7n3ySlNarV0/UULpr506e+ur27duixjkNGjQwJeUVUoNBLuRjNHKf3oC8ssOHDxcleezatYt27mw8YbmnIL8mcpjKar8azEjT2bPn6M4dbBZoV2wwJkryQFrDyH3NaqAPrcrx6QpTl2evX79OAgICCGsoosZ6sIQ3YcIE0rRpM57qSBZMAZCSpUpLDT21Z88e8tNPjUipUqVEjbXg2TIlR7Zs2UL+/PNPEkvCmTgsG/5QqTJZvmyp1x6uM2bMIBkyZCApUqQgCxYseC14wbx588jVq9dIgQL5eZn1Az7ThzME/l9X7Ni+nYfeGzduHH8XRvn99z/IqJEjpHry4p3+1Kgxad++vaiRA1KKTZ8+g/zyy2BRYz14r2vWrCWtW7cha/5eLaX9Alx33br1XD98800hkixZMvEba8G7XbRoMb+urHsF2NJASr3x4ydIm8076n04+EkFltMI165d4z9IwoqkqrJme+yh0cmTp/AZkLuEqcjmbwYYRa1du47PDpjSFLXmgMzzuCdnHDp0iBb4uqBpSaf1gFEjRufnz18QNZRnSTdjluUKjMydJUz2hODgizR27NiUGU3+TJ2B2SKa/saNG3kZWfeZcqVp06bh13cFM5a0YMGCfAaL/79EiRL0jpuZqSs++eQT2qFDR1GSB5KNIwm2HpBo2QxOnz7DZvX1LZmFYBaHdhMZ9CXohmXLlkmd/WAWj1WS9es3hF8XfQZ9xypw/2xQy+8Xz9obvJERerdx4yZcD7vSXWZz7NgxbmcaNHCv963CkNHECypXrgKtWbMWPXv2LK/Dw8N0WfvRcKxnM0JR67r++bNnTutDQi7Rvn370dy587IOsVzUUv4d7fuOYLnt4MGDlM2CI3QuZ993JTuyr6MTZM+Rk44cMSpc6Ri5J3xHAwYKmfSRddzRKCFrfv0GP9H27Tty+TVc3atW51jv6p6iqocBKfTNN+FZ+wHkunD+PD3J5HQ0Rq7uyVFGV/WO3L93nxmsOvw5O6J91/H7rmS/fPkyX2LNmzcvf54aUAZYmsMP/t+SJUtyo3f06FF+vUaNGvN3nDlzJp4N3pXsDx48oDdv3ODPZc6cOTRBggQ0T5484UtEjrKsXr2azpw50+kPltfjxotPjx8/Lr7tus041uNZazjWO8roqh7vr2vX7rR37z4R6rXvRv4+2tue3bsjtAHg+H1HtLrIsi9ZspTmyJk7ggF2JaOrelfvA7Lt3LGDGyrH6wYFBdFcufPQzZu3iJowtL+BH0e0OlfPXU892kbbtu1p/foNIjwzDADRZ9B3HL+v/Q38ON6TnnoNtDs2o+W6EO3REe27emRHf9ZkxH1o4Dva9x1lOXnyX1qD6fwG9RvSM2f+b6S170b+vrt6R1mAs3pMzEaxwW2lSlV8Ziw1vDKa+/btoyVLlabLly+nm/7ZFOFBWAVmi82at6CDBg2mt27pH91DYWJ93xsZ0ShhQIqXKEX37t1r2YgVMkLha3sCuC72Qho2/MntHpuZ4Lrjx0/gjTPy6B1yYW/oypUrTkf2RsB167PZ36lTp0SN5+A5pUmThs8y//zjD7pgwQKnP9jrLFWqFDeaBw4coDXZzFIbYadPn57/Ti+7mVHB39m+bZuo+T/379/jMjn7adS4Mc2VK5fHI3sj/PbbJDrMg31M9JfDgYGGZIRCb92mraX9Zj/TRVCw2jU2bNhA67JZLQZBsli+fAWfOECxRwZ9BX0Gfces5wBj8vffa7gOhi42+ncdZXTVt9FHd+zYyfspZnqRDZ1VYGBZp249vipznhn1YDZA8jUeGU04vWDJbgcb3WkvynFkYgW4DgzlwIEDvd7Id7VMFxUw0rVr12aj1c2mNfaowCwCSgAdDx1w37794jdyQGdp1qw53bRpk6h5HXwHncdsevToSbdv3y5KnoP3gxli0iRJ+IjbmbHUfmA027Rpw43dunXruKLQSJgwIeucHURJH4ULF+azR71AyUEB5GOzYVlgQNa+fXtR0o+RZXLMtrAUbHXfwaqPdo3FixdzBycZ/RVA92FJdNWqVaLGOegzroyRp2xjAzRs1WzdutXU+4xKRk3v4/la0f+dAV2I9jNixMgIAyBvdLnZeGQ08cBkNUgNXA8jSdngmjJm0JGZO3ee9Oti6Q77MJgFyWbBgoV8KdNb0D6wD4kZ5q1bt0Rt1EDJwWj++uuvooby/xd12APzhIoVK0ZYCo4KDIgyZ8nK3y+8Z2UARYh3a5bS1gMMWZ06daXqCgxEMLOVyZIlS6TNuDQiL5fLAPcoW+8DWQbaUww7Ain8Hyhxx71UWVy+fIXN+tqKknesWLGCGzs4QugF95sqVSr6448/ihossS2niRMn9mjAgu/CqUgvWE7TlJ4sozls2DA+u5ZJpcqVPXqOZuCLAZ/i7cSw0XR0ULErvhideQpmtnYdWWlAPjP3ijB6NTKCPXnyJPdkrVGjRvjf0SMjZkIHDxygn3/+OfdMxl7y119/TadNmya+EQa8a7W2A6X87bff0nnz5vEBBt7X0KFDPToPh6VZDVlG8+DBQ+KT53i7FCZzr/auSZ7xVoL2aPe+7Q8yAjvocsOpwc4HBZFUqVOLkj0JCQ4myVOkECV7gsDTiEYT08b5AZ+FhvJzqXbIaYdYsoUKFSIZM2YkR44c4fkxgScyXr58mQwfPpwHy2/cuDHJnTu3+A0hD+7f5+26e7dupEPHjvxv/vbbb/zMKlKLFS78DalWrTqJ6WXUHH9IDXafPQO75xtFKMBPPvlElOwJ4qWifdq5b/uDjMAOulwZTZugjKZ+2KyPHxiHsjx27Bhhs03xG3sZ9qhQRtMclNE0B2U09WM4IlCSpEnFJ/vy4UcfiU/2BYrf7rEf34sRg8SwIB6pJ2BWiDiyRYoUIWfPno1gMIEdZHxTeN/mAw+QMEEC8cm+oD2iXdoZf5AR2EGXGzaa/tCx/EFGNNho0aKJkj3BSNSXMmIGmTdvXvIeG1wsWbLEqSy+lvFNAs/Z7iD8od1Be0S7tDP+ICOwgy63/1NSKAR169YlFy5c4HuYWMpWKBQK2Rg2mtjTtDtYB7c72NPEfpydgXy+SgI7atQoMnfuXLJ+/XqSKIp0dL6U8U0De5p2xxc5QT0F7dHufdsfZAR20OVqpqmwPRs2bOBZORYsWEDy5w/LTKJQKBS+QBlNha2B40/58uX5cZDKlSuLWoVCofANho+cKBRWgaYJQ7lp0yaesy9u3LjiN/6PPxw5USgUr6Nmmgpb8vLlSz67XLFiBTl06NAbZTAjg3uVjRorKxTeoYymwi1QsIieIwtcr3r16uTAgQNk6dKlJGXKlOI31jJ48C/k0aNHoiSH/fsPsJ/9oiSPRo0aScuyr4H32r17D1GSy6lTp8QnhcIYpnvPYlZw4sRJUZID9r369esvSq9jhccVFEDPnr3Iw4cPRY0x9HrPjhgxkhw+fFiUrAeK9aefGpFYsWJJ80xFaDsYyxkzZpCyZcuKWn14KyMGBbjHDz74QNRYDzciPXqQfPnyiRo5wEhj2dvdeVazvWfHjfuV1KpVU5TMQY/37Lx588i5c+dESR5//fUX2bp1q1TPVJxl7tatm8eDP6My/v3332TTps2iZB2Ouhz3OGnS76IkD9NmmlAAw4YNZyO60+Srr74UtdZz/Phx0qxZM9KhQ3tRYz1PWAPDmcFGjX6StmwI49WsWXOSP38+kjVrVlFrLWiUefLmI337/kw+/PBDUWstz58/J7t27yarV68mderUEbXWgrZbv0FD0qZNa1Ejhz//nEx+nzRRlOSApeBRo0aTMmXKiBo5YCBN6X88VrAs8F579epNokd/T+r9Qj8UL1GSD8K++eYbUWs9CCmZO09e0rx5c2mDPxjp1q3b8Gf97bdFRK31HD58hLRs2YrpiNqiRiLsZg2BLCdIxorEyRcuXBC11sNeFs8gPnr0aLeZMsyMjI+sGLhXZLkwE/w9V1kGLl68SHPkzCU1wr+WY9PxmnoyiBilQYOGhhKbeyMjm9Xy9yqTtGnT8iS7shkyZAhlszNRihpvs5xEBjoCCd3d9VNvcJXlBNfC8w0MDBQ1ckDGHGYwIzxjGRlE5s6dq0sXusIbGZEtCDriypUrosZ6kNlo5MhRXPdb0Z70YNhoLl26jCs6q5WpIzdu3KD16tVnHeKwqLEeJPFlMz2euFjmy9qyZQtt1aq11CTCyIWITO1GMvd7w8KFi+j+/QdESQ5IsNu4cRNRksd778UwNDjwhp07d9Lhw4eLkhxwjxkyfCm9LbHZT4RUbDLApAHJt2X3VbRf6AmZbNiwkQ9KZOZNRRv65psi9PhxfUnfrcJrownDMWrUaFqqVBmpRgQJdT/86GMaEnJJ1FgP7i9vvvz01KlTokYOuB6Ml2wCAgLo+fPyVg0AZtoYtcqmf//+NDg4RJTkgHvNkSOnKMkBOS7Rb2QnG8dA8+LFYFGSw9SpU+mYMWNFSQ6XL1/mq0EyDSaulSt3Hnr27FlRI4fJkycbTh7vKUhYj8HX2bPnRI3v8MpoYlaJ2eWSJUv4FF0Wq1evpi1atJA6uoGSwejx2LFjokYOJ06cpA0bNpTaCcHixX/R5cuXi5I8cK/3798XJTlg5OqLJdKhQ4fRAgUKiJIcsKS1detWUZLDwYMH6dy580RJDliFwjuVOZBHgnJs2cgckGDpHEve0BOywDPVuyVmJrAzGFDL1g+u8NhoYgkCDURbWgo6Z73lh+HAcsuKFStFjWcEX7woPnlGSEgIf1kylpawNxP69Cn/vH79Btq2bVupDRP8/fffdNas2aL0OpDviUn7XI6gU/zzzz+iZAxPZMSyltl70+7AlsLgwb/Q/PnyiRrrwZ7ioEGDRUk/RgbE6DMYWFuN434ajBeuKbPfYKmwUaNGUQ5u0R61vm0Ghw8fpjVq1DS17bqT8fz58zR3nrxSt8Swx4o+iqVgDW91uZl4bDTnz58foVHKMJrY/DWy2ezNg0Yn6Nmzl7QOqBlNzGixqS8b7CX26NFTlJxjhdG8dOkSbd++vSgZR6+M2JuGsZYJ9k/TpEnLlYFMo4kZiTeOKN4aTfQZXNMsR6Ko0PQCZiG1atWSajDvsD7bsWMnt9c002hie8oKv4qoZIQunDNnrvRVr+PHj7/WBu1gNA2H0btx4wZJnDixKNmTW7dukY9snogaZ01jxIjhsxyGuH7khM6RefHiBQZZXE6zGD9+PGnRogV55513RI0x9Mh49+5dUv3HGmTtmr9Nu64ecLC/WbOmJEWKFNLC6OFM77Nnz0iePHlEjX5wdMKb/IXnzgWR6NGjkc8//1zUWAfONydImJCwAQl/53bMCckGLLyd2Tk/qT/ICOygy1XsWYVCEuhqmpFWsWcVCv9EhdFTKCQhc1arUCiswbDRxBKO3fEHGV88fy49Fqin/Pfff0rGtwgsddud0KdPxSf7gvaIdmln/EFGYAddbthoXr92TXyyL7dv3RKf7Av2FF/aXEnBsD9/9kyU7Ik/yOgv+MNg8+69e+KTfUF7RLu0M/4gI7CDLlfLswqFQqFQ6MSw0YwjMSuEt8T2wgNQNu/B889N1glfA/miRY8uSvbEH2T0F+zuSQlix4olPtkXtEe7921/kBHYQZcr71mFwgco71mFwj9Ry7MKhUKhUOhEec/aBOU9aw7Ke9Y8lPesOSjvWfNQ3rOSUN6z5qC8Z98ulPesOSjvWfNQ3rN+ji9mNL4Y/V++fJln/JfJiBEjxCd5IAO9O06cOEEGDx5M8ubNSw4cOCBqvWfq1GnknmTFDzeGJk2aipI8MDDctGmTKMkD10W4T9mgz/jCZUSttFiLYaOZMFEi8SmsM+bMlVuU5DFt2jTWGTeL0uvEix9ffDIPxECsVr26KBknVuzYbr0+sXySO09eUZIDOv4PlSqTaNGikejvvcd/rOb48eMkTZo0ouQZ3sp44cJFbgjdgff+6tVLsnfvXlFjjDlz55IECRKIkhyOHDlCqlSpIkquiRUzpvhkDsOGDSPp06cXJXP4QIf3fteu3cQneUAXFi7yLY/7K6vfgGPHjnmll4zICCP9dcFComQtjroceh8Deumwl2sKTKHzXJeys6Xjerguri8L1kho8RIlpWejb9WqNb0oMco/7hPJt2VmSsd7RDo2XFsmAQEteRYSPRw7ehTTB7p//35R4znIcoI0Szt27BA18kDbldlfAFIJ4hnLZsKE33gOSJmg7SJ5/JYtW0SNHNBPc+TMJVUvoR0hFy7ymMrEF3pfwxSjCcHfFoP54sULnoJIZvJtdEI2aqVbt24TNXLAs0UqIpkgCbbMnH3g6tWrPEmzXk6xZ2LUaLJZLS1brrz0To/E0Nu2yW1HAHkukbxeJkha36VLV6nPGNeqV68+PX36jKiRgy+Sb+NayHP8NhlMYNhoXgq55BODGRwcrPvBIe+dGSDJbeYsWendu3dFjXk8fvyYG+TI4P7q1KkrvRMi4TeUjiMvX760NDs9Rsjo+EbwRkYodE/yTZphNN9/P470WSYGXyVKltKtbDCYMIPBgwe/1pbMAvlYna1KoM9DP8gGyeNPnow40ER7RLu0Chgto8m3PZUR18LzlbkKBYYPH+5TgwkM7Wmy/5+0btOWpE+fgeTIkYPXsQZMnj55Ev6j4ViP3G0arurh8OKsHp937dpNmjZrTgYOHBieOQL12vcdefDgAQkJDuY5FNmDFrXOvx+V7EFB50np0mXJiuXLwvegjNwTvqPx6NEjwjo/uXrlCt8D0cDnNm3akVo1a5J06dKKWtf3qtU51kd1T67qDxw4SKZNm05KlCghasNkuXbtGrnM5ETuQg1X9+Qoo6t6R1Bu0aIV6dChvagJQ/uu4/ddyQ5ZNBnxTDUcv+8oCz7v3buPfPppMr5nq4F9XNwvfiK3GWcepc5kWbVqJZkxY7qLnxkkZsyY5LPPPhPfjiijY5txrMf9aUR1T87q8Td79uxF6tSuHeGetO9G/j76y6WQEHI/kpOS4/cd0eoiy75hw0ayevWaCPvFrmR0Ve/YZhzrIRtkvH79eoRnwwa3pFSpMqRXr16iJgztb+DHEa3O1XPXWz9x0u8kWbJkEfrq06dP+b4b2qWjjNrfwI/jPempdwR6qWrVamTo0CERsuho39UjO/oz+gxkdGzf+I72fUdZQp+Gsr7akmTLlp3p/i9Erff35CgLcFaPPrl27ToSePgw+fXXX32aMcjriEBQStjoHjF8GClQoAAPuYUbQYd0dF2OKcJcOdYjXJMWostVPR6udgTDsR4dH9fp2rULiR07dnjSWTTI/8QL0a4JUK/doiYjcPZ9V7KfPHmS9Ordh0ydMoXEeC+6KffEw+YJ2dEg8P+A6AhnJerhwPDtt0VIkcKFX7snZ/f6zMGYubsnV/VQQN//UImsX7eWJyDWZMH3ISeAgdGMjKt7cpTRVb2j7NOnz+DXq1C+nKF70mTE9fAsgeP3Iz/38+fPk8DAQF52RZYsWbgDC2TH/QZfvEjSZ8hA2EyT5MyZ06mMGKw5dnrtmmDevHncYJYuXdptm3Gsh6OG9txd3ZOr+u3bt5M+P/clq1auiJCs2VF2x+9rsqO/aLIAZ/cKtHpH2W/cuEkqVa5M1q1dw+uc3ZMe2aNqS1rf1voNBjplypQjU6dOJkkSJ3bbloAz2R1l0VM/fsJvJFGiRKTS9xUjyIj2gh/g2LddPXc99ZrsZ86cId269+B6KVasmOGyAE/uyVFGx77t7LnDwCKBe7u2bUj+fPk8lt1ZvaMsIHI93umQIUPY4OMqGT9+XHi/9hVeGU0YzIrf/0Dmz5srNYv27NlzSLx4cUn58uVFjfXgXuE9iiz/2suWwcqVq5jiekYqVaokauSAowjDhg0l8S3wOHYFjFe9evXZ+50lauRwmI1a33nnXXLq1L+ixjlZs2YlX3zx/xH16VOnIhhNTylXvgJfsdAGcDKoU6cO9zbUFKIMcGyoRo0abCb/qaixnmXLlrHBVxxSvHgxUWM9u3btIitWrCSDBg0UNdYDvdSiRQB7p1Ol6SWYipYtW5IGDRqQXLlyiVrr6dKlK0mSJAlp166t1D7jEhhNT8C+EzzDPN03MsqsWbPp8uXLRUkOvrrXnTt30h49eoqSPOCEs2HDRlGSx+DBv5i2f6YXNurmXrr411OM7GnCm1P2XuahQ4fo1q1bRUkOV65coUOHDhMlOWB/s3379qIkh5s3b0p3wAkJuSRdL0EXwjNXtu9K585d6KZNm0TJHnhsNDdv3sIbikyeP39OR48eLUryWLx4sfR7BWicMjshYCNXbkRkg87Yrl07UZLH9OnTvR6EeWs0nz59SpMk/UT6cRooWNnXLFO2nPTBZu48eS11uHEG7lP2NfE+ZeslOHIdPXpUlOQAnTR58mRRsg8qy4kNwd6DzKVggGtiDwMOKjJhMxJ+QD1evHiiRg7lK1Qkfy1eFGEvRS+BgYdI9uw5eHSbIkWKiFr37Ny5k+8dFSok5yA4wH7QwoULSe3atUWNHLBkmT9/flGSA5yXEiZMKEpy8EVfVfgWZTQVCg/YuHEjWb58Gbl69RqPWlStWlVuQBUKxduBMpoKhUKhUOhErSsoFAqFQqETZTQVCoVCodCJMpoKhUKhUOhEGU2FQqFQKHSijKZCoVAoFDpRRlOhUCgUCp0oo6lQKBQKhU6U0VQoFAqFQifKaCoUCoVCoRNlNBUKhUKh0IkymgqFQqFQ6EQZTYVCoVAodKKMpkKhUCgUuiDkfxe9aGNDoJl4AAAAAElFTkSuQmCC"></p>
<p>(i) positive gradient at origin <em><strong>A1</strong></em></p>
<p> correct shape <em><strong>A1</strong></em></p>
<p> <br><strong>Note:</strong> Award second <em><strong>A1</strong></em> for a single maximum in 1<sup>st</sup> quadrant and tending toward an asymptote.</p>
<p> </p>
<p>(ii) positive gradient at <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>0</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>.</mo><mn>75</mn><mo>)</mo></math> <em><strong>A1</strong></em></p>
<p> correct shape <em><strong>A1</strong></em></p>
<p><em><strong><br></strong></em><strong>Note:</strong> Award second <em><strong>A1</strong></em> for a single minimum in 2<sup>nd</sup> quadrant, single maximum in 1<sup>st</sup> quadrant and tending toward an asymptote.</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>There were many good attempts at this question. Care needs to be taken over graph sketching, and the existence of asymptotes or the position of intersections needs to be shown clearly. Many candidates correctly found <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi mathvariant="normal">d</mi><mi>y</mi></mrow><mrow><mi mathvariant="normal">d</mi><mi>x</mi></mrow></mfrac><mo>=</mo><mn>0</mn></math> at <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>0</mn><mo>,</mo><mn>1</mn></mrow></mfenced></math> in part (a). However, they were then misled into finding a solution curve through this point rather than graphing the points where <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi mathvariant="normal">d</mi><mi>y</mi></mrow><mrow><mi mathvariant="normal">d</mi><mi>x</mi></mrow></mfrac><mo>=</mo><mn>0</mn></math> as required in part (b). Part (c) was answered well with a number of correct answers. Often the curve through <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>0</mn><mo>,</mo><mn>0</mn><mo>.</mo><mn>75</mn></mrow></mfenced></math> had a flat central section and did not show a clear maximum and minimum. The asymptotes were generally poorly drawn with the curves meeting the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis and stopping or worse still crossing over it.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>There were many good attempts at this question. Care needs to be taken over graph sketching, and the existence of asymptotes or the position of intersections needs to be shown clearly. Many candidates correctly found <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi mathvariant="normal">d</mi><mi>y</mi></mrow><mrow><mi mathvariant="normal">d</mi><mi>x</mi></mrow></mfrac><mo>=</mo><mn>0</mn></math> at <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>0</mn><mo>,</mo><mn>1</mn></mrow></mfenced></math> in part (a). However, they were then misled into finding a solution curve through this point rather than graphing the points where <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi mathvariant="normal">d</mi><mi>y</mi></mrow><mrow><mi mathvariant="normal">d</mi><mi>x</mi></mrow></mfrac><mo>=</mo><mn>0</mn></math> as required in part (b). Part (c) was answered well with a number of correct answers. Often the curve through <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>0</mn><mo>,</mo><mn>0</mn><mo>.</mo><mn>75</mn></mrow></mfenced></math> had a flat central section and did not show a clear maximum and minimum. The asymptotes were generally poorly drawn with the curves meeting the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis and stopping or worse still crossing over it.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>There were many good attempts at this question. Care needs to be taken over graph sketching, and the existence of asymptotes or the position of intersections needs to be shown clearly. Many candidates correctly found <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi mathvariant="normal">d</mi><mi>y</mi></mrow><mrow><mi mathvariant="normal">d</mi><mi>x</mi></mrow></mfrac><mo>=</mo><mn>0</mn></math> at <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>0</mn><mo>,</mo><mn>1</mn></mrow></mfenced></math> in part (a). However, they were then misled into finding a solution curve through this point rather than graphing the points where <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mi mathvariant="normal">d</mi><mi>y</mi></mrow><mrow><mi mathvariant="normal">d</mi><mi>x</mi></mrow></mfrac><mo>=</mo><mn>0</mn></math> as required in part (b). Part (c) was answered well with a number of correct answers. Often the curve through <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>0</mn><mo>,</mo><mn>0</mn><mo>.</mo><mn>75</mn></mrow></mfenced></math> had a flat central section and did not show a clear maximum and minimum. The asymptotes were generally poorly drawn with the curves meeting the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis and stopping or worse still crossing over it.</p>
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>The region bounded by <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfrac><mn>1</mn><mfenced><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mfenced></mfrac><mo>+</mo><mn>1</mn><mo>,</mo><mo> </mo><mi>x</mi><mo>=</mo><mn>0</mn><mo>,</mo><mo> </mo><mi>x</mi><mo>=</mo><mn>2</mn></math> and the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis is rotated through <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>π</mi></math> about the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-axis to form a solid.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Expand <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mfenced><mrow><mfrac><mn>1</mn><mi>u</mi></mfrac><mo>+</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup></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>Find <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><msup><mfenced><mrow><mfrac><mn>1</mn><mfenced><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mfenced></mfrac><mo>+</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup><mo>d</mo><mi>x</mi></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 volume of the solid formed. Give your answer in the form <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac><mfenced><mrow><mi>a</mi><mo>+</mo><mi>b</mi><mo> </mo><mi>ln</mi><mfenced><mi>c</mi></mfenced></mrow></mfenced></math>, where <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><mi mathvariant="normal">ℤ</mi></math>.</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><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><msup><mi>u</mi><mn>2</mn></msup></mfrac><mo>+</mo><mfrac><mn>2</mn><mi>u</mi></mfrac><mo>+</mo><mn>1</mn></math> <em><strong>A1</strong></em></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><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><msup><mfenced><mrow><mfrac><mn>1</mn><mfenced><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mfenced></mfrac><mo>+</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup><mo>d</mo><mi>x</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>∫</mo><mfenced><mrow><mfrac><mn>1</mn><msup><mfenced><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mfenced><mn>2</mn></msup></mfrac><mo>+</mo><mfrac><mn>2</mn><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mfrac><mo>+</mo><mn>1</mn></mrow></mfenced><mo>d</mo><mi>x</mi></math> <strong>OR</strong> <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><mfenced><mrow><mfrac><mn>1</mn><msup><mi>u</mi><mn>2</mn></msup></mfrac><mo>+</mo><mfrac><mn>2</mn><mi>u</mi></mfrac><mo>+</mo><mn>1</mn></mrow></mfenced><mo>d</mo><mi>u</mi></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><mfrac><mn>1</mn><mfenced><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mfenced></mfrac><mo>+</mo><mn>2</mn><mo> </mo><mi>ln</mi><mfenced open="|" close="|"><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mfenced><mo>+</mo><mi>x</mi><mfenced><mrow><mo>+</mo><mi>c</mi></mrow></mfenced></math> <em><strong>A1A1</strong></em></p>
<p><strong><br>Note:</strong> Award <em><strong>A1</strong> </em>for first expression, <em><strong>A1</strong> </em>for second two expressions. <br>Award <em><strong>A1A0</strong> </em>for a final answer of <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><mfrac><mn>1</mn><mi>u</mi></mfrac><mo>+</mo><mn>2</mn><mo> </mo><mi>ln</mi><mfenced><mi>u</mi></mfenced><mo>+</mo><mi>u</mi><mfenced><mrow><mo>+</mo><mi>c</mi></mrow></mfenced></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>volume <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi mathvariant="normal">π</mi><msubsup><mfenced open="[" close="]"><mrow><mo>-</mo><mfrac><mn>1</mn><mfenced><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mfenced></mfrac><mo>+</mo><mn>2</mn><mo> </mo><mi>ln</mi><mfenced><mrow><mi>x</mi><mo>+</mo><mn>2</mn></mrow></mfenced><mo>+</mo><mi>x</mi></mrow></mfenced><mn>0</mn><mn>2</mn></msubsup></math> <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi mathvariant="normal">π</mi><mfenced><mrow><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mo>+</mo><mn>2</mn><mo> </mo><mi>ln</mi><mfenced><mn>4</mn></mfenced><mo>+</mo><mn>2</mn><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>-</mo><mn>2</mn><mo> </mo><mi>ln</mi><mo> </mo><mn>2</mn></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi mathvariant="normal">π</mi><mfenced><mrow><mfrac><mn>9</mn><mn>4</mn></mfrac><mo>+</mo><mn>2</mn><mo> </mo><mi>ln</mi><mfenced><mn>4</mn></mfenced><mo>-</mo><mn>2</mn><mo> </mo><mi>ln</mi><mo> </mo><mn>2</mn></mrow></mfenced></math></p>
<p>use of log laws seen, for example <em><strong>M1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">π</mi><mfenced><mrow><mfrac><mn>9</mn><mn>4</mn></mfrac><mo>+</mo><mn>4</mn><mo> </mo><mi>ln</mi><mfenced><mn>2</mn></mfenced><mo>-</mo><mn>2</mn><mo> </mo><mi>ln</mi><mo> </mo><mn>2</mn></mrow></mfenced></math> <strong>OR </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">π</mi><mfenced><mrow><mfrac><mn>9</mn><mn>4</mn></mfrac><mo>+</mo><mn>2</mn><mo> </mo><mi>ln</mi><mfenced><mfrac><mn>4</mn><mn>2</mn></mfrac></mfenced></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac><mfenced><mrow><mn>9</mn><mo>+</mo><mn>8</mn><mo> </mo><mi>ln</mi><mo> </mo><mfenced><mn>2</mn></mfenced></mrow></mfenced></math> <strong>OR </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>9</mn><mo>,</mo><mo> </mo><mi>b</mi><mo>=</mo><mn>8</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>=</mo><mn>2</mn></math> <em><strong>A1</strong></em></p>
<p><strong><br>Note:</strong> Other correct integer solutions are possible and should be accepted for example <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mn>9</mn><mo>,</mo><mo> </mo><mi>b</mi><mo>=</mo><mi>c</mi><mo>=</mo><mn>4</mn></math>.</p>
<p> </p>
<p><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;">
<p>Some candidates could answer part (a) (i). The link between parts (i) and (ii) was, however, lost to the majority. Those who did see the link were often able to give a reasonable answer to (ii). But some candidates lacked the skills to integrate without the use of technology, so an indefinite integral presented many problems. Even those who successfully navigated part (a) went on to fail to see the link to part (b). Either the integration was attempted as something totally new and unconnected or it was simply found with the GDC which did not lead to a final answer in the required form.</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">b.</div>
</div>
<br><hr><br><div class="specification">
<p>Consider the curve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>2</mn><mi>x</mi><mfenced><mrow><mn>4</mn><mo>-</mo><msup><mtext>e</mtext><mi>x</mi></msup></mrow></mfenced></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac></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 <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mtext>d</mtext><mn>2</mn></msup><mi>y</mi></mrow><mrow><mo>d</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The curve has a point of inflexion at <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>a</mi><mo>,</mo><mo> </mo><mi>b</mi></mrow></mfenced></math>.</p>
<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">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>use of product rule <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mn>2</mn><mfenced><mrow><mn>4</mn><mo>-</mo><msup><mtext>e</mtext><mi>x</mi></msup></mrow></mfenced><mo>+</mo><mn>2</mn><mi>x</mi><mfenced><mrow><mo>-</mo><msup><mtext>e</mtext><mi>x</mi></msup></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>8</mn><mo>-</mo><mn>2</mn><msup><mtext>e</mtext><mi>x</mi></msup><mo>-</mo><mn>2</mn><mi>x</mi><msup><mtext>e</mtext><mi>x</mi></msup></math></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>use of product rule <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>y</mi></mrow><mrow><mo>d</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac><mo>=</mo><mo>-</mo><mn>2</mn><msup><mtext>e</mtext><mi>x</mi></msup><mo>-</mo><mn>2</mn><msup><mtext>e</mtext><mi>x</mi></msup><mo>-</mo><mn>2</mn><mi>x</mi><msup><mtext>e</mtext><mi>x</mi></msup></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><mn>4</mn><msup><mtext>e</mtext><mi>x</mi></msup><mo>-</mo><mn>2</mn><mi>x</mi><msup><mtext>e</mtext><mi>x</mi></msup></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo>-</mo><mn>2</mn><mfenced><mrow><mn>2</mn><mo>+</mo><mi>x</mi></mrow></mfenced><msup><mtext>e</mtext><mi>x</mi></msup></math></p>
<p><em><strong><br>[2 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"><mo>-</mo><mn>2</mn><mfenced><mrow><mn>2</mn><mo>+</mo><mi>a</mi></mrow></mfenced><msup><mtext>e</mtext><mi>a</mi></msup><mo>=</mo><mn>0</mn></math> <strong>OR </strong>sketch of <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mo>d</mo><mn>2</mn></msup><mi>y</mi></mrow><mrow><mo>d</mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></math> with <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-intercept indicated <strong>OR </strong>finding the local maximum of <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac></math> at <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>-</mo><mn>2</mn><mo>,</mo><mo> </mo><mn>8</mn><mo>.</mo><mn>27</mn></mrow></mfenced></math> <em><strong>(M1)</strong></em></p>
<p><img src="data:image/png;base64,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"></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>a</mi><mo>=</mo></mrow></mfenced><mo> </mo><mo> </mo><mo>-</mo><mn>2</mn></math> <em><strong>A1</strong></em></p>
<p><em><strong><br>[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>Some candidates attempted to apply the product rule in parts (a)(i) and (ii) but often incorrectly, particularly in part (ii) when finding <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><msup><mtext>d</mtext><mn>2</mn></msup><mi>y</mi></mrow><mrow><mtext>d</mtext><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></math>. In part (b) there was little understanding shown of the point of inflexion. There were some attempts, some of which were correct, but many where either the function or the first derivative were set to zero rather than the second derivative.</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">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The rates of change of the area covered by two types of fungi, X and Y, on a particular tree are given by the following equations, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> is the area covered by X and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span> is the area covered by Y.</p>
<p><span class="mjpage mjpage__block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" alttext="\frac{{{\text{d}}x}}{{{\text{d}}t}} = 3x - 2y">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mn>3</mn>
<mi>x</mi>
<mo>−<!-- − --></mo>
<mn>2</mn>
<mi>y</mi>
</math></span></p>
<p><span class="mjpage mjpage__block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" alttext="\frac{{{\text{d}}y}}{{{\text{d}}t}} = 2x - 2y">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mn>2</mn>
<mi>x</mi>
<mo>−<!-- − --></mo>
<mn>2</mn>
<mi>y</mi>
</math></span></p>
<p>The matrix <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 3&{ - 2} \\ 2&{ - 2} \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>2</mn>
</mtd>
<mtd>
<mrow>
<mo>−<!-- − --></mo>
<mn>2</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> has eigenvalues of 2 and −1 with corresponding eigenvectors <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 2 \\ 1 \end{array}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 1 \\ 2 \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>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
<p>Initially <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> = 8 cm<sup>2</sup> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span> = 10 cm<sup>2</sup>.</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="\frac{{{\text{d}}y}}{{{\text{d}}x}}">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
</math></span> when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t = 0">
<mi>t</mi>
<mo>=</mo>
<mn>0</mn>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>On the following axes, sketch a possible trajectory for the growth of the two fungi, making clear any asymptotic behaviour.</p>
<p><img src="data:image/png;base64,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"></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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}} = \frac{{16 - 20}}{{24 - 20}}">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mn>16</mn>
<mo>−</mo>
<mn>20</mn>
</mrow>
<mrow>
<mn>24</mn>
<mo>−</mo>
<mn>20</mn>
</mrow>
</mfrac>
</math></span> <em><strong>M1</strong></em></p>
<p>= −1 <em><strong> 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>asymptote of trajectory along <em><strong>r </strong></em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = k\left( {\begin{array}{*{20}{c}} 2 \\ 1 \end{array}} \right)">
<mo>=</mo>
<mi>k</mi>
<mrow>
<mo>(</mo>
<mrow>
<mtable rowspacing="4pt" columnspacing="1em">
<mtr>
<mtd>
<mn>2</mn>
</mtd>
</mtr>
<mtr>
<mtd>
<mn>1</mn>
</mtd>
</mtr>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>M1A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>M1A0</strong></em> if asymptote along <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\begin{array}{*{20}{c}} 1 \\ 2 \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>
</mtable>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
<p>trajectory begins at (8, 10) with negative gradient <em><strong> A1A1</strong></em></p>
<p><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>Consider the function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = \frac{1}{{{x^2} + 3x + 2}},{\text{ }}x \in \mathbb{R},{\text{ }}x \ne - 2,{\text{ }}x \ne - 1">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>3</mn>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
</mrow>
</mfrac>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>x</mi>
<mo>∈<!-- ∈ --></mo>
<mrow>
<mi mathvariant="double-struck">R</mi>
</mrow>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>x</mi>
<mo>≠<!-- ≠ --></mo>
<mo>−<!-- − --></mo>
<mn>2</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>x</mi>
<mo>≠<!-- ≠ --></mo>
<mo>−<!-- − --></mo>
<mn>1</mn>
</math></span>.</p>
</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="{x^2} + 3x + 2">
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>3</mn>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
</math></span> in the form <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{(x + h)^2} + k">
<mrow>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>+</mo>
<mi>h</mi>
<msup>
<mo stretchy="false">)</mo>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mi>k</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>Factorize <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^2} + 3x + 2">
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>3</mn>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
</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>Sketch the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x)">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
</math></span>, indicating on it the equations of the asymptotes, the coordinates of the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span>-intercept and the local maximum.</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>Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{1}{{x + 1}} - \frac{1}{{x + 2}} = \frac{1}{{{x^2} + 3x + 2}}">
<mfrac>
<mn>1</mn>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</mfrac>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>3</mn>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
</mrow>
</mfrac>
</math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find the value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p">
<mi>p</mi>
</math></span> if <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_0^1 {f(x){\text{d}}x = \ln (p)} ">
<msubsup>
<mo>∫</mo>
<mn>0</mn>
<mn>1</mn>
</msubsup>
<mrow>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
<mo>=</mo>
<mi>ln</mi>
<mo></mo>
<mo stretchy="false">(</mo>
<mi>p</mi>
<mo stretchy="false">)</mo>
</mrow>
</math></span>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f\left( {\left| x \right|} \right)">
<mi>y</mi>
<mo>=</mo>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mo>|</mo>
<mi>x</mi>
<mo>|</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the area of the region enclosed between the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f\left( {\left| x \right|} \right)">
<mi>y</mi>
<mo>=</mo>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<mo>|</mo>
<mi>x</mi>
<mo>|</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span>, the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>-axis and the lines with equations <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> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 1">
<mi>x</mi>
<mo>=</mo>
<mn>1</mn>
</math></span>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">f.</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="{x^2} + 3x + 2 = {\left( {x + \frac{3}{2}} \right)^2} - \frac{1}{4}">
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>3</mn>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mfrac>
<mn>3</mn>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mn>4</mn>
</mfrac>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[1 mark]</em></strong></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="{x^2} + 3x + 2 = (x + 2)(x + 1)">
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>3</mn>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
<mo>=</mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img src="images/Schermafbeelding_2017-08-08_om_13.58.40.png" alt="M17/5/MATHL/HP1/ENG/TZ1/B11.b/M"></p>
<p><strong><em>A1</em></strong> for the shape</p>
<p><strong><em>A1</em></strong> for the equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = 0">
<mi>y</mi>
<mo>=</mo>
<mn>0</mn>
</math></span></p>
<p><strong><em>A1</em></strong> for asymptotes <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = - 2">
<mi>x</mi>
<mo>=</mo>
<mo>−</mo>
<mn>2</mn>
</math></span> and <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></p>
<p><strong><em>A1</em></strong> for coordinates <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( { - \frac{3}{2},{\text{ }} - 4} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mo>−</mo>
<mfrac>
<mn>3</mn>
<mn>2</mn>
</mfrac>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mo>−</mo>
<mn>4</mn>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p><strong><em>A1</em></strong> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span>-intercept <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {0,{\text{ }}\frac{1}{2}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mn>0</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p><strong><em>[5 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="\frac{1}{{x + 1}} - \frac{1}{{x + 2}} = \frac{{(x + 2) - (x + 1)}}{{(x + 1)(x + 2)}}">
<mfrac>
<mn>1</mn>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</mfrac>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
<mo>−</mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
</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{1}{{{x^2} + 3x + 2}}">
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>3</mn>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
</mrow>
</mfrac>
</math></span> <strong><em>AG</em></strong></p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int\limits_0^1 {\frac{1}{{x + 1}} - \frac{1}{{x + 2}}{\text{d}}x} ">
<munderover>
<mo>∫</mo>
<mn>0</mn>
<mn>1</mn>
</munderover>
<mrow>
<mfrac>
<mn>1</mn>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</mfrac>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
</mrow>
</mfrac>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left[ {\ln (x + 1) - \ln (x + 2)} \right]_0^1">
<mo>=</mo>
<msubsup>
<mrow>
<mo>[</mo>
<mrow>
<mi>ln</mi>
<mo></mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>+</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>−</mo>
<mi>ln</mi>
<mo></mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>+</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
</mrow>
<mo>]</mo>
</mrow>
<mn>0</mn>
<mn>1</mn>
</msubsup>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \ln 2 - \ln 3 - \ln 1 + \ln 2">
<mo>=</mo>
<mi>ln</mi>
<mo></mo>
<mn>2</mn>
<mo>−</mo>
<mi>ln</mi>
<mo></mo>
<mn>3</mn>
<mo>−</mo>
<mi>ln</mi>
<mo></mo>
<mn>1</mn>
<mo>+</mo>
<mi>ln</mi>
<mo></mo>
<mn>2</mn>
</math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \ln \left( {\frac{4}{3}} \right)">
<mo>=</mo>
<mi>ln</mi>
<mo></mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>4</mn>
<mn>3</mn>
</mfrac>
</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="\therefore p = \frac{4}{3}">
<mo>∴</mo>
<mi>p</mi>
<mo>=</mo>
<mfrac>
<mn>4</mn>
<mn>3</mn>
</mfrac>
</math></span></p>
<p><strong><em>[4 marks]</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img src="images/Schermafbeelding_2017-08-08_om_14.20.03.png" alt="M17/5/MATHL/HP1/ENG/TZ1/B11.e/M"></p>
<p>symmetry about the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span>-axis <strong><em>M1</em></strong></p>
<p>correct shape <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Allow <strong><em>FT </em></strong>from part (b).</p>
<p> </p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2\int_0^1 {f(x){\text{d}}x} ">
<mn>2</mn>
<msubsup>
<mo>∫</mo>
<mn>0</mn>
<mn>1</mn>
</msubsup>
<mrow>
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</math></span> <strong><em>(M1)(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 2\ln \left( {\frac{4}{3}} \right)">
<mo>=</mo>
<mn>2</mn>
<mi>ln</mi>
<mo></mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mn>4</mn>
<mn>3</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Do not award <strong><em>FT </em></strong>from part (e).</p>
<p> </p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">f.</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>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">f.</div>
</div>
<br><hr><br><div class="specification">
<p>A particle, A, moves so that its velocity (<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\nu ">
<mi>ν<!-- ν --></mi>
</math></span> ms<sup>−1</sup>) at time <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> is given by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\nu ">
<mi>ν<!-- ν --></mi>
</math></span> = 2 sin <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span>, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> ≥ 0.</p>
<p>The kinetic energy (<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="E">
<mi>E</mi>
</math></span>) of the particle A is measured in joules (J) and is given by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="E">
<mi>E</mi>
</math></span> = 5<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\nu ">
<mi>ν<!-- ν --></mi>
</math></span><sup>2</sup>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down an expression for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="E">
<mi>E</mi>
</math></span> as a function of time.</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>Hence find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}E}}{{{\text{d}}t}}">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>E</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
</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>Hence or otherwise find the first time at which the kinetic energy is changing at a rate of 5 J s<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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="E = 5{\left( {2\,{\text{sin}}\,t} \right)^2}\,\left( { = 20\,{\text{si}}{{\text{n}}^2}\,t} \right)">
<mi>E</mi>
<mo>=</mo>
<mn>5</mn>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>t</mi>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mn>20</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>si</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>t</mi>
</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">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}E}}{{{\text{d}}t}} = 40\,{\text{sin}}\,t\,{\text{cos}}\,t">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>E</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mn>40</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>t</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>t</mi>
</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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> = 0.126 <em><strong>(M1)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>The graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f'\left( x \right)">
<mi>y</mi>
<mo>=</mo>
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span>, 0 ≤ <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> ≤ 5 is shown in the following diagram. The curve intercepts the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>-axis at (1, 0) and (4, 0) and has a local minimum at (3, −1).</p>
<p style="text-align: center;"><img 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"></p>
</div>
<div class="specification">
<p>The shaded area enclosed by the curve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f'\left( x \right)">
<mi>y</mi>
<mo>=</mo>
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span>, the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>-axis and the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span>-axis is 0.5. Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( 0 \right) = 3">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mn>0</mn>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>3</mn>
</math></span>,</p>
</div>
<div class="specification">
<p>The area enclosed by the curve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f'\left( x \right)">
<mi>y</mi>
<mo>=</mo>
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span> and the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>-axis between <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 = 4">
<mi>x</mi>
<mo>=</mo>
<mn>4</mn>
</math></span> is 2.5 .</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span>-coordinate of the point of inflexion on the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f\left( x \right)">
<mi>y</mi>
<mo>=</mo>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span>.</p>
<div class="marks">[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="f\left( 1 \right)">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mn>1</mn>
<mo>)</mo>
</mrow>
</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 value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( 4 \right)">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mn>4</mn>
<mo>)</mo>
</mrow>
</math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the curve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = f\left( x \right)">
<mi>y</mi>
<mo>=</mo>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
</math></span>, 0 ≤ <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span> ≤ 5 indicating clearly the coordinates of the maximum and minimum points and any intercepts with the coordinate axes.</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 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 <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>attempt to use definite integral of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f'\left( x \right)">
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
<mrow>
<mo>(</mo>
<mi>x</mi>
<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="\int_0^1 {f'\left( x \right){\text{d}}x} = 0.5">
<msubsup>
<mo>∫</mo>
<mn>0</mn>
<mn>1</mn>
</msubsup>
<mrow>
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
<mo>=</mo>
<mn>0.5</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( 1 \right) - f\left( 0 \right) = 0.5">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mn>1</mn>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mn>0</mn>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0.5</mn>
</math></span> <em><strong> (A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( 1 \right) = 0.5 + 3">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mn>1</mn>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0.5</mn>
<mo>+</mo>
<mn>3</mn>
</math></span></p>
<p>= 3.5 <em><strong> A1</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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_1^4 {f'\left( x \right){\text{d}}x} = - 2.5">
<msubsup>
<mo>∫</mo>
<mn>1</mn>
<mn>4</mn>
</msubsup>
<mrow>
<msup>
<mi>f</mi>
<mo>′</mo>
</msup>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
<mo>=</mo>
<mo>−</mo>
<mn>2.5</mn>
</math></span> <em><strong> (A1)</strong></em></p>
<p><strong>Note:</strong> <em><strong>(A1)</strong></em> is for −2.5.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( 4 \right) - f\left( 1 \right) = - 2.5">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mn>4</mn>
<mo>)</mo>
</mrow>
<mo>−</mo>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mn>1</mn>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mo>−</mo>
<mn>2.5</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( 4 \right) = 3.5 - 2.5">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mn>4</mn>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>3.5</mn>
<mo>−</mo>
<mn>2.5</mn>
</math></span></p>
<p>= 1 <em><strong> A1</strong></em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img src="data:image/png;base64,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"> <em><strong> A1A1A1</strong></em></p>
<p><em><strong>A1</strong></em> for correct shape over approximately the correct domain<br><em><strong>A1</strong></em> for maximum and minimum (coordinates or horizontal lines from 3.5 and 1 are required),<br><em><strong>A1</strong></em> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span>-intercept at 3</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;">
[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>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</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 second order differential equation</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>x</mi><mo>¨</mo></mover><mo>+</mo><mn>4</mn><msup><mfenced><mover><mi>x</mi><mo>˙</mo></mover></mfenced><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>t</mi><mo>=</mo><mn>0</mn></math></p>
<p>where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> is the displacement of a particle for <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>≥</mo><mn>0</mn></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write the differential equation as a system of coupled first order differential equations.</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>When <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn></math>, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mover><mi>x</mi><mo>˙</mo></mover><mo>=</mo><mn>0</mn></math></p>
<p>Use Euler’s method with a step length of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>1</mn></math> to find an estimate for the value of the displacement and velocity of the particle when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>1</mn></math>.</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 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"><mover><mi>x</mi><mo>˙</mo></mover><mo>=</mo><mi>y</mi></math> <strong>M1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>y</mi><mo>˙</mo></mover><mo>=</mo><mn>2</mn><mi>t</mi><mo>-</mo><mn>4</mn><msup><mi>y</mi><mn>2</mn></msup></math> <strong>A1</strong></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><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>t</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><msub><mi>t</mi><mi>n</mi></msub><mo>+</mo><mn>0</mn><mo>.</mo><mn>1</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>x</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><msub><mi>x</mi><mi>n</mi></msub><mo>+</mo><mn>0</mn><mo>.</mo><mn>1</mn><msub><mi>y</mi><mi>n</mi></msub></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>y</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><msub><mi>y</mi><mi>n</mi></msub><mo>+</mo><mn>0</mn><mo>.</mo><mn>1</mn><mfenced><mrow><mn>2</mn><msub><mi>t</mi><mi>n</mi></msub><mo>-</mo><mn>4</mn><msubsup><mi>y</mi><mi>n</mi><mn>2</mn></msubsup></mrow></mfenced></math> <strong>(M1)(A1)</strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong>M1</strong> for a correct attempt to substitute the functions in part (a) into the formula for Euler’s method for coupled systems.</p>
<p> </p>
<p>When <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>1</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>202</mn><mo> </mo><mfenced><mrow><mn>0</mn><mo>.</mo><mn>20201</mn><mo>…</mo></mrow></mfenced></math> <strong>A1</strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>x</mi><mo>˙</mo></mover><mo>=</mo><mn>0</mn><mo>.</mo><mn>598</mn><mo> </mo><mfenced><mrow><mn>0</mn><mo>.</mo><mn>59822</mn><mo>…</mo></mrow></mfenced></math> <strong>A1</strong></p>
<p> </p>
<p><strong>Note:</strong> Accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>598</mn></math>.</p>
<p> </p>
<p><strong>[4 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>The wind chill index <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>W</mi></math> is a measure of the temperature, in <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>°</mo><mtext>C</mtext></math>, felt when taking into account the effect of the wind.</p>
<p>When Frieda arrives at the top of a hill, the relationship between the wind chill index and the speed of the wind <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi></math> in kilometres per hour <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><msup><mtext>km h</mtext><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo>)</mo></math> is given by the equation</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>W</mi><mo>=</mo><mn>19</mn><mo>.</mo><mn>34</mn><mo>-</mo><mn>7</mn><mo>.</mo><mn>405</mn><msup><mi>v</mi><mrow><mn>0</mn><mo>.</mo><mn>16</mn></mrow></msup></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"><mfrac><mrow><mo>d</mo><mi>W</mi></mrow><mrow><mo>d</mo><mi>v</mi></mrow></mfrac></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>When Frieda arrives at the top of a hill, the speed of the wind is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn></math> kilometres per hour and increasing at a rate of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>5</mn><mo> </mo><msup><mtext>km h</mtext><mrow><mo>-</mo><mn>1</mn></mrow></msup><mo> </mo><msup><mtext>minute</mtext><mrow><mo>-</mo><mn>1</mn></mrow></msup></math>.</p>
<p>Find the rate of change of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>W</mi></math> at this time.</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>use of power rule <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>W</mi></mrow><mrow><mo>d</mo><mi>v</mi></mrow></mfrac><mo>=</mo><mo>-</mo><mn>1</mn><mo>.</mo><mn>1848</mn><msup><mi>v</mi><mrow><mo>-</mo><mn>0</mn><mo>.</mo><mn>84</mn></mrow></msup></math> <strong>OR </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>1</mn><mo>.</mo><mn>18</mn><msup><mi>v</mi><mrow><mo>-</mo><mn>0</mn><mo>.</mo><mn>84</mn></mrow></msup></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><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>v</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mn>5</mn></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>W</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mo>d</mo><mi>v</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>×</mo><mfrac><mrow><mo>d</mo><mi>W</mi></mrow><mrow><mo>d</mo><mi>v</mi></mrow></mfrac></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mfrac><mrow><mo>d</mo><mi>W</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mo>-</mo><mn>5</mn><mo>×</mo><mn>1</mn><mo>.</mo><mn>1848</mn><msup><mi>v</mi><mrow><mo>-</mo><mn>0</mn><mo>.</mo><mn>84</mn></mrow></msup></mrow></mfenced></math></p>
<p>when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mo>=</mo><mn>10</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>W</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mo>-</mo><mn>5</mn><mo>×</mo><mn>1</mn><mo>.</mo><mn>1848</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>0</mn><mo>.</mo><mn>84</mn></mrow></msup></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>0</mn><mo>.</mo><mn>856</mn><mo> </mo><mo> </mo><mfenced><mrow><mo>-</mo><mn>0</mn><mo>.</mo><mn>856278</mn><mo>…</mo></mrow></mfenced><mo> </mo><mo>°</mo><msup><mtext>C min</mtext><mrow><mo>-</mo><mn>1</mn></mrow></msup></math> <em><strong>A2</strong></em></p>
<p> </p>
<p><strong>Note:</strong> Accept a negative answer communicated in words, “decreasing at a rate of…”. <br>Accept a final answer of <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>0</mn><mo>.</mo><mn>852809</mn><mo>…</mo><mo> </mo><mo>°</mo><msup><mtext>C min</mtext><mrow><mo>-</mo><mn>1</mn></mrow></msup></math> from use of <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>1</mn><mo>.</mo><mn>18</mn></math>.<br>Accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>51</mn><mo>.</mo><mn>4</mn></math> (or <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>51</mn><mo>.</mo><mn>2</mn></math>)<math xmlns="http://www.w3.org/1998/Math/MathML"><mo> </mo><mo>°</mo><msup><mtext>C hour</mtext><mrow><mo>-</mo><mn>1</mn></mrow></msup></math>.</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;">
<p>There was some success in using the power rule to differentiate the function in part (a). Many failed to recognize that part (b) was a related rates of change problem. There was also confusion about the term “rate of change” and with the units used in this question.</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>A slope field for the differential equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mfrac><mi>y</mi><mn>2</mn></mfrac></math> is shown.</p>
<p><img 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"></p>
<p>Some of the solutions to the differential equation have a local maximum point and a local minimum point.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Write down the equation of the curve on which all these maximum and minimum points lie.</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>Sketch this curve on the slope field.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The solution to the differential equation that passes through the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>0</mn><mo>,</mo><mo> </mo><mo>−</mo><mn>2</mn><mo>)</mo></math> has both a local maximum point and a local minimum point.</p>
<p>On the slope field, sketch the solution to the differential equation that passes through <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>0</mn><mo>,</mo><mo> </mo><mo>−</mo><mn>2</mn><mo>)</mo></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"><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mfrac><mi>y</mi><mn>2</mn></mfrac><mo>=</mo><mn>0</mn><mo> </mo><mo> </mo><mfenced><mrow><mi>y</mi><mo>=</mo><mo>-</mo><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><em><strong><br>[1 mark]</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"><mi>y</mi><mo>=</mo><mo>-</mo><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup></math> drawn on diagram (correct shape with a maximum at <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>)</mo></math>) <em><strong>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><img 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X7ygetb3OIWXSad8cIfb3CuDKyxR3esaIYvRmtkq9YYVq1xrh784AevmnI9rgbpYwit0O6Sq/a4vDr//PNXrcPv6aGTJoSu8nPeCqSHNgrq8ltDW9O0EcE6v+MYokPr5FevfvWrVw95yENWrYDW9DWPgA+e8idPdLn44otXbQS0+qM/+qOPogvtGBKPX6sUq3POOWfVRmZrvo7xZ64rr5zT6S1vecvqrne966qNqta6hU9CrqMbhP6CCy5YtQ5m1TqKHhcbI8MxNOA8/NC/+MUvXp199tmrNtKZpHOekGv5yKFzG72snv3sZ6/aE8al8o20kZmAvo2EV20Eu2qdw6qN5FbtyaKX45gfr3YzWD360Y9e8wU8Wsewak8yvfxcV9qRRw2JF5RfG7Wu6YX2uL5qI7XOI/mdV94CGvXne7/3e/t50muASlsD3sqhPUGsvvVbv3XVnn76tRCaSldtiF4Cv7Ub3eolL3lJpxXnOOatNII2mPzaQrvBrtpT0jqd7Eq/KeChPNsNedUGLt1/aCD0eEVW5ZkjmnajXj3rWc+6VFmEpvohIXwE6W0kv7rf/e63ajetTocPvuEhVJ0EcWiF9tS4utvd7rZqTwKXkoe+deKrdjNYx9f00AttxL9qT3g9Hv/jwuJfmGrC+x2oKd/vOq6xcJdyx5oD+UMHocudLUcY1XPtruku6W4d2bBUfmxxjZ9jlQ3ysDV35ZouLRC/RH4ryLW8aj+eRnlGFdJHnuKEyJbuXFmMum9C5Kf8nOMzh4f88rZK2Y9oxfHPqGsFmSAvWj41KgIyI3eOfNM57SbTn+hue9vb9hGaKRejQ3UBj8gDj8lGzaY1xEund2SJmzuCCl+07Ahcsz/pppLoVWXkHFzHd5EtPWEbIgPwyLXRoFFr+I0yR5AtJA9e1X+1PKtMQGf+mSxl7zr0Yz3YpEN4OqKF0KKpfPiaDHnHuhYfiJNnTj2uQBPeAXp1FG/1FO9NPCu9PAlo6eKJy9OAF+Xeh8HICy0bBfZJT792HLh0iSxEDDOHnIJaArR4hI/jCPEJgUZk3pDzOZKTlwAv+kZmGkaVEUg3j2leLqj6VD5LgKZWnsrDdJACT9oUYrt86JbKR68ig7lejTaNaRfkQe+oMuYmuAvsEXRIOuvaOJI+B6YN2sipv1+5zW1u0zv5NMYliM/QxYe7/BgbgsisNvCr+ePKb7RNPFr+S77wDs0mVFlCOlcvc8ke/boJoReiT3QZUfMK8pqqMN0RWbFlzLsNyQ+Oruk/0qmb3md4vyR9hPzpg2L/LshT9Q1SpgZcbMz1JoTWkezo4Zre3kGAqckqK8fAAFa9Eb9L5lIs5hYFNG4djcAZU87fBQalksbhuwooTkAHzoXRadtAV3qjc6xyp+R7YaLwxgoUXcQ7zkW11bHKB9e1A6jItTx0ShmM+XZB/sjGJ7x2ITQJ4TMH8iZ//B7auTxCaw7VXLb5ah1/0mCOTtKXlFlF6nr8jo9jID71cxvoEPpN5b0LqX+Cjqn6dRevyI6vQge7fFM7s/BwFGJHwi7Iw/7In/Idvm5ibizyVP7R1ZH9jjXPNkiP7vFd5SukvDeh8qA7PoE4L+rBO5RA/hFuChl8HTeO9OPgcYjVNF5KzKncR0Hk5ZGRPI7cJxSy5ZNGjDqVfcuLfaYhQIWYqhRHARmBcyMJ9pHpulbU4wK+7OBPFdp55OzDxoDcqambfQF/Nv7d3/1d9ycb9y1Tp4u/1Sxe5J1UO1SO7NtHfRnBn54CdYSmxPYpM2WYUNviHKAP+MhiAIsu9JP6kBHJb0ApWKignznOOnPkXosyHL/vDjDgeAWuMzwJKAQ3sbzF3zfI84SUQj7Owh6Bt3KzUiEd0j7LEX/lp/KbvgE67NPGk0QaLBtNL54UlBmZVuycZDvUBnXAJwEzCN5bZSXQSYBtbGTrYeqo+mDJtqWiVhHlndSItAHpbITUpePCkTyWhqvjPQlwhkI2Aj2pjj7yjCROokMiQ+UAvj3uAq9I+RlFAFvzSL4PkMc+j9+pM9HhnxL4sb7T2Tf40I3amvZ91pdAGZLnZn1SHb16mXpzEjaC94DaxtJ2n/z0fPvb397rt6XOm/QWL+hjIu+4bTxSRx+DPBov6SBi2FKExpzsvh9PA4VkjbE1/yfR8QpWE5G16/E0+XN+WLBNg1WGRx0tbdMn9UXZeRx2TV7i/6mA7Z4AT8ouctQXH86RfZS6MBdkqJ9jHd2XbHUzL2NPAnzKtsP0MykDwbcidPe0ta0+yGsRho/45DvuunPkZyANlTMOW8CHofPRhGVkJwEO94GGQt/WCR7W/imYKprb4R61QtDbY7HjUTv5XYgMN2rzkK6F467U/5hgD/io5iRBro+qdPj7RsqNPNMNsXnfUE8NgvaN1EfTKPqapfUz/kDnq2f9o4+4tg3c5JUeWUtl7sKRWnYM8oXXNiNG1IqxxCB08nuE29fb6RFk+mrO14VV7wrxm9KWAh+Pb3OekPgiso9SMXzVl6eIo9gRPab0yTUZ4xz9UWR+LII9mX47CcTfvgZe0g6PAjds7dCTYC2/sdyPCzpLa9HdQPclY4Tp4cNM3QCf8E22ILG9yKabcPh7V+aL2jltfymOZQi3dDUKwwTO0OjnggxO8Ln1YbdGXgo6GrVoQLVCB+ISjgP4eJFHnoqxrZJV2Uvkx/8JplH4NteHRfSYumHEFkHjsRTQuXxHkVkR2Qn/GIjt1kOfFPhP+Rl9ngTI0w4tGtCZ1bY/lvsS1LIb+ZCZKb+jyNiGKl9gn/dzbF0iM32anVXdLMw+2BZhVx/piUVfmnZxnFjU0Ue4I+ONqjVYG/ssvQtxBj5GdlmHL2wzUJoOUMeb6SKdho5/yQ2DHPrGBnPUCmTKBoVjXtD8mQIQAvLRSzO62aV/EPk58oEN3PDCPyt8yN7GT97w8JJKo3OdsAl4hs45//Ej+iVAy+940d0xOox6x2/ilV2+aqZDeKB1nrIceYyQjk5+fMBR3C5ER0c00SO61LAN0mN7ZOPrKWkb5KkBLT6RvwTo0fpSGH0WD+A5B9EBjbaoPm/TQV51M1Oa0d3RACwvTF07LgH5U4s72OUFtznskWdkyYNee3bNhiW+xEc7ztNYbITa7kegiyy6pz/Lds1zn0L4zgBBX4B+qe+24UgjeutCbX5pN8KlSjGcU+zaZ+c9Dp7jDHIs5/SY49zXgHbcy1TAHJCTu6u34jbk0jhgrBiupekIyYudOVfx7ENuF8+5FZt8QV5227Huu7/7u3tjAfZJw2+bT1IZbOz1Ld/yLX1nwui/y5d4ByqWvdDtpT3avwvka2A6Gbsvut4kWxqYo8+mYDoK+XUu9uK+8MILe565iA/Y8MpXvvLg4Q9/+LosdwFdgsZpN8Hzzjuv39jF0WuXH8lXl5SjXSvR84dNx7ah8iXLTc5Wy+eff37ftXEJ6MCPvlDXlmyQ5hrfXUgePJTBox/96N4mt9FGdwMSZemaD9htF1UbrNX6tQtkCXTwtbNydF5hIKa/mJoSQxv5thh42tOe1nUSv6v8KuhMtraInyckIfptQtLIt5vr2Wef3fW3rBL0L8pjtGmEPsZ22XZD3SbvMDhSR+8Oa8e4LPBfAvkVnMptdDe3YoKpDbQcZw9un/AvKVBI4amYuYNO8aCnLzDr7oGAVuN+0Yte1LdHtbufvHP1UCnk1akoXG/l81LUiAR2+QQ9vX0IpHPwLoEOsW0boit629OSWb/cmwP0God92G2NqzFG9pQfxAkqdB6JdXBGMbb31citN14CPJTfPe5xj4MHPvCBPW7TeuURdKGrunTWWWcd3PSmN+288l3IroYJobdFsL1MMmDxlLsL4c8GutNB3VbXloAOBgl20Ywe/MqGbahlRRcfldlN1Fa66t4uZPkhHuh9WKguZ7fMJYguF198cZ/qGOmNrK2hF+g7gg9tK6wOHfYFsTqpP1AG6JWrMCVvBPvle8lLXrLeftjUjTjbFuO3yyf09mGVgdscmYvQFJiNZsz62By7anfO1Rd/8RevWifTr+ci9Hbd+7zP+7xV6+z79S4e6FoHufrABz7QdxxsHcTq8pe//Orxj3/8IvmATxvBrK54xSuuXvjCF653vBshTt7oR4fob8e9q13taqv73//+q1bpe/xckCd853d+5+oOd7hD38nSNVnvf//7LyVrROQ72v3xCle4QtelNfa1nrtAThsdrZ70pCetrnKVq/Sd99pTUY/fhugkn/zf933ft/rcz/3c1aMe9ai+C+ImHVyjEVpD6rtXsrl1cL0Mr3GNa6ze+973rulhyvbo124qq/ZE2XfP/MIv/MLVLW5xi16PlIM8lUcb6a4e85jHrHVPoH97gli10XeXzxfoo3/yVVR6ed72tretrnvd63YblEVr4J3vO97xjq12hEe7sfQdNFvntmqj8V4nQzcH0UM5tI6i60B+9N8G6fEVX55xxhmrZzzjGd2324BOOdl1Nfq6tnukclAPxOG9TQdp0UFonezqzne+c9+Fc0T40c15ReTbDbc96XedInuX/ByFdnNefcqnfMrqjW98Y+dnV1f1VLvcBrR0kK89AazaE8GqPV2vWse+aoO33s6SZwqVXj28+c1v/lF2ynMULLrt5i6To5GYxzcj0aZUj5uLpni/87nLZR7MedOpn2+CdC9kzPPai91x6ZrlVgk6HyMg8KIE8Bjl01MA6Qlgfb1Hfj8WwYa5OpAhb2uQ/bN1+6a7m/MjsCd6jDyjC38bhXsiQp99zvlw18gBD3xNNXgi8UMI9gf3ZDWHlmyjdyMW0072E/d46jHeSHBKB/ISp/yMnuxlLxjBeirxVFTzjbaDOOUHnuYsXyPX1JOfQ+TH6DCili0b2O+HJoxA2w334J73vGcvx+hP1qgDHqkT3m20DrY/2frxEhutZflfLcNNaJ1JX/ueJ6I73vGO65dxc0GGp7lXv/rVfSRvv3P1aEr3EWjlcTSfTB9PNeJ2tWc+TjtEb427p/t2s+jxMFUGI+IjRzuSZp/61PNAeurtaFds8OMoplryRDCVdwpoBT8qwi7LmwEPuoiTvg385QlH3bSnvr6BbG1LmUbGNpDjaUJ/NCf/EuwuiS3wWOMXgTyyzXFoEANs33nLW95yPWUwxzDO8GitIVm65MceLCsbK8Y2KECyfMyA1v4g9E9FGkGOiiy96kj/dvdd/yrQXOBBjgbKFj94oULhIV6co3yjT6InmE/kD783O6X3JrC/jR56w/jqr/7q/qMZrsnf5cf4SIXUuZiP9IMR1gnPhc6YLL/YY05TJ2O7YbbsAtlCGpUbHTv81iofzoUpFh203xt2wzrzzDP7FOQu+yOfbAMdP9qC3m+l5tsOHa0lx9sQP/r1K3Oy5sWzhHcs810wN2/qzy9o+UGL6LgL8ujczc2bfvRLX/nBDGETwtt7MjdZ+hp0ueGbftpFH8ROR3VB56g/mKJVP02HabOjf+JL6eecc84s2RA78MPDj/yTbwpUXXTTmLslAZn0VwboTSPh2UbonZasyNsEgx8/PqJPJb/asYt2Fw69qRkjGEMBI1N3rrkNDa1gflpFy108xmwzCp15VCMvHaWRpTugUcTcAsYD0Fs1ZCTJDdG/ytegzZlphAoBpMvv3YBGkpHoEvvxFVRO9mcUhi+eOp3oUfXJDUU+Nx83W/PaY8XYBvLR5ytD/L2HyGh0W4dLPlrlRr6GkF/gmdI3QJOjTlZHjYfGpOzQSHPcZkd0l8/TVDpXQDfKljebmvkVq6SzQ91Bo9wEcdK32Y9f6o95WDZkryDx+OHjJqK+iI++FfLIr3PgvywuiO3bfFCBB1l08ULck0Xa0pTcCrQCP3i69StMyjO0m3SQJvC/slN3bb8g+LWv2g7w2qRD+ID64GcHr3Wta/W2jUeVT0e+UncMzGoaG/jTYgSDJC/Cq8xt8qWhB/VEXJ7wycJX2cAmPui1ZTa4EbnJWxzhafVVr3pVH0yGdsqn8YMndIs6/MoXO7bVw6U4lt0rdRhGApl6mAMOBIZvcuAIsjiVPBVSZwihX8KnYhsdeZ5YdIKmGGpBxf5gqhCnMNKRET/wiwqdyrpNtyV2VFT5jkYSOsw5j/yj3iD/LturPCsnDA50hK7n+q0i/HbZLN/U7pXRHXbZPBfRSeev0zG6Y1tkjkh+WCofbQKg9yLPE4YOYg6/0Oa4pP7yn6c67UJHGHlL7ajYVibaRZZRm6asnWD1AxymPgEZkYtflmnOeUqK7o50NPjTb1iZmKnlbfTkaYeCJwLXx9nRH84jBZS3TrTexXeBEQojjWAJ0HBc5vUPgzh9l/NBujm7jHQqXMcOYS5Gujp6keaGosLwzVL/zAEZNbBNJ0+W67moNiyBUaAbS+QdxsboflhU3Y/CpyJ24GdEtg3V5qXyq5zoL87Xl0vLAuKHuSBP0AZ19NE/x8MieghTvPjUk+eY5rrSHhY61irbFGOWj07pswluDp5uTMHkiXMOvRumTl7bXyJvDg7vlYZUOI9wh8FhjOEEHaFKVgvguB0TsNH0jhFobZzHCXyF2MM+lS5xm5D8CUtQ6Tz2G4U432VjpUuYC2UnaEBeUMES+or47GMJOgk6OXrCnYPD2g+1DARTkUv4he4wUI46sUz3HZbPEqinpipPCuqojp6tS2AKCo0fqV/iY+8h2LgPXx6powcV20/tuYstBdolRsVpnGHOLljqmHQSCbtgKsV7gX0junh8g6OMTrZhtN8juI4+aXMR+rmIPbVCL6EPNCLTI96RKBc3YfUvvKLXYXgfhiYILf3yLcQU5EtdPgxG2sj1YjfluATol9gtr7I0h60sD2vHEpBjlOy90EnII8Ocu7qVG/guJI9VVOi/4iu+oh936Rs6A2Y2zqHZCuwSTuFIPUkcoNEtqWAxZKlBcUga92GxRL5Ga6ohI+zocJwY9fEydl+yKmI7+5TfUpnRdwnSQXhKmoPolKBD1+hf85rX9OWEPrTykQ/9p0Ze0W+JXUcFPXwxvQlLfTYHZHoSPAyWlmN8aXStLE8C6o0pzkwx7htk6HiXzFbEh1aReRqwInAbqh3O1W0DWOfHbeORp24UQNZgnxTMZZ2UPIVnNUJWlgT7KIzwS4Nd0viOAi+1M9d6nC+ANoGsOke/DdLH8Bu/8Ru9k3nEIx7RV+1Yy271Ui0P+WBpOYXuMAitoxeGFUt0WAptkExLTE+i/GKnKbiTaoc6eT/ekReV+0T4s089nRpAVKSO8YsXsFbpXeEKV5j13rLaol9jY8rzONE7+iiaMBfJn46p8tgVNmEqb0KQOfqKqfzHETjcKGJToU3RzAkjEudYVzGoZJXuOMIIthnVQ6ZA5oaKbWmBeLK8UK8YaTcFMG/qwxwfSN3hDnfoN2Ijy9FX21Dzoau0FTXfnAAaqpeGI6byJ1RMpW8LmTZVb0aMeSvGtDFUTMXrCJVlTdtnyIhe2xjTKpak1ZD0QD3LgARq3hqATuqQj/Cc6+jVg6m8I5LmpqlvS108NHQdCadwLJPA5s1M30RhFc91bTxV+VwL8uY81zUuNBA6UzfmeF0LUzS5Tp6az7m0kS6h0oDlnO7Uocs0QY5C5ZsgPfE1DXI+0gjm6HOe9MrfMaj55IlOU3xrXOwWx7ZMhSU959JD51jpcl79kLwgPXlyhLxUG3njk+uE8KjxOnYrvYxedW6WMYrTsKRXnjmvaQLkOudVVyH6JA/kOrSOKWchebw/SN7kC3/XOa+hxtfj6BdxSRfPNtfWYKcdTuVN/qSP+XKsQdxYvgKYw84NtuZPnoSk51xI/5AQWiH5oKb7XsD7spqnynCMHpVfyqfmrwEckw7OTdukn0lc8lReSXP+hje8oR99T5D0mnfULfHAn3m3k/ik5XhYXOaRtp8ckNHkNlRF7NKmo3D3c4f3ctajtLlUDdGcs8cZlV/FEGdNrC/6FF6WMMmTF5+WUjnmizkjeCMIKwtsncAp1reqMOZDPc67I8rDgXTy5aqRo3hrt203QC5eAt75aIsObJAn86vm2ZzTEz2dNCo05LHF0kQFR3fx4K4szVpqtvvYin1092GKCsQH9PTZtRul/KZQFLT10FXPyPPS1FSFikse3fCig7k9H7DxMZt1guynu06VPLqnbNhMT3S+/M1cJJvpgw49OrzoEN35wVE5xO/KnmxxypTe4pUdOnWB7uxkszx8QgflTJ4XUfIYtbENL/4zOmaz9fB05zc2KhM+tmbZdwdGfHZS9aWu8le+fKIDJNNHfXyFjs30JUudYTPfiMtoHL0PYPgoZZo6wz51xjppediTrYnVIXrytzxsZx8ZQHf24YUu9ZYv0dHPkt74Sl56ouMzA4GUDz2k0Ym/XNNDu/TlbvTn17RDeqChm/Ihlwwy+Tb1KHVSZ/fe9753vSacvpmiUHZZ166cyVNW/MxuOkV/dUS9ZpN6zqeQusyetBUrV9QHPkydxJ+/6Kms40N9AXnaL/+lTvKpNL5iD98I/IKOfewBNmuf8qvzyoBedKKPuPiTH1yzT50R+I09drLV3qy48WEXyM8P5NEJXLMpvJWpdo8P+5Qf2wP55qPeFC6hO1RHT4lAJ8eJCpuhCoCyaUAcKb941yqXggutApJHfvnkca6CuXau0PB2RKMwXMvDGSqiGwxaQZzKRhb+ZLMJP7LCW2dHF/nlU1nprkJyPl6uVWbydADhTRe08uCt0J3jozLhr+LJg5drBZ0KSAa95BFHV4E89uVrSbwg0x3RN3EaMH6A1jk+0UtAJziXh1x00dOLUX5245QmzpGO7InNVR4+4qO3NHHKh958zJ/o8CafPHHKK2VMBojnB7T4KpuUKZtTfvKgwU+HoVF4R+QaD+Ugj5sRe/gRP2XoC0W0+MQvyrX6hTx60Vs8nekkL7qUfeShkw9ffsBDObPREkvx5POLdHRsj32u8XcOZNCJzYCOTvigi9/lw09etPTSeehcoqu89HdN55StOOUozk0AX3zESedDvFP/2CmeDvKQjzc7EyedDmTRJ36NneqRY/SPX8PLNV7oxaGTHh3oxKfSfGiX+k23+NU1PRxTl/FxrL4nz5GPxePBD3i6duR39rhmn2PonLMH6COOnm4Oj3vc4zqtbUlyA6SPfGjwhthMh9QjfYG8fCcOqk3z8dEdff8yFqOKXUzlF4wQ7PNhGZFOMApPITzR5Vzh2ZbUtT1j0HNI+OfcEULLIQpPMHIzL+bTb04caeUX6JUjoHXubnzBBRccfNu3fdt6NCGAo0JTgAoF/9DTVRobnvvc5x5c97rX7R+soIncakv4Ooca/7M/+7N9tGLPG9cqnUpJljxkqXTgXFzlrcHZy92I1TSGOECf/Pi6dgT0sUUDetvb3tb53PrWt+40kLwQnujYDeJyrUzs7c8HGpl49FWXwHl0kye6eUowcvShicoevSMnvMgzGjICYzNkJC5eQ/RSC9Ab0dkXyF7v5ERmReKNEtWNbHcB4qVD9HAU5zzXQH+ydaJsqJAvcus5xKfKwTSUJxT8+ajKrqh6Oc8TpekD5RgbkgfkC8/orW6Jc7RixMhVe5I3doYWDYhX5uk4XeOhLeoX3FQhMmIvGa5rmsB+6fayB/rjK06Qrp6K06lWXSrIV95nnXXWpWRDvQ597HYkQ/kZWbNfO5RfmvwVVX70sw++/YL0FfoM8WjjP4g8fAN8EvSn6oDtpkF+8TX/bvyDvFZS5f8hwAiPVnYQZCRFYthUkB6nOVLeyNX+zRpXaHPU2JJfXEJkA7ke022oBMlfaR3D07k052jpYFMsNxvXIK98oXPMHd21ii04R29/ET5wDfJXuVPXjqnEHmG/93u/tx/xS16Q7tqxyhXkBUcbg9nFUeeSdDzkr/Jju/PQ08MjqkrqkVd6gnwJiUPnmgzX6On+mMc8pu8F7rFVGkSP0IcuIXncYDRwG4zxZ+pTaFKeAvmmZzyI+sEZAwT7fWvgfjBCnMd3kD/H6F3tSDDq08CUgw3edBR0qPrTQUgcOnAtrwGHGwm9TLGRIS3yhPg/QXql94MhdBg7mMiOLgnhD3joCPFww5UG8oU+8hMvj/Jz7qiDs5KJP8I7R0G+xFUe4evJUD2qvqt0jkb1VR/5yOZPR/XY4AnCF51zo988BYSnEH3UQ0ttTQ+JS1ry4VF1cYTooQM2IDCtR5fQQ5VX6QWQ7+d+7ue6HfkNbXkg+ZI3eiTUdDuAWjoMoY8O84Eu4RSaYn2v4xp2Ifle/epXr65//euv3vSmN61aYz2VOg/oP/CBD6yud73r9f2fW+XYKVu6fG3k0vc0//M///NVG8GuWuPYuNfzFORthdr3fL7BDW7Q9yRvlavHjWiVftVG/avf//3fX+9xDXi0kcvqdre73eqcc85Z78EtPWETkk6mPcjtR49XdPid3/mdnk72iEorvP71r19d6UpX6vvJyy8temyDvK1T7rZf9apXXbWOpu+bvcuP4S/w/8tf/vJehm0k0/fuFk+vUb5rgX1/9md/1vcybw1z9T3f8z19/217uuNFp/h5igf+9FQmrXNe/cAP/MDqqU99ag/2I283q0vt5Y2mddx9P/pcA/vbiHR10UUX9TpwnetcZ3W/+92v50VPByF8AtcJeLTGvbrjHe+4uta1rrVqT4VdL/TiR9qK0LPFPux+F8FvE6jb6JNn9MEIeeR/1atetfrSL/3SVbvRdJ7KYJt8wDt2tkHX6uu+7uu6H7XlOXKVN1noybMX/l3ucpdL/SbANh3I4ANBvnajWX3jN37j5J78bNQu+HdMc00eH3z+539+7xtc77IhkE9oTzK9HPRH6P/gD/5g3e63gXztv3XUfQ/6xz72sR+l4y6Qrz3c8IY37OUQnyTtqChd/nJ4OWa02xRadNdpcnswgjBFYS4LfeI3QZ5mdL/jy2ckb17PFq3b6Ea4U8rvpVkrxP74n7srWyrI9KKkVbRTMf8Ac8Re1tz1rnddj6Zhji/YYSRre9i73/3u3Q8B++jhDj/ahU4wKvBC7FGPelTfy9wLSfpLiy83QZp8vuB7/OMf35cp2rHPaEn8HBi1+ulBU172MLefu3lxmLI/+jhKNwKn94/+6I/2eWUjMVsVe+wN/SZdjOyMXv1sYOuo+5SPR31TX+bF+W0sxxGmeIzAPSLTn/wnPelJfb4fffw4InHKzsj5Xve6V9e53XD6U4113mRP1ZcKflD37KjpV438BCB7TPsoW+nKcwrVl4JpP3PDlpzaNdEUY+rCNqCVR3174Qtf2OWqi2yMjClIkycyXPNnu2n3X6cyao/uUz6sSDpf+ClKW5Znvr2CHFNFynoTTztF2i5bPVaGsM2OCvm84+ADU2fxHT7kbeMj3dRh/JKXsPHNXKg3Zjf4IDKX0G/DkTp6j+mMyh7WcxEjTN2Yj80uhoK0TZBOjvlL85/egNtP/TCbOSlAhWP/aY0b8E8FCeij4mXqpsIjor3oUzBJ32UHqAQKlex2F+/580iXzmoKsVPj1NGYk7efOVSZ2/whn5uXG3UbQfWVAvJrbLv8yDZoI7l+o9FJ6ezrC6RdPNycvfC89rWv3afNdBD5AeXwSFmPEM93bqy+iq37qOtgMr3Fl9vAVjdHnQN6Pxahg0SHPz3w2aSDRqnT8bGW9yPqYabK6ObHYIL4rAJvnSM57Lc1rfPIFZJvG/C28kgd8KMjbrZsEB9fbgN9TbmYrlIP8vJ9yu4KPjLIU+50tErGgCn7yZO/S3eIrd4LuHkbdOA90rp2Q1V3pvypPNhxpzvdqduNx1wd5BPoQH/9ER7s44+kbwIZ5vblUY+WDjzDX/2xf713nq7HunAUHOplbGBEzMEajMq9q3IECgHMpSrc7KdOj12VEy2ZdNTRo3cXXyKfHHx09G5W9XdKR9vJIkMh1JEKei8DpSlc8bs6lyCVUGfrPYcfbKB/0tpjW5cVXapOoRXor8PPlsZL7DfitLTOC2iNJzcZfEYfVJAv3ehKw7KqJXZvoyOzHmPHHNoKdAH62FzPK8Tv2qYYlvgussIjuoc3GPErw9Tr0T601YfyzkX4OeKjg9LhqjeejufaEv3VIU+n6mEGNHiMOldoF9UPBi1sMfCb2w7QA3qDPu9ndPR5qqt2yMOnaPLEEkgT/HCIhQDaI/pt+lfEjxZm0IMN4S/eubCJH3rvhzxdeyLyq2v8CHP0QC/wqRumOX43mV194RIcqaOnGEcYHaVSzwF56BynHLhNPjqdI0dmiVP4pHB2AQ+I3aGbkiuvTlHlZV/yxgbXjmjnykfLdxC62MCXRnoeH5Ne9SJLXnF4JI1uNd82RHdHINONLDpssyM0kR3dnW+THzpHHQKf1qmeubrH1xCelXbkI89UR++YcxjpNqHSOUJok0ZHg6D8UlDyV8hTsc3n2xB5jjpCneCSdhj6Wo6Orjf5JHSeaNWbTLUk/1xb8AicRw91q7Y1kOZGBnlyCqTV+hg9Nuk/InqET8rM6iE8PH06buInrycJU6F+GtLiAHnFb6ML5BO0w/Sl7Kg2HhVH4hRlPLZw0hLEeHSMXALOgDixFu5cpBAgfDYhH0lUPUND/220U8AnvgsfIfF1RDTFe6RzvdSH4eFovl0nkbhdqPpH7i75Y76afxdtBf3kD030XcIjmGPrJmzSPz7Z1dlW2UfRo9JaMaKjWgL0qW90B3G7/CldSD1AkzAXkRNZoQ2/Cp28p4asqJpC1X+pHgI9+CI6LeFjiTRay6zTpy3RAdxYvG/ZB458y2BM7oBzgUahcIwwt6KkAHwkk0fM8NpGN0L+jBh2NUjy0njIqHZWPsJcJH9CeMaGuv66ygP5YzffHUZ+6NGolEbXbBxlTSF00SPnwi7gj8aok425nkMbjDJdJ85xLkKXMBfyVvkJlY+jdw7bUPmE7rAID9MOS4AmssMjOgmbEDpTfqYcc52wBFVebZMjH/VTyCCvAk1G+cJhkb6IfIF9+pmpQay6K9DHuxpLO/1A/lSftgnhAfI5jw+W+hHCr4bg8F5pwIhC+ST5JEBenHESYKNHt3ypuC+kUBwzX3uYwl4KMswHklkrxnGDHGWmsaormSqCfco9SbCDnQL7ThJkWhRwku2CjSfV7rV5gwPvM04C1T6+HSE9A0Df8ujw+Z+eh0UWKewDvVakcibMRfJ+zud8ziK6o4Acd8+lo5fDgjwvePbd0ZOj8gimiXK9T5AhWI6mo3e+T7BHR2QEZLropDqlkwL/Gf3pALwXOEnwrXcCJ+VTtppqUJYnAR2oTj5fr58EvINg4ybE1+94xzv6DcHvIyzxPztii/Lz9MC+fbT7I9UKCqnYOt6pR6pdQL/UqMg8qcJWcKnQ+5IZP+BPnkpT4/aF8HfTNAcamfsEeWQcZeTzsQ42WjkC+/ZnQI53SSdRhpGhDE+qHFNPrVTbl30jX7ZtGtFDpj1f/OIXr1f7TE3zzIVVWOzbJO8oOFJHH6MsMzyM8xm01Cj5LSEb3/TvE5YxWinCxn3I07nXkYDVKGPcvsAmG36psPv2J1lCVkxF1kmU4UmBLR75LVVUfrHtMO1jKSzNy3TCSUAbzA1t3zAQ8XHiuChiH1BmgjrKxk3ylK/tQ+h21ate9Ug3PvIy1592cpw4Uk/CKAqqXEfplJYY5eZy3I7Yxo88dp5Epwv0INMT0nHbuQnKMPbtSx4ZeDuCOsPOlOc/FbCFTUafzk/Ctvj1qO1wCSLzJOwLjK73aV/qZqAcN0Fe6ZZTgm95DuOL0MSf4DjqclQcyWuUE3w+v7QA0HHUYRq8O7sPnSI/4bCI7E08rGmvK2GCKnsT7RyENkf2nVSDBXOfVk+Qf9wVLOBjNpHhIzOPqK73Uan/sZDyY6uPb04SZGbHyJOAsvN1tGW5xwX+q6FCJ+9J3hYR+6ovZPKjAOropqkUecX/xE/8RD/3RSufhHYOYmOO3gN63wmJOy703gTTGnZhzD/VCW4Dmjg0jb/GbUIcnkckNLlROM5FZKGpI+cp2fTLI+pY4NEXj+gRXsImJK9HvujgXBwZsY3sbZAffUL0ELbJh+gK1Tbxc4Fe/oToUHkH4V/LUJ7KI7onpGwq5IMp2Y5ofOnpPPxHHhA6CG2OOc97i22QN7pElnMdE4x2B/LFV2TRueohPde7IK+6kh9MiQ7otyH5HEH++G8b2CIvmc7RC+L4LHXZdXjPBfnoR3+pL1alZMq2InYI6OPX6LUElRf7xnIMpFvX7xsiafYZir9TlnPkow1v7ZB9kb9U92040rDR3c6WmrZ4jZFzEAMUqO1pBYXDsds6N3ScokIrdM4wR/ac5zynzzHPReQ7mn+3sZWvbcke7SDDJ+bewFfHOxeMTn/4h3+4v3lP+lgpRqSTU4lig02lzNG5tlwVD7ps4yUtetgq2V7iqSDb6Kqezj1BmN996Utf2stkLtBGHl35Eh8vlSKjIvmVnZU+rukgTgBzsOYpEz9iqn6oO5WP65TjJj/EdvlzjDzTLs7JStwmoI0djho5GzzlboO86kHoEtBmm9xNuk8BrXZE9xe96EV9S4M5iD/5S7AFtL134r9NoLsBnm8iIPn9cpOtv5foDvTH4/Wvf33fv2j0u/btXaA99+WtkFecoy+SvRzN9RKQry3a4A7Yt2m1HfvoSg4f5CtvGy3aalq6ergEnpBsOe5HS5bS7sKROnov8WwKpWEucWoKCo3OyVK0VJSxEEdwIDpBJbWPt8enJY6pDRi9PfEzCpnqSPLJ/pRu7up2YPQTZzV9W0WXTzqbhVe84hWXullGN3m2+UM+eay2sPthNsmKj7YhedjNtmc84xl9g65d/q8ID3pr4DqI2DGHD3qQFx87Wj75yU/uDYhOboQjH9cC2sh3NIVgkzW+fMpTntJvXsm3CbnhAj8411naidK+KbCNHuJD9c9gwA1XfTII2obwReecLnZPtKe/nz6MXjnuAj0sWdUebfWw7QvSKZDjJqGT9Cn/HKCperLDwK/u57/LfxA+aJT9ppuUQRUdR56u0aoD9suxLzx/1PKdA3mVuzoMeIR+5OPaTVEeu9d6AU8P9JnOIX8J1IU3vvGNfUfWfEtzXFjc0VfDjXRVKHezOQUa6EzRu1H4dNimYtnUaxvIQOfOrvPlGJ2sDxWMEuciNqgcRj/3u9/91qPoEeJ8xJAvcQG9AlbxdI7u+naxnOsD9NFBpXjrW9/ad4A0ygU3DZjiF1qBH231bOfCc845p++3Aei26SKN7YLRo+1hrRrwZLKrHCIbrZG7R1db4+JxrWtd6+Dss8/uZTElXxydNVYdsc5QpT7vvPP6Dp5PfepTD653vet1X0aPKT7kGzl7CtOx2n3SroM2lNJZXv/6119/eCJvkHNHNxIdo4b14Ac/+ODMM8/sO4Fq5HZhtFGfMh7loxWk6eB9sm4EZotlOrANLxvGVYSu0gv00Lmx3+6RduG0Kye5CVPAI0ftQDkqP09lbnZ+kGUTbYVyDA+bcikXnfU22uhvBOop1LVOTV32wxm2bs6ASb5tqHbgZylztkquwM8IW7mGJoifDNqMxv34Cn3YNscH4ac/Qn+b29xmfU2fUR7wlxs6vW5wgxv0OOd8Z+fSKZpdoLPBiq+q0R+Gx0Y0ZuuN9xO2IenNiX2D/Vvc4hb9Byia4T1+DvBA30YOq1vd6lar9rjUr4Vt8kPXRq6r1tH3H90444wzVq1i94365wIP+pLrh0vaCGLyxzJAvuSPfoLrdkfvP1hBjyXyQ98q9ur2t7/9qjXOS8nwgw5VVoVraUIbya9uetObrh7xiEesf6hgzD8F+VqHtHra0562+oIv+IJVu1n1H5vAgw7bgL987RF5ddZZZ62ucIUrrNoIspdJfIj/CPFJaw2o/1jMl3/5l6++9mu/dnX++eev2g2/60C+EDtyDMhGTybZbTTVz/3QR2uU/QcqIgsc/ZiIupprej7+8Y9fXf3qV+/157u/+7v7j5jwO9mxo/IJXJOh7B/60IeuvviLv7j/YMkrX/nK/oMX9ENPZvwQHuEnHg823/nOd161G8uqdTD9hyfazXPSfyPCk77tqbj/8Mlzn/vcVXuqWPtgFx950LfOenXBBRf0H4Bpg7d1GWyDdLI+8pGPdDl+8MUP4NCBD2LrNqBLvjZoWrUb5Ko90U22RdfRa0xLvB9+8cMd5MOYbxPko0t7mltd/vKXX/tAPVOnpvi84x3v6D8y0gZn3d9kalOXu9zl+g/o4Bf75iA6XOMa17hUe14CPMYQ9Ntu7ogJc2Hli/XCaBrTU7HzYcoHvY8NwB1xjnxfj3m08ejUOoselqLZ3h8xr3nNax60DmMtV/wII/fm+PVdNnlNV9iH3Eh2im4T0LdC7L88b4RiNIc/H+KTJ6QpnvGzNE8j8hrBwFz/yWOu0z7mRpH3uMc9+mgC7130yWMkS573I36+L9vjJn2TP8QbrVs94UnACMqPlvAhHWJD9Bj1kS6fJxB74ZtmQa8OeJEVHUb5rhPniN7P5nk3cu655/ZRGBtg2/YFaMlQ9z3l+b1hU1b2gs+3AfLYOnoTpMtnisHTjy1uPf6jjw+WwHRNu2F0HfjAlJf6tQv0aB1ab4eWCfKDcol+28DHWaQgvykLK0Y82YRWvLAJ7JSOl/roaczoeKr86KmNtJvAR9kWn2vP9pvJNWyTXyGfn9L0uwC2BAFP6nkXOPLxLgCy2gbYoB76jQk6oFsCsxRk50eEdpXBEvRtik+dz4LsMULB6HBVej/+oZLOAR6Cx0SP/36VJ7R4bzIQDbkKnCNM4bhGr3IvkY8u8rOfu/ixkZGlArm55EevAb3HLJXOtA/ZS+Sj81hoXtHL5ditQgtpQGOBk5trU2c6TbrJV+e0K80IechwA1N+GrppIxUb/20dnXSgPxnkpjyq/PgpSLxjbmp0jz1V35G2glw0fO2In/z1mPNcZ5vi7/qu7+o8xMUOiLzQBXQSV3UjX5wQ0MV11UmjNdUQHQK88OAD+WN7QvJWPaYgX3iRpb7Yc8VNO7qE5ybIh950kzpg2kgZsoH8bTrIU20wjebatAM6fB236UB+7DBto5x0nK4z5x1oJ+q7epvfrwjoQp6Bl4FPpmHxhm3ykw+9/oScDPzE4z3Vt7gpmmr04tXUX8rBy3T9ibqNfqTbBHn1B95Tmb40iNhVB0bgMSK2L+NUgIGPBBSsDoNDKgidEgyJ13Gi50j8EqYQGumZ382v5aPf5JRNesivQqDfpCfIZxQxgh5GgH6NhvwlQIuv9etZFxy7VQw3EIhNVb/kc+Q7FRvNJvs3AY3K5Kjx6PDJmWNL9I0uaEInji5jfahwc9WwQd4luuNPloZljt28rJU+4umPVxpX/OYYXSHnyVtpdyG+dqRH6MXFZh3e+DJUnsh15G80QuQHm/SQJyFA6yatY1RP0zGN+aYgnSyjVu0gfqu6TgEdOdqgti+vdoBHdN9kQ0Xk4OdpxlNd9WmF6028k+YGoD1U20c+myCfJ5J80Qzm6HXeEJsdtRdP43TVKYunMxustnLTBTxTJ6YQPaOrm4z+AP1cvedid2kMiAI5cgoHjc4PqiGBvAIeOSbMAQer2OEjbMIUz8SlQtXGOkI8++oLRseEyE8jmYvQJYSXo04AUknEBZGbc34gG23iav4pSJefHwXTR/HBWFYjwjsyhdAmQPSpCG9pbKw04TVFNwIfNG4Wz33uc/uKob/927+9FG30CEa72C1/5I82CJBjRfSUVnkkr7gsO5zCFG3ki5uDKitHL5PZxdbw3oYqW/7QzKGVh6yRR5D4XZAn8vDDRxghXT3VkY+dZ3jUIE7YhqQ7Rm61Q9sSgEz5DDAEHb0bW/01KumRHT2SNgemwjydT9l/VMzXYgMY5ZHPcQoq3djIEjcnLQH/ONBIPCtUKqboYNRtjJ+iqSCvrqfddJfexWcT6FFpTOWkYo06ViRuKm0OwpttKtdheFTs0kfZsUuFNhepwVTbN9FVxPdojb481YXPJuziuUT+trwpL740utzFaxNGGeET/lDT+NXRE3bN41jzCbsQ+jHvFA9tMAOgyDwKKu8qT9Dh6ugzTTmmH4cOIy9Tb+lnXEszmvc+w9PMrW51q/X04yaEZ8I28KVZjm38Dotj6eizBImCNYiL88f4im1prsdgTtFjzhhfMaaNMmrYhfy2LIx043XgusrcFiqtcyPV+C28EzbRjRjzToXQk6cCa0DkTuVNiB4JNa1iTMs1GaZuhNxchIpKNwY0jnVEDtE7/HaFytN1RU0bw668iTO9OCUrIWkJ2xAeNf/ICyyxdNOr+Sqm6Kby5rrmnYKnKAEqnzFUPttCpRmhfpqqy3uxipp/iu/cUOWrT+qoqSlIvCdRixjAi2d08lbakWfFmFaDqTfLVWGkOyou88iGU+eLEWW8BMlvVWp8nKMzlu5OrJDcATnO/CVnqSDiHPMIaD4MrU5cHnQMR4cv/tJ9GKSwjUTJkC4fHgmRFzq83JA407VAHjpOj57RAdDSxdt4ae7ukcc+OuAD5KWQ2CO/DjT2Al65SZGHf+TRQT7nXuiwAW9HNxnx5MojLz6uyZGPLuYUYzM6cuhEB/LQSY+e6NB7CQVo5ENHJjq8hNCFFzo6yYeOzdJjozzi0bkmmzz+oyffGA2xRx58HOOX8BaXOU/8xUknjyxfJOsAjK7E4ct/ZJDr2moIdfTa175258UG8ZnuEeIrciMvNqd+0F05y4uGfeqTMnSMzfSxjt6crXzo4xs8qs10EYcu9qVeocNbXnnY5yhdXvrLJx4dX5hm9O4FLfvZE36pk+LJQxc9HJWHvGkn8rMJL3Ty4CNINwCSxxMahE98AWmHQB6d5eOL+Jk88ZVOnDz0JI8f1FO03m3xa+oIHfB2ZFv0lIc96WfwSlmjYxc6x+jpnA7Ofc8ir2vBuXxexEr3cZs4QbnIE/mCa/LEKa+0HSCf/uTTi5559+hpF534JZB/pMn1kUb0mDCQchRPx8AwBSWOAYxWUQROAXnlEZdK5drXhQoVX3mdi5dHXCoKGa4F1/KQ4xo0No0dQhcd6FTp8CI/lZwOCgHSQOVRCRInjyN70xDS0bpmHx3EOwc80eEnD5nsE4fONTvxoUfoEhcf0xNf/NHTQZxz+eQf6fAGOkdPkEdc8uDDVnno5RrEuXaE+Eq+VHRxKivd8FU/8DYKq3TqSxpW+OONX3SQFt3xjq/wxpP/yBBS5uTxic/YfS2sszUo8PVy9Ax/vORVLwDPlBd++NNRnDRxfJ/yIldnQk/56B5dlDOa0An4xu948wcavkk5008cfnjTE40VMXgHdJYPT+f4kSmPo2twjp4sOpCNBv/wI1Me/Mijizz0qr6XR7w8rtFJx0scma75i9y0C/7CG1+85I2NdE+58mnqGzry8KIzOvqyQ17n8kjDK34G5/wlDWKPfGSpI9GJvuSJc504vNFFXvwnzct/AwfLceUjj97Sqq/Yl+v4WF7XZEan6Bk/uOa748bi5ZUVDAVKm88CThOkOYZ9VT50NS75alri8AHXzskzaskSpsjhQHdI5/iEv2vn0UlwXuM1rPAa4c5OXubpw0PhoK80zukR/SMDarx89TzpoNKQlTyhc538jkF8xgaoeeKHkaZeq4SRh4cjJI8QXauvQp/8rqXxizg0EP3ECekIavlVHqChpSzDq8rLuTXolr8+//nPX+eLjiCf0byvRa23h+gV3kF4QvKA+Km80Un5RB7Z8mnEWY8NyR+EX8onceEpgDw5D2oe6cljbfeVr3zltd+glhfkiCaIDsnrKF+Nk3+svzpAslKO4BjaXFcawNO5UOmiR+LrUX3QLtwczGOLA8fom+vonevAuXxCzQORE10hna5ReOSoQ7bY8DX2ox/96PWIPLwdIbwjJ3lyDjUeLRsFc/WuU6eOA/9Q8xaCIgl5YcGoKJ3K4VgVFpd8EB6ukz/pNYiTJi9HcD5HCaHPYxFEfmgh1zmPXo74QNIDvL2MzUunqnuVkSC/PDWANJXCtWNsAUfyo49psEobOZBjhTh58RFqninaehTcpOWrMutRSN7o6HwqLXYkLvkcQTr7+ABqWo7idB6OeEUmyJMyNwIyctIBeJKI7Cofj9CJr9cCRAfHhPgy6ZVn8iY9jd114qN/8tcOQXzS8JU/afVcHkj+hORxHhrBtyy55guInytPocbRAaouyRO78Yu8XJt60eZC45g84FhlBeJqfPIlDmo8kGMqLB9XJh3kiQ10gJE+eRMXewRxsTNwrgxNs4WHPG9729t6mvX+SXONX2TnKC0+CfCoMqO3azYaUOaGepw4NLfq0DrHK0RJ50HSKlIZx/g4wjHngfM8BiV+TA/NGBIf53Oo89iSwqo0rsnyuFjjhegfiIvO43WlCxKfc/xce2SNTlBpE0Irn8pS8zsXRrmVLgGqbbWSJb0ea6hxUEd84QHSq1wjQR10RXiEPiEIj/Amy7sTHZnvCXyZmWmY8MoR0LkOnxwFSFrNk3zBVJ4xTgCP99E1cTU950Ia+4iqcw2j3OQ1TZXzeqx5E0YerlP/QgPhMdrh2pO1sgxqp5W2lOsxTMXjGVT9opvyJa/SCOApMu1A3QBpriuvBHlznnTABw15GUgEpgAtqwQ3VXnShwAe1W7n4V1lgPNc5+gJwpSZvOFxXPgHzx4SFPLyi1FLUI2OUbuMky4o7BQIp1RnzwUahY0+ujsfga+53rFjghQIXnjUQl4CNNV/5vSqf7ZBPvIdhaXlAG4sZKKd8sFckM+WKT2qXzyeKr8K6QnoIccR8ig7m9nZqdGmYre+9a37CDM8km8uQreEBmKzEH3ZbnpqLka5OQ/vubBBm7oAynGT/zahynN0PaWDayH1Pumuc75UNoRvQgXeplAzqKyQV31InYvssQ7ugvzpD/DMOwRw/cpXvrIfLev19azz6p/YX21P2hyoM9oiPunfltBvw5E6eiNde0+78x0GjHjDG97Qd73LqHIXFISVBZkvt6TMnic6qrkgJwVqFPTYxz62H12POrj2SCX/lNMVCh+w47AF48WhfWv4EX3mIJ3v8gmd6WDFhRdFwS66CnndyDyWLvllJHSRD8rQHjgq6zaYBrNygn1pHBUqeRrbFBKvcXu89rjrMTrTFCCPQMc5nZ68sWMJ0NE1+kamNe1LgCa8fOqvk3E9B5GduV1fCitL5bEEaH3Ra6tineo2f8hrKoXMXLPBk8xFF120bl9LgIdty71PcT4FfKcQWeqfthT6TXymQH/bMNj/CNRR05qxTT/j3KACxCUA3Yz4bbWd+CXy0dP/2c9+dqdb6r9tWNzRR3FHneOFF17YtwLQ6JaAER5V7H1tq1ujvF2GxXEadOT5kQJ3WvRzgUfs8Mhvzwq6bCoYH+X4WANqunMVwwtB88S1YUzxCaSlM3ODstXuy172sm6DwjaHHV+MfMiQJiTNI6UNwnzIIW6XH0EnIC9+KrSXTPe9733XPzA9yg0iv3bGznPDxIMeqfxB9HJUfuyUJ0d8rZawZM1Ga0ZuUzqEb47oyQ/cZAwcbN8MkRsZrlPXkhabXCtH9OpV4ivkCR9pScffuR+esPeJDfeC6FoRWnzoj6f3DE984hMP/tN/+k/rdfhTEF8D/sJNb3rTvt/NPe95z94mdnX0aOlAtmD1iI3ebFJXp0anIE0bzEgaL371UlwdYpOy2YbIjv14aAd8N8p27UNC03Qj4gMjYvRucnjDNhsArZBz5S6go3/qqlkEAzJ57njHO/bBH9tjY/jYYM5W0eG5Sz5Elpf3bHdDcS3MoZ+DrmWUTNiEmkYJc+U2M8omQkuAl1GkpXAehfLyapf8NAijXxXL/tc+Qzaqmwu6hpcfmrB/dnbtmwLe42hRRdJJ+0UYo0l7XsiDdyrZJiSP49vf/vb+yK1xGSHhbX3ypkJOHHoNRIeoY7z97W9/cJ/73KenT9FVsFtFRW8E/83f/M29Irth3e52t9tKT7/4iQ4qJjq7BhqNsMO+7KMP8GQTkKVTUXc0Crs/kmsfenvz2zclq7hG0BkvN2YbQBlooLeZl33kr3vd665/8WtE7HLUkWm4Ro+Pe9zjuu/sXGgbATtiekJL51ORa3oIOrWnP/3pB2eddVZ/QffQhz6014dtT7l4xD/qkMGG3xRgv3O/j6BTq76uiB0VngKsAuGLb/qmb+p6bNuGAfBmg3Jhq9UkVii9+tWv7lMTm+ogOvHab24InkDcXO3nz5cp6yn9g/AX5FMvzPvz5Ugnrz5iakCJnj+9s/F0/fVf//U9Px7b5Nc09AYJ6O0CCnTRx0nLj5Gol/bkqYgdjupd3dFyDuhBhqNpcE/0tb85DnRu1WAKb0JNo5jRrM7JHLZPk3Uec4GXjsHRFgoKkB675HOgxqExaaju4l6MLHEskOXxVAMxEtWZ4zGlg9GqR1R33KQLHlNNl3zjN35j90OwS5fwUJH8cIpOOtv0QkacU3w0SJVAusfspz3taQeXv/zlD+5///v3Shiabb6Ux83STU4Fdu1HJ/DRmGLfFMTTQYds5Kuz5hs3G7+baVptUyVNZfYLQp7kNKLIsi5ZZ2frADddH+HIi2Zs3EZsOiP0ZNPHfL3tnjUST2AGHxX4BMrtWc96Vu+MyVaXvuiLvqjfKMhmQ7Zd3lQGtidW/z0BeDdApps9/fGStgnsItsghR/d9PJUZdtsA58MGqaAPkFHrRyMJHVU6rKRvU5e+jbEt2xQHjpqNwu+DPAY9XAt3hND2q2XiG6YN77xjXs9Sp4p/wXSE9R5O0HaFCwb/VXQVXm7wfP1CHLyJPkVX/EV/ZgwB/jrxwRt0TXfxj4DGnFTN6Fqg/Zg0KGO0En8HB3kcdM0FW2wQtY23y1GU2S9Sb7QBIjaiKQ7tpHDqj2qr9rj6nqz/zloDuz0bQSxao+p/YdLIn8X0JHVRhCrBz7wgav2mNp/9GAObYCH/O0RaXX22Wd3XnSasl98G7X1H9aoOqJ55jOfuWqP6f1HUGL/SD8FPPD1gx/f8A3fsLYfrfh281rbOfJzLW+rUP1HR8hvN761bkkf6SrwbaPeVRt59x/caDfM/oMX8cE2SG8dQ//hjosuumj9YyHRtepQkTjhwx/+8KrdaNY/siGtpgOe4kZIl5b06BweU8EPy7QOrJ9XOjpHZuu01nzDL+cVoWtPUt0G54J4NPK7TpmA+CB5/FDKne50p9VrXvOa/oMVVbYgX8KIxMvXOqDVVa961VXr5FbPeMYz1j86MqX7COny/cIv/EL/ARq8Eidsky9f6ozjueeeu7rJTW5yKVs20QfhA+2pdtUGS6t2s1mXSwUZbWC4+sVf/MWP4um63Wj6D8n4QRu0CdsQ/QQy24121Z4G+g8CoeXLBD9o0jreVXsC73VlBHv9kIwfUlLf0McHcyCfdtgGCqsXvvCFi2jnoH8w1XCq278Ec++CRgBNoT7SNQoyMpoD8gR3z1aAfaN+TwPiyN4kX3ozfn1uRC+v/eiNqsaR3ya0Quh05rYjH+3UHVReI0AjzIwS0dLDtIlzoz+2b9J7CkZDpmjINVXhGNuC8Kt86QPo3f09Yudjp7lPVOR4KhKUnScbTyRs4INtIwny+R7okPcJ5G+jS9mil9+IVdw2muMAGUbX9qO3Qid6HBboBTYos8pLfI7xiTyjTNetY1nXeVjih/BHo/3licB0kxE1vnNsDB8jeiNRdQltdNnEIzqjBdemANFmT/s5CB82+MDMB19+eEQdyfYIgTzqaet0+xND9Rc+0v0sqbacxQxzfADoBfLZ5Edp8K/6scvTLjvxrW0tbcLUk+kjT3V5oppbrvJ6yvMyl3xtuso4Ko7U0XMAcI7CXWKUkMo6ytsmH50O0iN7Hu83ddKbENkCWeiDUba8OjT88zgtLmkKWXyOc4AuAT96QDp783Sbpg5CR17y4yHM9UHoq+zwgOqPEZEvJH/47KID+Uy3abBecFce+wC5x93RB6PNSVNfTKuZwpE+ynSt7YhLmc0tO0iZ4YMOL0cdjTaB7xwb0aceBPjg77iJR/Q39ahDy0c+sXVbPaiQd7SB7OjvOpAuTdABVhni+Dx9UPTepP8IOkC1QdDx4q3++B1ic/8/9EM/NNnXoa2oOszRgzw8IntKxlFwJE6cEsfMdSokfwxxHWzjw+nyqtCpoMk/OnoXUiFGZ1ZdINfy5bzqKJ5eI59tQJ9CDa2CjT1G2pG3SZ/4PRV+zLcNU/qD+F12xO+VxnnluQ3yGc2G/uMV22wW70XlNtR2M9d3gfxoU0dCnzo0F6HFR3DOrvDdBvrLqyzVCbKrLnMRmanHEF0qomN94qhAL4z9whwkP750YYejduj8zW9+c8/nR8OnZEPocgMPluoB8e1xYn7vNAGGCX5SUIHPRXVWnDwHyWu0m5ennCJuUwFsA9pddBzuha2XTeTE5ugS2Y5zgV7DiPzwTWXx4wowpVuVl3RHvOaCjCrbskaVeg4ip3Yq4bMLqbyexjya0uPjDdXm+H+EPH6UYptPpB3l0Tyy8XHOl695zWt6O3Q+x7ehDVwrX7TbdA+UoxF9eDhWfrtARpU16lPBLlOtpkbSTgJ0udEsaQcjor+Aj+lM9vlxG1NJVuNM8SeXfHSZwo1N2xAbcvSEZJqYfZv8cFh0blFsroIjPIYvpZOfMVXuXB46eY3kMLQw0tTrxAWuTTWkAdU8jtWGuUj+kT7n7IN0jBXSc6xhCUZao5DYNiWzYqStYRc0EvyVXe3k5tB+LGGXveyc+mGcoNLv4jWF0NSg7mSqYS6m+EDq4S6Ys64d2xyaEbX+C/W6wrV6aopmRKUVjoIqW+dtVZT3F1nJtKlNRn7Oc70E9enouHEkr8QgL0OP6uA5IEuHxPG7Ho2PC2R6WVp3Itwn2OelU0D+vpGRi0q8T3n46wS9/LNU7iTqzD8GdEh21DwppMwsb80L9X2WI96CMlSWJwE2WQyhLZ4E1FU3Tl/ZstX7ln36FAwOvFRfcrOei2Npafl5r5PCPhyxCTpeUyke3fZd0EFG2JDjPpEfVIF92VjtyQ3lJGz7x8JJtof48SQ6o+Cky44/1VPfipyUjUbYvtkwyvYy1pHd+5Kvj0lHf9wyjlQbKSSYvz6JzjcyOCMfyog7rkqHz8iLw32IYhnmVPpxgzyV2ch3XxVqtMOojB/3JW+EqalMT8G+fXqSYAs/1pHnSfiVXB8g1kHCvhD+2mCWQU7JFHdcuuBjCaZ57H0jOlvqaPbAFJWPsEypCMdVnuGToyliNrqpHZffgiN19BSklJckx63YFMgjxzRDZDvGUUfFFC/yrJdXwU7KRp3gPjve0U6P4HkHsS9Uec6P8iLyYxXxH/vU0ZOEtvCmN73p1NV+wT71UxkakKQdjhA3FX8YuIEZUG7aA+m4gDedHV/wghf0waSfqTRrMQdH0c1qQkvHo8Nx4sgdvQ5CgR/FQLQ1bINKpcA5JZhDtw276I1aln4QdRRYBRPsWya7U34nZV9GLv/UUDs8q0NOCspOx5vVWvtG2or3ZNphfap2zPlShHaKB78aWe9zCnWU//rXv763DS9iydwmt9IdFuqPfuYoPDZhcUcfYxSuuyxH2HtiifMrj1SSWlm2QR6rfDJftoQ2kB9CU3UYId7eG3UvmyA0QvRJ/DZEXgLUaSh+DbbxCn30zvU2Gqj5BC+aNaDQjekVNa7mSRw7qi9GJH6kg+qXXTwSgho3FcI7eaHKcz5e57yiXo/p4SFkdYh01yMq/6kQ+4VdSNuzIZupBTThvw1juuvUwymdg3R6VUbNv0T3Cvk36a0TNKo2JTb2NZEV3SufqtcchJdpG4MR9He605163CaEpoZNdoyoNJYc+4gwq9NCn/Sj4NAjeoIp4zN8lYtyc6GgQu+TYnOLCmkuFLZ5bPRG9z599mQxF+SnUhiR+HTcG3YYHZq7rCOkksX5eBi9xYaRfhNCL6hQVmkY6bpWocfKvAmR6T2CLSVyTa9dSN7MO5Jvxc+2l8HVfogfBWUgfUq+OGmCqSk3zvg09CBf5S1UyJc8CYnLMXYJ4EhW4gR5oqsQu6KL6/AbkTwgPflCx59XutKV+nX0T/6KyBaiMx8mb6XfBvnQqjfo3WRsWue90lygx0fZW8ud9zbiR4hjpzI0Tx99ydaO9AljGexC8pqasW35FLKcM/KD+EmcKVa7moZfzbcL8nuatlGhZZX6NF/DZqp4EyLHUftBG8Q3c6De+J7FVtfRXzgOHGnqxty8Hefq/s9zEOUZZe/liy66qFfOOUbJkwrIuXZgtGOfTmouqhxOfcADHtAr2FSjEudzdjcUdKFNwfPBOeec038wofLdhtDirXG85CUvOXjQgx7UX/zgkQ5wF7+kq9z2MBf4kX/CYxsyv6pTsIsmO1Rs5ZKKG12DyBRPDh7inPuwzFa39lQf6cKvwjX5tRPGxxNbvs0Y+QRJQzOeazBjWeIbyBOZziHyE4fe+SaEDuQLf/E6Sx1lzVPPwXX4R1d1QT18xjOe0XejjC67IA/59qC3zfD5559/8A3f8A19L6htqDqgd2MwL32zm92sb7e7yf7ohKbqyKe2nbaLpXOYoz8eAn52FaXDCD7VPtSxUa/IkkdfgkfKYwmUg60ybFPta1jXdpbdNYit/vB7AH44PNdz7A9MhT3+8Y8/OPvss9e+hSU8NuFIHb09WXSCHJLKOgcUF3RQ7p5+TCBz4GMhTsHoVUMi8+KLL+70S15+kaNyoNc4bEtqU7apApVPx8fxqVCplOIVrApoi1zpcwsFD/nZb0/7s846a739am4quyCPTlrF0Egf8pCH9FHPHD9Kd1PwROUXcfzghw5SJ82Xm+jFp7yVn/30/SKOX925853v3Ecz20aS6Nhn1CWvhon+YQ97WH9E/uqv/uqDa1/72n1PEXlHf9LZlrG21bWXPFo/JWgfdnvZ+OETnRw7PK2hT3mH3ogTvbK/8MIL+2AFPT58aT94Px6iUxnlK/eMen/hF36h1z/lR56tmu1pT397/HuxNtIH+IANyQyU6O7LSz50bSotN+JdUCZ0tXWzG7UyPffcc/sGX7uQcnb0Azg6OWVhOeEmyEt/2+oq69joqZQ//QgOiA//XZAPP0/GU7+JQJ4bYZUXRB+01r37YZAlsgNPI+qE7ap/7ud+rj+x2Hp6lDcicpSXraevfOUr97hddCPUKzbkh5zQH8aOKRzqE6woYMSgczCvNFbIUbnRaOlGLxqeXfMYJg6fSjvSARqPjKZM8PBDF1MvMaqTRj6c6g6qo9XI8ZOn5gttHlFznaMOXofhhx7YANKq3KDyTSNnh8alc/PDHSlcqDwqr9A6ovcDE6Z+/LKPl3HoRvlT126UPu32NOHmoGN1w7QpVvSoNPXcDc4++H4wxXSR9c13u9vdegdnysLUzEgPrj15PfzhD+83CA1bh2a3Qf77mq/5mt450SF1InQ5otcYfe4PPqLJx2ym8/K+KJvCoUkA9G4ynjykk5XfQwB7wXvvxCf4hQ7CR8fuV8nozxemEu2oiFYHqT24yaTBhkeOys6I2a8h2TFRHeY3Nzo/QShkanKqo08dYKebod8UcKNQDn4hys1GGWibFeSjCXLDFufJWB1yw7KDZKYran6IDaAN5UW+TtJTqbrstwWCKnOU7zoBLz8YY+sI9Sh5HYEf2VcHhDXduU7WIJD+Kc8qD5I/CA8+Ne1iOxdt0SDGDSdPliNdRdL50iBC/a5ypVUeNQ1ck6+81BsDT3nHvjBllfIHecQl7ZLrquup+tMYLUIT0o+N8arduVd3v/vd13tvS0uoqPGCvMKTnvSkTt8cvN5HW6iodAmtsfb87RGp70f//ve/v/OrmKKroXWSfT969K3DX+s05hPXKuH6mDjBfvptJN73wm6jjY+i3RTwoT/72+hj9cd//MddH2n4tMJe5x11ko7evtzXvOY1V/e+971X7clqLX/MH11rXOug+r7bZ5555qqNivt++nhGhzF/DfT+3d/93dXTn/70VescVq2j73Foq5+m9HCkZ7tBr9qIuO/7re6gRxPbaqi0iZM/dGjIFkKfc3RC9qN3njxtZHgpHo5Vj2pPbIgObRTebfAbBXyJrsp3FF91r+fytJvFqo2eV294wxu6D+mCTp7ITQh95SMvmsc97nGrNgJdtQFHl4m2hpG2BnzwaDft1Zd/+Zev3vKWt6x9OSW3Bumx21Ff0G5yvS5G/znyk+9Nb3rTqnXkfU/3UX7yCCmjpDvyhbr0FV/xFb2ck1557ArytpvkqnXyq9ve9rZ973m6kIf/FE0N8vjdijPOOKMfq35j3hqSjl4/9EVf9EWrCy64YJJOnHxtYNp/CyT9hiA+Pmq9cwmXYPE2xfLmrmL7zia8j2bqBzAw8qwID/OJjujH0ccmkNsqZ8/vMQeMArdNN5Axprn2+A1GsnXkFP0E9hkxG3nWkar8nmi8Ka8vT0MXHkE9Z4N0o+G8aW8F1GW0Auv2GWnKV+kC8R5hjaKMcvKYD2N+cqJvvWZXRptGEfwnj+vwgvALnQA1Xn46wZS+gTRy0cTe6AGVhzy1TKDKD6q80Dji5Vy6KQVPXp5+pIUmvCLPUUArxC5x4Qt0Do16mPjIlO5p0ZNG8sWW8BcXGUnnk8ipZRD+QXgmPvp4wq1PQniRkXOovMKDrujVJfmioyPU68S5bh3MOr+RsHakPgO+yes88hMH0U26aWDtMb8OVfsD13yjvjvPCjjnsQG8b/N1sCfbam+VCaERL9DDtf6ELyynNJWbl8Ke7iIjtCm3xOOh3RrRt5vm+skjOkLVI/4ER2Uozmof/VnKIggP+TyNexJsA8X+q2yeYFLX0BSyhlN+bwygXwRVoRHySnekGMZeXJiXinFxHEh3TcFUQEfXEB6Axrk0x+ghDxrXOkI/7acjTAcrPXnCI41GhQm/yETjGi/nkQOxLwEfNySOV/joyFLJVYDYFV7SBUAbvYXqM9doBefygXRzu6Yk4s/EQ64Bf4ge8pDhcTr5R/7RAeIv87vk8ZUOP7bIJy7nkZc4vCE2Jw++1UYy4hPwWOxGZWqDrqPujhD+NY+QG4R8yUumOEfTM+qHF+XeIemE/BZre3rs0yx5JxS6HGM3OIa384T4gD38gAbklS6efL+F7GcBw186/eIXYEv8Ik4ecZEF9brGVV6Ca/PTpn8y6BInLflG/gK0UWu3ha7qdXROOlRf4IeP9wvy5ncF8HEtLeVIB0fx8kgLopc8VVe0eLlOnE7UwAi9zlxc7HGOf+peysi5NLzBeXQQpEe3yDOdaRrMR1Le2chnarHyrXpVXo7i8RT4NOfgGB3Cq7YlNzurfkzdxR9BeMivjPjIjcm7PctqTUXq7OkxNXVzpB8eoQzjvOXXUZhX1On40pIhKpwGRzGVghwdiTuyhm6OE3SiGp48Go6GrPIo3CzzysgVjeWUNlIzr0lXeYxK5SGPXhxGHj54k0cvDpKHnu7eCpieWSZGB/Thha+Xbm4qfs8SyKMbmszjksdmIxoB3zSEfMiClxAdFBqdyKMPOnzJo7c7O91jH178TE9PGejwkpffyAMFHpvZGN4qmVER3ZQDG/EwilCZbS3Bx+TxM3lpxK7R4ctmdNE9vtLBkQfo6IaOTXTIvLPfjKWH3/rle7q7duQrclM/+Chz8GzmY7qTRx83DaAHvxmFyUemvHQn05H9ykqgG1vNCXv55qkyfHw3gcZ8Lx7yxWbyYjM96El3OqvXjuZ6rd645z3v2fUEvsLTtbJgWx2h8o8yFs/H4ujrZTuoawJoT/Ezvejn2gt1vx3MX+zjhwxI5OMv+djvOnWLDZCRcHRFRw/lon7HRvH09FLcj5irp3jirQ4oU+XPd3g5Vn/Ri2xx9ORTMsjMuxV248lfgjxmENiQ7wXIEtjPZvR0UPZ8TA9llb5HHnR05+v0T+LIk1dHbyB5hzvcoQ8K6OXdi3xpO+HFH6mT6QvSdtiQ+kBHtOynlzJnj7KRR7xzgxJ+1s+whZwADdCRvmTypacY+lqU4P2QtCa64Bg6enQEW7XBMQqOIM4QFKZAadeMF8RRlHFkueYQBYSf8+RxLahcKgUaL4380roA4uSVh3x64YU3mjhGoQHe9AidPAKgo6886OSxskZlVsEAH3TS0ZEjHz3DGw9xoCJGHlr2oXMMXXi59qJRQ/FIhjde6NglHzrX4uMraQreUcUhL/6Ljx2Thyy8pPt0/ipXuUq/mcmHTpCP7tU+fPk5ZVp5VZvlqT6mZ+hUTitv/JADWsALQsc+tJEH4UUfemmcVsx4semxP/6IPLqhj52O0pzLKy156OEn3O51r3v1lS9VBz4RYjO4lgc/+aITuEHg4fsMMslAhz6+QicAXilTQXpsVl7o5ImvXMfP4tDir35ayaQzAXFslU4352O7cE0/IfqnHZIpLnpByofuF1100frHzEEedHgL1W58YiPdyRbokjzO8Xdkd/QUdKhWI+lEPbWIi14pv+gArvGPzRAdyK900cFNyQ+jo7OCSbyO28vldMYpM7zYJ84Rr1pH0KIBOsgXnSBlk3oEBkA6e6uv5MO3Ak8hMl74whd2n1iMcte73rXffKRf5jK1/z6mjl6w7tVjMaVdo1/CM3lDV+lDl2tHoyVTRX6bsfJNnhEj33qETefgmjyVWSUILYz5YBNPIQUapHIHzhW+1Sw6em/fR1SegbhRPoxxrmveXOuQjK7rqp0geXIO4zWIG/NWPrFVvCcIHX0+LRcX3yRfpQV5pGkIr33ta/sKC37SYHUOHufN7zpaRaOsTLVpXKberEu3vhkPnYjpKu9Y3CDe/e53998/NjKS7slGY7N6xc3IaI5+Y/mN9uaoM7LE0/LPMb1eOycP36SPkCd5Rx5B0k1J6SyMVIPkTZ6cb4N8tRwqLUgDK3X4mb9h5F9pEucYP4Zv0qBeV3plZvpNZ2xQWRE+4zlUXiPfMe1Rj3rUwXd/93f3pzv1wTsPI29PfDpeesf2ykOoMmt5Ji3XU7IBX3XR+yQdtxvSiMi2WstqQYNP6/x18BC5n/iJtZ6eOm8Jh0YT3N8Et8eGvhJm3yCv3VH7KhsrXcgWt0+Q8aM/+qP9V/vJStgXyPv1X//1voqgjQL2bh/+F1988epDH/pQly3sA/Eb/m0k3lesKMvEVbhuHXgPOecLKw1ag9NC+tGv/v/wD/9wXzXUOoK+guUJT3jC6iUvecnq937v9zotGVbdPOYxj1n7MscgMqykOffcc1dttNZXXQit81y1J7r1ag8h+jhWxBbx7UbUzxO/T8TOz/7sz16v9tk3yGs33fWKtX2DXa9//etX7Sni2Oyr5dVG/at2w1q1Dnb15Cc/udtnBZF2se92GD1++Zd/ua+CI3uUV3V93/ve11fdJd8c3S49RFmIJqDfkTLvt29EhjvXSYFMI7Tm1LX8evc+bkSGabCpu/o+wB4h5bkvhL9RtpARiDjHhMB5q9h93brVJKbsWmXvUxS+qDbCu8c97tFHsM69dL3LXe7S50M9hSmzOXUlOpiv91Wnl37Wo3siMMon48Y3vvH6a1F5t5UNHTP1chKITkadjvuGcuRX0xfOTwJ8alRvymIfsKDEuxX+y9428elJ9TemmEzxbCpD8YKXtaZanfPLHBzJAs4gyJTNJuWOExxOjkfpzOVFLl32BYWtAEYZ+5DJRvadJNLxwj4bbhoMf6ozQfzoSL4OWr3Syd7oRjfqX6paieRjKNMFtpvwolPZ63AdbeGAr5eDaDzaerEVbCur1CGy+cFLte/6ru/q0yAPfOADu94WAODrfRT90tlNgR6m3k4K9KePwUFs2SciTxtMvdk3yFQnlONxtTs2COC9mPOb3OQmfcEDeeoo+45L3i6oN+RusjG6ph8ENLP0a5kOjeb4HvKY3BTZGEaMaeP1FKSR45HFI0yl2UY3hZE2+o8g67d/+7f7xwmRKS9UuhqWoNKFl2kC55t0qghtkOtddCAPGR5RPbqyb7TnuIBXys1RGGUl3VSMx+fP+qzP6tMn17jGNfrHaaazQpcQna93veutLrzwwp7+p3/6p6u73e1uq5/5mZ/pab/yK7+ynrpJqEhceDqPHH4xlXb++eev2gi/Txld+9rXXr3sZS9bP9JX2oSlvqx5Kk3ippB0cujy9re//aPkLsGYPzym4vmnlkVQ89f4KYy8cz0GfLSJdrPv00Xkig9NjjXEB9sgnb8+/OEPr9pTYa9rL33pS3scGXwa+7bZMsocwyZIC2/n6r26G7+GNvnqMcH1HBx5RA+WREET3ENTdH33SZ4KcYmXrym7Pp/KP8KLPMujwqfym4voV/UUch2Is8wqKxnyaOWYvNFffO60c5H8oSXPVNgSPtGbHo6ul9B7CWvkYqSANnzC67gQGz2iWs1QebfG1eVZNeIFn3175POi37SMjecsYctoJsGIPnyMop2Lw08ZgTh5NyG88A5/PJzziykdL+q8RL7tbW/bnxY83tv8y8qf+D681H/7zQTbZFfMzTci9nt5T++qz1LglbAJ9BRMkZlOCSpNzrfxGUHvCrR86Wjk6iW5pzr5BPE5F+gkLjKTZxOk89dP/MRP9CkhL7PPPPPMNS/LIrVF2FY2Scsxes0tg+S1SsfTfOiqLXgLY/13PQdH6ugJoog5TQXiay2P1FYbaGjbnIw2DqWsvGlc2yC/OfPwxyPxc4EuciLbEY+Rj2vL5XQ6VmnYz8NumVZtsFl6GlfFNttHoJU/NHMqF8gfnaP/YUAef6pgbFTxM+2xxI6K2FPpo6tGpREFyWda5uY3v3mfsjF9Yp7dMsXYFfoEQOfcPL0leK51PuwZVyNswrY0wF8Zu5G8+MUvPvjGb/zGfgOwx4w9fnJDSTnKr87MReQ7hr6GOfDxjP2K7JvDXvouRWTnHKbkR09lyPbkTXxo6LENo33Ow4sv0QvK09SKL5vtI6Ockzc85EOzid8UpKvzBhJgpZT8eGnbqUfbeIB0Aa/kxTtTMdsQGr40mPi+7/u+vvomN7jjwpE6eo613MkXXT791RkaiVmuJlB0ytkxnkONCjQKeQQ8t0EeDc78oLyCEX7ugkuAl4LUaXP0qCfQ1QcX5ojNA9PXsr5v+7Zv63aTq0KofLWg5yD2koFvClcnl/RtQBd/kY2eT5fCEjL0npIsKbN8Mf5IWU1B+qaQ9CkYueTDkZShl2FnnXVWp7FM0qZn2ZxKPkFa8sduR/GWhxoAsN8Xsb4JMF8f/atOUyF8RkS2dD7SeO3waGM1T0L0pK86FNAhSw7nAP9ad5zHRmEXUm5eJNtoL/RT9uyCOiREn4QK13T2TkA7ZC958ZNBAnphDiIDrUGAp6T4ROBj69p1gHYLvfDCC9cDBfminyPdDTwT77gJScNfXpuY0UG7Vs6eINVV+bbxgaTTX18BeAmbQKb67enWINlSSf2A90P60l0yl2BnrxBlBYoJzhWAArH7IqdboeCDE2+EPV7p/FQA+UaFw4NT7J6oA5Vf/C7j5NEReuOPv0d9I2z0S0AvvLxws77aig06gfgE+RSGc4/tPot251WhjJ6Mhp/+9Kf3ddqh34XwBjS+5lS4dBDvazmY8oX8QnQTVAo3n/POO6+vHEieishMqHnIcbO1Rl0l51++FUaEThh5OXfj5JOp8kheo+Gs2NAo7KOvkek0nvOc5/QytUaYXtFDGTiGh2PqlmurbTROo2xbzHoyYEfyypcyj9+kJS58a0h8dMA/sArHWms7HVqn76MaozFlgSbTmeEbXjWMshzBKDlPJwkVI63OyFrvjD6N7tXHjFS3ofJSp9niIzSdqTgY66FrQRkqs8rD18leWBsA7kLoHAWgNz+6JoM/3VAMIH2B7+W4gV2eekOXMlV3PNmwZQqRJZCtDG1XYe26b3P4zM6kyoB9nhSTdxPoGZ6eSh/5yEeeSpluw4E0fMnxNaw+1D45XgbXAc5xYGdHH2G1QYCjO6fHax8umT+Tz48maMjXve5114U4BXlVZj9UoCDRzDEKP4/2RjF08smyysAxc0GOhks/S/ZUmmxvi79jAr7maHU8V7/61XthqBAalgKx9M/nxxq6/HORUZMPJHy2jhfeeORJYapy0U9gM/Ch/cN1lnQ0ypwCeQKe4atjMPr1dZ2O0bSbhrKtktHLDdpTjtGjm9PrXve6/kiNhy1mfYrNL7XsI5dMj+GmQIyM+fSCCy7oeezhrtMy9Wc+nL61PMDox1a66hd9v/M7v7Nv7pStfsnw2wD2Ea+PzuHDb+qcpwX7mfixFTdqcVbWuNkZnbvx0TX0jng7KjP13hfFblBk8YclmXTyxaiOcuopDw9x7CCDLDd6NN/+7d/e+alLjkbpU6ADnviAcx2TcnOjNM/sSP9tQJdyEfiAXKNLN7LImYL87MvIWlmpSwZNvpY1bSbPJvoKeQT6+tpT2WqfYFStTlleq64oW9ufZGUaHVM23ov44En7RB//TCG68ZOjJ3SjaTcJvPlSv8BGiD5TwAvkUa/RxO6kbYObJX/ZUkK9P+OMM3pcEBuPgv5l7KnzSXBWOsUozQnITNcYQVlnbPRkCsA+5RqMjYFUWLSjk9DiZXqHg/2qCuPEM2iTU9HIowG4sXhUN4frs3WdJb3mAA8VU2PzQk1HqWJnyV+VXwtKPP18lWnkrEL7oQq0GmmWme4qFPL5VfADF25aOqt8wuwR1c0lvCq/+A7s/WEvffPq6I0GVBDp6ayD0KTzcq1z1MngY5rD05iRBV46XTc/qHw0Bi9ITV3wAT8avfqa0B4xaEyt6UiVUXwWvT3We3oxilKppdGVL+tjsr3p+TZTPOHhJmPEjtYNX2epHDPKo5/OwVp7I0ON1ihVx2yOl891CH74md/drAQ8BHXYDYL/3QBSDmQL7PUEqmNOR0p/Hb2XtGj50kBAG9DpAB4pFx2iUR//Z8pH58V//M+f/Ghv9/oCOsBHoIsy0Ll4unZzUR/c8Mjlu7qMdQR6vN38vJdR7r5w98MddJDGZseK+MKAxIiXjvzmh1cM2vjdKDV1cKQPwifnBokGDp4o8I3NbE0ZGFRoJzr/pItXrvaz9zSh//FULD0+D1xHLvvpK6gD2pEvot3wvNjOeyqDsLEMKvCKjta3q7cWD4iPD6ZQ7aK/L16VH9uVB7020S7Fzr2BU9hgjxJrmFVqcJSmoHW2FPSIZZRs1ObOzACohY2GAW4UGoX5zMRtqhQgnXMUoicAnQ16e94sdYgCMC+MXqXBTwdrlFl50d9I+T73uU/vNEw1GMXoyH7yJ3+yVzq/6lNHj3OA7zOf+cwuU8NIxYSMVqaQiqos3CR1CH46zbRZ0vDJMXBtOsVUmcdr52wGjdLIVker8ocufCrSCRq56tw1AnTqgPLnV3mgNo7w8cThpmgajL62lnDj8DQjT/SmG57VBhCvvHSC8uqsdbjKhO5GtjpsvNNJQspHx0emoAzk5ws3EHxcu2HgZ6O3AG2CT/Ctt0bvmixy1Cc/QqPTMPDJ5lTRgW/orHzpz1/qljrknB8zmhTQpf2MkOYm4WZCV6N4NEa9bppA9jbgwW5PIrbBUC46+chOWeRYIU694Vd+t6GapyGdLVviq5FuBD5gCkUn5+kYX6A/erooGzMIbDWATLo0eZShG7BtNXSQ4Su9IjrhR1f5DPbk06bU6Qxw6LFLf5AHH+VhoOV9ketR9gh01UZPhnzIFwYDbhrHhqbQTjRl+nrNVqB9XbFjqyB93W4bxaxax9E/QW+NpMfbeL/dldfrUdFXhJ/PxNsdutOKG/ONQCMPHfBuBds/e28j+9nrSUFe9H7s4K1vfWvX0TX9BdcJ7PHJcRt99c/2fYZt+wVrutH7vD52zgUbyPFjEW1E3WljO93YF1tHn4ink20g2l2/r+9Hv8v+2EPX1lj6p+v4iGs3m/WPViiLNuJetVFvTx/lu47/5I+OYz6oNjiX/13veteqVeC+Hr09NfQ1/ORE/0qTsAljvk0h6+hDk+OmAPSIThVT+RP4xPr61ln3Ndl3utOd+tpv8bELUg6OU/Ym3yZIR9ueXlZtFNrrYht09e8ObNWBd+rxNpCtvN/4xjf2eph2WHXdBLyVG1nqU7tZ9zXuUz7bhNiBhz7Dj9m0G3e/jh7StYd3vvOdve2xV51nLxvlEdTn9kSw+rVf+7W13XiMdkgTyEk9bE9CXeZrX/vaHpxHr5xv84c0urQb/OpFL3pRLw9xsWMTpNFdPvT6mOc+97l9Gw92xJfybeMTJF8NweyOXiC4nkdBlbmNLlePetSjurH2GkkFhygcoJcmPjyTx/kmJJ+OqT2yrwsjPOZCXrRVfsII+djko5lb3vKWq/aI3fc/aaPavgcKenmEuQgN3Uc6aRoMXaJbReLQ5TiVb0T4JW89//CHP9w7+lS29gjeb24asDxTQDdHZgLe7Qmo/4pQG5Wv7nvf+3b+c/U/DCJ7qqPfhdAuAVsEg4c2ou+diF8MMjBIPYXwrmEJQkMWvnyrA/z0T//0vm9RbpzStkGe6n/HnG/TKXntWaSDjW3VxjmIrPiNHVUfwc3npS99aW9zaXtt1L4eoMlbaR0rvVDhmp54uhm3EXi/cYhHL+RcuQnhtwnJj28GT+IEtNsgv8GqX9pzk9Z3tif9S7W98NqF5KshuMwj6yviDfBokUeYqaPHaY85HpnNmXq0NsfnsS55kj/wqNIUOXU1LWMKTec+ZeQRyaNuMCVjE+RrTjx1dekpBhj5sMMcqHzmfj06stHUVF5Cg/Q5kJ8deZQf6axY8OgvX0IgfxAe8WPNNwXpkVVl4pFpHC/DPPrysRdgmTcfMeo1YtTT3KuXn6ZavLz3LofvoOq1D5hr9R7CS2LYpnewy75tMOVjKspKKLK9nLOqI3aGdw1LwJ/Vv61j6VOqpu/UU3P0yjNTMJsQHuqPvEH02aaXNO1d29AWycFvqT21DwiPwLn64p0Km7Q57c+uoqa+8j6KvNhZz6f0aZ1xvzbFbJrE7qSmnZMXogPZzutU1BTkCQ0/ypfrbf7XB0m3FFib8zLWuyn1dOp93yb5IyIbQrPzZWzFmDVO5hDzoUD5qmA6otHgxOEpb3iHbgryCOZQ09FX3ttoK9DAnPzs8bKKHdX50RtyPocfxI4g+idOx2DeNrZVvqGt8kK3TX6VN6KNKtYVNOUFrsdym4voCfjd+ta37vPAthF2ZJ/yk4fMfSDy3WSs5DGfLW6bn44KtgpulF5050tLP2XoRhffLq2zFbELbdqbOB1v3kuIS9gG+oR+zLuNFp12r024qaBfahOa2ALoXePtvOom6KQFL5lDN+bbZQN/aV/eC6r3XrwqI6j6yycdb/VU2sirIjpA9MhxF13ykdmeSLpPyYvM2LqNDyRfjhCbFrXiKF0DptbtGlVgqtDd5TXeNOAIq0gcHjnmfBvkUdB5UZIOag5tQLYQul3Bnd/IzHnoKn3O5yI0CSPc3VNxRr5T8qbyjUieGkD5WbHSHpHX5ZWym9JtCfDWMB/zmMf0pYhWllgnbUsJN2rpR5WxDbExx30jDYw8K4+MPP1ojfbgC1or1DTk5DmsXpU2ZUa2NdzqTerVHP7Jl2MNm0AWOXn5jtZxF90IeaOrkLjwqrzJs0rJSqvUm9A4hpdjDVPwpKW/MuvgRSzeUGnw0seop7CJV1Blj/psgzyx1w2IjeIqreMuPhXJX2mOpZWlgp8EyDK1oWNyvm/ZKoGGmqmNfYM9lqoFJ2EfGWw0ojgKoqtjRkRmBvNjDkbyVuz89V9f8hNwwb5tPGmwx+osMDXgRscXvnzMh3XxOxyH/ToGN1VH/PbpUx0IOZ4g3LiqLfuC+hS5RwE9LWPEz7LqTMuMkM/ybf1MrvcNMviSPo7HiSN5LY43l6ujOAmQqYLpKE7C+ezTSeWHyE8CpqYia58y8TaaMMLOU9hx+VRF9RGXtdl8aMmtegI6PdNhZJ1EGZ4UYgvbrTEHcfZHuu9979unOnxzYhnfcTZk5UiOmwpfC/usN7FTGbq5uE7YF9RN3xnYeOwo0LYsi/Z0edZZZ6071tFfbNHJs3GfvqzwXk5b3Icfj9TRRyFTDScFMnW67sT7rtCAv9H8SRR2ZNQ19ftsPAE/akhHBV09EgvZk0ScD2C8RDOKckMxj5x5T2EJNMwEvHNM+MdErY86JLYK4nxW70Ub+GAM+IlPjqp3OqvUm337Ib7OB1lk7rstqp9kZSplLuiZOunpw545Bok+DDM1g68w1kO2+KZh14vY40D8yb686zxueUce0YO7JEX3jThEJ7G0wA8LjchjuMe4k7ARst3uPisXkME+U2EaQjqMwyC8BKMgWyLAve997z6axVenJz0V2jm6JX7FJyEyP9ag49DpxrbY/qIXvag/1fgi3M0vdhwVePCDL4bdOPaN6KwdKsuTgCcHc9jeJy2pL0Df+MVqL9c+bFPnt8GNQEg93Tc86fqym6ylA6BdODI3StmS4KQqGAdYwpm5s32DTEvWzJufRGGT4bP2nJ8EVK5dlX4b6CnwlcpqtGSli8akYekM0ljkcdPUaPMUsaRSxycavhFaGv/HQocfHbQFL/DrKJfeOv+8pzA/bM+e49CbDEFHH/+cBKwsqu9a9gl2WRCRtrEEaN1ozc37Gt8vhVn+mvq3CQaw2kbK8LhBr5QXGWT5qlq9Oe5yPHRHH0VUVE7c1liTN4YJafg5ChVTcSC/uc68BBISD5voKpIn9LlOqFAARthGoM6n8k6d57pizJfH9hrATSyyRtR8MMUjAXKsqHmUm0fwVPpNOgVjWvWfm4U9va1qsHeRrR08eZEh1Lzj3G7CHMhHX5+KWw9tr6TKL8i54+jPsdyhnk9BWqVznuCajOQzMBjj6WzLCNM44mw9YFOt8IB6HkTeLvj0PrJSjhW5Dr+pUGVHl8RJD5zzeQYIY56R1xygqfQV+hg3Sm1xLMvQkVePNY842ziDgUj8VDHS8yH7nE/lDyKrynWetBzHMIIM7SX2TeWZA2QJwZE6esa465mP3NbRQ1U6DgF3MXdq13M6GU7QWMyduVYQ7oI1/0g3Ao9aKF6cGWWquFPw4sZ7gRF4oBHYQBe8BQj/hCCVAMS7abl5ZQpslDXFA1yTxW94xqax0xvpKtBY055VRfixI74YacM38iDnNqWyqgTtox/96D6Sxy91wzlasrx4Cn1khG89Jq0i8dbkqwum1lzjL0whsqTz1+izpI2ILAj/1NPEJU+CTskUTfInH1lsN6K0pwy/nHvuuX2/I3Ug5QbRjawg/KPzeO0joshJ/oqqyxSUO1pHT2byRcYIuns3l28h5KU/em0JfXy8CeFdg2m/rFiqwN8+SXlZOYbIoQN615HvBmtpq2Xg8vogyREqDwG98kBnjr4O8DZBOp8B/S2RBPKlQY6bQJ52yEYfiNVyXw6yEi7BoTt68NhsO1qPU9sMqcYKafgqgw9JbC7kEbDeLJK3InEKwNwZ2DFQp+KGEYx0IziVLPlUCo3NjndTUMDyoYHokADWSv/gD/5gt0d+eWt6MNKpgPIawdtL3osiYF8q1lQFQ5/46KWi+cEQm5aN8pM/AepR/qQ5F3RW4VNReQI/CqYqTEfoyOx8mU3HEgCN4AZQRy6RDzlCZNQ0eQMdDV/xIx0qnxFVVng4ik9c4qeANgHk1SmkMYd/YORZIS35QJ33tadyM18PqZOVl7gRyZMAKQeInnTL+RiCkQ+7DHxs7pcVJ0nLMcgcvXiyyNcnWFJrd1EYaSrCOwEPPwLiKW1E9U1CRezylEe+sgF0bDKFKN1mdFbvoB/L23X209cm1VE2ylt9BvXaOTnK0vciXrbjJa522FN6B/IK+V2Jo4CIhOBIHb2RtM7Jrm3bKlWMdeQMThCvc0evssRRSRMg5wnSVcTMmbtRuEYvLfSVzxiALvJk7/M8lYx2gG1uM0oYg+1urRM3gmNH5c+eXDu6jlzBtXlOWyybO7RbnXzeCUSP0CZA/CSoOEbR9unWOLNUUt7IGm0SQi/Njy34apQeVoaYUkhDkafSyc8XtnT1bsZ2w7YXsDWrjsGunvh4jxJED0EeUy1+js8PuBgo2PbZbn1VL3ZVO5Mmnr7WpPvdVi+S2etY5YQGQucIBibo7XToJq8OkA/bfCVIt0ulF8y25TUN40Ml7yTqdCKfOI68hPgQrXpn2wn7oKPhU8GoLqPi2DXySADlZRdTfNjkK1zLLSsNHuCILnEJRrz3u9/9+lYD6mXqUfLVo6BTt9TZubajc1SP6OJpK20y+ccgjR4pZ3FugHYWHfPKo5+h4xRPsvjUVJgBQG5A6Py4kCWV6NTt6CW98lB++iNH9PoYNsbHNa+QuOhjkGNHXXUJb3Spd5Wu0uQoL1nasrZ13Fi0BQIkOwVtC6pgVHqFY725xw8do87KY4gCV2lUYiMvcbZmle7upXLantRWw37QQcUxSrPdKsONFBUW/qY0PBbh5dFfIyHbtqy+QsTXozw5Ruh0NW/p8cu0iM7UuU6ZDE8D1np7aWirYXL86AI6L2t8Jq1yiZPfHuMqgEc7UzUqhMZqPlqaQuUDtB5f3UA8ebAFnQasMvIRXfmQfLrZlx4PflExTU3ZE8a8JHno8LJnPMhj9YY9TvjJNsMqMN62xzWizJeE5uDF8Qt7jJr4UOfIv264fg7PHuYaOftVcjq7+RjV0N1LT52Pmxv5Om08lRu9TXGxH71fXtJp2FqW7raDZrNOwJMHuToGgwX6ywO2iLVXjOWYbFdmOhx08tFVHbKlq2uP5DpugZ4Cvspd/dCA+I/+yoqO/G0tuwGCGxL9ddRk8JPfF1Bm/CZdnVJnyI7f1BXXXoCq78obP/VUg+d/Nzw2Kxf+4jujSfLNF+MDygP4mm38p34rU6NKvLUjPgT09GUX/XQs6oLpP08J9mDyA+vqF9n2FGKXvPwgnj3s4jejWDdPPNTl2K4c1DdtUjngRUdtiK18QT9tl2zbZvOh7Y7xcB67U67pC/gIb/QGC3xpa3NtSmfMPnoqQ+2J3vwMnhZ15uq2Os5X7Ladgb6Eb/PDQOqxH0OxFfZd73rXfnQTZbcyUj7ql31m+NPsgD6NXHqBfoC/0GkDdNdW1C3tyWyCAZZ86qXBkoGEMtAnqOOCL3EzEGOfekqOcuU//ZEbpTrkB5W0rWNDq1SL0JzfQ6voq1agqzvc4Q6r1tj6rnHiW+H2tFa46/PmkH6UR3At3baid77znVetM1jHoXGsNJWP61ax+45zT3rSk1Zt1LKyxekoy7XgeirO9fOf//xVa9SX2l2Q7OR1dN0eB7tM19XGJz7xiatb3/rWfavd1uldyr7Qug5deEtvFbDvEtkqbd8ZM/nwsROho2uBPOmRS982alu1Tn3VblBdfmQL5CSvo0CmI773uc99Vm3kvWod36o1nlWrdH0Hy9CHJvJjjyC9VdC+RW7rgPpuia2xdl3wap1Q3wKWDqELz/CQx5bP0bV1JD2uNYZVG9GsWmNbtVFz59M6hm5fa5h9R1TH1mH13f3wtGNqa8A9/m1ve9uqda6r1uhX7emm05Oh/OSTv+qUspSPT5Xxwx72sL597VlnndXT5BOie3SWljLlB/5oHXfftlu9eOxjH9vj4vdRNpnoHO38yoftJtl9gE48mtSrWgbkSr/Xve61aje0vq20bYa1J7TSo7MjPqGNLs7xbIORVeuUV60z7duGkxf70Ar4yS8+5wKfa4d0aU9oq9ax9bYsTT4BTfQPrfPIyHkbsKzaTW7Vno7WOsdmRzLaDaLrG/0F54L60AYHqwc+8IHdhtA7bwPBvpNoG/xdii56OZfP9sD6A/VNHkflkXyOoQ2da/FsaJ3+qg3Oet2N3WyWN/ZW/4WfIB59uzn1uovm8LBrZcIlOFJH//CHP3zVHsPXlSNGCfU812hyzWjbcT7taU/rhVjzCqGf4oNWOP/88zu9hu9auiM5VVchssNPPvtf6+zpXnVznmtpOkE6hlaa+O///u/ve3mnIow8qnxxOZfWRiKr8847r9+kQi/Uwk9caIHe9LnwwgtXr3jFK3rnIo4+Ne+mIF0HI6AjT0dLXnxX+eQ6tK6dy4uHzkkj0lm5ASSv9PCZ4pG4HHNOnzRU1+EDNU/S1T8dfTp+OrCn6tJGo32bYvmjQ3g5RoYj2WjJEOQXkldIXK6loRWUQ47oIzM8QictvNxg2gi1b9+sw4w+UzJDL6gH7I5MHW/1TeiSv147ymff8yc/+cn9Zhg+0jfJTpCewN8GHraDDj1UHuEjzrHGC35b4t73vndvZ+FR84kjR0haAp4GAX7fwM0+egnsaiPjVXvK6faFZ2jDX3jkIx/ZO1l00aGWk3hHCE3i5bNn/kMe8pBeDjUteXMeuTlGln7sW77lWzof14cH2oRLcOipG8fm9H70eOJRz3EKySOEvhnZH9c8DpqScAySJ/zqtfM2aumPT61Rrl+YjPSVtsqt8a1S9cejyB/zgvNWiD2fkDxAf/LRiks820Z+SQtaoXe+eakc/VsBd/tMWwSVFi8ByMnjnbgqsyL5IfpUtIrW+YQueTbxIxeN3xf18sm0gikUUwRT+QNpArvZaXoqvoKqW/hIkyfXSRdnLtNjemtY/bdjbR9dX55Ff1MF3uX4KcGRT84TX5Hr5I0uUNNGHo6m9kzDxLakBbkOP9NhVuKoF17um19P/koX8B/elYf6XL8YF+8oVPnjuQCJg+QZaQNx7YbYz9Xh1on2qb7Ux4qRvvKuukPKLwHkF9jMP/JUnniIx8O0UtopeFehjvotZFM+KY+K8DIlpl/Rpl3rX6Tl69iKKj+y1Gs+Ma0EVVbygPh6XelNbZmi5stR5nxc4tNLcEqHJuTQcCcSPBIZCZ0EWqH2KYI//MM/7OfCPtGc3+3L9NK+QcZv/uZvXuruv0/g/xu/8Rv9kT8jnm2gE92e85zn9BFou8H3kdwu38QWwUjUCMw5XksQGro6Kh9B/ZvSXZxR69IfHjks2OmJwMgyPtklUzp7fvAHf7CPPk09GbmyaZdfAb126Al5ThkeFXQSTGt4opyj41FhpK8c/ZrdrjqTenHRRRf1Hxe5/vWv3305t67xn7ymbkzpud63jWRoE+95z3u6rKOVIV0TLsFH394WICOHptgR7j7z0YzvwV3bC42TABvd5ck9CbQy6fIc4ST8SoZREJ8qy20wkpcnH/pYyuYF7C7U0Q15kbXUPn7BK+XvXNxJ1YdtiO/oZNQ5F7HJD2DYNteTCv/Wkek2hF44CSgzMo06T0omKOM5dsZnftydD/0Wgja1BPEpnFT/Vm077v7mSKUUpbw53uX840Ac4W26N+0nAQVsJUsex/YN8jy6qaBpUPuGGye/tlHMzgqtwVx44YV9FYbVHX7tfg7YkRu1R2HTPIe1serpiA++u3TfN9IG+MiqjLlAR3fBem9TIZaeWuUx1zfstyImOuwT0dX0oinbkwC71FPydvmEblYRWVklr2mwXQOYCvTqlFU5gvOTgJV9+SDsaOWoHSRcgiPVitzp6s8G7hMcoEKbB1tScEcBmZZNqWTOhX0h9pmrczwJkOPGovzS8Y6gF38LlkFaAiavEZNRnbAL+Kq8sdGc7mH9mScePKMv/Q/D6zgRffgp89dzwB/8KahrPtH3/sc6fTc16TVsstPS2Phj36CDpzsh1/uEMrc8k39GG8kW4h/wbYt83tsYiMIS3+CnjrJv37YFbvBunkeXx86ES3Ckjj4jES9BTqLjjQOsZfXi5CTAPo3OcUlFOSz41AuZfcuKLx2VH7lpJBVpPMqXH/xSkheN1sgbtdJTB7VrBBJ58uoErW8OltgaOY5C5Dvu22e7wEZB5+zDnrmoNrlhebFsJOkbCeu6+T+848eK2G09/UmADmRaNKBOnATIU2+8IN0EefjKtw2+0TEA8ZWrmwTfTvluE+RlnwUnJ1Wv0sb2Ie9IHT1wrA9OVO59gwMUgEccdz/XccqSQlyKfCC1TxkV7FtaMQ8L/nNj0dnrMEeZ8bF4H574qIZ+5uaX6ie/+qLhWdng/KTsPAmkLvLjUaYWjUB95Yyf7RH89izwkxA5FeKs3DmJdgjKThtUlmRHp32VpdG1D6R8oLQJZPP9BRdc0PUxGPnyL//yruthEPsOS78UH/7wJT8lyIYjyVQECadwpI4+joUc9w0FaNmaqZRgX5ULwtvddp9yAvYZzSnoNJ59YMqWOvcdyEcXDczXevJ4LPbOIiPRpfCi0iPqkSv0xyD4j198cXlY8Lkvi29xi1v0ax2X+me0L2zy2b7rTIWyM1++5KXzUcEvm9phbDcCt42BDtqeM+LSN03RbQI6Axo2HraeL0XksPG4ZS7mloYfpzladbFUMTwY5Bh+le82eHzT4Qh4wBy6IHISosPUaEi6T7DzUqY2pNDTAa3zoJ7vQq28juboc74EsQPdJlrx8rAjediWJyQIfdIdvYBFZ4Rk24lteXNeIU4dQcdXh537jH3V1sMAXQ1LsY1OfL4xOQz4iK9s86xc/GCJJ0r1QojvA34A+wwdZcBVbYp/N/lYnHZY624QGiG6zYG86RNGnm5wto+whcVoP8jPN8973vP6tKC9lyyiEM+X/ILHNkSuo5A+Zg5CE/oclyAvYwH9ocE9CaewuKPn5ASgkI5iyvmbgEZAkyNUvpsg/Q/+4A96Q0rHkfi5yE2pOhO9+BoH4oxed41cQpswR5/kHfXxAVDOc9yG8BnPp0Cvqptzq2CmaFRUHbKpq2c+85n9ZZgOX150I029dp4ANf9f/uUlPzxS9ZiL0DjmfJQ1B0vyTiHyK5+c89tRNqYKb1M4WcZqLyZHYbQ1dc9+LKlLRwFe0WG0MRC/bY4+Osq3BFOyQEetnm6aEkNnXyU/xG5VnvcciV+K6Mw+gc+32RE7hchDA0vke6dgpiKyDqM7oEsIeq2oCTVxRE1jiLud+V1z9DFsDvBRIXUiKooXq9vkBsnjsT9frxkB6DQc5yJ86EBv8ukxNTpxraOzaddUGnpHnVd0GPNtQ3jUkUymRZxvq2AVkRkbcl2ReHKUnaPgxhLdXVtZYzMqL7VsuGXbYfadddZZfdTEF+ZMo+9USFpFGoNR6vh141wkf3SfkjMHaOKDpUArRHbkRzdlZ3pxG0JX6QPXeDne5S536ZtkveMd7+gbdgkjTWx43eteN3sEOoXwxQ+fqfYAdJNHGZoiqRAfv2yi3wWDuBH6CvVSZz7F043Aj7Cj5TODT/mW1C/5yfFEgC5z9OLn2MHetI+03zl0gX7I5m361PjwMIjcSr/49l+JNX7bctpVLh3FHKQSc4aVBbZp1dGOyo1AJ+jo01HYYe77v//7JyvHLnAmOi+88sv8Y8Ugg51pQCmAqqeXZXbFiw1zMPKxSuD5z39+vzZqcRx1gdA5ChDfk69z9mIVEh8aEKeTVqGMzs8///y+y6LdIj0Wu4FawnbVq161/0DDzW9+8/75OB5PeMIT+tSNn8Iz8pDPrpe3ve1tD77lW77lwA+P6JBMM0Q2udHXUbxHaHI0KtdJS3q9HiFNWWj0GmTy1GMNiatHPKzc8uKLvOha5SaMSBwdNOro65g05ZZH8CmMfKus6JK6LvjdXfGPfexje0hnF9nakWCHxMo7fBNX0yBpeMT22GF3RS8/Ae8K6fQy1WCUnThB3Xrxi1/ctwUe6UaERiAfT23JSiPnFfytfhthQ/RN0I/4mUbxdumUf9dUTRAe9DXAsWyYHHU838+M+lSEXh47gZpys0LI9TY6CC3wve0aPJWELmlzUHmNWNTRR3iOplDsnzzHoIoopEOytahGx8m7+IROIahQzu0F7uOIXZVqRCq3UZA9vLPSZYQ4L2Qyh00mhN6NxvasKleeMnb5QiVEm8qrA7WixU0H/xynELmhN09qLxc8bnnLW/bVGujpQPc0XDcSo8GMdiyNtCe8SnnxxRf3D6DYIC9ajQT/zG1W2x3FK39bMSsDP7xiy2U3B52c9xrWgbsJk02P6KIxmIrQgfmhFNeRLchHds5HuEEoN3vteDH8Ez/xE71O6HjjW7SbIE2DsirDFtX0M4oiC229AQkjpLlB2s9eo8SLL5KGRp2uiC2CJyhTMrb5Zgf73XSUmzKL3PjclrfmpwM/zBH7Ik89Mvgx6LIyik1eSuZXlbSXKUROdPa7qtbxqxs6XbR8WiEODZl8jpbPdG6m+Nzw6UK3XcALvbzk2CLYDWakNa2hDbqxyJ+AXl57HvFdBhGO0paAj/jMDAU57IvftvFKmcHLX/7yPmgkP3HbwI85CgZKbliRl/QlCK9K2zc1C9NgG3N5pTNCJ61jUWFVRAbPAR6CzsVLE/tP++WXKQUr0JBrRKPzVTHsMY1Ww9P5zEH4uNHc4Q536A3WF54qElQ70pASl6M4jdNoWMepw/S1p/Tov8kOUIE0bHT8oPEaLWsg7KtfG1c+5NJJx2IPbU9U8tuP2+fzRtjy63z82o48OhKy+Ad9+Lm5GbWYZsgKCj6wd7nffuVTnSjefmxDOp+97W1vW3duRtV466hNoaXha5hsjDyy7D8ueBJw7ckFL7Ty2hzNE4S5ZqtW6Fv9CfgJ9OC3iy66qK8IcsOzL7rP3e0Dzq74z+jUj4tYhRF4TDYC1qhsyOapTTnq+PExklNP6DDlf4/nyt+SRjc7PuBDv4twrWtdq+exiRZ6fEKLl5uKFUw6NHvSs11d1llKd6P0leud73zn/l5Emie1rMKR3w+2KF/TivyOt/qncyJTWfGBF+fqVWyJTwCN/G4OpujcsPHw4VV+3yHLJ0c6/NwYIoc/PG140rL0Vh1Sl8byq8BDiE/J11GiH983RKYjO3INnh6sUnJNf+WZBSKj7lMIH0+5+gMDIr9J4MlUh40nPpvsoHuOPtDyhOujQthGV0EH5Uw+v3kBnwHWXMSOijV9S+wb6NSwDUlvRq3OOeec1Zlnntm3q3U9F3jIb2Ms+9G3yto3ERK2IXStofRNuNporsu3F/cS+fK2RrV6wQte0Olbo++yxY98IlN+5zWfjbLueMc79m1Jk88xYRPwYPONbnSjVbtJrfe/JsORfZVXhXR7wF/xildcfdqnfVrfQ7tVyL6J04/92I/1LXtbBembObVC7sfW6FYPfvCDV+2Rum+W9da3vrVvDGdTKrTsV4ats+z8hQsvvHDVKlzfJ90GXXQTYrs8dGujuF4WrbPuWy+/5jWv6fvc2zdcnDK2ha093umlykU31/ZUl9/GaLYcbp3tqt3k+sZn8UeF68iPLmy3p397oujb/SqTdvNb+689jveyqv50DB9229itNa5V67R6mbRR2VpGBTrxCeFhc6/WWa7aoKHb8LznPa/HhwZGGvF0J5+9rbPs+/Tbftqe/PyXvI7iBH5rHfGq3YS7n21z3DrYvvGX8lUmoRHISahonXzfXrp11it7tttLH21kxv9CRXjRGQ91ud3cVm2Q0etAZDuOMiuiHzlsaTf+VbtxTtLJw1fVNnnE3/CGN+w+UfYPetCDLqX3NvmBPELr4HubTHvULtRt/LbxkUY3mxGq520QdCkdNyFyBfnxaDf2vlXyNrpNqPwSgt7RHxb2b9bRtBFjd8ZcUEB+nYD95FXOKeWmUCuH/dztKb90Bz0yONY+3DpHvDYVing7yqnAyeOI/nu/93tXbaTV6V3PBXo3JzdKnVCtFOHtWM9rnAr14z/+412nNqJePeUpT1ld9rKX7ZW9jXb6rpJt5Nx/kIJ+GgfelX8NGqpOItcar84WLz9soiNSCdFWhF8N6OUVwk/gI51xG8n2m3sbJXX+6fDbo3L/ERs3Cp0u+viVHMfIcB35SZOXne973/v6jUwDjX7p6EMzQhx6dn7wgx/sP8ChsdNhKn9FZNBBfr5TPo7ik2cbpIePEP/RybWjdDYJykYZtyeadR7ytaeU9RyQ0UbD3V/aYORHn22ITPINPNS1dJBzEZu1X7/t4Id4DDim/M4u9cL+8tFTsHd7e3rsv4tgL383q6rDHF/EFr9v4cdbyHftKG0XD+nyutHrk9qTTqdLuc2BfH5YpT2B9d1y59JVoBlDcKSOPs6gWHucPBW7GykktBrXJuVGSEOnc9MxkcmZ9BA/F/KjQ49um1x5dZZGWvK6TnwCXksQefFBlS3OiF5c+OecnOitwfs1pMtd7nJ9hJxO3mj2jW9849ovjnjGTnAUYo8GlJGS/C972cs6L52vzjl+EuYg/Guo+is7jeHZz352r9hGlORpsI5+7cjIzA2If4TQj/yrXfLk2nnSd3X08oQ2/spR2AU8IwudOm10HtopmZsQXpUmesQXnraUuV9Xo3d8Y5tix7nyYmN4zKWTD51O2Q0NH3WE3UsQPrELH3o4H3VxI/JLaAaVleZrvuZr+k3vta997boOSw/m2BQ9hPgEb7YJeG4D+vg99MIc2YG8bpRuZOGxFHiMIThSRx+DjGAotwSpXNUpwjYDpaPR+eSuWXnMRfgkbAO+pmYyPZUQudF7F5+K2Bk+FeLYl/QEFU9DMqXgl3hMq+jcP+VTPmV197vfvY/w3QDxk8+RHOUylo14wFc+j+AaqnPHM844o3e49kcXJx8ezrchvkioCA/BjcxUQ641JiMxUzcarkdwtrHRTx76JTM/z6YMYtdUiK9ynuOcEX2liYxcz0VolNU73vGOfp74bZC+LU/KwFHQoV3nOtfpnb0y4kMyjfCV31yd8awh2KVP8pv6M71ENr3o4TgXeJDjGBtrqGCXkb+2SI5rPw+pnphqS/1FN9f+CjT4RrZzsvKUsssfsT++iB6OcyCvwVudOVgKNGMIjtzRYxbD5qIqEh4J25D8Oiajwkq3i7YCj8it8qeQwq+FN9I5X4KRXgic17lZ8gVzmNe4xjV64xY+4zM+o0/ZmEeUd4pf9MpxhHxkRI4O5KJTP9Zw85vfvMeFX/jPxZi38tCx16mVxCfQx3y5zswNx/SOuf0rX/nKl+rY4pvQ4JNjeDrO6eiTP5iKmwP56eBGhj5x2yB9Wx5peNXgp/eMZD11pTPiC8dd8oLIjZ5B4jchNMpQWeZ6G80mxJ6ETbKlsU1Q9qabPAHygSfY0Ml3WESHnHuKyJTWLttCm7zRZRcdJH/sW1KGFeFTQ3Ckjh4jxuTl4UmAHBX6JOV5YZn3CPsGGTpcNhqlmL7Q6ep8jV5UbC9W6SOdfmOhzkXKz0iJTCNmL5PI8A4hle44EFm1Mkf3EdLE08l8+d3udrf+oi7TU25yRvmm09LhV75ow9fxpH5hKnJjm7AveGrjB/7wg9j84P1Ebnz7BP61LPctD9R1A6AsPPDyV33w4/p0gOPUI+V3EjbiLShTbfGo8sKv8ti+7mgGGo/+uXdT7lTM/kAWWNLXOoF+vm+QafliK4BTMfsDWUIr8L5M0Ucgvoq0hK919Ac3utGN+nJEH3hZ1tYa+aLlV5vAn8rPb5VaiujjKcv7gIzjBDvYlvKjf8o1kKc1sr5E0/4tF154Yf8RjnPPPbcv97M00lK+r/zKr+zLU9uo8hTlP/BrlfxUzMmDbEs39wX28VG72fWjbxgsdVRfTqIdBvyuLFNv9wnl2gaU3a++PbCkVd30ncZxtIEp/NVf/VW3cV/8R6jXx1FvUh61TI7U0WPECRrlSYJDdIYnBR9gaFD7rszAlzo1H6z4ZRzrknW61hgL17zmNdf+Ps4KyDbfM7DT+m1H/I/T5ujbRme9AeE9xV9HGX8Lzm1Q5YtJe65b3+/Hbqxztl7ZunOdnrXc4XdSjbMiMunwe7/3e/18n1BHfG8BvmlxY3QjPAmwVUeoLKfK8LihzpOpc/eBo3PfPPhGYB9gkzp6UgNKiB/T9g4LtAnBZR7p64RDgvMppYL5AKYydj5WANeJG88rxngNv/I2avFRQSp1zb+JVxC9Eu8a/1zX/Dlnm9AeFde04RN6qLQj3yDXjpXOuZuXjzVsJ+BDJx3YE5/4xP7zcj4mAnz5fG5lGOWDuKqHpwNf5trqwBetz372s3uZxsbkG5G4mq+ixtVztqJRhhUjDzYGaJS3m64tGHx05aOcj3zkI72D9zWiryo/+MEPdj6f93mf1+nVUSNBXzueccYZa16jzs5H+UH1c/JsOwp///d/37fqmJJTsSnN+Zg3SDx/+ADN9htugL4SVk+DbfRJi36OtT5W2pzXONAO1RPtAsb0YKR3rHmrj2p8IE4e9cUXv095ylP6R106fPUBpG/CFE8QX+WO+fjDjSX1dMxTaXchdQ4qXbWdLL70VW7SDgN0CWs0hodGU77PJ5kzc2yNan0UpE+dJ0/iwivXjvLUY2iFOieb/EnLOeQ685a5Rgc1LvE5r2nVvpo/54nPdeUzpgnR3zG83/3ud/df/zfv6KMdK1C8cE4+CJ+qS0JkjWmJj+wAz8i3frd1in2Vi9U7eIRPPYfwqzwTEl/TQiOMurmudEmvcaHNNZ296MwKo3e96119San3FpZltsbSX95+5md+5upHfuRHen6rUR772Md2Pq4jt/Kdist14sa0eg71uo0GL0WXPDkf5dXrMUivSFxs8aGYeuOblvCK3BzHIE+VmXM8K+0Un5xHfk1L+lRcPQqhrXFT10CWpZXs9KGgF9Di1N/kTX7H2DPFOzyn8gqu8RYXf1Q7c0yeSluvHdGFfwLU69AK7Anf48SxTN3YuyPXGYU5b8quz5MX6mjUsRnWz5uh/YhOfOWX/NL+5E/+pI/kkgedc0G6uObcnh/cKaNLeDsmD/mJxwt9zf9bv/Vb671Yqn0QHjW/NCH2VIinD7rIs0/H9a53vb4vic/O7R/07d/+7Ws/Vf3Ri4McIeloohvkvOqV8+T12E/G1a9+9YNb3epWPX94y1NtRhv6nLOlxgfO408hfP7yL/+yvxeQP5BOZqUZ+QEetmywRYH01jD65/K2kLDHyznnnNO3Ivjwhz98cK973at/ym7/EMCXjMgJb3ziiypTftc5QvySPLmWJzCdoU1Exsi3HhPwiQzHyI1vRx87ph7l3YUnv7/4i79Y65K86ndoElf9IA5GuhwFqEdpytHURvKhzzFIWvwEycPnVQ+2gOvQgXhPu3ZRFXePe9yj/84x4IGX/AmRH/ocq6yaljhAH7340vRN7BqDPJBrqHonz1i32JNrQR5Bn+bpU9xx40hTN1HSpmKU94jjUc4cs/1HOMg+GObybLyl8isw8606TnFecnrcxEtHZ/8PFUgeR7zEeaTB32O6PU408Mte9rL9aD7UC1O6kEfur/7qr/a9WDLlotDEmd+Xh/PRCHjIp4HSyWOwghCHtxeUdEnlEkcvldzjubz2C0GHL/4qCF46HHR0Q2dqgS7o4IEPfGB/FGW/HSP9mhB+OkKdGd1DZ07a/jDs4yvy6CCfxhwfpxz41pQF3dlCN3zYrJMwZYOXzpFf/b6mqQB+w1+lCx0ae4HwEd2VI17Khr50UKb8EL+T4eWSGyX9+IWd9ERHR52yCs7HdMebzfjznfyuHZWdkIahXv38z/9894mbFFp7w5jSkZ9/1EP79dgXxh429kDBg57xFb341qIC70TwYBN6NqrbZPA1v8iDhv5sFoeeLWxmH7+bVvAYzvbwxlcc2XRTtvjQg6/ki4/Vy8iTh0/ZzS/y4SGOXHvheEFJL4MGcvDme/zE8zN+4vlU+ZMpjh3qaS0femuHykKb88IXH3apG8pMXY6/0GTfoegfP6e+0R2dMqA7OvLRVR2Ul3qEjn3eH9mvSF00tUgvdZYf1AP7FCmn+CW+l6beisvUH3+K00bZA+SJx4Ne7KIn/5DJHvbxJz+45h96p7/QBrQ5dYZurtGqG/zAH+SRQb48Kfv0kXzn/RM6ZX1cOHRHrzAYR2mFZpTFQSqQeA5lFGfLS3HnCk2aPCl8gcEgj3P5Kp2GJl4ch3GiwsaHw/EURz4H0Yv8OBsc6UheOo9UCEfp4shGJw50uuQrAJWFTEeFjTY6RJ6CA7zIkwfwT8VVsWyIpvIaxRvV62hVWBVQhUdLDtAXf7R4kImv6+ggxOfxZ+SJTwfJrvC+6KKL+gZbd7rTnfruk3THJ34Zy5TvxLMNXOOfssETjbiUhTg08YvKLI6tiYt9KS8N5qd+6qf6aFwH5p2Fo4anY1EeGgs+VgmFTt3A0+ZSfshcXh2Ghm6nTB2WfOb58aAX2eqLI93xondsZgs7xaHFHx3fBmjQisPLTcG2z/KhAT6SBy++Eo83WnZUH/OFdHG5lgbRA2+0KSs7T9px8o53vGOvm7Ws0QO6+Dl1pJYZ0A0//uETwEseIbqq83SLLHaHV7WRv/CWJh8fiZNPusFGbEEnjzjX4j1xPvShD+10flDEIgV24e8oH37hLYhL/YPwF4cuZSFefnF0zDWop679kIn8eKGXD63zWkfiFzdWviMP2JI6I2/44ymP69DxB17JE8SOQ6MJOBSa8v1jnVYIfY63FXKfV3KUJrhOXM4TWiXpa2Pf8pa3rH74h3+4r+EWH7qRT87Rtbvfeg25deb2u2l3zX7dGnqnrfRC1SHnju3OvXrWs57V9/zIvGoN+LXRS5cXvklzTp92V+/zwa1z6tdC0uSLfLzwsS9IK8gebn3rW/c18ezgS/rQY9TdMXJrWq7Jap1LDyO9EB50yJFdn/3Zn92/rm2dYaetNKMsITYI1T5HMnJMftfJIx5Na0D96182i8crelXfhV/lJY6ebXTV91hhK35tVLXmUXXAp43M+j4qX/ZlX9a/EeD31jn1vU3ak9k6X6WrcqdCG332jeHaCLXTJj76qc/0wSdBes7VYXTyhF58bHaMLpV3jav5+NCmYuyzl1D4yhPanNdQ452jEfDj18QLzisPMhKfNLS+crZnkXpd6cYgrV7jp07YGkObQmfTNja1TrWXF79Fn5GPozrtXUXqcnyQIK7mdx39XLfBQ5efcqVTaJI/9KELj9j+xCc+cf0lvbzRIXThM/Ig66d/+qdX7eml53EN0oWjoA812vFSYS486tgutSm0pnXnEdyZgtyNkqcp3ePQXXjhhf2xJnk2HQN30dztmjP6/tG29AR5EyqiVz1vju3XtjT1UNMqdr+OjgKQNY4O6A/yeAw2yrBtL57sjg7RM7ykPeYxj+l7x4Ptlf3GpdUu9mc3oofICeLL6EV+dOADj9Hve9/7DtoNq29z6ocfKqJPkBGufcM9Pt7whjfsoxH6twZyKtc/6A10iG2mJWwvbT2/qRO8jH6k0St0kenaeejJ8SRhGad5ZaNscfLFVvlCH8Rm00iWWpqbvu51r7t+v8EXeER++OFtVGZljqWZnpY8hdLfOn1PVZ6i5AtdZFc9arwpAuVoiZ/yUwZARyO1Wk+jT2jlMRWhrExDeLeAvnUI63R5468E1wJ+8ni6UX7K07SM9zpsYI/6IF9sgqq/8xrwM8VnasIyXr8lcNZZZ63po4u8gbTQOmrPaNUNq2OUB4ROQCN/EHrx/GWfft8FGDGLV0by+GbCUtq0Q/oEzuXVhv2IkTZt5Ow6aZFTaR2jlzSgf7tJdV3kFU8354kTILSQvHyv/WlPQeQlj+vwyBHE+2ZGnyievPA/KianbqrwEQRLd/SYqKPV2CzLo1icB6MxjjEazHVp6Cq8PcR30QrSda5xuiVlfkTDPuSRn/w5dxz5gIrow5srXvGKfU2yygXhHWRuVeUJFJh5tcc//vHdDyqhKZhqH8QeRx2rBo7Wo6hGrmJYQmkP80c96lF9ekjj9fgGlZ9zlc5Rx2SfefT45KMZ89A6bkszK23sMaeog7SPvakMnYtHb52tSu6DLJ2EztDUGB5C7EiDVAby+lUsH+q41rDI5acpf4NzurNPXh0+G0zDqEPk0ikyY0P4iFMebkjm2/0Ognc1eby2dziETn6dqPlPnboP0JS1R+l0sPZAd05vdSl00TuylVuO5Kg3Ony+/9mf/dlOYxksPxi80ImvgvADstQfvjM1oSwz32xpqLwp69A5JnhH5AU0/3l5r2PV4Wcu2HszN2Ado6WoEF70rPYpO9Nkliv6AMmNS9n4bQDvjWJDZOfclCY+OjVlpj6h94Mw3vuYvojMhLStMeBjrts0ohuWcrBHvkGUqSZ1Ey91xxRIyhfQasvaobakfntBz3b5kpecHNFA9APtRxvUH/ktAe8T+JL86Alo0FdacO7mry/TH8nD3iq/5pde49ysvaNTNy0kSFrNMwuXmHYJTpEd6YdHOEUD4ViNqN7FtgEtPiqGiso5ftwicjfJRyMYyZjL00H8+I//eJfvhxJqo9oGPDQiNymN1I+n5IcKpKVgQD7zwzoTIbqxQaU2h3zeeef1Dzdivzz4ZISpE/MrVH7kws3AqEXl0UGruDoe88wKGDS8NE68qj/og86d3xy3eWCdO/luVBqX/GMlpC/oSHQGniy8pNS5GuXSR+fjJRQ6P8bipqOxVJAvnW1GgBo7+7xz0Jl68aST4FP+GHVwjUYDor8XcOz1wybvete7esftF474I6jlETvokXMyPVlq6Oi94OY/9YH/6aUz98MjrumAlv6egjQuOoj3C1m+KXAcba8NFj1enmTcPL0M1dnTxaoQnbWP21Knqh9co+dD/uYPT0ZG01746bD9kIZ3DOir/YBWOXpZyA/46ZTcNNRR5SkPP9AlgyiIPgK/GKi4aaKTzw/XKHc34syXhy4IvbJTZ9zwPVXr7P1ql4HG2KFPAQ8+UBeMwpWPtnTmmWf2usB+aUbY6jg/u/YBYdWHD/jRj+V4+nazyMgfRv+NoAce6Axc/JCSOuypT5p6isc2O9Drl3TwBp9Wr4nfRhfIB27M+lFlZuA4h3YSEx09Ies5oIRtSHozrM8l+fEP85ytwHr8HOCB3gb79oRvlXY9f7VLvnytQvU5uAc84AF9k6vWUDq/JTD/2B5m+mZhaIUp+eLN25mTJVsQR6bd88w1txHcpewPH3rao7oVWJ8Xbo1/vSWw+Xhr5Vvhruc6HfFG5yhu1Eda65T67p34oBEXvUIz0iU989mtgaoOfT/+NkLuvJInfPAd+YA46ckvHz3aTWv94xniK210GnVNoBcftxt/3/mx5qsInwR5Iptd9jU3Jx86ebLXzUir3Mhlux8K8QMtyql1En1DtXZj7nniB7Icw9sxdshjXtZa79e97nXr9yRQZQahFRf+ZKFTtpkjFkYkb+jjgzbq7fa0gUPf8Ev7QC9fZIReXHym/aHDA6/YE/6hqwhPwZz0hRdeuN6nRZAe+m1A712HLbe9a/OOyrbnbeDRy0G50Yu9v/M7v9PbG/4V5NjV1XuY9mS6lh0bdiF5W+fa3xeSF1pH8rbZIY1+dKCv912RO0d+fOz9xNlnn73+oZ9dvluCI43oW0Xp5+7qjnNH1GjBqMSdFz1E7ib50bM5r5+3BtpHjR7Zl9796ICezng4isOj8iFLwF8IyG+No496old0wEfwKOgObSRiFOLpwwhaHjRC5RmgjS9HfdBET3rxX+Q7hvdYFmhAmikkT0GWUnoiC4/IiUxxU/qJh/CsdIkLXdJq3tYoun5CeDlGh4TIDw9InuRPgMQ7hs51HdEH9KnlKqgPRlJPfepT13yt3TYd4Gmr6uI8ecThV+ME9iUuCH3yRMcxT/hBjkH1ZeuE1jwC7zDUNyvhPG1Jjy5VnuCaH1IW4ZO80cN5kHzowg/GfDmvcRXo8Bf4no89ZZk28XTjKdtTWvqX0DivPhHXOsYe8JCefAnbIK+gPyJLe4jd1f5tfJQDf9AhT0JzZAfkk6VPpUOdHTgWNAHrO0rCXOROZ/XNErrDgozmzNUHPvCB9SqbfYON7vBk7fKPu7CAxsjWrzwZWfmpM6PcjJg2IfYZ9TvGv8cBvOhGBz+X1hp2H8WSY0QrPTL3AXZEhpGfJ5nYd1w2TgHvjOhzvQnR0Uqw/Dxda+T9F538epMRcOpCbAk/x1zzpy92XZ8EyFG2RuXkG13a8ZPu7Qa3rpfHpU/sNHL15HFY4MGPqQd8+53f+Z3d75/6qZ/aV9zUPMJx2rEJZJKjjrKR3H2CPDYpN09X+7Dxo4drC+Bu05TsqzCacqdi94embz9mBH8SYJ8PrcydRv4ueO9glYCXa+ZaL7744j7nl7vzNj7SMsI2yjou4Cd432BO1Qt0P6zOPnODXgRJP7YRxARSZo7CPmUdFvTiG+9TzLl7KcY/VqF8yZd8SX8vY40/XwXxm5DzPKVCzbtP2OvHUyZYleS9i5f++yrblONhEZ0cW8fWXwZ7mvJ+xQ9+e+JMv9JuAv0FtLZ4Ekh5xr6TKEP1io1HLydPfAmXoFsRoxLmQiexlOYoiCzTH1aKnARim4LeVdgqhRfElkq6Mfg1eC94xPOV4y5fSc+XcWkIx4HobhVDG6EcPOlJT+o3lPB383R+XPI2AX8dkNU1aUQfK6j2Ky9f2+p8vGj2YtDLNv4zLWL1lBeA8gm1frBLJ3VSiN4ZGJBv2uMWt7hFn7qx4ED6rvo7F+HlxX9Wqh0G8TWYHvPyGD+d/NWudrVL1Y/krTT7BDnqqNVV9DgpuZF17PJagR0aHi8EL0n2/XgDrUH14EMMjznH/XgzBTLyiOqc/Ir4gP2mlC572cv26Zo2Wu6P0qEJnespJI/AvvA9LtDPyyJTNle96lW7buAx0VSKaYnjllnBLjoIpjaEyItv9gG8507dBNFJcM5H6tu73/3uVXtS6z5sDbJ/aPbKV75ybVcNHsHDx3GfIIOO+eERgQ4vfOEL+wZvrcNa+zv253hYoFdnyFoCOgj0w4NfTYvx6Sd90if1BRpJi76CaZ0//dM/PZEp28imW23D+0DsI8PLfAsjMj14eKBNuARHGlLlzmNJ23FOM2xDc0p/lLb06aTgrm652XinpQu0Qulruy2psuTMOmJLKCtCV0cpQSuHHuRxzMc3uT4O0NWPdND19re/fR/BOwe2mWqIHvtCRisew5Vf9eXHEuqoKkH9tlzSlM7DHvawvuTOi0NTcz788sKXP/m5dX7rD5bCb9+go6ml6Eq2pX7WnKtPnixTX4+K1BPLQvlgKaIHHpaEPuABD+g+8gRy3/vet9cPabUc2OQJgt/3DfLIN5XCRtcnAS+SPUXAccs8Ug3kDMgj7EmAA7yRPonGA2w076nyjZCmcasMOnmrazzuW1NseuIwb87xE467oH05mi/+NKjIENrIpZcfn+7zhk1W6kxw3HbuA/GTDtyKCuvUzRU/+MEP7tNf/Krc7V3kRg/y8Sd7R5uPG+Fv/T0924i4H9VB7xjo6NuL49aDjMPUF35R/6wwSSfv6+YLL7yw6yqM7Zvu6mneQewb1baT6tv4pD15revNceJIvSVncIKXjirXSYBMG1UdZW5wCcgzz6kBj86X5kMXL+98vGJe1s/xySctYU6hJZ9CZt9xw0s5ZeXLUB9XVXmWgbqRqWhzdD0swlsn6B0LffYp7zjBX1mGy2e+Gvahn+V/OivvVXRUfGse39fNbgxzy/+oIMNP7GmH9CPXkS6eSC2YsE3HcSD2KENluQRolTtd+MrXsObmvdvKElZBXaxw7R2Jl5UnBfYJfHkS8NFbPtw7Wp2hb8Il6B09pjXMRfJm9DIXaeBCzh1hm/w4wJ09NDUcFpXHJvlGGdJVOI1Jx2jbAF/kWvNr7a/H0PyyEV1r2IakJ290OIpNeOho+EpnpBPA2/SNzsq5Y2RE5ib7jwvkphHH3uMAvXfpXm2sYRvoV8vTOb+pD46+6PZiWx145jOf2f359Kc//eCWt7xl/xUk2xuQYSSaG6k8Qs536TAH9PJVqGMC/XzxasWQa3uwkEePTI8cBvGFdjDqP54nPU+NghVBvn7lM99z5JexTCdC/AxohfiO36VvQ2wcdduF5B+Pc+QlrxDZiduG8M8xdu+SuR1oE06hCVi/EEiYi1bQ65cIzueCDHQ5R5tj4jdBPuvMrVOXV6j8lgKtF0qVVwWd2OelTOS0RtJfbt3sZjfrL+XayKS/sMVHkG/ksw01P3n50ngpn4roiY8Xka2BrG584xv3OCF85WNLa4iL5S3NK5DnpZqXTrlewgfCK3Tb6KVNfRmbtKMAffTPS7sPfvCDq6/7uq/rXx7zufpx29vetsfzsfqhTFK+h7F/BJ7K1Be1Iz/xXmD60rQ9hfSyTn0/rNzobZ25RQqVj/Ma5GMr/whPecpT+ovX1pGtbnOb2/SFB1Wn6BX/gDjp8mXRwDZU2eExB2SGhjx+08/s4pH0yIy+ud6GpMdH/BHbd8ldgiNP3TQei9e1o4HmjH73q3e0XWjGr78cS/7mmEPfAcmlO77Ro8K1JxYjCefR12Zs9puxjM0+4F6iVF5LgTayjcTIOKxNEFvoax0/GNmJD39wnDNKqsD3MDYGfKT86HZYX6GrR2EJ0Kk3h9GBLHTonavH2oBza8Dt4eQ9jc2xxNlTyI+h+C1gP0ohjg+OUm9H4Jf9zyvEe4FpPyUvwO3jAvSnx2FAZ0E9EkbwZ3zq2G42Pb92kv2G7nOf+6x/0JyO0sWH1nlC5Kkzee+1DdW2HOeADEAfmRA9NkG+aq9yZdMS2aAd6mvwiA1LeXQgSQgao/VdSGhCRG1E0h2NdN/0pjet3vnOd/a70FxElruer9/s0eFOjcc2+dIEd3YjFTyMKl7xilf0UWLSdyF2CkZZ9jW39BDfkZ5ORoRZ1mXZ3EMe8pA+UvuSL/mSvveGL4Nf/epX9zzyh/8mJK3qQfb73//+zoMMcVN8kj+h8qjp4gX7hnzyJ39y/9KTbnwePwvO+TBPZYlPGCEOD08w7CYj8Qnb9Jam7IzoLQfMl4BJTwjNCHF4pO4YxTqPzKQHrqdG9Gy1t4inJ/pA6KRvQujVmw996EPrpcVC9BKyBNgX0Q996EP7slsjWSN9I1m/zavu8HvVPTJyzPmIpAnoyVcH1SPX4Rm96PFpn/Zpqytc4QrrfXSq3PAZryOrIml5Qqn5BfvV2HNGm+Rbexfd5CY36W3mcpe73PprXfShSeC3n/qpn+pP7XQPX3bxdeIrIl+Qz35J5IZ+DiLHCL7dDLuP6Bcdt/GJbEF9eu5zn3upkfkmhAbI4Lc3vvGN/fcx8oRY88yG7AmncOiOXl4V9e53v3vfCIhD5iJybAx27rnnrr7pm76pn+8yShpam19pICqLTc2ucY1r9EY3R/8AbSqhdcbW8sa5FfL90i/9Uq9k8t/znvfsDbaNRPpn7grHZkrWpquEteJvAhlpkPLTHQ+NwIZYeEqbskfloUfSneOBHzoN2o8fxD4dfBsFrd773vf2MhLnxvbWt76150enHLMmWjoZwpR81/jYfEmH5Satw0/+2O+8QlzykKUxffVXf/XqQQ96UNeX3DSsTbIhvPFQ9+jQRs/dDrKVoSMeyZ+OPjyjSxt5r65+9at3n6MTN1UHKqTRkQwb06F/zWte0+ujeOl4uJFB9JX2/Oc/f3WVq1xlvQ5fp2+woONzwwtfegjRdYQ4edsIuctWf/jjGc94Rj+mTjiSjZfrbIvwhCc8YS0HL+fJjy9btEeDn0wHVYSnco/N/C/OtyRnnnlm3/7Dhn500lbI/ZzP+Zx+cyYLTXxTddAOv+ALvqAPAqQlnU9/9Vd/tddj1xXoUn9t0nfta1+7509dngNy5H/BC17Q24yOXpkI0WEbyCbze77ne1ZXvOIVuz5zZeMtKEd1mp/i16QdFev96OujSo6bIL3R9p92s7TM0kIvIT16zAH65oS+TtUnz+itUc5j4Db55FrRIK/HONus2l7Az6d5DIRd+jfH9SMd/ISfxyV7YOdlVqUnr3WIfQ2vlQIePdlpH3v7dZPPB14w2UqVTvHPJj2S1gqz7wtv727nfsQCT+ue8zOJ8lU+dLaMk+9tkWw7A1v0ehFoszKPw/xjuZrVUNZ82zf94Q9/ePdZa5AHrTL3R3j5ybdyyGO9zbCk8wdftorWaap8vhNnNYJvB0xT2CucT6w8sW5bntGP4JpNrSPp/uQvqy0EPraXO/paD0Ye6HO0Vzr7TU21Rta/mMYXqnwrpqyK4pPEsc3PCbq2eobP0FrxNKV7wLaUy+d//uf3Osz3lltqA1a4SLMVtBehFb6oNYVin3rbK7ROpNMrA6tiyEVv9QkfkFX1yLl45eTlqukQW+ta7WXFhhVUvspuI8rOT1mGV+t8+tbcVpFZKaQc1CMv620LjIevaC3FxFMcm4SUCdCDDywrdlT29FEPbF1uqsjPUr72ta/t7YvsM844o3+DYFttdTj84kvlwWe2t7aRnNVh+Fb78eGbfNMShIfFEbqze9/73v33HcRtK8sKPNQTP3RCtsCH2qU6iccmPnRPui98+ctPcyZuEx2Qm2O7WfT2ZFksGyrtNh6z0ASs7xoJ25B0dxsvVmyQ5RFt7t0LIsdIqjm0f3GI3xzZ7nRebJHpMceXikZ06OcCH/ravKrdYNbyEyrIk9ej2JWvfOU+GrvPfe7T48i92tWu1h/D3dFjV8IIcfijNWLxk29GPj/0Qz/Un1KkkWdKIXlH2IL3q77qq/pGW1e60pX6C2FbvPKlkattZ/kGH3oazbebWNcvso02jDhtD8sHRrv82Dra1Rd+4Reu7nKXu3Q+8o524CEOP4GuHjX5xk/18Qneo+7hg6fRkukieRxf//rXdz/e8pa37E8X1Zcj0McO+fjNyNMo6na3u12f/opugMfUiD48jF7f/OY3d90vf/nL9xHnqHtFeEQPo7j3vOc9/QXs537u5/YRoVFgpjMjR4hsupn6skW3clSnWkPuod0E+hetygitcozMCnF0N9WgDNigHqkPfGGKxpNu/EG+aUH1gbx2k+j1Qj5l7mcI73//+3cefJAFD+wjq8K1IB1PthrFt068bxPtCUVdIsu0oe245adDnjIc4wt2eKqzTXS7MayfduOv5EvAp8I1n6u37WbefYa/vKPfNoE8T0fKw1MJWvYpp+iyCdGPr9qgs2/ZjEac4zaglYcN6D3las/iIneuDdvQO/oliFBGtLv26oEPfGB/hHM9FzHOjcJ8d+ZZdxkUOgVp2sG+29/xHd/R92cXPxfhc955560fdzlafNIE5+I9Ut30pjftDYTN6SjZ7rdmFVClTxghjp0qpUZmp75MqUjDQxgreoU0nb3Gze90IT+VKjxMN7WRSO887JEO0nPEN/k9pgsqNd50qjpUhC66katRmZv02Kyz0NA30ZGHRqjn7Sml72uuTGzJUOVUhEfSQ29OuD219Jtb/AmOU3P0Aj4C/Q0eTCfpsJSP9NBX1GvyE9xcdRSmR/gvc7TJF5k5l+acrubW2whw1Z4m+hRHG8n2ARR7TLEpi9DWUPXHx7QdudqGm7gOWB2peU2V4W+q0U1Ap+5miU6HG39GRtU/x5yTa0DgJqeDvuCCC1btiXQ9NXWDG9ygT9+EX2jDMzzoYCpQ23LjSPnVgIe6rmzQVUi3h/utbnWr3h7ZWWk3oeahh/372xP7uj8gU7zjLj7yuNne4x736IOV2LeLTpCPPP3M137t1/b2M9eGuThURy8wTCHrHMxVu14CPHRWOmm0ccwuyJOKqQFnJDyHNuBEIQ0DbZzpKE2cCqfRmPNTeW9xi1v0gpfumKVluypCBb6hd6y0ObLJeUJFjZ9KS8W84x3v2Dv5vITdBp2cBiQfHsm/y6dVh9CRTYeRNnnl0ZB0KvF79Ql650JoKhIvb/Kjd3TD0ymGB8ibjj7XOdYQHurDmLYNNY+jOiPUwccUjxoXm9C96lWvWu+nk2CETn/tLU9a8sbO+NGghQ+ci0t8gmsdqT16dMTpFMOn5hUqwsuRbEdtg7880XkyNRdPX08HOt6qW2hHkK0+qPN0r/krMsDxJDymoXFD01HiN6ZvQnQS6KqDNshQr1xrE7VdbEJ4oK06JH4TwjN5DR7dLFIHjxOHXl5pzsjcuLna5og+fzYXTW6nsUdO/V1N82nbgE4wh906ij6na14dL/rMReSZjzXnh7bSOyeHPj539+Mczs3Dh9b8YZazOZd/G5KOT+wkx3lkO7YC7/Pl/LmL5xTQtRtQn4sF8527+MReQF+vt0E+IWUf2/hI/IjEtQbd5+QrLZrQi3dMWoW01hC7n7xb8E7Cz9CpA8rBuwV06OcgOkWm+Vi2L/V9aOjQGnp/37ENkYvGOXvQ2p7C77eaf7djpnc+lmSa+7Z0U/BxnvcsbI5c9LHBMfxzBOfaq7158DVnHlS6hArX0VVZkeeHdHww6GcyvQMBX2B7/+CnIOUJ0I1lgp84Zeb9CJ1GucFUXajwwZUlzqOMbSArdoGl0viEhzqqn0m+TZBGP/2BH3upWKqP/tDy8V32LsV8LRpS0CkgQeXMbyrOBR5p2Ok4wzdOn0Ic6gWTSiEv+jkdbYW8eAmQIyTNUYX1YooMLzXtZ5KbSgpiboGEZ4LrUW6O8e0UKk09hzQsDRgfL0et5R4RHRK8AORPCM+k7YI8oak+GWldJ50/BdejvOpPaSMfoGt7kuxloVP1JapNxJyD+lD5TCF8HSMnNK75v+o3F/iQH39uQrWLDP6IXIMP5fa85z2vfz3qBaMXu3gKfkj87LPP7p3CzW52s4OnPOUpPZ/f+cV3V7DlAF95ic3GqTyCtIRce0Fr8OO3j62F9yIXvDw0IKKrvGnb8V/4jJA39keG4widn4GZweEUIhPmllnkoEMTHtGDTuG5C+iF2A1TdmyDGx478Zgrdy76TwmeOl+MFJxjDIU4rZ7XI9T0Su98m5HSdWhxJoRnpc8xcqoOEHmhSRze6KyGuOENb9hXQ1hp4yfmxoIPLwjv8HOsSDokX0WuyU86WZUOKq3zUR+y7ahoBYQdC21qlTR04YcuducGIS0NLvlyXpG0YEyH0FVfhCbHqjuMfMIjxwBPIy2NwqjLCiIjSKt4xnqBVhla2WK0KS46Vd5Tx/hoKj1wnbyOgIZPdcpj3vAA56MuNY9jrQ92yAQrfDyx6XR18EC+FS32WzLw8sM3Vn946jZStVKl+tsPsFvtpJ6o59LIAMePfOQj/eZpZZXVLM516q7BzcjPZFol5gdsrBJDl1DtCGo81DxjGox5+QpiR80bVBrHsfzAeZA0cEz9Rxea0I1IvLwp/8ov547JkyOMdK6VNx1cJ99cRG4FPv3YEj86dSZCamna1COLdCEGM8AoDNIgY1g1eqTLiF0ecX/yJ3/SK1o2NhMnTZ7qRPHCiDFOXhBPH8FWw5ZT2qHQ9AdZWToX+ugYush2HZ7inOcorerpHPglPrFsLcsrIXlHX0HyBL//+7/fG7algxomPyV/9HVMcG10zLY67REbHIWg6hAkHT+0lS780ClHdv7N3/xNz5/yQxdEpxyn4gR8LCXU4f3t3/7twVlnndX5hQbkJ/tXfuVXDl71qldt7OjH63peUdMgdODoGthKN6PPsV7gIYgLv8ip5+EVX9c0CK2O961vfWufwlJfTYvVfKG1ZC9TpaY4tKG3v/3t/cZgczF0dFYXTI0K6FKeghuIQcRtbnOb3smLq0/XsQlCy35IOho2Rf8cIbrCyE8Zu6l7EjGyD608FeLxCWq+yMpxlFfTQT2lq3YRetfq3mhLReIheQXn2kBkoxMP/CTdDVZbhKTNxagHxL4jj+iRq2jWaqsInGKO2TyqyiUoJBWH4q7Nx/3VX/1VNwo9OpXFKFpBOhen8iWP+S+dlpHcr//6r3dnmK9kCHkKRefIwZypwyNPxXakj7lrkI8MdHRTWXUS7NEArC//nu/5nm4T24yg6KxRkEcn8thipES/P/7jP+70rvEijz1k05VeKio6svHDi05sNjqVD2879Llxxi/i+Mu5OVby6K5yiEOHB92tg7Z22drw+93vfj0f+yKPDnSrN0odIXnyyMvHAp2je3xc5dE9NuP313/9173M6EV35ZDGyb74ypyzSm3duznxlDNeOiEy2AvOBborK3T0wc+aY98xqBsXXHBB32aATtkBUB7yjOh9c2BagQ7RXR5lGB3oKo4OZOj42MIneRfED+jVITbTm93o+MVR3WOjdf7kiVM/gDx+j4/pF17kp5zDiw78kvoP/IBWunh1DJ2nFiN46+StpXcTNHfOFnJAXnRCOmGQTo50x3RIN7rRjQ6uc53r9M7V1BD+4ulABl19T4BOfRfP76mn2n3qCD+qewYyKWu6oeNb+cWl/UrjKwGNdi+fJzf9TNoFv+RdobJQT/lFuYrDn87q90gnDtJfkEkHeipD16ZA8Vdn2IdXBmL4qDtpY2SLk5bBr2sy5Ul9oBP/xy/qlKczflGPyZVvCcisR2ATrD+YOgzC0Ac8nKYSc6RRhk5HJWAERxgpMFacgnMujhNi/Ic+9KHuTA7gWE5TAPIqNLw4SENmgMrHQZ4oVCjy6YFXPhrBm0xyxCl4ecRpRPRkB1p6vvzlL+8/E8fhOhGPw+hAwclLh9ApcPrSU2VkP17k8YOKwRY6sY2uEJvZgk7lFkdnL/Lw5id6sk1HgacOoMrjB/lUTHEPetCDukw/gSeOjWxW8ckjn/4qkjh5lR8/8gs/sw0v9HjLiw8d8Aid8orNdNcJspkNKj3d8EILeEHo+Ald6gf/8BV+ZNlLyEc8Omk3XCNW7x/4Srn4oY2b3OQmXT/zwz664893vvOdXX/6kO3DMTdPLw7jY+mOKa/ozscpZ7QZoLCHj+SRl690aHiIYwO/0O13f/d3uz2u8eZ/euFNV7Q6dTo4ypdyTv3gB+WMl06drNy4lIs6iR/++NGBnTpBnbPgQ8S73OUu/cc8vNh147nCFa7QOxIfb9GFDjpvefnShz5oDBI8AfnwiV/JYCP/OU87pKObuvZCf3bjS3/+UtZs5Gft1zk6PlHX+Ct58NfJ4qVOso/NbCQvNtOXv9Qh9ST9hWPqEVn460PwV054iU+dlJ89fMrP8pDNHj4Xp56zV77QSZMHvWvy2KeO0EkcPV1rL3QX5zz1IfUPb3qynZ745AYh/1ERHscydUPB/M4pxuIF1zCKEJ88yR+FOM+1yuIoL+fnCCpUnC8ejTQ0kLic4wM5Skt85DrneHO9KqIXrzoalYN9KohKuHZcO0b38AV8AzpJExfdo2fkJ13hx2YVSuNPwx4hf+wL6GK+1aO1OddsS4w+ulY6R/LE61xUQBUOxEF0DJwLY3z4s02QDskn1Hh+ll/5gXS68EF4hS40AT5010GAPDpxv+h13nnn9UYSWfiAUa2pm4c85CGX0ik6J198Ao7Ski95Ij98qn7JoyNWhl6gVjrIdfgGU3olDsRN8Uq6o9U6vuhMuVb941/AV5r67XeNlYNOzY049UOeUV4gXtDhgbpDTkLkg+sK8eHnKG/NE95VD6C7Tt3Nw02+8g9N5Ne4yMp1zZdrkC/x6hb5OmvytYv4LPnlq3GhD9iV9CrLtTRHSBrwpxuLwRzE9rmo8oPwPpaXsZzvDovpNmHVqCBxoZvKA+JjuEJ3HjrnaaShp5v4ym/q3BHQ6wjsSumHiT32Zi4wHRD+u1B5juf1OqBn0nItpCMLKh9w7Tz0ysAKDPOutpWwnULVt/KPH8OT7eKSFr9NAZ+aNl5XSBPwzbW8yZ/0kV/NX49AV+l+sBvMO3tJ6cafR/qaH28dvSc1L9Shyqx5K0adKio9OE9cztUZbSIY86eujgh9zqcw6iMfvzhGZnw4toFAuk5FXTdFwTdWlslX60IwpVfNI67mr0h+wKNeQ/hOyQycJ/CdfEl3HO1LWtU5sse8kPwV9JE3+Std5RO9a9wox7U88oK0Mb+08GJjpZ8DPEaEx3TJLABGHqkIiSBxNQSb4uqRkTkPP3FxpGBE7/EZ5BWX9FyPfB1zDjVeozQ9oJM3ajf684QCeBmB6khBfrJyLr3KCyr/oBYyoAsv5zl6/EzeAE1CELnyGs3p5FUQS/M0+PCsiJ/AUQfBPkcI/yor59F1hPiatyI65pyuRkoeZ8UnPSG6BeGdkA6Szrb/fc973tPLTCfvCUgeCB/XeFTd2Zp8EN4V43VFdN50rq7YtyU65Bg9NnXyaIWa5jp0Y3rOk8cNLR1+aJIehFaaOmJqBvwgjScRCD1EJoQ2MN1iThwiawrhIUSfhMiB8Aifmg/IMzUC6lHyJh1y7lh5h0/CFCo/R/L0Nc6h0uZY5Sc9ASrPtOnYKc4x8dqF2YNN9eMomG65M0C5dA46JmCAUI0dQ4BW0MmGj/Q4AB8Gp7BCL10nn4KGHJNnSo+aDmg0SPLNj5nHROOlnhe4EB4eaz02VkQvkEcjiR5B5IVPjilYCE21h17hHx4JNZ9rvnP0Alm8F2aZY47/BNcJATp5NB4ynSc+IToLgD7XzumTUPOOAWITnUOTc4h+9KhlXyHd044Nyswhm3c33yyu6hZdKpKGt7yg/CM3cZXHaBNEX8fQOU+6Y6Y1nIdPEJrqh/AIQhee4DzXoa/XbvRQ81X9cw2unXvyM3WjDbz5zW9e6xZEP3nD6/9v7+6f99vquvDzb/SDYz/kMBQi0Tjo6ZBxIyj3cAThHECOhAQEkdx1DgIjqXBEwIDgQIH4AZKOmiVkOmBRiUjRD2UxU+Y3Z2pwpqbv0FRfR6eZ67sey/N8+zrr7NvrfV2XoOc5s66999rrdbNe67Vu9tprr8tRGj4T3YNcs2vyE/mhrdf1vvTyhN55Oh6gnylGDSFEn+gSuejJTrnCKCvHGsILwkv+BMg9GOkc0dQQPcJ3lFHj+KNz7xMMYMhO2AM8xhD8gfZHQiMYw24FQ1BCJlUIj94KNfFrMNo2Zy4tuXTIyGIL0JEXY/s7M4+v5ratC9ZouDcanH4VuSdeAdHBdfjOIbwh6cIL8kZ/DmgEculqdPyZz3ymn/uIZQr4hSayRhlTcSNCXyvSSDOVd3HSOZpeMdWSigDuCbleg3TpnFSqSjfqMwX5SGV0FBK/BPcjO9dokzdwtLx1Ceikq3ZJ3JoNKg1UXimXKRskfe7l2otacb4wrgjPKdDRnL5ylIYOiY8+MKXHFKQLH22CgRU/EV/5waiTNOQ6CukMUkZbgRaN1T2O3j3kPaB7S3CfXnT3lIlGiF9tBXo7ip4a12rovem3DFEDo1C2gkEYQQ9mW1Jbomp4xwKcg0fOTE2YZrHNr8esreA49HWkv8/JvXDx35XyAnRJ8BWgF2t0jo5kK0Rv3H2u/j3f8z29gNNwLCFO4Wik4mWipXFW+9y4ceOq8rg/Am+divxyCkfzzxzT8s+/8Bf+Qn8ETMcZHvi5lgdbB1hsZXmiryw1lqlUyd8c0EunY7HlrvIzX06++CpvRPgrO08RVjV5gep9iDzN5bki99mBzXRsH/7wh69GQrDGA8gyQDB1wQ70iX2WgE46R1OWvgj1BSteyR8eedE8B/qj4zeeIjVO6GsHOgf3BauLlCN65WhtO/+t5V4RutiZn/A/K2yUv7okji09NQvuT4EM5VjtRXcrf9jT+y5becvnlC4V0cs7A4sIfORHF/yAvmR5CWvV0IiUCZ/3VHL77bd3fmtlOQIPXwvb0lu9Vg9NB24BHeirLeNTbCjve3SwkscUsn+C0zaeEruWV6bAYlhfy330ox/tlS093xbgg4dHMZ9vW9blow0NDri3BEuTpBGsGbeMybIwTj5HG91BGtc+KLrtttt6nDlK+4zkMaoCX3mr+UOvcVDJrF1XYT0NcAzpkscg1+zmPicwErdfyDvf+c6+r7rKZn7dPfYMfeVjtEHe5z73uavgC1hziVYK6TQ0vF6y1TlriF46B3Rkkq9yJt+cDaJrzgP0+LA5+LpS5fIUYj01W415D8KTEzs+7GEP63v566jE6ag8qSXtFJIHcryIVYYaBenJH1cOgRUw5q91ggEeluRpXNhfA2fZnnygFaSBURc6gDJSue1Hb+26vZfYL6NCT57hM/IQZ9SvLOzbrizoQwd5C6ZoXbMf3cm19NQyVA2zzkejqHzG+lDP0Rrg6LB9J8JG3iuYhvUlrO0/6MRH/QF66mYFGiDPcshf+qVf6gMnI1LfSPgD8Ky7H/MQyF86A8uK7eljL3lTqRpatg6t/KgjU/zw0B55qey/KXwZDFN6B2QG7KlxR//N3/zNfQpU/eaXfGopD4AXXX2/otztJxTdl+hGGLSweaaS99Auoim4Gc0YV7uqNQfvu+X5izB//dUcp8dvAR6t9z00B+/bctr9Er9WWIu7trkv2IK1dRL9n5LslPfTP/3Ti7vEQXSPHPr6N5fmPH0PaPq4J1S4tvvhf/tv/+3qfvSw97g9zNvI7GrXwDn9W+W/om8V8vCe97yn7/hnD3P/+PP/3Pvn0e7bDxt/NCNag3VoI7BD66AObeTT97RvDXrfkdAWqeJs9ductPMCOuGXY/SQpzaC6FsLt4ah78VtT/jWeHbZwpifXDuyWXPKvp1ua3T7ts34x5YV0kdu65T6PvQ5b08xh1Y5Dm3M0ePw2ALp2lNN9z/y6dFGUveRT+7c7pXSKLd/9s/+Waf3J+/4VdqkH5F7ZNkO2P8q+H+A1sD3OLs3Jh9TPNyLje39bmfU1sn0f1v77//9v/d4fBxH4Bcd8VHWdFD+rWHteWmdWt/utiI6o8HbfvjPec5zui8p99awdB+w1TJebfBy+NjHPjZZt9H/1//6X/tumrb2ffCDH9z1t/d8e9LvepMzlfcK6fBXBmQrp9bgXtGGXhq73aZuVCQ/z3jGMw7+X0IZumajJfnhH13t5ukvH22bLI6PCks8wH3y+B7b+8c9soVR1zngodz9L8KrX/3qLn9N7h5cq6HnDJxEIcXxtgAPGWm9d9+HOo5d+U8hha9xQGMrXhWjjWJX5Yc3OQql9f6HNnrs/6NpL3TxaawrxPtjCWnIcB+9/d5Vzl/4hV/oW5lGt5E+wEcaR45g3+4f+ZEfudq7PLqRIX8w5STu6yjQOPqjiDai6vvbu24jxH5EG/roVXUU8JIXf9GmDO3HrtG3r3wqypQOlVfs1kbV/U8v7Cmuoro3InTsRX+0gus2Ku2d/p4/kcFPA0C+v9Wzh/t73/ve++gtzVRDH/0d5dUft/DlO++8s+sE7if9iNxDTwcdsLzbg72NAq/+WCVpR7hHT/fI12H67+OnPvWp/X8KYpvoUhFaSF7x0HDzU7bwPw0GECMiE83/d+920XxGeXzd131dHzRoUMW5p2wiK0AvTgdDpnrgqMNKXaD3VL4rko+f//mfP9x88839j0PaE0XPy5hvNuWn6gu6Cjz+0T/6R317Z3lHK03CHGIHQZ3RsfH/DJLkP3VzLS98wMDTnvj0TB7W6ALp+JA/8DFw3EO7BbvW0Sepx4lmnP7474WFYB67PnIuAZ+WiT5X7BETLZ4eVWDucQWd0Iza5ZsyMEXRetFOG/opRHdHUz++HDSvatrgcY97XL8X/av8VuD9sdRjteBREI/mGD2QDWjymDilv/yKpzd6x/poXWlaIffHYRBf7+GTo+CrRZ9O+8yfLtLiP9JNQTo8vPiSN/rEhuEB9fE38Y41AL0BHzS1PJLGMbonjXPxHp09mm+dBkSjfKKPR16+4fsHvAXxXvYrZ4/lrpM+PKKPMuET9gqK/edQ6SHlypYe3Vvnc/WhjfixLMirQJ+0Kf8p+0NkBq7ZwdSJKQe04vBkz0AcuaMuQesc+jcY5oh/7Md+rOsjLZqqQ3hHrjTkuIbwdx0eUwi9KdzWqPYtTegeWdUH2NNUEbnm6us9cepo62T619L13pr82IPNfUDWGtqrrVXckyb1dIlPdDDdo02K7y3Jr5BW/rxrseWEF91LMnejCdiNZoAe9DqtoHpoGb337jpG+j0gx+ObqZvIDb8lRJae1sivFUT/h6rEh88I95Km3t8icysqH3LkT9yUTuKi00/91E/1fHg6cH0M8E/+E84BciLLiN9jf5VZw7GIzSofcVMj+hGhpd+xiFyjO1MpVYdzghzlL+yRJW3S4+GpqjUs/UnXyJot4hsjxEljJHwd4B/dl0CeuiuMeXQdHlO6rgE9PTIKz7mnMtNQrkeZFe6RixbdXlR6voPHsfkYQ7A+bFqAXkuP5YXFXuiptvZ2QdO3HyvdVnrpWoH1t+I+LnLt7XgzRj9PmEIrgKsRzLlBhzqan5Ppnh0LwUvgOd23QP7IEa7DZw3hTQ7fcZ04ZZpwLPA6lk90ua782PE6fI4BeVneK2xB8gxoPB16mdwamv6SPXVjhLSJn7q/BymrNXvRRR2k21T+wuNYfdDWJ9HwEpyvQTq0eBwL+ZLH69p0CtfyRgpRzO6HCmAr0MnUsQVjL4i5VSlzIM9Ui+V80j//+c+/esxzPccDnWkRqxGcC7BEsxfhE9515cfoZKl8pp8sLwQrDK6ji719PBrX/J0DdMfflIYpO+fJ51cDTqWLjnpqGeC5QGe2tQLOYGYrannzM9Mv9pYXZyVR7k3BPXUwq6SOBd23tAMGI6YorTIa9QqPLXymgKaGgI/ac2rJDhUj/VagobuVc77ncX5qXIujRl4vBtfpyfZCRYoxYtgpAysgFSCNo2V0GnvLp5yn914qnMhZS3dKWBZJ94QpWMbmnvlA+ZHHrRj5pgwd5+RdF2yXUEc+5F3KrpeAvCgLSwMvCXLT6I7lO4dqd+n5uP9f0OB7r2GZ5BwftJm7PhZb9Qwi6zoytwD/5I+vrslL+uuAHcJjj02mELtWPkc19MkYx8DMyDiNxVYkU3sMFMW9vPlf/+t/9fM5qGzS469DMmq1l43RpBdzeustsvHwstgLNtij7x5EV3obrcNc4+2Fj22IpVcx0aZD2gN0gpeXGonocA5U3l68efHkWnDvjxLkJ/8GBeeyaZBR/JOf/OSrhmmrzKSL//hXKmvA8cxfDc7BNwDK8lhUPdf01ejSzYvyrXm7Lrzct0DgUv5pQUReNB8js9pzxFENfZDKa9XNJVANsFbY7nPSzK+//OUv7ysyfI1pLxiOjNdaA4mPx7djGtKtqIVa8zWXR2/m5UVnZR/xvZ1sBdkqERsJfxhYK8uvFaQ+1PK8BJQ/mfzhOqC7OuMjQudWtc2BPGkvVXby6EkjA5Jzg4yES8FMRbZ8vk57o0wSgmu3XgrbCHOp5z8lKK+h3tK5SMtgPqu29MyXiP5UWdzWxhEPvboOoxrulBj5GmErbPGjo7GzCuie+VSd0HUqHLqMys7ZmUH09FTlpV8q0pjHr3XIo38nuxTYlQ3N717Xlhobe9p7irX82ZPzWLfljxwdy6UGeQZmniCES/iLPPJRo+xLweBNWwOnzuO1anYaBp/Q73kJBDUjOV/LnPsKgNw4W+KnwEEZ7zWveU0frdqmQAOPPrwg9KNDgzgvgayPHuW4Ttx4by+iS/KVygv1aPTt35Sks88KyFPo9wId5/Iy1vkan7l8zsVX1DQpA9hCG0gb24xhL46lm0Llw4ZZOQVTMsY8HIPQpvxtSlb5HgP1BD9bivAJW1SMPoF3jbuOvIrwmbKNOAsiTMGu+ai0x9gh6dFGxta6VeXslZ203h/6JyrnW2TOAW1CcHRDTxmNuwqbxnMrYoBkEBgHcj0H6czvZo6eY1Y+QYxtG1YjHaP5l73sZVfpU4ChjfwcK4xANbAjRlrn1wWdvH2vhVRlZG8T2/Tagz2d3l6EBm+PxOyxRf9qsxq2ID5iOa6nQLzkKzz3YpS9h0cts2NkB9Vfkg8+ZvCzhe91ZJNXbZe/akz8MeDnyslHUzqrt7/97ZODOPzVwT27xu5FzYc8Zh+pJVzHnhB65arhlcc1nuN913xAqP6xBraXX/To1uTuwVENPQUERpeZ/JntHoSHUCvdElLoZFYngMoP3LNe/p577ukN2Y0bN+431560lbbyBNc6iexRXzHySsj1MWCLulS15lOFs1pIGhUx8vfIio7VkeSNjfDbyqvy2EojH0mbxoPMlP9WoOEDsEd+RWyKNhXyGD6hr+XkuPZtCbrIdNwjX7pKFzzjGc/YXQ/nYPGBTf5szmZ30RGRf0rUPAmxh6NG3kZp2po5SBce/Cs+shWVPj4RP03ZLgEtOkfp+ekWukD75KtcqPk/Ba7V0Ju3tgJmarS7FeaHbXcsY4yyZBgyGc8/CpkblFYcHfKxSOCej6PwNcXxrd/6rffeuW+hJR9WupjmmZIvf9KN98RFJvlkXRdkaHirk5Dhmn52m8xLWKhp1hB9hVoR8Ibc24Kk25q+Iuvolb1OLU69lVd8xYgLj2PsThY6PMZy3QNzxrFj9NLYWom2BPLTKOzVn4zQ4aEsXWsMc/8YhA4vwQdUeFulNoVTrKOvqHrLV7WLe+phnsTnQG9TPMe0SfjKL3ry5c83LYDvGuiLvv7T1x7Im7ZI5+o85XAKHNXQpxGy/Mie2v6df6+zygCj+ALvrrvu6hV2LWMpYEagA3ofa73hDW+42g+cgTWIOg/zixoUOzFrUKqTpIJ4NLMS58d//Md7fkbIlz0sTKfg7zrH8DB/b127PzBRUFP5SBw6AV3oo7POxlaxtdOIPPrbi8O0lQqYe2jDGy/yE1chrbyLNw8ov+Z06a6jNAJ1T5rwjewK1+JNTZgK83UuG9a8TMmHxJFBTx98KXu8qsw1REfvKnwAZztd1wIZc/JHsLNtcf0zE/tHhzWEt/SWt9oXPgMNNgDTHvG1KbhnCaa94NmBfLQCvlugztii+uMf/3j3XX8Oj56vyEd0CehXbWQOHo/IFWI7x5tvvrnXM9tA8xH3lRvQP41RpRH4gy2HbZedjjByK1yLFyKfHNtw+2q9fqTIPrZgtg33FJ/oZkqQTXXgdB/TTqGm8XGkvfTVxTqVsgZppP30pz/d9wzy3mup/CukE9hNe5r/10g7ewrs2o++QsY4qv2T7Ss+t2f1HGRA42nzpG/8xm/sTpVeeC5zZAoacUeVCb2G69Zbb736SIVTPPOZz+wNtH26fVSUl2PoOFY6ig984AMP+sQnPtFX4/hLutG40upETG3U5WviFYzOxJ74rv1Tj7W+6BMqyAZpbcDkxZKVDaaXTC35D1Qdl5GEVQ/S4yG940tf+tL+BxVWA/zLf/kv+9SURobzc6y67hePIHrk6L73Fiqwx3Lz/h6Jjc7kk7xUYjShA3ECO2lQ6O6dgS9BfbFcZVW6AG1G8tL/zM/8TH9KsbrBfvK1M56ijy0AjRflKgWenoT4EN0g6XSOdT/6xNOF7/jzFDa12il7oK8BD/Rk+xtHMoziPWlq2Ozxnj+XT/oKtJ5sdLR8iC+YmlBuqUcjTYA2UIesKFOW8siG8sLfR18O0Asf+9jHHvTe97630+ss1Wd6xI/w0Xm45vt//s//+W5fZSBIAzoM5WA/eH/Co+Phy/a0NygZ/SKIHuLw05D78x98LJyoU8LqKhsbkFS7Ah5szo/VRfJ8U5D817RTyH312X7yrr2Mds1P+cQWPnTwTkOnn4UfW+gCnRRf1MFZDj5Xfseg717JUBVbmCsYO90ZZZo39qcLKZQ1oCWTM/iIiXH98chWuRp6lcQyQ/SveMUr+vwkcAj83vKWt/TpDQ0JvWrlJVvQyNGdc5naIX80rsLj0BpBMgNOh9ao8vbbb3/QE5/4xF5JQ49/lYmPa4VIPyN3eTEX6o8WdEbOvQ+Qv3QYKRt/sGFXOzT+hUY+OZSGSqOtktJTvI9K5EnlxENAl3M8jX44sk5CQ6tSajh1VvKSZXM1DyAf4YGn/Hi011ipXEZTphAyGgqSD0d6q7Dkaah0Vp6ILBf1r1WZDhhlQ/iQrYHxBw3iNDT+cMJThnISh96RbcbdK9mJX9BF3tlLA/H+97+/Tw0uITzooCHwz2L+uIZNDTw0thpdA5joUG0BaNnSUXn7NyYN3FOf+tQ+ovNidSr/Qc2HgY2RqCc0dUr805/+9D7Yye6qFcm7UasRND9gBz5p2sBomC9Jw5aeutnEKF0c2WAgpQ1gX3HmmP3LlS05dLr8OS/6gV7VDmjYQP7VRX+Ewgee97zn9SXUSSudQE82psvIR318yUte0v3KP1ypi3iTvwb0ykFboaPQWRm46sTF89MMQudAL3bTjnhKlA/AO/lfgnQaejvpCvxhrEPXQhMwu+PZiJqmGbHv32wP71bh7rPLnWPlVWlyrznWoT0i9f3kWyFd3cenpp86b0btu+bdcssth/bo3P/8oDl7j3feCpknXu2rPvKJrq1hOrzrXe/qe11X/asOzgW8c42nP2R4/OMf3/80pTngVbqE0Ncgvjn14a/8lb9yaD3/oT3m9V0c22j0Si9ykhch/FoP3/8kxR+tkC8ODds3h+x72ONtn/022r/iIR3ZQfRIkKZV6sOHPvShQ2s0+57e9tMWn7TkhIf46OpIDtmt0zu0Ee3hX//rf33FG0IXHuLpX/PA/vYRb6O//scdkS24H9rQu5aG7VulOLQGrdvTn+CIq+mF9pR0aBXnPvQ5x0ewF78/6/jlX/7lqzyG3vl4Dbmmvz+r4E/+/ETeah6Emj4Bn8hXdnfdddfhEY94RNfB/fCI/Boqj9iUH33uc5/r/01gb3/7vI9ypU16/GMD1/y4jd77fz20wcPhs5/97OHXf/3Xe11qjVXnTR6gaR1Ft6uy90cr6NRrPMlKGUN0zREc8aHvwx/+8MMdd9zRfSn0Ses6x5znftJ85CMf6Xva+38IPik+MmraxI1Bvu2Hr01ix9gGr+gD9ZiQdOz9tKc9re+WiicegescQ5dzR0Hb1QYHh49+9KP3kXkKHD2iR+MzfCMGe2Bn5JJ7eISPY1P8qmdzv2Wiz++i/5Zv+Zar3qtl+D7yM6qpOqI19WEU6Cs+0wboybAaxUtYnxKbBvHYJT2eVUdpPbr6uzM9NrgnTeQnH+ijv/uus57X43bSui+dx+bISX7QuRf62CKyHJtj9Hi00cVRXn2AY5Rl5GjUKh3eSRc9XccWjq6rPOdJG10SR74Rtkd2I7noU9PjJ9Q450ZSRtX5O77coyNIA6Gv+QwvI2ujYCPDpI1MCA95A08i/grPyMu8rpGlkaCnocpbvFHpX//rf/1K3/CmX3T16M9mnoigljlI67zaAsLPqNKUjWlIiE9HF3At4CEOf+cCmJrzJOaJih/FfyqkTXmGb3jQS3pTQsowixaEqkPOHV2HNnEgnvzWAPYRtykqT1Dhxz7SxkZ4VN7hV/UabejaUwQfzz7u4ZH7lQ+wK7gH0nqakVdTRdKLk27UKXyDxJPviUJ74EmE7SNPmvBIeYWHeNd0MYWpzvhrVEjaILKjd85Tlp4IvO9EP64QvC4m5+gpP4UYBZybm1cp8km+RpWBPBIKMsnZzJ+a59ZQyZhHc3EerzyiupYpc3SmIjxGcnZHcRofj2B446EB99huHk0jjcYUgukDj+D0cNTgKHy05Hn0dHRtesGjOuPKCz3bCKQ3ViqsOHo5N1XisSqP9PTBhxx5kFbDTw+2E89xTIvIQ/5EAA8Vma3IQ+canTjp6OMllmvyOYFdCTVUCt9UFQdGRw/p5DG6sw/58ogXu2uE8SKPnTyuAxtwTJtXeSyWhk2csxX6TCGgUabsTg/H5Nn9NEz400v+5FmDQ1e+II2jqSZx0pNJd/LoTT5+7KYBlz954Tf0EtjWuw3TXKZNXJuq0LkrL5XeNIQGXiMif6ZSNMDyTHd+JS0d6Opcw0UeHdhFnuXHOT3dl2f5Qxcb052uOhnTNmxuqo3tpEEnjgz5kE/p6cA+eNPLOb+Shu/GLtFTuStDOuLHn/BiL2lMWZi/Vj5k8wV2Jl86fpCFBfwqMk2dmIdPBxtdyeff4r3rorOpPfLkm43pm4ESW+FFHjp2wwtvNGyhrPhp6g8fcU4fPMmS1gv66Mk++JraQss24t0XT0//N8z+8iOkfJR1dHAtoFOO7MEnU38NEOjEdvyMz/Dv+Ai+8sjuaS9cSwP+tIRudGKr1AHlIC6+rA6oS/hFJ7ZS1srvT/yJP9H5scOpsIsTpYWcU5ARZFwBKuA4ojgFKFMM6pqBGRCttApWQaCRLmkEce6JE/CUcXEMG94QB7bqBczZ+6NshZs0jviSqSIJ6KITQ5OJRsFoaMBROnEKFqRBJx4NvcSpHPSUP/EKW7xzcbmmO/ni5Q1dbCd/9OFA+NPbH6iD9xAqEF54cBRpAa04jRM90aEnj9zYwTndyVFZyEUnn+LwVpbySzd0eIvDizy6kyGNOPeAPPbFH9+qAxn4oMVXHJnkxRciD1QEK4O8a7Eyx8qov/E3/kafQzenT1cNFZ54hBedNO4qqWCErWPBXxplRV7sHhsrN34lf641qrExOpCG/dCRHx/CC398gT3ExVb44UU36SH5lSa6O+LtiAYtupRXwFfR4+c8uioT/Nikygx/99kdbXzZUR7DCx154uQDL/Pu5qytTmKj6Ep/IeVa8y0NOBfHdhAd8RfHZq7Z1TH5pic9Ui+r7u4Dm9BbPiF6SSd97EB+bO86OsXO6JImdYVurukQ/ilrQRr85ZlN4g/0xaOmQV/LVRAXHYA8eYudYr9T4eipG0An6Iny91vgOPJcgoyrgKGZk5/7et+MWqQVPHZ5GWM0ZsSfEWbVaQnJS/gF4ozMNbBGDOll6TymrZjjNwX3OUnSGYWlV5dXK3DkxUobowZY4xn5QfSGxOPhnDx203AGKY8lOZEhTXglfc6TpspXEVRwU0TyrcORNhU4PKCegwrlqcaSRPd08OKsurLVhRE13u6F1otK0zyW4QZ0mgM65QtVb5iii/9GHvl8xgIF8fGVrYjsmoclxMZs8pSnPOXKninDIHrW89BC7oG4yHZuNQn7WaX2vd/7vT1eg2qwl5FrsKRz1Qfm0oqno6Aj03B7AgJPcksyQscOI6boIqvmGTTE4ozS5+jG/GwBGvLYu/LQAQl5gV7LYwumdIne12roOXScBU3CXsh0pZvjQU4MFJkcQANhtYIlYqZsjOhrIcegc0j+az4CcXpZ/FKBEj+mrQhPmEtTEX7yx66c2/U999zTVyGYljBCrZ3XiC1ygKzIcyQv9gqPymuOb+UDNZ34oMqSP7IqXRCaMb5ep0MUnL/61a/uTz83btzofDO9UTHV0I+oMtBXXcZ7I8RJk6M8gvP4aq63oNJvoUkHacSp0Q1GenxzHV2FqTyNMGViuaMllt6LhE945Dp8r4tRJ9fpwGs9nIK0wlRDWekiQ1xocj/3AvEjbeig3ltDaEZ5zpNHIXVyK8K3Ivz3dRkDwiRzvrmeEriENMQJa/A45ZEpDaLdKb0IefSjH93Xnwb4bjUWuXEM+tdATq18OS7pWu+t2SNycp6RhGCURrZGqspckg2hTxhRK8E4kq731uTUsoPIq3SVn/LwuOoxFapu4TOGilpGzr0Mz2fjeEd2pRt5QNLkXvQOPd7JW0WlS0g6ITBdhFdQ762h8tyCyM40wJLuOeY8Oo5pK9xjYy86vTD3/gHUQU9T1XZrqGnX0lc9QT0wNZL48X4gTv5zPpe2xtU0ifPkyU+n9JRGfE2/FaGptI46avJq/KnwBzXwCMSYpm7SEJ4bDKChZxTnDKNx9whpVYDKXke9KYw1pDClrUG8F0NGjTBV6HOofJbgfmzpyME0vjoYH7N4CeyvD3VqwZoeVfaU/EpvPpw99+QNpvjOyQPx/MRTWBolmEs/h6RnK+9lrL5S/sciOicci9QBZcdHYa9NYa8O0pPjqaUOSuYgbc3vmrzkwd43ys4HUWCeOQ0vbOGVNGtp3SM3upJrmi6bxUWnHKewdG8J4S9v7AmjrlvkTyG8K32CaRt1MW3BKXEtjlGWYjmHpQI8Bcwn5225L8m82HnsYx97ny8ra1jDWrrx/lr660CerDiSD08pGn0ffsXGkT0VtmKJNudj/DEYeUR/7wK8Y6n3c57rOSRNKsNSeed87nhq1DIyz+qYcG6Qa4CzhDV96v2p4GMg9v7gBz/YpzOtUKnvdU6NyM05+9brHOdCfGQr0ATOTYMJKdeKKmcPKl1o65F9ddSjvC0IT7QJwbUa+hjSEsu9Rj0GUd5LPEuuNIJWZpDtSzRHRkovfAownCVPWUJ2TqRg5M+5L/XAHD0HODViT43u0tz/qZD8aSDSUUOOf1TAD6e+SD0n2PA621avAf9HPvKRvexM1/gCNwOSS4AcAzxr3C8BNtTGaOgv5Z8Wl2hrtGGnbk+vxS2PiBqmczcSQAYDeMQx1eATAOemNm655Zb+yHwOR7faJssazw02NR2ls7J2XoNoJCVvp5aPn6DCZlro3E6Nv9GgqZtL2POSiO0cPYJfEhpCK7Ic2ZUOpyxLPDV6llqS4WPHvCe7BNRrq1/G7Q/OCT6apZWXgKcjdfG6+UOfEOxu6Kszh5G5szT6W4BWeo2XI2dxvuacofOy8td//dcfdPfdd3cHMDevocroYktviE/4VV1G+a5NDXl6GBHaMaCZ4lPjcp0QOja1bM1qG2tvn/CEJ/TRdvKUdFVWzreiyhM4szlQ52uoMh1Tblvkx1+MCDUSaEJXeS7B/YSkr/6zFaFNudewFdEhspM/+vhAZgnSh34M4pcQ2hro4MNF88rOtwBd5I1HoULeBCvawGZnVjPxGxhpEypfYQlJMwb5EaxrVyfwq6jyBGlDtwehyTEvmyFlOwUy0TjGF10Le0CeAQJZeOB3DFJWVefeetQb9eYUYkyQKYb3MnaPUmTIiKMRuRed4buFj17P9gV42LjJki+9L8NG/6V8pAAiDx+jaJVkhAbWWvY0TOEbPavOI98KdKFNuqpj7pNnKkXn5Vz+pB/lOJc+TjUlcw3oBB2ZI6zxIFMacnW4KvoeWsHIxZMKe6PHK7TOt0B6/qfcjgF6euvkHNd0nwIa/psyCAw4bAWxBLTyjz60lccapvS1IobPxM5reXJfujH/U3Ti8PYuzHcddLcizEvwpHe/8owOOR/z515owf2aZswDP9WBiq+IDEc+4WnfU6O4rf4E4UOOBp6PZrlq1WOEe5HjqD2kB9T8TKHmhd4aevRjHq+L3SN6hQkypJJ4hDOi3wOZz3pnn6vbMc6IWeaWMugeuYxvN0K62HvDLpI+d99qnKSjB4e1WuHNb37zZD7IS55H/il8hWqbBDs3+vgIzZwuaNyLbEfpIXFsYXQG9n7J47ggrcbN6Ebng98Y1kDGjRs3+ks1nWzKIvlcg7QaeTsX4kPfLfLlDzL3aasHZedT+thgzm4BHmRIp0K88Y1v7DuoJn5J/giNwfve977eQMb2e2HFD/l5Iko5rc0lS8vedjq0WyL56Lbon3yi8ZWwr4ctcWZTvigs8QktnTVqaRgr3xyDeu4LbeXlqbOuBBNX6RyFxI/+FVu5lzR0t2bfx28+EEw68eqBIN0UwsMupbY9IA/tVqBlR7Kt7tHQ89Wa9ymQQy/2/MIXvvCgO++8s9s197ZCWu2IlWRZtXUqbNeigSGEGI9RPvvZz/Y9b/ZkCBhPR+GlTrZSmCvAQEGS7TN4sl/wghf0KRwfStkW1b0tBUsOXvC5z32ubwlq7s8c4Aj5sqHY+HUcHvKgc/C1oD8bMDrjHFPO6LqGgENo8Gy5rMPxxwP+zAJvL7ntcV5ti7ctEew58qY3val3tL7CnOM/BTy8+DEisyUup+bQGro1++FNH6NyLxxNnwnyviQ3dODpS3o29/GNLQ6UpzJZ8yP38ZLWC3J590JeB8tm7i3pUcHvvFxkR+vea6O1Fexmawb5iHw21NAsIb7q5Zs9atQj9FuALnby7sigQDladYbHWhkGviD372sGKOnwNPppMCHHOhhQ71zLI/+XXkOsDE3nWBJsIYF6apMuNFM64Rf+wC91ehYf+AqWbUEa7QN/8+HgXP4MfDzpGzxmQcNWW4D02iONrCcyPqqNWuORcqejvBu4ZK695m8NaOQb/akXf+z645FkOEcfKnEOzpLtCLaCAWxIZUToTz/yufjII4WVSmAaxSfYKqnRHAe1D7kd3zgKLOkRw0tjsys7Gtocza6XGvNUoEB6fDl29ODYRtUqqCcBleOOO+7o+chL25EPOoHjG8XpHOz+aVdAIxeVTiGj03hrdP3TjWkp/BLwsMufwLHth++PU3QQ7KmyyQdHkX4K4n3l+JjHPKbz8y9FKq18aniie7VVED009HSwl74Ow8jSy0D5T4M9ysdP0NB7NFaZ7F1OtmWyPsjx4htG2kA83oIRrBfV+LGnj3qyrFG+kg/Ti74qrn88kmAPF0815Gus7JGUe2uQJv+joPNVZgY9oAOKLlNgB/fYj8194Wv3TJ/3W12yJN+9BPby/wGgkcvmd3xAGaVOTEFH+xf/4l/sjaJRtD8v4QueUJSJctSwQi1PvC391cjb/8bI3tp68/YGXfYaMoBhD5uNsUPowwPYgAzTFUaybICnTutVr3pV3waE/rEVHnlfNfLRIPsvAHr4M6S6TUJNW4EuR7r6EMxoXp32JO1JlX4Gb/GlKYQPPzTg882Ll9ZA9pL88Z4nXD7pqcl9cvnyHI/NaMx2oQntx2aAQ+vZ+x7OrbHv+zFvBR6toh/e8IY3HG6//fZDa5zus8/0FNxrhXG49dZb+97YL3rRiw5PecpTDi95yUsObTTX728B/vh85StfOTz60Y8+tAp+tZ9+dKhw3Ubch9YIX6VBax/9Vhh9//XW4F7tvY3/VB7wYaNWgfp+4/asbp1W34u/OUjXSWijkUNz5h5aBZjUJ3HkkUuf1pAdWmPf/x/AXvkjXQX93E++v/jFLx6e/exnHx760IceWsfXec7lYwSdW6Xu5WAve3mMHSoiU3r5bRXzKi+tozy0SnpoFaTvjb+kewWe4ffKV77y0Dr8Q2tgr/gmDVu3UffVdY7SxIZvectbDm2w0fdh3yofpEXfGslD6zwPrZPudv3yl7+8yodsQTo68m174qsPWyAPCez+wQ9+8PDe9763l8NznvOc7l9TCA25Ah3YUZmoS20Ad2iN7aENqK70q3RCa9y7jz7qUY86tBFo9+s2Eu7/jYCXEFrnjugqxJGJh/9Z8N8Q6oP6lPQJyrh1hN1O+FawN1q2+8Vf/MX7+K8wB/fIwU9dY7OXvexl3QfF0YNcadYgj5/61KcOT3rSk650DP85RLeks/d/Gywd2gDyyna5f130vW7ubfM3QXK9SxPeR6RGI0YgraA2f52IRyuMPpo15eKfk/Tc6bXG3kt6wajMCIhs0xymbWxPa4oDzZZH72TX45nHNHvkZBQ+Jb8Zu2+PavSTF2xG0h6vjGyMHPCM/uEPlQ+dc5SGrs6TxjkeRpbm6DylGKHlSSIIn4rINM9KX+k9Yc2h6tgcqs91KztPA8rFKNvISboqe4T79MHDqNhjq6c7NEYi8hNEZvRLfnPOpuY3yVYeSyOoAB2+gh0ETWHwj2w+J7jn/c34D1OhC/iDJzzTdJ4MtsgH+ZHWU52nW6NpOrSK38tujk/KMbqwoZEtG9jKI082W4AeP1Nopj28sDSi9nSxNAVQdQA6pDzkSx5MbclDrVvut8a1bz/hqSFPg9WurqNX+DvmHNxnN0+l9PSSN/LQxXZo+LanDsf8c1dAb08hbF/bofCoMiuinzT8R3vkqc//W0DVt8obgQ8dTL+C0XyePGBJvnuOginU1lH1/f+945mjOwpNwC40w1wd9XZGEv/u3/273qPtQTNM73kd8WqO1Y/hH4gXpDP6p/Kf/JN/so8iyRTCYwukk57eRgKhD4+Rj3j/emSE5jy6JP2U3nN8QHqotI4CnTwhNQc5fOADH+i6jXwq75GHY/LjOIfwQCOd0YxRtUAH8TXdHMLDMTbBL3EV4eUevzHyS5wQ2lxvRWQJ/EmZOtY8TI3oA9dkSy/vwphmCehq3h3p4MnCOV4JFbnOvfCoum8FucInP/nJq/Pos4TIrjYMEp97I+jZOpXuq7/yK79yZYeaPjzrdYXr5LvSjjKdK5fW0B9ap93TVrifPI88Kp8RuZ/AL/GRN8f/+T//Z/dTfJf4kCWgESrPJeS+I3pPRPwmeTkl5rupGaSXccwcYObX90D6jFYF1zmvaEa4mj8zn29e1g6VTfdOQ35ot0A6NHpcISORGirI8ZKrvuwlT4ApvcdrIAdCFz5J1wq6j6aNCsyZmucLTUXlXXmEb+wxRRuER4InMmWRUHUUlpA0aCJ3icY9ozgj+KDSh99WVBq6G83Fr7ZAuvCIT2ylhap3zbunI74zh6RzFMJnj+5Bytp/jeKxxQegys55kPjcq5AvcRYEOJoTh6Srx4RcV7iOjCprlOlcXjxBGG3P8Zk6jmkrcj/B0wQ5KYPW8PfgfAmhRyvkegtdhZex8hj7nhL7WucBGmFKedmlodqDGEKBrIHxvCQCL04z3bKlMKeQCoguRp3Twz1LGb2Bj6waTgV5tHLDx1LtyaU3ODq4PTK26ibP1QZeZJEFid+CKifnNYwQh7/O2qM6/4G1MtiKKlvYimPpAjTR3bmyy0q08Fvj634Ne5Ay80KfTY/lsxXkCV7ketFqUUJ8dY/MlPsWOmktgLCsdwpsfWr/4aNZ1CDModIspVuDAZApxHPgWpbRODGupWnXyeASyOC83uiTZXWLUZvKdCyqrlsKx0ofDf25kIqqwoD3DhlZpzE8NSJT3nVkOuottrgOkhf5UoZVh1Mg/KCeXwrJh3zKX67XdHH/uvqS5R3BOcsvIEO9lEdz2hp5o3p52OuvW/PNP713UBfPlcexHNRB7cwYfy54yvWu4By4VkOvUKcVyEwAAIt0SURBVFPop26QYlzBckMvmJ797Gf3Rze9OqNcpwDonbCGpDlW1hbozY3oTRFZ0saeGsQt+u0FnpW3c+Gc+QMywBSVpWvBqeRWW53DbmtIHZAfywSTr+R7DnQ9hb4aJg0iPc5dlvh7ErSklrx/+k//aS/XPdiT72rLc+Vt1Ecbs/Z9yClQ5WpLhVOje6CM1LAVlONYVr2cQzm6mMqwWsJaVl/QilMAv/u7v//fipeAOXorbM4FdrQWWQdmxYW8idtTFscg5W2O/tyNffIjGAHKI3mnHiD8YSKV9f/+3//bn5IuhdjWvkjqIbtGl3Mg5agMjejJ8i3FOUfbOjGr3qyIIvsSkD+d17ntGZiO9u3AOTrq+zX0eyDzwjl2r6SLDPtq1YdElhx5UcHo5s3y1dwlYKniOeXJa7Zb/o7v+I6+lFNc7HtOkOMlVKZS6HAuJE/kcWrle+78XRryyJaW3V4S7JivRukgnAtk6FAMvnww5n0EeaYez1We5KmHwjl9tIKPknduewYWt5xi98opXMtiqaimVfYghhNq75U4TwmOVi7402/nT37yk7sshczBNBZB6K6LKR7irG/2ZHEq4Jl8O1qD/u///b/v1/YUl79zVRggRwByPEnE5ucEWWSY99QYKstj8slmsZ9QbbkVyWuOx6LySV4crekOpmSIu67sILLN0XuaOBXfOeDPXwx+lKONzsT9vb/39/ox4ZSQL4s+tsxhH6tD/Cd08pYB0B5U+XtoPUHkZeypO7POjZMk7EFoNIQKYivQMEDonddKKpMe732Q5U27qQWf6wdeymSOnsNdF5EfXSrowviWWZ0KyXcaKMtGwcdnlqpy6Dl9ToHwDW/zkN4R7HUuOiYPa3rW+z62Ua6wRjeidkj40D3vbORrK6rezo8F+vhg+KiwPn5ZQvIQG14HyYsPvtJ57rHFXuBthG16Sr2wSkycjwgN+mreTgU81QuDvyW+0kU+1PM1SBvejvxKHp0v2dP9yK3ylvScAl/WtgV7dF/DtboNmZeZrBDZihjOMR1EjOQ6FcdXovh+3/d9X5cB0ngJpJKj53C18h+Dqs8UHzKMQpcKey/ICT8vYeXBlsTi64vKc4HsyGdbYYsNk6bqv4WuwrynxhCdwO5bUfX2dfStt97aN9myz8+ewYYyjfzrgC7hI7jmsxqJJYQux+sg9jDldwmQp8wMDvitaSqDFLD3DUSn6+YtMPDjo6Y35tqaWgb02+NXAd78CB8ynYtbyof7kRkdjpGNh/xFVnidAtdq6FPYX//1X39VsFuQtI6MaVokGVIBwctJDZ4XvTYeU8ih83m4R+MUuFHFnko+gmw80sFUuEcHMk9ldHzkRfD5uJ0L5dvISJ6svHE8xlm2Au/k1+fWHGwP5CEvq7baJfk27eY9C/nKfw/Q42N0Z9T8+Mc/vi9H1djEd7ZA/vGSh+vYuT7pxR/pYdO4NZA/53d7gJ5N/Hl3dDgn2IscCxTyfsdGZOKzC2fseyoYaGlnsuXIHNxTpvzqGFtoR5QJv9bGmDPfAnLJI9eofK9fA1nq/jmWO++2RIysIDXQtiL11ap4cTkm1Pg4ZAJj2F/C2niZq5nygRS6t771rf0DG/dCx7lcKxQ75b3hDW/o+5zgL1R5OZe2nifoqMwtfs/3fE+fJwdpcqw0UGXknGPYo8R2tdkbPXT1WPVxjqctVcEulfbwcE/+HGOPkQ9ZXlJb0mZ7Xnzcy/0qp8Y7Rmf7v9iXxn46pj3wiI2lTXrHqkPiBcte8bD7Jp7KM/cjBxIX/jptjaFOLjpkdJhAnyoPHBPI1KhY7eGRN6PZ0CSA9Ll2Hr0c7cnvT17wiMykCU3i3K96yS/at73tbb1yV77mryPXtfMaxONlfxq7JfIh11Nyp0LuebK1N5L64ikidHQb5YYmaTRmdn9Vh+3npAzcTz6rLpWHBs25ckxjqhzlmT+HNv7gfMxX4pM2gQ50Ue+F6AP0dV51yTH82MAOlOpI4se0jtKOtPzTslgzCHYSzeAyfptQaYRcuyf/9hsynT2mT5q5eLb85Cc/2b+KN7WZdKfAtbp/c9cqXOZbY5AKmRAPjjWzCtV+8EaTGrdk2iobL5ZsceolLGNWKEy0+Kns5iaNECM7/CGOqGEJ/8SBjdGseDE68ZY9skKPzgdhGoLwJVc655aUKRxbD9skTfqkIysInTlGMgUObSMwsBGWF00cRP4hMiov5+JtBGWH6Y985COdj/upBLG383qUxj3XbM6R//bf/tu9s03DUHUGaRMXvq4Foyvb8Xr6ig+EBxtHlmvH6KHC6lyMYBw/+tGPdrvQAY2Q8qoyQ+/IXra1lY9PfepT3XajHEdxgF/sCZFjBGW1iLnl3A8NRD7gIV+579zTnq2BlQfaNEo2ipNOmupvEBnibaJmsMKPxVd5zqWLzs7rtaPRNN+1j7uQKTG6oneOJggPkF/pvSPSWdlCW8cB1Y8TAnR466j9lwL/Uw9sSuYpi2974hJSb2I3dDXENtLga6Bku2FbJrsnLsfUj6pLdGN3efn0pz/dp0L5ZvKZ9DlWe9AhPPifciCHTelen9ig0sKYL9t103HqSUC6yBJCk2tT0uRrbwxuT4nJ/egJX0KUM4pWySyvMs1itQgFv/jFL/YeleFNeXAIe7jYfZLhzOcxhoaRc+pBFYy9rDmJguYw4jnel770pd4IC1746DU19pZb+pcj8QxrKZQj3TRe0lqXqjGwssXOcEbAXu6qIPbUNgrHR0Mt3h77nBdck23+l/Gt40WnMHVE8uhPM9Bkb3W75+Xxi8OFzqOnCiG9vXrsm200ylbsyS5GyALeHMgolU09abCN3fvwtpxVA6lxZVcNrdGUUbpGwyOna3ZXcVQ4vFReDZKgXCzlVEFUckvzYkef0rMx+/nEXZ6VNb00yB6h5U/eVA4yrG+WR40FOnrqSJS5vcZ1zKaIHNld+eJrHTa76CxVBGUhXkPJf4Q0OiqefIM8Ghx4CjLg0HnwIXqbRmBfQSdiLp8NPTXFdnyB3Ic+9KE9H9LwYTbzERB5BjHsHp913zVb8A02prMnWvzkxYBDmcifxomedENnQCI9HdiFrvybj/JD9uH/8qk8HPHij2yJDtQdT3NkKC/fXtDb0wnb8UOyUw/pr/zVAT5JNt3RSk+OeuHJFk9+hp/8Rg/1CV3qk8HI29/+9j7q9n8Igqca5Wvwwrf5Jfl0IY9t6M5W6MnnM9oBdcriC093yomP8QUNrnTswm/5n7zQgb97J2BmQR3D137w7GHxhrrDT9lZWm2TEP+WP/zVP1/e01V74o9/+L1Ogx34oHxrB6QxuOEP4unEzvLIJ9RvZa4+0BNv9UF5KYdMWaYearPoqT3TyGtL5Nty8nTWwnXRtynm1BVzjGs6DuuPglUyHzQZ2cTxZUQDwjkpm2t8NWCci5He8Y539IZJw6dhkEbDevPNN3dDc/DoomCdS5PRH7kM5w8/fDGLL5nA4UBcGgoyHTUQnN6fjXzLt3xLnxtXSKDiSRNeKmkqlMKlP+fSiDh/1KMe1fVF75ostOxAHp1dO+LF8bKEig7+NMI2zf5EBb2AVochz3RgP/kFcYAXW5DBTnSiJ0f2j0kcHw1e0Sv2E6+CiXPOuTQSnJ6jKgsyYyv05OEFaCFxGr73vOc9fWnoK1/5yi4HHUT3+IJ7nFq8ssNbI6OjVTle//rXd3u5r2IZKab8EvxBibTSuKeiqahGlDovMgLn7OI+G+PNBgKgVx4aCI2PgYdKjU7+yIjuZKMTpxzo7lp++INBj6kcvusJwYd2o63wEtgEH3Fk6eA1jNals4EGAOKz+NR8oScfL+DvtiHgC+qHxsqfsqBBzy/oSg/X9JcncWQ4yr9OWcfFJqZjYk/8HUNHvs5YJ0A3PKWxRS//8Uc4YEChbkojkIM2eTJFYqyJp3L1B0Leb7BN0gh4a3TZy1OQfIeXcKM9lWi8zQBY1JC6A6mHsT06/NkPb3LkxV8y0sPfQxr06EDQ8VPp0KAF10B2YADjj0P4ssGvNPjRmd3xStuAl0AfaUBHyubaM3+uJH3CtdEyeZ9tNYU55J5jy/ChFdChPfr37W2b0rM8anxCG1Ud2kiyb63aerq+Najj0572tP7HIq2x6nHkSA9kCOJbwfQ/Pmm9Yt/WU3zFKK/GSWvLU/LbaHdxa1Fp6dqcpZ9La1vmNmK5+jOS0FQeI58R8tAanr7Nqz8MwRtN8uw6cXOosuhIn5/92Z/tf5ww0tW0NZCh/NqI49A63MNv/dZv3WeLVGnm4F5s10ZxhzbCu7JTdK8h8QK62M51c/LD3Xff3X1gCWjIcBT8cUUbLXf5H/rQh/r21ZERuW0037cprjrknoCP7W/lv43ur+KhpqvxgWv82uP+oY1u+5/wuE6dSJoluK/c/eHHD/3QD/U8RdYaLZATW7SO8dAaqkN7yuo8K5+1ED6tgTu00eWhNfg9TphLL8RnlcvjH//47tOf+9znrvglPYz06mF7mup+p+wTX9MlDn92dV3Rnmx72bWG/j5t0RqSzlGdfvWrX31og4ur9kSgf/IwB/fQtM7+0J5KrnRc0yFpIscWxdo9f7Ai7pTY/ccjQVOw95TpITPC2QIiBfRo0qMZzevVwdy9c/fTo6Eh16g7NHpNcfTY0vPh0YzY+bbCuaKrtPUc71E+1LzmfqVbAh508DRg+sAIzOMb/ng0R+lPR+B6jm/SOwKe0Y9dtuiDVlCGuc4TyBpCG3vioVzQ1nKDpBUiK3LQA93RpWynIK1RpBGjpxYjL2XoqSp5dl3LzUjJ1Iw/bQ6qbikPgeyM1rYAbfJF99iCjh7hnbtX5Y2Iro582+h7Kf0IdAJZQAf0wpItR+AByid6O+IT3gEZ6i9b1Tx6EerpztEIeckP0UQmOFd2MNJIm6N79T5d6v05HiOkTcAj8nNsHVhPp21b4hUe6q38snnsJX60XeAevo5k0oHMPDXN0R2DozlRwuM5hVSkWmBbwSgqexzF9IGjj4ZMA8CUgT0We3HBILDHmfGTnr6p0JExJUs6c7h52TdXAHvyj97bfXPA/h9SxU7lBLLYQVhD0jjiK09x9DmEd4KOhoOFfsoOU0ALbIIGbWyTexXipDNHniW1QujRhn4O7vM7j/jOzaebokhlFJTFKH9KnwANHcYOLvqtIXIFtjeAMCe8FfIROeng9yC05vPT2a7ZcQnKEY+Uy5QN8DfHrR66H3m2LgYdK/otiM2TPtcV/NPUknoz1rXIdozOUzxGJF+hS/nLPzuakpPHKX+aQqY5QfotOgTSqhemkPDY055swfHecC9kSCO1BzFCCta1DBqlOfdy0P0pQ4kz+g0to0ylW0JkO4ZemCoc1156Gr0ESZ8A0WcOeAsKUPCSUKOgYoivjXMa2zjNGpK26iSsIToZRR6DUV4CXaZ0dw/kr46S6ABocr4EtA95yEP6Hygb1fMHvJRB9YutiN7hsQehq7o7905nD8LjGEQHA656vZdfTS8vc3zcE5Sj4H7SexGswTT3br6/pp8C2shwhKn0rr3M9KQkXdIkXepzeGxBlZnzlL+jhlv+1hC5oY8uuTeH8V5mOCB6nArHeda9YGTKeum0BzFqgo7Ciyi9trflXgDBaIjAyzKPxZXHXNo5TNHN8cjqlxTgVNiCOjKwwkC+rTAIfQpZ4+W8jvKnMMrO+RINhC7BKFLljG5bMfKpYQrh70WjF7/xHyE+ERvMoaZXHjpI56GHnOd67liR9PXeeD2FShfd6bXlg6mKysNxL9jSC8DQht9eRH614RQf8pShp1FpE7yItfpIA5k96ud4VP5TocK1ztPL3VzPhT2Yos05edqZLZA+Nst1PS5BGoEs9nOeunIqXKuhBwp5K30dqKxWReBlJczco1KMZoSd+bNzg0wNrzxex/hoOQJYVmV6xrWlYMlX+HuE08ifulcfEblZBhb9zoHk39FcZh75c28v6Jvw1YI8bsuP8r0UYlvvKc5ZhkHs7kmwPukG/gFO/bSUE44p3xHaCMtPrXy7VJmrh3z1UvIMtjT2e2dItuBaXhEDqLTXcTBz4JbIma9+0pOedDWNMTqIa8F6XTIvAUYXVOJTFDj9f+ZnfqafWyZm1FD5um/ZZTDa4FQIX7LrS9RzofLWOMRnjnVq+id8tUAe6SNP5pIvhcg1n3zOMgzIUh/MYWvs08GBexp6/mTNvDXrpygjPLz7GOWdC+Rp6IVT1f018BsdyzkGeNdq6BmDAXyMkpUUW4CO8WRMz+/tvDh/LKLyy2geySsizzSKeVo8xAnnAnk+1vDC8rqy6MtOpqmcv/zlL+/xZMSRkr9zInZ0pIcKycHODfKUqbIT4NgBQmwWu321IDrVp9xz65hy9IXvnnp4HaQcM4edPKq7Rt723QGr5+iWwdKxkC/v8XwTcQl7kmGELY/H+uhe+HbBACHyT4lr5SDKmHPdo1jSKnxLKn1sYvWE0by4OYSOI2kMFYC4PbL3ggzOzIEVwHWAl07RB0Je9DznOc+5H0/X8ietinGOvIWnIzmx4zkcbAQZ+UryEvL+MMBf9r63ug6UH/9k00vYM+VG3rhKyD2DtFtuuaX7r5VAQMfrAF95ZFvn5wYZplEs35WPS8jUJghse51OcQon6aryzzZbwWgyouBsqsQJrD7RyGfaZg7kePNe5wbxO2dBmKPf25lNQf7Mz4MlgV5mjSDD4+I5EXvFZvQ4RUe2FcrOI7i8knldu361QZ4sH70UyFOffE19iTJMeSnH8QnCPX7u/50Nxnw1DfS6jm7aBR2LQdClkHcQ1+2ktkKnmV1yTy1zN7cUWO3l6mPqVqD1wso/SOH1lKc8pffWa5VeWnP0llo5r3pA9KtxI9wLXT0feQF97DEy1dCHPjTjeQVawX4n4NHWnGMa2EpnKoUuU3C/6jleb0VoVEorJ1SkNdsH6BK2ypYmNlCBzCeHPjxyvRXSa2jS2ITHFkTWdeRD6OuTqHK15juY0inpq/zInko/h6R1tNcPf9oKNFMBossUlCH9beehHkJNr5HiTwYzpluqLZYwpYuAN5neZ00NjkAaIelzvgehy7n3SFk0sFQ3omdk12thDUlnqkg745wPbaHdiqO7DUoobAppmPY4GHAGOyeqpHppe86sZcx9BtfzpcejA+Om4kizxUD41EKJYccCpZ/8aZjIqPdDK7hXz0eQRYaXzvCsZz2r88IfDTjKl/y55zz3gpp2pN2K5CENpFVMdBNPzyVUebG9EF5zwBuNtHxFiJ3ci/32QPrQVFtsAT2q3Bz38AByhfACPMYpjRHKVvrYL/rkuBXRHT++hX4r0EqffCeE3xyk0ZCbv3asuie49j8B+NgQTdwa8JUuR6jXBnjq4mgf9yuSXj5SJlshvboA8qbxJW+UMcL9tCGeAvCJ7luBzuAX7d72dA3Xej7IEio78+0xqMx7Y2/aBmzqleWSa4ZhdI835s8Y0jSHr/BsJUAH90dHmEIKAY3dHe2aZ0Mo8RUc1SoYDf2c8+NDD4XkncNUo4OvPNNTfi1pBDyjh6N0WSs88gDpyLJLpA9S8JRuKu0cyJAfq3/wkP+t5Vdl+Sjmxo0b/Z3DFqBTNhpBozMje5/L51+J9uQh8C9kGpK9tKm8dgy0+6Yy22qDCvSWBmcVEZ4aB5v8LSF+xh9syes6DcXevNhl0YZevrTeQyuthlP9scOiOkgP+VizBXsZfSZt8lPxXd/1Xf0p3Ts4mNNNfKXX4Hkx6evifCBFhgHJ1JRY6KVhQ+1Rpkjn6uwU8OGT1v+rx9oY009zeldII692/uST6sTWtiiQPxuj2RX01KsKr9XQy4zd/vR8e8AoGqrsDueFB6w5WCqnRkqBOLcdqg7DeSpa0i4hhWBFzYte9KLOB88R0miUvCQJ78C1iuKFkz3tb7vttr6jZjqtCnz8wQOH1Ft7mtFAaGg92mYUgaf8Rb8R8khPOzvaj92WvF7w7nFowDt/2mILVTrRYUpmRbWzcv/Zn/3Zvq00J10D3uhUZI/8KoZKabdGW7+yzZr8ERoou59aorsHkcOWyowtot8esIFVY3YATWPluGUdPXn83fbA9rTfW4aBPChHvrhUf0aQpyH1lbYdE3WYyob+S7rEB+xyaXtlgyT5V5Z8mf9LY0dX/xsgXaYsR0iXgA6Pd73rXf3vIfm2HSvzxGSKcW5FGnr+a+DpWxx22GMLkN4uouxBlgEVPyU7/jIH9+lpRZ2OTScI9NoK9UFDT77BwikxuR/9HEYHMBIzGnrVq17Ve785Y6QgIca3j7VGljNosOwLHoMuGdU9I2wVLNvb2vzM/s15zFqijy708KLI3xTaRtW/XNEB30ovrbzlC0DOZKmlkZiG2z8EcU4F6z2Dxt7XbRrOkY+tHVQE/2al57f0jA01+Doae7JrtFVc3xSgGfMjzoj44Q9/eP/c3ByoTsN2vp4E6mqWSjeCU9nLnd4aGnn3LiI2nEPK31EH7QtQjZQRCJ3Ez+kN7qs8lnPqQNFbUmZU66todg7dkh4Bevx0ts6z8iQ6gCcW2wP4YhNynz/7ylojz/a2Ps4WxVV2PQ+SH9OOfEljx4/sJY6v8vUSP7Qjj8jgK/Kss1EW/myHjWLnJSQP9OajGmrz5r6QRR8ZS1AGFkLYWtkWJOxIj/gwTDX8+KoHnurUY08U2U+dLxvIaeCVrXrq/YEn2VEnHSKZGliDJQ0deBqw3bAy5avqJb3oO/Kgn4GGkbQnRO1JtleOHaaATnBfGRo8+bcv/4Nhu2WdIBunc5njA9Kpt/7ty3cE/uJyazniK8inuuyjMCvy9k7fxCcrovOuhh4QhiHH4Fj++mqtgQB0Ms4g9i1X2Rn09nv3g0e/xTB5QWIkqRc1IrecbY0WTQpXZ2E0onFUydCnka/5kBZdGm6PlPZst4UB/R/72Mf2/GvkrXowJYNP8grOObQRCjk6B1sfcEaNrUqmkMnSQKBnF/LQVn3ANXtrlDSurq2h5qgqPX5kj3QjdBj2jDH36cmMDpa5bikDkE7D6mnHNJD94K3Ais5z8pUZyIPOhU38Uxbd09DBmv7sxW/kVyPhXKev0tb8TzX0IA1b20TPhno6zcwrV9lLerhHb09mpk7sh85X+CjfCu0Uj/gWek80plDoosNYkokmkE65odHxaugMRvjFljoprzqa/C+yfOi06Oa9mfvkOQZ4sjH/sX++r7stLuDLfJI+GkmDDzroSHUKL3nJSzptdMJXw0ZnPkgHdVkjj6+OUvlENp2E1NNAe8L/+bBGVkNdv9OYs0HVw1O0Oq09e93rXtdt6QlDWYbXEh8+7anSoOfFL37xVXuE9xxdhXTaE9M28hDbXxdXspuAzWhG7qFlqu99/fKXv/zQGtvDF77whcX9k9G4j6454eF973tf37O6Vcq+h3PLYL8njbRzCJ9WAIfmPIfWGPY9qFtBd/o1RHc6tBHk4TWveU3feztyEyqkb6P3QxuV9HTy/Wu/9ms97itf+cp99q4eA0Tn1hAdmtEPL3vZy+6XTsCHLPxbJbnSc9Qn/NzPUWgjp0N75D284x3vuNJpDpWHtPZgZ4tWwbottyI8lB/Z7emqX+OxpHcNibf/+Vve8parPI/0U5BGesE+4P4fgW+IxztpvvSlL/X96HMNZFYbfOpTn+r/hdAapCu9EqZQ76Pnj20U1v8fAT2b4J+0I8RFB6GNSPt+9m3QMZm+InID9HT48Ic/fGij40MbfBxap3NlgzmET9WjPR31PfVf8YpXXNUNvCtCg7+Q8/BIPF/+D//hPxxaw3xojf+hjfzvo7fzf/Nv/k3fh11dqjzGQK/f+I3f6PXOdcVv//ZvH9ro//CZz3zmyu4JVd6I3HdE++xnP7vbjQ/Jc3jQawlJo45r2+gaG2wFHv/iX/yLQ+ukuk2W9J4DmjEEuxp6CAOK/Z//83+6YTkG48xBeoaTcY1A66kP7XGo/0lECiZ8hTlE7pe//OX+5xqhxXuJriI86JECjdyECjp/8Ytf7H90gi5BfNK7rvRJ41y63/3d3z084hGPOLTRzeEnf/Inr/6co9LmSB+OW3lUuE5c0iRdztfskbSO0vqjg3Rawh7gI4/ocoRRfmS6rwFoo+wrfd1Dm0Y6YQ01HT6hdx64nmro65FsgU9UnRKmUO8n36Gnhz+yiB5TPOq9BH6Bz1T6JUTfNjLusvGhU2TMIXID52hCi1d4j5BGR6ANCJ+pID9tdN4Hdq961auudArfXOMHNS5pBHWdTaca+sgRH/s5D/0cwluQ1+Q3vOSvDgTnEHo6yodrNLm3BmnQ+dMYbU147AWayE4Idj8beBQQPFZ4ZPdY45G5Cbg3xf3R5Fw9xng8NW1i6sK8osc898JXmAN6aA1Ff6zyWAjjo9wSIgMtukzJ1DDC42heGI/pHEf9c01fR/m15wcbmackO2nqERy9BMr5CGkFcD8h8cLa3J404e3c9El91N0D6cmLLR0B3xGxR2sQr178JqDlCzVuDTUdPUK/JQ+hS3pBueTaMWEK9b700R/kc+3ldHSs/NGvld0cWuPQ/8MB3/CJjDlE/6DmxXmOI/ixeC9u1cPUS3FjwM90FpiLB/TxheiYfNc4x/Bx3/sQ7w4iL3Cf30mftsB56OcQ3gKbxX/IIkP+2HWJR0CuOhSZjrCFNvAuIPkb83hdLHvCClJg+U/IOch0lP/ABz7Q4xT+HiMAPmSqkCmUvTwgdFvo3fdypL6QEdacKPmlrxd9HMZcqPlY8UuQv9DvQXRL2ALpdGIclY5x0HOALPzTye7N37lQbSbsRaVzVOYZGGxB6CufvUBnhcux9FOY84XEpyNYK0f/iwxe2mZlEB57yp8sH0t5B3ZuH80xAzJ6LtnVvRqOATo+U1/gH4spXa5lsRjBy66lQnNPsELHW3UG9MIB/Z5M4YHGygZGGTNzDqi0nNMoO/nYgugmWNUCL3vZy65GLkvIVqxbZR2L6OdFlJGZzlNjfy6QxZ46PJU2eTx3Pi8F+QF29JJ7K0YbhM9WhHbv5oJrmCsX8XT0JKgerjW8OqCsqvu1X/u1fuQHeyBf2hlPx3tp94J+ghU+x3z1vxUpd7LAE4S6eA5cu2tUyFYMzDkFJDPWOztaOmSlRM3kFkhrNGDliyV1MdQ5QSbnyocbWyEt3ehqJYNr/4i0hcf/e+9HIXvkHYPYz0oDIG+t0l4HKStlZ2rDNZl78qn8VXqVXTAFZCqoTiH8YYNedlncqs9eG4xImVmu6/w6vILoNKebclCGv/M7v3NVjlNwj05Pe9rTehrLLmEu/RzYVL2wOuecPhqQJ3+C83P4VrWto0Y+Tzx77QPhV0NwLYthlIKcM4R4wZdqenPnlh+hNbqVqa1A65Hf16B6v0vB3Bu5e8A55FHlA6N0S//m7FShU4EtaU+FlOU5ET/RUGucyVT+e+WiY19PWZY0Wtr5hS984axPI1sgH4I8GX1eEmxidJ1O8JxlSZa6mw52CcqcPi984Qv7dbYZTv3YA7zIRXsJyJ92hsy9uh4D+eI755jWvFZDTxkG8DHEUi8rnbW5YH3s1gZvDl7iehF8KXgHkXX+WyGtPOZ/PLMFc+KXkMfh2Pfc8HRlOm2LbtcFGaY2rMEm65hKhM5I3sc5Bg8esX1gdglbLSHy5clc66UQPzFqzkv1c4I8vqwMl97NgXR8mT0EHxfmRfweX9P4ff3Xf/3qNwanAt18oGWaUR4uAX7MRnvsshUnGdHXrxHBsSprdGOXShXgjW9845Vz6MHSoFXaHOfOvZDR0KOv8TmOYSo+GONzL+fJV+JhTDMVOIejrwUdfVQVB608awB50pGNtpkLMBcPsdEYn/McpROiY02f86W4pRCEvwZCZ5Z7o465dqwQh15D8Qu/8Av9Aycv+nxcEh+EkZ8AOVb+Nc1SgKn4MdBBmGvoa9rxeoyHLTqSJ53RfK5hTDeGpAnm7tX4xPFNDWEGXON9cJ4RuOP3fu/3dh0tTki6SlfD3L3krUJ8UNPCnjKugRyNvBD7TqWDehTW0k7FA1uy6VQe9wHPhN/HtUf0gk+c07CphOA6wafZ5vJ89egLUuAo1SEhPPL4WeOSRryv7DJH77oaVvrEJT5x4BxyH8IbxCUNuP7yl7/cp4vG9I6pXEJ4Ok/eTCl44rFdAYgPfc1bDVUWTKVJCI/IDR293Bud1HnSuAbvEaQPxjzlCM5zv9pyvE765FcI+ELmPmtIOjRkJB5C72ge00ZeOkRfInpazHuGyoc+6SzxjK2CUTfHyHNM+hoPyWfuCeJCA/krwehTecFIM3UdO0e2Y/jV66Q3qMj9pAnCJ8F16Gq8UG2fNEkfXs7NKWeOHqTPvUqXMrBHvaP9YJIeaj5D5xgdAvVeXUza0AnOK5/wSBlD7kVG0o3xOcqfkCcQacIHQl+PkLToanzsCpVH4kzZeh+YvB8PvBN+H7u3QKigrKAxM1enNzIF4OWsz55lTJw9XuyJYasDywutuNC4WFtuawBpZMyOcSqIFy7WzKq8+NgWQG9nJKiS2yDJ3Jn17WSgQZtRBkPZx4ZTGOnRSYHZnVIDo5c2wrAhFr0UCDpHvHVc8iXOuT07NL7WuJLnU2dzjTobUzqAtw2Y6IC/+WOf1dvHxv433/7t3955WRmhcmik6GCHPvmjszh5t5WBF0/0lm+fkqNjMyNhtjIlJN/mEdGxh/137KWDxnJQvNBxINd0Q8OmZIvzTQI6vMmTZzaml2P+lCRlyn7i5E/50Y3dPHaqiOQ5ZvqJPLomz+wnDT1VCOWDl/ITlyky5eW9jvJhV6N38slTbu5pODzOsxcbGkjgbWuO7CNklZeAxv7o9Eqe8TLVId98hQ7ynMdnPMmnu/wpb3kRHzo2FucojSkGvPmH8/gjW2WDLv4hr3iThw/f5h/yGHnS4oVOGeAFyoCu6o4y9ITMPpYvuvbEqz6yvXJ0zvbo8COHzOhBf3pYRSMtG6J1rqz5mDqgEaKXvHtKpwM/tXWCPKoXsRde4qShG5+0PcaNGze6DvbXkYYvsw2dlEH+WMSWy67jk/xPOnZWD8XHXnyF35DnWlp5Y1NlzS5sSAfx8SH5Yi/+xp/I44t8Ur7kmY3Zl9+wCd78O76snMnkL9Kwo2t0yoFOzpVD6qr6wubsJR/0oqe2h8+wAZ2E4/AHDXzrdvrvtRt6BcUB7O3AaHmRkMdzld3eEZT+m3/zb/YCcQ+k5TgK0jlertEJwMj4cgrnDMfA0uFFFt54pFFXKaRjVOnogLd04siPPLzJ4rxJxxmSzrWGEh8FgA6f0ElLh+RZOkfXNnfiQPJvuV0qObrYIPIqL47AmdPICuRKgz/50T16BuwQe0onT2gSFz3RsZU0nEze8Mscr7SuY7/Q0QEtOuUhKJvQSes+OvdCp2yip/yRnSk4NBCbyq9GSUOtoVChNUgqmg4XXz7n6dBLbpVTY+9jNPfk1YDCMlyNNrtKY6Op6CBd7EcXesoD2ujAh8TJS/QUkp/YU9pqd9DYqRORJ54eeCUOnTg64I+XdOTjRX5NExsDPcLPufLQQHsHhh5CK59kSiM4r/zYIuWKRhrX6ATgb+IE8uiqHF3z0+ggH3QiF410zgX3fvEXf7E3sk94whP6vj5sL5/Rkw54iWOD6kfKUHrlihe4hwYt+dJFz5SrPEkT3emFThmygfvsnjKkC7rUsbRZaJMueqLFV1zKldxckwPSygt58RFpxLkWssCk2vN4oE1ovy1DtflfRZI7arj01hT0MiwZcC/3zaXapMhbd7s9yoCgIEHFVUnNszIYuhRYMprrBD0jA5LH2fSwdFCx4qiVPuc1DhSka8eMrOpGVIAmI2ky8Q+de4J8uufcPflzbuc9fD2ZpMDl2z30KUw0+IoHjUQqijRQ5YnX+WTEhVa6KjtHAW3kJJ14I1N55rQc0HnkVUQ21KNRj5FSNtSCpI1MAcTJO/6OsZmXc8rNC+HkA+iIdtQHrZU2NjETjJaU/6c+9am+E2m+QQAyBQ2L7ZRtXhfdHEHnQYbGo8qL/PAKjSBNztFrgJR19X+jzDQGED7uAf5gdKks2RA9G0BkVNTr6IaPkahz5eGpBiIXTWRDeIhzrhzUQbLpKz6yK11FeKiHwHcSV2kjo/IxrrSZoLJ405ve1NO7n/YgiP6BfHqiki6dW3g7Rr6jtkCdrYPKoOoTmvAggz08laDjk+7FLytdzkMbevqZTeDTBmvu17wk/XiOVlvmiYX8Rz7ykV2HyLku7mvNDSA4QQXTeDNqVcg5xRnIbmxgVBu4Bxoqey/fc889vXCqAWLY8B35g0Kx6sILXh1ODBq6hJEXOXEsFdV+9rY2tYoDKq20Gb25pnv0pL9HWLJNX6H35a+GyFSOeHPzaENj9Gw7VQ5vx0ZOoaK5nwD0i7716D4dPP7apQ8fj4/RC51zQfrqoNU+7pvSsA+5x2SNbvg71hCaGqSTF09pdh90HtR0FWnEwD06mF/nG0boeEZ+bfQr3Pd0ZOdP+vsIzyOvf+zS6bmf/KOvZV9DeHmkfs1rXtMfo9kudq+VO+kdqw3B04YtrnVYaPF0jL9UeqhxdNQhmOLjM5D8Q00r1PxA8qqzfe1rX9sbCnXCfcfYYImHNPz/Fa94xYN+6Zd+qTfe9Af8g8pDSD75si3HfRRoqkNDhaeQvNSQr3f9uYdj+NBJh6nOqBNG/na2tL98+GTUH4iD8KC7pxq28MJXuzTqUO0BjtLQF9TjO++8s3eeQdInjDzQAx6W+hpw6JSg0iX91DmefM5Ur/oQ+lNh99RNDOLIKDJmNJ7RdOC+0bythD0y+zMAqBk0mvERlaWHGsSMepNmCoyqEBw5lf3oVSpr89N718KtSIHkvnky9J///Of7HvH09Dg1QkWSH/yNzjUs/qzDHupGioJKwsmks0WsucjMI5ufB4UJnFLjrpPyMlFDa9SZEXrm96bygT87m5IwCjQXraP0whdNRle1LKaAj1GsBpquHqWN6NGNdhsR/mxFX9vDku1RHKJz5eOcTDYwLcORVVyjM35kisb+Rx7Jl+TjIY/f8A3f0DtZ6dlTZ+epMqOgVEaQTlmP+9E70l/ZkY9nnpC2AA/pyeYDtqnOXDFfUEZzeYl8T0Ke4HQW9pFXJhBf2QIy+ZN3YabgTFspS1jiQ0959TTCrhpWHYT3HqlLdBzzQHd8PUEoR/5qmk1b4OncADBPh8onZYGObtoE+mr0+Z92whOZxtm+9KY8zavruJSptgUtvugMvKpO8qGT0NkYWHp3p70xpahtWIJ8C/jr7JWDPYPU2zw189MpO1TgQQ+DBvpqT0KzRFehbXnHO97R/dsKpdhtN/6gf24FeO+xZXAXWob6sWXq8PrXv/7wxCc+8dB63757W0VzmMNtt912aI50tX0tmlZ4/ejatp62xv2P//E/Xu24F/5zkKZVrENrfPuWv21kd2iN7hV/AY8pPpFL1zaSO7QG+HD77bcfWqXscdGtQlxriPrulc5/4id+4nDzzTd3uS960Yt6HlpDcWgNSd/KuDliz0tr7PuOfc35O+/o59hGHj3Nb//2bx9+9Vd/te+s+JSnPOVw0003HdoIu29XmvRjPhKPJz6t0zs05zw89KEPPTQnP7QR4mQ+KvCMTs25Dm0EcnjlK195tUXrEtDiHd1aRT20Eenhlltu6eUifkp+pWsNxKGN4Hu61sj1/P7pP/2nD2103uOWEB7S0Z++tnX9xm/8xkPrcK7s4wjktkZ8dpti9K1xOrQG5fD93//9vVy2IvS/+Zu/2cuPDVuD08uFz7g/h+go8JmnPvWph8c97nGbtqh1PwE9WyhH2+y2p51eH1sDvCgfQivP8tEGDIfWWR2e/OQn9y2F5WNKl8jlv8pSutbY9zy3cWNvE1ojfvjO7/zOXrfJCB/ndnLVLkj39Kc//fCYxzzm8PznP7/78c/93M/1+vhf/st/6XmiF1ny0wZDvaxcV9DhTW960+GbvumbDm9+85uvdkYla60MBLqpRy996Ut7Gfyn//SfOi0flS/ypuxQIY26rN63UXm/Du8tkO63fuu3Dq3T7XnYInMWyBLuxa6GPoIdW293aL1OV0ojp0AYh4IcR0G1Xr03dq23v1+m22i872f//ve/vzdOub+UudzXuPzbf/tvu1H/8T/+x/0a/Va0EeWhjX56R6UBp+8cPXmcnlPLmw6mjdQP//k//+fOhwPKe3RjAxVfvgVOs8QbT/rjr6G1T3570rmfveZAHkf/yEc+0hsLuuK5hPCNvsrPXuw6ZNdbEf1VSGWpwXc9pbvrxLM3uyWthsL/Gvzoj/7oqu4V4cn+ylJjTv8q33Gqoa9AT7aOli54VPolkCU/n/70pw/tieRqO2uNtyOs8ZBnfvXc5z63Dxz2ghw6f+xjHzu89rWv7QMRDVTkb4VOqj1lHl74whcefviHf/iqkR35yI84tkrdEaTly2xgX/yPfvSj99nfH+hpQKJu/Kk/9acO7en4qlFHO1cXydJpGKDlfvSgswGjMlAnxa3ZHKSJbu3J+vCkJz2p84gOZApr/NxnK9tE+08F9XGLDrnvKL2OwuC4PaV3vU6J3S9joSnWH1O8NPDI7vHbI4bHwLAzz+RlmX+dMf+YOWCQphmmv6j06OpRDEI/97jiHtke09A3g/ZHXY86aJrz3JtyGXT3uExeQqWt8qUlE8QnJJ+5BunEe/zyCOjfmjzK4k3GiPAIHT7Rrb7cDP8R0QvQsYs5TPKEObqATAGt+Uw0pgGmdJ1C1b9V1P6onRdQUG1a05LXHLmXmzTuyTMdTHesPW4H6PATPGbLv8f22Cy8zcOPL2MrxJkr957AvxyhE6SbSl+RMpAfU3zky5fpKdM60WGJh/tswp9bI9NfAqLbCrR4tEapH9Ur0wfsuIePPNBTWYDpm+hf+biWb7qSoW6LC+o54JmAzlSXaSrnptRMFeGfNEE9l8dAfPQhS1tgCsnLZNf8t9LOAU/p5NuUUOto+hJQwB9fyDTUHE/5EJQ/HbRnSU+f6DrCvaQR+C89TL/V9nIv8ArCY7sXNFQGCtj8mYZeJcMwTBndfB/4P1lpq9LOFUbW1DOEUPnPQRpzrqBSmgPcQldBPrrMSS8VonsKPDpKJ1Sa0bBeNEF26OQEUwgvOqRSOnr3gGfuz6HeQ1dXeSwh+oa/aw28sliSNwJd9OTcWf8LI5/IpJ8KpWF1Lp59zIPubeBAesFgQyeR68jbCh2kOfrQ7wFZKqYGi19pOKyl3gr0gjI4xgZsLeDBjt7X8Ie9NiAXHzzyrgqP0X+lkTbvWtJgBtGnnucaP385aVWJc+/HwP3kW/yoO5smVKBje+WPZmsjD+ShYSu2z1Yuoeej8dMlnuGjc8Wn5mOrLqA9yjuiMf97gDYh2OdR9yLKY8QJOHV6XPf8UbDeiQEtrRyBTropIywZBh0j1CD9WkGMkB5tjmjndOJYRoReAMtrCjFyE0JrNOSljnN7jwA7LAHfWih5QhFX40dE5yA6RJ8lhDfZXorLZ+i3otoidhSmdK580bFJGhDnoa3p1hBZKYMaR589GH0pfByX4D5aedHgJG95mRlbTNkkiP6RtZR2Durfxz/+8f5UFX+LTbai5lt+nOMx2tJ9gZzcD21oHGt8kGsv3ulshU3ic0Q/6k6ebynMIoz2CS19xntLiC6hR1vzg58w1s8R7sXmzsNzzMMUKl8dp7YjPPYCXeVXscsTIrxmpDJXcM419O7bf90IZ1TadYyJh2PCEiLLWvd8IFF57MEob4l+vLeU1moc+uiZs5QsDdocRht4SoqMHKcQukq/xRZJG7BpHsPHEdMaRt2FlO2I2EHZjevuYYvuI6r86lN7EPqEoJ7PIWnIlo80kJ42gy180NP7GBuQO9Icw0d6oeqS+Ckow3RoEPqc12OQ61tuuaXLMUgUl4HiHKRh26l07kXXHLcidkqenQvgqSZToYmbQmikG3ms0VXIWwZcqRPHIPIr/31WGUAxGTMvCRgrDGtBPYZY531dpSvWDH4OkGneOUusliDv1s+zywte8IJ+nKqEW3GJvJLhsVeFTXmeC2SxURrD5O9U/vGHjZqnHC+ByPU+zFPFJRCZgvLbWobKHuhqqiTfH6yBX/JTUyOXxN78XQc6FR/8nUPWSWq1ufY8NvmIRS/99Kc/va+rPqXSeHEUL5vMnV0CCtrc77h2dw42lgLz88c6iY9w4BjaPaCf4OlD+eX6XAhv87pePIJKfk6Zl0TKyvSJfVkuBfbTEPqeJdfnhnro5bn3LXt8lJ7S65CsNTfFa+28EfUS+KcneYPKcw5GAjpaYODlqPNL2FRD7z2X/J1a3rUsRhkhX8OBL12dW/CfEeKplI68PGaRs8fJjgH+HFrjNJcPaaKHD298qHHTTTd15xTWnLgCn7xgPpXd1iBvHhmTh3MgeUn5CTm/VD4vAXkR6gdH9XgOpO7ZcOsStkwe+Xbq9x650vI18/TOvZBNXU4YoS0xuDNAOKefBvQyK5EB0LmAd4JFH+bpj62LlVdCcO2GnkK2EXDUO9uS2FTHt33bt101xqcEmUbY5gfHzJwLRth69ymkQBwtG/OVoJ35jn2Elh9z9IDnJfKno+Zc5+w4w1d+TIN5QrpU/i6F5EXjkK+ELwW2/Cf/5J/0p4lzD4DCWxkqy2PAVjfffHOvx7YOWdNZvix/9UL2EqAL3fIu6Zz2DHRk2hrt5qnlXauhjzJZJWKfF0st7SqYUZtwSpCpQ8lLw3MXAP5G2CrvKMu1IO9GHD4Bd/3c5z73fmm3Ap0Xo+F7LJ89yKjllE9fS2CrS5XfHwbkST24NLK89hJlCBkc7AX9BH5n2wgDRduA8Is5yJenpC3vyk4BMnQu8dNLQFupLQX5PSVO0tBbT88gdqVTWN6oA2NNOQK62oM7F5YKOsDT6FpjH974jGEJ7kd+QuTnOiDPCgrzg1Dv5Tw0H/rQh3phfed3fuekM44yE8b4fCcwBWmTvp7vQeTE3l4AZWngHpCt3MNvC9CYKvKIykbokoecb0VoKo/Er2GkDf0e+RA6ARzZxOZcQe6NiLzQ17CEmqam9V6M/7HrlP9NYeQVn6h852AEqh6GR7Vd4qYQ/dy3/5Dze+655yoOql2c6xTMz3uBO2WzkWa8twVJHz46a+8gttgSjQCVxxoiUzDlay1+jT8VrtXQaxgoYy7LbnNGonpcH0MEcbyKZIBT1cYdvxhrCnEEhV6fFKqB9xonjXt4LMkPqpwUqOkPa+2NUOg38qn8Iy/nQc41ujmfc7Lwc995teMWaIxiTyFluQZpquzI30IL5AhshCZ64yHM5XcORl3hsVUHkHYMe/IRoBnly5+nwCVEVuTiITjfi5TjXqAhMzoIdE9+5hCaWrcjP3zmUMvXuY+nHLUfeOR+5es8fgM5QtJFZuQLaKPXFqgTdeCSDyrXkPQJKcdcb4W0Nd+nxLUaei8PLCf0t27+Exb00JZBLaEWpjltOzjamY6RlxBDGIFmjp5B7RSY/2YF8UuGCl3SONdI2/p3hDQeLY2yK88qS2Nj5QD9baEL1RkrpJdXe8HrGMkOwnPLygJ2+7mf+7mjP7DA35r/X/7lX+55pwsH3cIn9jOiu3HjRv/0ewvoKVifbJoBj5/6qZ/qS+zE02mtkalAw3fsPqkM9iC62CWRHdDXstgKu46Sn9UZgk5MA7YENpTWnLP6w/6xvfglSFfT8gHl6P1Ybai2wvbCysDHfltsIH+gnqPxErjaL7rBkh7uWbQA8pA6hk94pR5oa6y242vujTaocmwFYWdYeu1B8mXloA+5+Gm+uF2C+3RhA7Jtt+F9XeK3wlOu7WJsX66NOCWu1dB7BLdftMbevuAK4C/9pb+0yTBpVDiZbUk5aAy9hurMGjxbnKpwgOcW+bUh9ZhttZBGJw5W4fFNIabQpMGjOpw5Rr3/N33TN13dr8i1I13f+973dptZ0QOxhyB/kTXHR77/zt/5O/29wNYKWiH/Opu77767V4iq3xqiG6e2VbMP5PbIlzbpbVErD3yJDbaMoCpUSoOM7EuyFfJAB42HikU+bMl/hSc55eAlGtrYRhkuQVplYHpAR6GRQytU31xCdEXLfzWWsDUP0gnq77ve9a5eDzJK38JDGg2i/OtodPzhuRUGbWYBlL1thgM2YMP4Cp7qYZ1KqbKklw4fW3fbZ8s24VuAB1ry1Ef/kaBOuK71cg4pL0/ibPmJT3ziSk98t0Ib44W6L5zjj6dC349+LJi5TNV0MqCR9qcJMqgHta+1/aA5i8wvGcc9lcy8vo8nHv3oR6/SpWCNgqRREOhVdn8ckrXu0sT4wai7oIL404y3vOUtvWCe85znXI3EooOCNg9pjavHcQ06eY7ppFRUjQUZGhxPAFYd1Xzk3NFoViNrH3e9v8ruQxAywJYEdd+ZygfIMZ9nbxUVPP/QlRc58jbmfwTb2RDOem862H9bOaair0GaLCHUaZmystoKYuvKR5wgTn7Z1FMZPXTU8NCHPvTKB7YAP4/XfI/u9ufPtFe1gXIe96N3P3ngwyqZlTJ52SdE3yUY8Vk1wh8sF1QGfCb70VdUXngDX9Co+AMXq7WMIt3bagM8+Q5f1Gnmv1iDJf1jA36kw7I/lSdS/JRDdITKBx2wq/wauGicPNVqoOzXIx8pgzHfAnt7kvQknpG6wYujxs7Tqvqh3Pw9onJFo7wyYxBe5Bjs6HTe+c53drqXvvSlvbzVqTkbVHplZkT9hje8oa/u8Ucs5GU6Gub4oA8f9N4h2uNL3sf8L0Hb4T0f+f48ZQ/tKpqCfZvMGubgXjNID62C9v237cNtL/Km1OEd73jH1TaxS3C/FUzfCtX2uLbyxS+850A+/s0pD2300veytj3p5z//+ftshTqF5A3/VqCHZtDDIx/5yL4Xe2vs+nakzdD3kx+d8G9O3Pc8b87Ut1cW7D3dCrVvydwqWN8Sl26jLpGdo3zgZ1the2k/6lGP6tvktkavb1kcXUY+0Sc8WmU7/MAP/EDf/7uNiPoWq2joOwe07gts0Zzz8LrXve7QGot+byvoSJ49+pVDq6RXus3ln8w28uvb0jrHo1X2vh+8/xWQp63AT3o2f9jDHtb3U0++kg9y5/ajj07ybT98vsAPKu0ayLetNF+0RTC/xiPb6YbHyIuO4hxtafyMZzyj/z+CLaO3yK0gB48PfOADfetu+aHXmi0jXzo6t4b60BrVXq/9t0LskGMQOmXYOu1+Hz0726b3W7/1W3u70Ea2k/mWnt/Yvtz20LaY1n60J/q+XXDrsLovqJe2MEaTgNYRX4HurYHvWwyz37vf/e6+RTP+7o3yR+AlnS2QbRGsPeLH4thUIHOJj3v8WH7UYf6MPnquQRoy6N0GvX0bZ/SnxLZhQ4EeptH1EZmex6jwV3/1V/to15YHSbMGozpf0LYGrn8NhqewROs+GAmSbU7dKDz/k7mEytdIwijWP0qZasoLIaOYZvR7U/0+yDRKaQXRp5ae+MQn9i/6br311i7bCCrr0B/zmMf0HRCtDIiuFXpoiC5GQ0aR9gTyzzT2xsnWz3PTWGgFvOhrRPi85z2vyzS6p4cQWVMIDzBCMpo3KjUyHvO/hOjYKnanM6p0TFmOEEeuEZKRVq7/3J/7c/0fxjz2Rq8tkJYN5N17DR+rKSeYkj8F+hrFeiIxutyTf6CD8nrsYx97tSDB6NPX4ktAR0flxBZGgEa22eRrD5Q3Hv4lTF2yTfYWOyZNfOlxj3tc9yVbmNCDfimjKZCZp0i+wJd9KJl/ezPfPJYDWfg5Knfp/fWepwi2x1Od8nSkXnsqiq1aZ3K1yifl5NwUDZu//OUv73XSE438hG4O7kUf/3LmSZzueTLNXjdLPAL6mML0FEFvfPfC04zVfeoC/U+Kloneo9QwB/f0PILRqD/JMMJtmeqjer2oXkxYAj6twA7NMH30Gp7Cknz38TZq0tP7o46MwIQlJG/o//f//t/932PwSc8vfkq+OH+OYIQjjbRJ54i+dXZ9NHLXXXf1OOlGPknvmHPBSEAeHOljlES/3B/5JD6BPmiN7P0ZS5UzB/eSX7SeQPzhgn/wEb8V0QEP/wpkROd8Kf/uSZM8x55GUey8ZySDp/TKgGz0ow9KMzWiTxoBD38k0zr/K31gzMMUpCXTyM9oVBni6anFvfAYebkXWdLj4Q806LpF7gg8Wkff82Fkibe4JZAjSOeIRrm0DrfXj8SP+iRf0iYPyU/y4s83/sf/+B/308F1Qmik969W2pH3vve9VzyThix2Nepmn8Q7SkuOUNuC3Bt1H5G0v/mbv9n/UCj6iMPP+ZQNKvBA95nPfKY/3UW245p8kAYP/uuJnkzXp0T/45GGe5v938dcDw7Sut+U63NYz3/+8/t8mnk1o/OtwCdhT+9Frvlt83TmsdMjnwvN4H2kaZRglEZW5NHFy2ijfHHeHWQ0UNNtBVvg6aWYedPw2MtnK2J/c8RGxPnke+9oJHyW6NwH9vQ015y5j15q/uR9zRfQC7aOzqgTPFla1ptVEuHpfMsfj5DtKA/H5h8PvFsF7/8r7Amt8hrlBtEpOuwdzUX23/pbf6v/2bsyxG9O3ilAnjl1I3dP2HttBvQOvMDUlnhS/omf+Imue+XpqVq9aI1vf5eTPJ4Ctewik49loYSnDfFL8vBQ7sruWFuYpfDewxMmPqcsv/0aFVhpYNWFl2B7GnmIUfcahUE4l0bpGIPuBXleMnuEdB445xw6OPAH4Nm64Dpgl7z8qfLOCY0kewrHYqtTKjOP2aMs9FsbOC+7DC403vxPI2HFydra9TnEF4+tXKGv/mhhwF6Ez17ET2677baLNPKAP3vv7ZSmQH/TTvJuGliDOepvisjHROriKWSOILvKdM5HyV0rE/pLf8wgqcI0rFVIY95PgWu1lNYwa+ys+HA8BnszxZB1Hf05jFJBXgpbqPKcW5LpaH4zhXxdvcaR7rnBQYWpCrYFW3TNfX6iI+PUx0ClEnx9bFkmH7zrrrv6HHvsfyzSYO5BpZFHOujIDH62otpuzY5zoIc55dFHzwXy1MEMSo4BHgJ9PYmpQ1acGb1PgV11LtXmpwD5CYFzPiqP0XEJ0ek6uhnAGixukbcE9AnBrpqRxtxRsH4WMw19ZR4B43XFVJrwr3GJz9EjlaWQHuHE1fS5X9MnJE1FjUuaei7g5xG1vgTKPY9Znmo0kJaHSgtjunqeEORcmsDUBiRt7uV6DFP810L4AXkql0ai3gfnlT+M8gLnY9qci2cfMkz51fKDSpfzKTlGc6bRvMhT8S2J83GKbSrcS7ogfFScyjeovFO5cl2PoanpYeTpXD499te4HMeQ+KQNv6k0wdR9dBYnxK713ngO0k/JSgDHpMl1zuH3fu/3ellC0uZ8TDslCxxDZxmlumSZJlTZzr/yla/0upg4qHzqOUzpsyVE7u/8zu/0EB5z/PhNznM/RwFdeOZevU67oezksabbgyW6vo7+3vMrzPUmYUQxUxbWT+thX/jCF/ZRr6Cy+apLzyytXsqHFN4o22KUYxgFiJNGxvLII42C1PDoSTXoVshY1YIvWR7b49DmzuiCBi986CDOozx5ZKHVkHmrTq4GQh410lY6cC56clzz1dkKFS8NiflWemZ9MTrzab5otAbcXj9WD6DDl57moMmjr7yA++bxOY804BovBR955pTpSyfy5I8d6I5O/uiNDi8jOfLMYbKVR050ZLOpNHjTLbxAOnTkybsyIFMjJY/orTpAR5agTOQHT7qTF92lF4ee3dkweWZjeip/DTMfUV74y2vKnnx6iMNPnumWSpAOSRBnnp5eGgl5tmrJ6hnrrwV5sxrIunbfNuAf3ekpb/yBLPajr3c/dGcnge70ZKuUPd3ZT3mh47PyjJ9r8owIU17kyTO+4gwS+Cz7hRd9yDOoYCty+RC70FE6tnafveiMl3j8vC/yrkX5pKzJQIM/HekhLRplljR4oZM2fpS6Kt/ei7F/6pN8Wz/vqOONj7CP8qK/fOMVefJAd3mKjyhHvF2b7jINx9a+rUn50B0PNmVD78HQ0d218kTDXuSLd59t+Ah56F3j51z9cY8/KBPthXi82IHf8Z20PXR1H392RMc+ZMk3u4lDpwz5ERsAu6FjR3mRFo2At3Iglz3pJH/0FK6L8Ng1okfEKIz47ne/u5/bqZJDy4RrcK6wEydjKqjrjAAUSuJAOpnlCI5koXMu80kjPQcKH3HuC3iiAXT45xqNOPTi5AEP1+gg/NGFv6P7jujwia6+wHP+zGc+s+cXnWPsFD7oxOU6vIG86CAN/njg7zxx8icOX3AuDq/Ehc51dEic/LoOXfKMd/g70hVP6ar96KhsHEG8c3SRjz75CcKLvOglnfjwFhe9pOH8KoepGV/uvv/97+8vGh19PakBkB98NFAaGsAPHxVSRZNOUPnIYwv8ky75w8vRfSGI7dCCNMkfPoJ80Vu6AF3koZFOGnSxg3uupZMPNnBNp/hC5MVvKkKPFj/3HcXFxqFNXLVz9JDOtXShc5500gAeuY48POiIJrpGL2nwc42X+xB5yU/opLE0VLjnnnt6Ays+eqLBC7241F900gXuVXnhL6BHJ408usZbGtdkSJf84COtIC+O0kgLyYt0eCVNrgGfGidNeIe/a8h5wimxa9VNhKtcvn5jaF+GWveakR9M8UtcMuqY81S0xEGuK12ODGSEILiP3lFIOsj5GF8RuqpDhcLMSFqPXfXSwFtzbARp3a97Gilwjid6caETnAdV95wbNes80UafyAx94sXBVDrHnAv4sVnki+NcHFfeyBcqD4gMEDfqD0kTOsfkJ3zQCeLoYiQDSR84p5d4tst9R3Qgnh/wv5tuuulqRRRIG9lgZKbTyKobiN6QuCDyKo8p5J70KZPQqiOeDILoNofQ57zaNXyDKZ3EfelLX7r6piTpp+jwrr4VjLKS3jF+7Vygo4bLCFX60VY5D7/qe+EP0oVWsOrG1I2nZWvSYwf+oF5Iky/ghZHfGFfvJb7KDBIPjgKd0cRP5yAtejyr7Mov/B2rLEAD5Al5egufrag8g/BYHdEjDoMo6gMFjbyXhuamNRKQTMhwDTXOOadxDolLiIykcazpgXPFEOJDkzDyHK9riAznkVODeI1u8pg4wdYDaOxvA9EpvBwTF945T3AtPudAljwlfT1CjRv5jEf3ck2XisjQSdd0lUfOE1xXmXgE7odP7oem3qdHKk/opanplDGHD33lg16F8OhrxVBtUCOj0uQ8wX1yR9k1SDd3LyH8Y0fBOd2y6iZpxhD5gope08WGQs4rbe4l4AH5UKfST9HBVJq59NIGuSdOGYUm9ypNZIiP7+U6wTU4Bx8corOaKo2mozg29fQWGe5Fxlyc64TEO7pOmjFd0sZPE1fT1xB+kCMol/BKOseRFzlVnvOU6R7gW+VX/EHruYAQU0rP6itKeNGLXtTnrsw/pVfagyiWAk3cHMgH83LmyVJR0FTDHGOkOZBpGsCot8oxSqSHL/HYYKtM6ZbSkmHeD6bSJa4el/hNYbSxvKUMjuFVy28LTAGZ2yWLfavMNT6pBI7KxVexeZrco0PS47U3z1MY+XhXsBWj3nvyAclHdkwcdVlCTRebwBq9diDvTCC0W+VOAb2FHXjbtyf8HPmYOp+6sQQ0NWxFTa+zJi9+ugY65hg+aa+2gi1NWW2RtxermiQDhKcx19uqaD7bVihebOxVLsbAsx7X+LhPphCjQnQLovcWRG5ChQLPizsyBLzNFTu3fh5GujmMsqboyIycEZwn8WyAfsz7GqrdHDlX5hSn9JkDXTzZVX5LkIbO9M18pbjQ5ih+KaRzsPLGUttch34LqizYSx/IC7rwcRSnkVjCmD58jgFasLke39nKJ+kcBbRbkQ43NMnDdYDeMlm8dZTZQiF+5iWnQWX0nkPys5ZuCfIl8Nc1RE5sUK/3wADI4DFt27FAmxBs6nKiOEJvyDV8vgb1KJWVM3uNGqOYd7PFad6wV+VGRIYpo6xk4QR6eSsr0ohMAW0Nke88lUOD5+VKTcfpzM97TAXXXtjYG8Ojln/1QU9v6cNTGJF46TktvaOHAI71P2NHuI+eU9hL3YqCKluInDnknrf8W7dynYLG2j7mplAgcoXkZ0Tsad20fNiq2RSYihzaNUiTxp18L8UjL/R4LyE2s9pCgxLfWZPvftI4yruPfDTskckv8k9BS0DP//mu/I95WELVwZMNf6xTWFvzIp080CEvIiG6TCF8rSqRVz7oPVZoHaWZkp+43K8B5MFTGj/xp+Hi6eLatEbek420SUe2oC7HL5dQefEJ11Y0qRvyl3bG/SWEh/ZDWWgnE7+G0GpjHH/lV37lamSfcF3serYg0OfJ8LSnPa07hpdhMmc+dSuiPKMqzA9/+MN9pcRapnI/c1rOvfTy0YylZfiloxj5uCdUJxTEcVIvVe3J7fGp0jr3ok/DhLcXgNKQZ36YLkKcpPIdEd00UmQJlnOKR5PRg+s0ZKEJXAvSKgvbsqajjczosQay3/rWt3YbygOe5G4FOX/37/7dbv/aUE/pDeFdXzb5uvV973tft+kUzYjkP8EyXzZQhgH5a/mguzQq5A//8A/3gYbrNFZzqDqSo8P+sR/7sT6VF3pp+MsSUlbKjv0MmPCDKmMNaOTdKriHP/zhnXbO/iOkU26WQNqq27Jleq35QWSogzq47//+7+9liE/syg7Ok6eK5J1sgwVpxCXeIgdQJwOyrMixCIT81JXU58hyT8dju3R+sWSH0EQ2XjpN/31t8Elm/HSNT+ht1GjLaX4RfbZAWlOQbIhemYo7FTbX6hjEv9gYkRnJWqfs33nsyxLDb4G0jOgxzP7LPnax/heWHAzooVNgCJVTJfN1qq/q9Ph0nELliwcHY1QOap5do8vQWYMcSJu1w2gUwrOe9ax+7d7b3va2vvTPzpFW31jXbZQ4h/B+xjOe0fW346f90DlYkE6Pzo4ViVcGL3jBC3pDZfc/nS4k/2sOppJ5TH7Ywx52tVQWDWfdCpXgqU99al/RYo0zHuTjM+oNuedpJFMbT3rSk3qHY/00ndb0rpDWrpH8ScXMiHRO/gh0dlzU2Oo0zceu+R9ERzLQG/kpQ3zck4+1Ofrw8HLx677u6x70kpe8pNPTac6H52D6yktMf7ZhNAp0WONDB6NI34H40tgnNfx3TQd0AnupB3Zf1cDdfvvt/SNK8WnAx3LItbIyWOLD6p89eu64446eh8yL201VXVKm5GREH/n0VF6Onoy0RW984xu7LdHkQ8454AHxeaNoX1l7uuGX9OCneCTtFKKPQH9PEr4D2Av0VhvpqMb/MrguNn8wJbO2YfWRlOWUtur1zz4KwjIoo9stlQSk4wif/OQnu2NxFI9rwZT8wD0NpALQOKtQNkPyMYyGp+pQ+aSRcURjJKmD0GHYXtg+IZxdoz7KN1XF0TiZikFvnRQ7yDddpNHY+biKE5s/XsqHgrQdqYqugzAy1cD7dN7o1rRYHKzySRx9VHCVVIUw/cGGWVM+0o1wT2dhi2bTH+xiymjs6JYgnWk0oxcbv9lUDM/cm+JDjidADRF7yoNVTf70gh9lVcUWsAX5RnoaGLbz9DXajV0NSvCHxEuncyfTH2c4V0GX5KOB8PB0oqG29468/5k/82d6PH/IFruQYxB/RC8P3vmYn7WCKy+X1yANffi9cpQH/nfzzTd3P5aPJT5o6UFvK3Y8VXhC4wMGX/gGlU/oPN1qfHV2j3jEI7odTaMJYACovow6uBaUP1kGecqQPDz5tnyoB/KhfpKnIVbXdKzJm3jvJnTUpkzQPve5z+1tgjy5HuVXyIv7Bn06a3/E87rXva6XgwEJ0AGW+NBD/b9x40bvtFL/E9ZAD1O5tpg2aP2O7/iO+9Bu4bGIJmARrbe72nbzCU94wqE5xeE973lP3063jWwPr3jFKw6tYPv9rcCz9XqH1pv3PytpPWePE5bQjNnTtMb50CrW4aEPfWjfGpYua/LRhb6NwA9PfOITD2103rfHbZ3O1XaiOQbStx7+0Byhn7dG/NAc89AMf2iNc9e9dTyH1tj1rZPbCPtqq9IlkCPg1xrp/qcX/jTii1/8Yt/iNbouQRq606+NXvqfOLhGt0YLbEZPf57y6Ec/+tCe1lb1rogcf57ijzdss+saj1F+8uPYKvPVH1ZIqzxbJ9u32d0rX/rWIPT81z++CR/yprYphujUKnSX788r6LYH5JD55je/+dA6/p4v163Sdt5bgMc999xzePCDH3xog6n76LgFya8/8lEn7r777l1b3ZIn8N3v/u7vPjzkIQ/pNuNL+I76xG7szl7SxA62OH7hC1/Y/QmPrToAfq1TPrQO7/DMZz6zb/0tT3i3EXv/M5fWAfRrfOmh7FqjfnjpS1/a/Y/dR33nIF309mdCN9100+Ef/IN/cNUe4C1E1hKUuz+eEX67bL2+RJd70tGhdViH9oTa/7hkTd4U0Iwh6B9M3dvmT6Il7kePRkYoRqteFnjk1/vpiZuife4sLyy3QG9tTtBoIB9AwFLPFVWNCM0rN+NcPeLouZdGYsmHo7xIm8fA2uuTUXWQ3qOUUYQRiycQj4RGH0b1RsE1PVtEjzl9kg+8pXGUF6MVozkwygrfyr8CH7RkmvoxAjGiTX7m6ACd+45oPZkpS/RLdhxBdqssfQqJ7sqDXnhUPuSA9JGbEaNro0nl6qmkjiSXkPwLXp4ZHWfPm+RfGvPnc9sUoxVnNG3aw9bCW+UDenniz56q4hvKwpPKFlviwYaeijwdKIeq4xrIlwdPZuznKTVPFmyxBrSAjxE1PWwFXEfOoz505rOxtXQpW/HqpxerRvxb84In4OPpxJ8CedL2N52t8e1PZo6eFJIWb/nle7Xct8iML7C9Ok5XTxeutWX0IEd5Ltkx6bSHdNOeuR7rwIjId5Se/5ku8pTOl7eUXUXKsSJ2WG3oZQLM4/oXJA1Q5iIBuUCprYrJlBA6qPzmjBNZlnN6nOKI6EK7JB8dmYC/c3RVhylIp+AVtrT+Y5YdNIgal8RHh0DcUj4iG0a9TIWYEkq6kXfgfiqXI3p5wcP5El09olU5OTe6Ob1HoI/ukRmM8qWNPI2gAUOmmdBK6/5e+RAbRIZzdkjcUkMP6OngnvLcKh9Ch4cjufzF9ODWfxpCJ8SWe2wAVb5j8hB91iAdpAzpHzq2Gu0lvWCAkHnzqm/ui5uin0PoBO9t2M97JP/cFL+u+ZEOwr/KzDXMyc99fIWkI4MdNbriMhCd4wPqT/SraZfoyHfPUWB313i4rnndguSnIrJXvSkFnz+/9ifctQA5lZFYLeg1oJM+GQqcR7E5MKZeXUHgEblrdBCdodKRO2UkEC//Gibn/iEe2iNu5zFFV+XModIlfeLIk88tiGPVEcAW+TUNeSosbJUbhA/ZKQ/XU3YRJ5CRhglCkzLZA7wqPX8c/WoNoRP2InLZP/PRro2Mg7267EXKTMMk7/GDvbykp6ujfCSMkEaIn4YuaV3vlT3Ck4lG3lO/gUF8xpNTMOp4rEz08ZvqO+QJa4jN8Kg6VJusIem0beiiw6mwahkCjU6s2ZYRuzQmM1HOIwsFtwIdHgLDOiZuyTC5Z5rI8rXQ5riEKtP51PUUxHO2vAD2Rh6NlTecL3zCI3xynEJNK0SPhK0rkCr9yG8J0lR4gtBZA1tuRWQlVPlTOuSepzFTXs7j0KGZopsDfin7hKrDVlSaPXQQWgHQ0ylTJ1sQ2WkojtGBHb1IDfCITmuI/tGD/rmeQnT0RJapxtBEbtLM8ZhDaPGz2IFfWmZpxGyUb3WPvIZ/ZORY5Y3XI3K/0uZasFvtltG8e/QV0NXjEsIzR22MPMrfkrxjsOpRhFoVQnF/Am0OVk/uOpU0vfq5ETmOkX1ukOPRFHxYAzoZ7yiOGQFugTwq6EvkkS3ByOXUzjVCflKGfMg1mZfwnUsg5SVP57blCDZca1hOBfk8t2/ib9klWIEDtTG9BFI3LtXepM6fw382WcxLJgpY+51RRxRytKztEo5NLjleytRHuHNC/rLsMGujbeRWbXBK4Jkv+uJo54R8eGlo6iaN77kQf/Eo7unoUhX2UogvsGN85VJg17UlvacEOcrwXPWQ71uoYTrsnnvu6VNjXu4aZV+iXoCXvFm4cQm7ejqyF/056uD9ahohCTGoDxu8+Hze857X42vGHfMYfm5EJ9NEjq7PDfljfKN663Rd+1gs906J5Cd2P7dN6S94BNeB71k1dQwiT/7qHP0lyvHS0NhfEuxqTvtSUH7yGF89Nfi+wYftvy2+MGWjkd+7IulYpK3J4OcSPqqNURdTT06JyZYkmVKYdpGz9O3FL35x71GrAs4Zw+qTcxX4CDLN0RthX8L48uVjCqOX7G/jq7lzQZ7izM5PXeBTMDeYSntuefibo+dLl5B3SciLYHphz3/GXhdkqqsGZI7nBnkaYmVoAHgqxH6xIf+wzNPRh0g+KNIWXQJ0MEW7tpXFKaGN8TSvDOX5lJhs6GWSIMb2Sa7rfKk1QoFbZnWJRhemdDgnyFMAvsqz1tbXrB6xxF8qz+eG/HlK+qOUpz8MsF0q6CX9lFxTG9lv59y4hI+QIS+mo5x7P8ZPTadcCinDS+QXrOyTv3P4zqxXMLL5N9sc6LnNS8twzXycOuvZLwUfKpk7uwTYwQjUaN6o12f+IO/nqFTsmJ3r8L+Ek3lEzgcilyhHSw+zRvlSlehSkCd+Ym/4S4I/ZrXWpaAMvW85B9hRnh7ykIf0a/XPChwd2iXAL30vpHO5RJ0A9V1dNGtwapmTLRUDy6j9OxSkjaOA8CiQc2l9DbZHMbzRaVjyoUHC2qOnNJ400kDkMSchvHN/CklTAz4JKmr4ypf9WOwnA95TiAd8atgC6SKjygmP2HSOZ9IL6B3FCVsRHQTn3rEAR9tbjuFT5c/pXlHzKNT8bAU6/hP5e2ilj9yEMR9rGOmSly02TNr4QC1L8VtQ5Tna7ymyt/CI/mM+nK/RJy1Im2vnS7S55xiZkR/a3AMfTZmmZR/7FflyeETSjyG8Y1uhIrIC6aUVcq/en4M0kRd/xMNxC30gn9qa6LqHFpT9GILJhj4jSX/GTKBNhUZCkBm9Huwd3eIvMIYXLUbortf40MHLUStFnMfAPmEH59F/CclLDBrjyhOEh2sdSzZ7MsJwDXsLMiDbqNZooTpV8lM3wxoRvci25tacXmj3gnyrQ/CSpy12q6Crshsfp+kypY/04Akwc/Qqb/5MAs0e+WA6jQ6xI2zhIb/ybdqRHZ3TZ68dPfX6zsJoM3LZ0YqRNdCZTPkHOqDdC77EDka7GQ1uzQf56p4tMDRSW+nYL+vo2dCWB7Bmw3ov9kLjPCFp2MKT5pOf/OSum51zpxYMhE/o2ZFtHU1JztmUnFFXuqDx3oqPmqMP/zWgBd+loMd7Ky3IG7vy5/rB3X7QI+H3cT8LUI6R7Lvg4yCw6x+jjUpzblvU+rx8zwdTtWA9kv3gD/7g1YcCWxB6OnFSDmDXN84Aa3xq4Sat3QbtSukpxlxZCo2T2NuHHNM2OhQ7TVrb62kHvZBCrWEO9MbDPh54x7ahYX+Y4lH18l8Ar33ta7tTJX4Lwtecrn8J89eQHKs2lltAb9tE2787tEv0bCTQVXB+9913960JDBjQbmnopAsP20Oj54tT5ToHupPF7/7aX/trV++Z1uhGaKR9LW5/mIr44hySVz5km2sdbmxT87EFOto3vOEN/YXlnhVp0pBlCkYe+CRattmClIEBoT3p7QOFdot89w007M+js+fDoXGs53bGpKetqC2MGOGedBpJstVV7xZtge5PYVK/5hB5ykM9sIul3VBrfdhSJugtTHn961/f/9CHLlv8OVAHtKe2Hdf+Lum8jDTyf9AmTGohU/kazZdhdaMrygMlKGNVzt45JWkFTm6rYD2ZUayCWgO51tFzboWiYVYoNgLCc8mwaOmvAOXNsi3O/UM/9EO9sdBgmR+PHskTBwMrKTiaPafR2PuaM/3zf/7Pe0eRjdZio4paaPg/+MEP7o+h9s7xQRq6yMscfaUJ5E86Mp7ylKf0Bkr+OXMc2r1KP/JBL86Um39CUgZoUlEqKp8gcXSxJa5KJf9V7sgnEK/clB96m4jpcFTILYgMYG8v64yejYJ00LVS5ziF2NAKJ6s55CHbXGwFHrYYNurT6Wpkk3d1YwkpA0+mdDBQ0fDBnA7iEyo8adobyYArHSbMlUEQX6LD4x//+N5RGI3OyQ9iu9RBCzW0AQZs8eUpPYPEexqypYi69+Y3v7nbwFbonk7wzmj82c9+dk+v3usQqw84J09nZZvg7GnkfyNsX+w+HsnriMoHD4OeT33qU/2DSPqlTJaAL78D203zxa37HEF0wEP+DT6yWeOcDffifvvRx3B6FZXXH2NkOWEUT0EbARuR+0cYe1NMGXIKlJfWaNaUyEtf+tL77AO+xoczS6NA/DuVL3btdOdFBh50c7/ySUFwHo/aaO1nr6E3AlCwt99+e5+myq6WcQ4jFqOOF77whX3PentVKwijOT23RoItrAzQeZlamnpBHZ3oZ178pptu6jzsxU53ja7Hbw1YNvwKTUX0woMsnYVdG5UBJH2OsXcFHTx2o7GXt4quU9fYj2lhLo58+4+zr5FX7B8dA+eJ1zhIL88aKOWpw6aDF9/SxMFHueJ11Kasfvqnf7o30vQ2os/e9jW/jhqIuf3oyTOa9JeGOnKdL/lbgI8GTnoNzJ/9s3+2L/3FV7lmCi7yRqCTFzrfeeedfU95A5ZR/hR9+ArSk6uRUX46z9ig8hI3QjlkP3qDNnbK/wpMya3g6/jzIXm39NGgwUthdhwHTAE9YnvLJ/m8RvVLX/pSH1SxpY802VBHyo7+IIjfOLdfO18SNM7+cMSTnYGarVrY05Pqd33Xd/WpVvxHmwJ6ejhq0HV06rql5PkHPfezzcOcPaRxz8DRk5XvbPwhT9JP5R/GeO2QDko90rnReU7mMmoHf2++m9D7oGX60HrUQyukQxtp9/3RW8N/tbdyjm3U2feMth99K6SeZivwaIV4eO5zn9v3gW4V/YqvMIfIJ6859eHbv/3b+/7d6MkP/RQf161S9T23W6U63HHHHZ32N37jNzp9c9qrfa6ji2NrdA/NcQ7N6IfWkPc46cijh72nWw/e9+T/xCc+cWijib4/u3Qjqm7u49FGLoc2iuh7+9uLGs/mYPfZc7uiXjtvHdehdTKH1vn0Pc1rWdVQ4Vo6oVWeTt86ykPrtO6n90if88STbx/wW2655dAeWzt9dK9IevfQtIb5yo5saC93e4GLW4L7rUIcWud4aE8Bveza4/+hVc5Da6iuZMcGMLUffezk2DqCQ+vgD7fddlvXbQ+iT3tcP9x6663dj/BsDcf9bFAR/dArb//xoAzt5z7SRefAdQ3h4X8NWoN7+MxnPnOVvyWglUaQ7/ZkdXjyk5/cyxL9HCJXGnToyZd3ZeB/K970pjf1ukO3pB8DOvfx4Ifo24i9+6H6fdddd/W92aVrDXdvk1oD2mlCq+y1Qe2pttP534jU5dgggcwK19LRnf9rT+wJnzIUL0zRVrjHD9Ou+K+NyJ6SmfgEENcGnf0/EZQfvUba7UCX8Pu43zbFTWDv2VrG+4hTbwl6lgQkemDzkh7/jYA9PhrdbAEeHs0sQTOSMHpIfD2OIFdAaxSGzihOXHRbQrJq9Ge6qKZvRr1PD+pcnJGm3tXow+iRzPBJWtehr6j8nVe6nDu6NjKRLyMzowujkkofkFFpyYXsx25UiE45Bmiqbu6hFee8OWmfPjFFR34tx8iKLuTVc/SuPTYrCyMwEFdl4iOINzID+mbUF/091UgTGVD5tArQnzj99aAnTpvseULjj57K0Ce/4WOUmkf66ED32A74omkLG2lV2WsIP1NobGA/ennyVOZLzqp7hXy4F7vQxROuzdDm6Ob0Cr18tkaz10V2yNNGEFlzcE+9kg9PP0t2kLY1gp2/ulR582P586RFh8qnyg8N3WOL6Jvz3DOl4aUssLX64R5fSD5DQ7Y49+Nf7k0h8W1A232fDwOdtRPuq/tLkIZPm7LRHtIt8ulT81/z6tw9184z/cgHQldpr4P7Td1gbBrFNrwK22NYDBnBgsI1t2iKwrxZHrkFlU7QCKClvMZLoQhpZD26y7A0jhyk9aadVhpOKy4v2vB2bg6PPI/9ZDByGg+QjjNISwYjChpT8sVpVMIbD3GO0ojDg5OYgzcl86hHPaq/PAX6ScOhpKGb/NGh5kdllweypPFohlZcHCH5xUcFZyuPwNKLkw5fuoaOXLzEO9f5cC7pU/lyjrc8i4PoQD96oSePbCHyql5o0aUCo8PTffHua+BDyzbOk2e2kVfpHE27iJNGObM7OcognT4eaJNnvHRI9HJtRYTHdTqhMeVGBzQ6AOUY2RpxHZkpCbqTl/JyTbaj/Mc/44/S0JlMdHgKoYtvy5vpCnnQWFrFokwAL7qzV+wgH/jIp7zhZ7rO/ZRNdJBOGjoAHcWHlh54m/bw2K9eQHSLfdkydCAudqZP7G+qgozk25E98UtZ4GVqxTFfx5KFH92z62NsKM8C/yFLPFnSpJ4IaCF6pmx0Xv7GVHp7bmmQ6SPQL34U/5OOruSlrMXhR74gfeTJOx7ybeCKh3dlaMULbCy4F7vgI861qVT8kg5PINs1XchHV8teHLnxK7aLvU6F+zX0nEugtIacYMoII6KMbYoZgoEUnoqc5Y4KSIY0JpyFcVUABpSOoTKfzXFUEunJl8YoWhzeZHAA/2YE5ocZTYGQR1fpQEXDW4fA4IzoRSr54hiXfPO38ksHxk4jIV/yYq7PtacbL53Ek0UvkD86ePmGF/kqioLFiyNm/wqdj3k86cnDi104uvte6mkQ6E5PaTiQ/NEVL7ylNfoUH3vSnV5kyB898GVT9/CSZ3pL55w8ts4bfjRo2Rt/eZA/epIlcFaNrEpJJ2mVCx3IQ+eITnlEdzzIk3c6sSlZ0ikbaZQ3PqH7+3//7/fGy3sQL7y9CzDq0aCrDPT1bsT99tjd7WwQ4Foe8SDL0wI668zJrP5Id3F8BY04vsfGsRXd6SnP8qeMyBb4ozh5oBO74i3IC7srI3qgI08ZslHyTL54cWRKz8bxx8iTDsjiR2jpxUfoqTMzh63DUp74s4Hz2Jmu8hj9+Qvd2IRMwFu5opMH99Qd9PKjPpEtTnmSx15kqQP0V9Z8Jf6tTOmPB3+TN3Zgr8iTT3UaL50zGanT7Ox9mrS+TjcYRUOmesH2dHE/9ZCedJe/2FkcG0sXuzhGT2Utz+SJo692AD980Wm40cknXnSgJ/vhwaYpm9Qddge8IP6Ajh3kIS/w2YBOaE+FyambKmRJmHRp5DgYpRWS+LBFn/P0aDCmE1+vIdfSJrhmSAXGQCAe6EKeY9JX5H7lWdM6j/zI8SgmXiPjhVP413w5l7fwCdjSdfIcVPnOwwfw5sCZAhkR/pVmRHjjFV0TR89c4+E+B+aUGWGFd2hcO475du1c/iqdo3SBayH8VALXZIYOL+eBTkaDrVFyHxw1NFYsWVGRqSb/QOSej9nSmUDkecn3D//hP3zQHXfccaW/kPuVf+IgOiUvlS5IXLWLRkxFHv+WMfcrPUzxDF1s7n7owXmlk85eN75g12nkPjiGB58E1wJ/yHWOaQNSrnhHj9x3VI7SqIfSJ03kBokTkg9BPIRfEJk1TidrKa/VOa94xSv6Ch2QrvqfECQucB1UHUY6cfhq7J3zudC6jr7SjLriIy4hvAVpxriazmCDTT2tJO2pMPlXgolyTAamIIOQ9NI6rwrWjLkf4yR+RI2vx2Q8jgljAzoFNBBdcz2CjMgWpLdkzvy8xj5zyJU+eiVPOcKYrl5XiI/MnHOk5C1x9TxlUuWFvqZPXI5BrtE7T8A3vJNefBAaaRJf+Y6oPATXqSTOR94Q/hkUSBOdnKNXAT2IWgFlpY5G9a1vfWtfKaasIisw0rU+2Rw9kBHeU5jSu/IMXeyQ+wk1fbVVIH6q7JLOeb0ekfRJI7BLpioct/IKwjN6Vb0TDykL8iDXFTW98wC/mn7kP2Wr0MiXpxvvoIzovc9Al/uOrudkB6EJQjd3jpfzdCauk2aE+MojiD7Rs/IWlzoB4e96Ts4xuH8pNRAQYUuIIh4Ro3wUTIDEQ81ATZdQ46UVYuTA45LHpyDpp0IQXlNphKpXjM2xxFteKn8w0kBo6ek81zVdva6hwjU5RvQVladjUPnmHMa4mm6MF9KoOoeaPmkSl7KocXMhqNeOHlM9tua68gl/nar4XCed4NHaP3xZs62B98L8la985dWun0J4BzUeT0jcGALn0a3qGD2SJsfcU3HzGA65lwAjvxxzDrk/hjENyBO/cQ+28kpIWnzCI/cg/CB1wehzqh5C0idOCH0wpoeaXqg0ptRMz+m41VFPTrkfWTnP9RigXtd09Rzkz6Ai1zDHV4AcK6IPuB8e8e34o2mise6fCve1/JHg1FMZPDUUrtGKx3pzb+cCwwu+EZAvDYgCuEQeVSLTFRDnOCfYVMfJnqnA5wQbkqWBOCZ/9FU2vmWw4sYqmSc84Qn9OwLTecElymoEmWwomDO/NKyQY59LQF6VoU77EmBTfupdjAbR/LnBwLnyi692xpSRvF7Cn9jTNGr86JS4VktCGUqpeJdwMLLI8dIjL6ZOjcgwKjO3a97anx+IO7XxK2rhGrk4z+PxOaGxrWVHj3Mh/mIOOS8pE78VfE168/vf8A3f0D+WczQACM6ZhyVUO2aZ3iUQ38mL10tBGebl4rnBVwRLaNn5xo0bZ82reuHdQ/J3Cbsqx7E+ngqTc/RbkcZPr67y1sq2B1EhDjtXUd2rQaWfw3Uqu3wZPVimZudOqz/I07mcYpSNP/2EnOOPt6NHUnnLdZXpWghN8jmmW0LlQX7KD7248NwC9MGSfPKCNPBptJMX2CMfTbUljLTSeNQf19HTIfomLhh5LCF6B9HHEf85m1S7TcnbowNe9PDUaRBkpLsFaMa8Q+LmdEie+WneB9T0c3TXAVlG146WfFtGahUO+fHb6yL5Sn5iV4GcPTJCu1U3aeWNX+bJdM53jsG1OVFQhT2FoWOcJZBj5OIxJ+mnZK/xmUN4etnjaC8VjeApRy4pQIUa3XMk02qf2gjOwf0Eeu9F6KpjiduKpKXjkp5B5OlYzH8Cuqr73nzElnvpxkFC9N/LB9JxQfLj/c4S2CJ2i0xxe+wfhMb89d7B1lRjIi72WIJVIsdOwe2F8tKBeZKzmsm24aZwtui5FbVMvLc6Nn/xh73lSc6Wen8MrlVCFOJYGt7rgrPL3BbDKOAUQAolxtlj2CmEh534wL4VXshonOjoXg3HYI4uPDWC6QSEEdExDkivseFaQ/iiMxcZG+6B9PjELmtIebFl3gkkH0L47UFkH0NXj3g4j023Ag0bOgr0YEsLFJYgnRDdHYW98iF5964Mjz2g80gzFVcRfeM3ybfjHuBRwxLw5jPqojX05NlBdo1uD2oelIM2Jkss9yDl4biHVv6sww+22GUrdntVhEcBipnLrqOaLYgzCT5C4DTilgzDcORYY25+UPo0iF5g5sVJ4ipGvav8McA999zTR7o2XfJhSN6G13RrPJbgMY3OdE16Rzzzwc4cr+TbqJHtalr3hDWgN2LxcRYeqbB7IL2P17KCZk5fyD3BFFjesWic8KDPMWADH0kp+yo/xzmwGf35XT6kgzW6ETot/k8+oNfwW+YZTPEUF5vLg/OU4x5Iz5fY0cdj+IjDdysveVAG+KChhzCH8LX6SR1hQz5Q/RC2yE/6BHpH9wSQL8to5dO0DdjULqjpaxjvjaj3Ipcsg0kfScnjFN2IykNb4aVq4pZQ07CjAQJ/IB8vWOOxBbsb+jgA4RoKe8H7im3PiDKKq1xGzq95zWv6rnM101NwT0egAEynACf1MZP9tD1ZMA5dRj65jowaQOFq8Ky0sQ+/yu+zeo6sU8kctjzbMdPOl74ITuVAH0dxTCFNQRpfar7gBS/o8/8aidDS3TTRXIcnjXvSy7dlhb66i9ylClpBXw71ute9rn+YtIc2INPOme9+97t7OSyh8qZ/Omo7d9rLfGvFqJDWtw4+oPEUFNotPKShh47uZS97WW9ExG2hreBz6E0lpLPA15eaS2AP6Ywa7cyos4C98qU3EvQxmCO+8aWtvPi0ZareYwDfWELKkp96ojeytgLK7pf8IHVBWIPt0O2e+vnPf753Fmhqp+s6+qj3gg8Xwa6x0rjviC40Vj0pW+1LBpFjvkYa99VFu4Cyp7LdMmWLVsBHR2Q/evnZgtjSkSx+gN5Wxbl3Cuxu6AOZMpq3ltnWqq63QgYYWAH4xF3vlc+o18Cg5HJOsBmRxsLcXf1CbwTeiVeg5JOrg/FFoe1V7aWhAff1nfS+9gXH9OzJty8tNXI2W9JIqLSRLazlxZJNL5Xs16LRT3o8OHycfYR00ugQbH5k9GCL3XQ4ZDsKc5BGB8bmnMtePjqyOZlzkGcfsKhM9CAz+i1BY5BHYk8vRpMcfI1uhPSWVWoslWPVf83+gJ7PWKqnc88T4R6wn7LwBxrJu8AnliDvgFanZ192DXV8dA/MXbOjDj882GLNnvFTZW87X76ckfnYKFaEr86VPPWDHfyBio3V0OMtj0v5kY4vaOh/9Ed/9EEf+chHrrb/ID/6CfJoJZNg5Q0YqEgffQ3UlKPBJ1533XVXbxv4pjRsXSEf4qKjc1ttfPzjH++zBvKUdmaLLeltiasBoH2P1mgCuoFOTB7UJ+V5Stxvr5slUDxKyZSvDRnGnvUawtxbQ5xQI2/TMKNSe2jH+XOcAhl6WgXAsfzphk/i7QNNB7TSJARkKgyNon2rORdan89zTiNKO/7ZR5qzWKlhlPLN3/zN3QHykkvhG1FYjcO57rnnnj7C9/JWx6XxtPKBvKV8cFxfdXrx6tNuL9Ks8hHPMbOxEYz5yDVntPunIjTFpMLRNTaYg3v4aOR00hoZG9hF5hJthXT2cLcdgQprbXutTCOf+I+0bKWBszTSuQ7P7oSmdPbIty8Imf5f4OEPf3i3YypY+ChbZVz3o0/lJk+DxZfZwpYKS+U2QuU0yNA4oM9Gezb6cx4dxjyRT0+NLJ/xRGpf9/xhxZh+DrGpP1/RWPJTX46GxxofPsynTYeoG57ubOBnILBGm86a/vaIN4pWt3T+pq7YQdnM8WED5e/PazSMRsEaaE9p8oW3Bg8P1/KmftDXvu0GN8pW2Xsq8uGcekgPvqWe2p/Ky1u04Vkhjh4aWDy0Kb6G95TGT+UhW3Ws2UNHi04bggeZeI8yR9BBIMvX3erRrbfeelWGsCZ7DbsaekoTSCmjH5v0M7p15tnQaQvwMCpGr3HyBwHZUKlmboR4OhiJ+0ciDax/Wbr99tuvGjm6hb7yQccxPGKr9Cq03Qw1LnhwVNcqp6kkHclf/at/tVc+ToBeXvEkR4OikTO6kHfzxEZEGmn7jSzlA9zntOSZ+vIIyyHJk4fMYeNR+cRpxDlnNyNism28Zq+XpF+SH3BijZIKo6HKFNVWsAVfUB7yreELpuSL07jIu8aEPDTK0lSZp5yt8tkJvQ5eA6OCKBMgJ/KnGnoyHOmvs7hx40bv+DQ6lXYL+IJ/JsLTNsXAX5RN+Iz8arwy0MAaofouIA3LVpBLd9t0GLioT+LkbY1P0miUNDQGXhpqDfAcrXj5Y391Io0vOr7kCZkvZYOyJT70ZCcdg/rnXAei8ZYfPq3Rppu65YlDeXsCNHLWGJNrQKYdMni65ZZbuh098apT9IusOegkDVjsjKmRzaCHXfgYzOUDDFb8iRJ9/vJf/sv32Zhsjs79wLknezML5PPlqu+S7C3Y1dBHmCOje+TVUKq4GqwlQ1bIlFG5Fw8aaYVcDbJmGAWHVoH6dyu9fuhrqOCYCo0DPvGJT+xfU6qUHNqIREXR8JhvNOdMhsJXyObLVIgUfvIpvSkYTmbfFXyNTHQ6NT8jxMuLI6fVuODNjvTDl66hH/m4Dg/pPA3paJSBDiy8R7ogtMmLx2ENoaOnhDm6KaBXmfiDRrY2UiMf18qBbVP52JVcfOheR8FrSD41tBpzlUt5Ajl4wlRDHzhX/tKqpBoWcVt1IEdHRW9+orHBK74SPlP8Ikc6cg1C8lQS3deAPnYgX0NpOiub4i3xifwE9RAdPurGHC154L5yjHwNsnx4QtLwG7RI494UKh9+wHc8mWqs1U/nfMs9gzQDLoM8/uJJ2Mtn8QZJBgkGXfKuTikTdbfmbwT5guknnYoneA29slOu6NWv5GEpH9oz/4rlfzzyVCbAHF3liYdpaHVIm8Z+6YTn6Pdg1wdTkhLqqIdlDEcVKSPqLUCHh4YNTXWGpUxFrkckI1eOAVtoyayo2a50/gfWiykV1vsH0KGRxQGkTfrwpL9zIfdqnkZU2TlHi8Y1h2ZPiLNMQVp0juwS3dAslUWVj94oiVPHsfeUY/RWlkD+lIOSGblGbNJrYBOfe6HdgsgX0MfmsZmj+KkPpirYTgA0CVsQ+TnHGy+jMw1VdFjKE7rUBbTpBLcCrfSmLfLHJ66FrfmA2IA+eMzRJp86d4OSOt2GR6Vb0iFlHtpcg/PYzVGDzk89PaobVqwZGIjXSKdTgfBbQ8otsgAPPNWD7Jyq44U5vvgIKTuI3nN5r5AOvQ7HTAJ56JXBqbCuRQHhMY4MCBqHNEp7gBej4FENGIPPQVqObfSVtGs0gE66eqwBxHupBj6Uci3IX50SCcZreWEf8bHTGsKDY0UeZ3Z0z3EOSSPUirlGF0hDT3lTWdNAbEXSOpItDzUfFa5jH0GlSLrIzf2tGOUCXlPy55B0+FQb7kHo6BB6ndkepIGI/faATLbIH6vELsfwQRN+c0geNYjKUdqkZwvAR1jiU+lg1Ldey5cnd0/Q5Bt0Gf06z55UI/0aIju0yT9ZjvInJM0cQqfs4r/ht8efPYUY/MCSvGOw26spTwnHBD3RHsUqrYC2hiW4r8djlNDmuAT306CkYiYolDRApm7A5knhmyCdY+BcGgivdF5x+ClUnpVH9PMI7LiGyESXxi7yl+ir3NqwxA5bgQf6HCM39q0Qj797mZt3LV3inTtuRfIQujRyzrfySXr6O4bfVkhb9XAu7Bn8hO4Y+cB+sSFUfnuAJnZwPgeyNPCZ7oy86I4en/jGVoRPeMWnaj5yzk+9E3KsK5zothVV51pvHd2TN+3MljqBPvkOT8hxCdGDTQ1gz4F9HjWAgpTz37F5dD83FCQHy8vKUwBPhrasyaOTwrLyQAEL/pM066zPAXzDm03z6HYueQH+ys/UFAcjs1aqU4M89lSBTIWxec371zqSD/5jzv5SUG7sahWZ8jw3yFN2mQe/BPinpZSmaeSVfIsowIg+oNspgE9tZy7ho947eD8ib6fKR3Cthl7mNQzCqRWbQmRomLKV76mAtxUD8LSnPa0XMoML52z8YLSdToUzX8K55A8i75wy5VPwss6qK7JcR4c/CpAnj/v+YPuSiG2FcyN+ogy9HL0EdJ5kppFnYytrNMRegoo/NbwTOHU7swT5Sj6FU+JaNYxxBasdLgkyTzlyYWD88vLVXGccR8WRvzT850IqqALmxJeouOFvpQYH06Gdo8IEZOGfPAZ7nTrpHcfwhw155CemMy8Nq68uaQNlmPI8t1y+6Uk3q4k8SaiTVtsY+HkSp8sp/Td1L/XkXIj95M+3PPzn1PKu1XIxPqUsKdRQnBtxJi9lFPKpwDnkxWoe51YuyE+cRkOvMbwUMnVzKXgnYDpFfs/ZmYF8WTpGZq73ID6QygGXtNUSog8bWhZ4aViayW8vYQ959EL0lFOoSyAvDX3yp876WJLf5juGU+advNT7lO05QXdz/eeQda1abV6ekX2JlkbxEjj13CAHkRcfrZhb9WUhYzO8fFlGlqVW5waZaQThEjJNpWSO/pxIXnQqOmrXe/NXdXSO3tPYJf1vDtGNTqf+hH0J5Aqmi/jxXpseA/bWqVxigAfKWD20lDv5c7ztttv6uembUz7ls6c8CuSkbM8JbUz2rTp1GV6roU8h+1jnkhWNMcyfnQp090ENJ7LaxkihIrvJXQIK2Dr6SzkXeAmU8ju1g43A39JDH5joYPfmM3pagmqOWOVQwS9lqyVU2/lq89KwyR070OPc5UiOOmjK5BIwTaS8x+2fbVnsxb4X0Wv5Hu8lfeId07AL8sdXq02FpM1xLlReCRXjtU46A8qavp4fi6Mb+mQCMgKNQjXMxaMXVNKEXOe+Y6XPOVTjh1eeMBJf79d0NYjHy2ZEGvhv+7Zv6/xBPMhfHqmiH1k5j/7Ox0CGe1WeoxF07o304iHHGh+5SZ9z92q6nM/Fuc4ca5Yl5tG3pkuoetRA9jhXm1DjAjKAnORBupRdjslTQuVHnhUYtk2wP4ldDKd0qNdQ4xzJiD1dJ26kyTFh5BH6wD18c55jArifvCbM8RqD++M5mNZwzrY1zRjcmwrV9lN0NUhXz9kffU0zhpHGkazQJv/hkzSO/MZTvGm/QLz0pqw0kP4VDiIjtAnVLuC6glxxsUPShU+uA9fhkXvRP+lDG7ifeKjnZipMF8lrpTkFrjWiT8HYn2Q0WlAVliYh1zJp98E0fPUeuJ9zYAQfFdS5QWnykdEItJGDtobEy4PN2Zz7IlZ89JbOdI65utBJDzWd8wCf6CK+ykvQqRgxxLncR+M8X4zWuBqADl74mXahQ+TBmHY8T1p0RiymGebmBhM3dY8ORnTV9o45r3LFOSfT1E0GB56UVNLoFTs5QuUB0liV9NGPfrRvLGf7CXuU2O8nMoJKGxvlOucaGTaENDA1QMo4CF9BGjZkg9xTtuboQw/OK03yFxuS4R4kzSg3CC04l44NbfEw2m/UAaJH0sm3csh9+te09RhIY6sBDW98UZrUjaDKrzycj7rlfnR3nWD2wNYH6qK0QtLZ0A3sWgmVFzgXV22ca0dwTgb98dUe8VEfE4K40EpT6WoA/uCpHKRDm/Q5T6hxbKmzNoOAR01zXfyBxxwBDuJx0R4NowEF59XIictRJfMoZic684uVR47SVsM6apQUinP3VHzbkVqDWmXkPLSJEwL3VFL54ET2+3Cf8XOfU+ecPAXhUdH63TRyNSTP0kb3hFyr3DZ1s3lTRgAJGsIKcaGLjehnM7R3vvOdfSRT6SE0NQ6qPqaq3vGOd/TvBALxFXPlmmsNrs2cwjeoaSLfNb3ZMx9M3XPPPX1PIY1t0oe2ol7b0oDfWKFgIYCXgj7/Vxahh/CDOpdceeFj2wu2oFvKPWVY00LyE94q5Fve8pY+9Ze0jmkgKlLOwF5kScsG1oeLg/ChQ9KDeCHl6jz37bqYKbjEhU/okr7G40UvZWgzsso3ZZp0oQP3lSMd0b/tbW/rgyXnNR2ERwJaAcy7q7s2Y1MG1e5jYK9ajuFj+sa9rJoby04acbEvRH5sLG3Khy6vetWr+pNizY97czzId09++IOXw+iqLpEjLvGRGXr7Hd1555338adTYNemZlCFe3y2AZie3dtvlZfSRheZu9MoOxen10cvzrl5WnvA++DK5mju2QXOCF/vplc10kDPEBxLg0AuA6lMGkr7Tnsh7Es5lR4fBsfHyN+1ToUjSUcXQediLlWh4GO/DHKl8WWj/Kg81tfTQz45Gt62dOWg4jU24FwBShN5jmm43dcwKFC85f/Hf/zH+2fd9u1gF42VBjx2km809HUvHZynAUcNrT878IGXuPpSNbYjLxVEGjZO56WTxsMyNWnIjDxlJA4/vAQ6yA/7Kwuy2cHoWjw6MtHRHU1GzNGdfdlGevp84hOfuFpR4fGVPDzowZ/YSh5c4+VPYeyWSSaeOjpx1lXjp8NnW+Xvnkqjc/C0Fl7xD/nQ4fITT6byF5+Rl3RA7BId6MZ+AnvboliZmfaTDm8y+WfyzI/QATqyU6bK4IMf/GB/P0QfQRq+Q56Ab8pMuYhDK569NHJ3331334XVSFQc3eMn8o1f9KBX9JDOey9TYTYTw5sMtldG8q28pFU/lCtebC7Oi/X84YZN+dgRnTTkk4cXmfgm39K45n8GTqZPbUqGXrmxLR0FNOTxHXVcvvE2yHPPH/gYgNn10T5DGkz/USDEdjnGt+Qr7Qq78FVboGio6WGTtNiJruikoRd7pl6Qr8z4jbpAl+c973m9TqON3aUhDx2a2CVlyIZ8kQ62K8ZPvAA5HoOjtinO8Z42EjGSygdGHMw9CsuYSssxGFWByJg4aRWkCm57Xf9y5I8PFJRKwglUeg5ojSx6vNM4cjojMaMQlcyjm50s01ArCI/y6LOzpXNbmxr52OCKXI2FLQ/si4+vtCkMO9ApXI2GhlFhZZdDvHQEdPY17Y0bN/ronlxPBQpIHtCqCKGTZ490ZGgE0LOTPbhdywfHkQY9O8mzSii/7IcXB83aYTS2Ns2TCBnywcFUCrzIVjnIjY2Vg+ko/Pwlm61eLSNlPzopL5VQ54aefHqQjc49vFQEjShbybs07OMFmTh2FaexqTrIJx2UGae3H7yRmUrO8emdiqqjJUNjSg86q+Ti6Mf2Hus1MkbGOnD2YQuy6MCWngCkVyYaDXJT5uTrbNibHdiSDrGf8kme2Q8/5cw30fMh69jx4//SKgN5dF8cnfHiy3TLoEOj4j8J2Ed6dYsu/C508X92ZDt6kCGdODT+YwG9VWOAvxCZ6CycIJP+0rKNsmZTNmE/eeBbeLJX6pO8pvNyjjd/YC/1BT//bWAHUY0ceoMy8vgVe6VxVgZ48RF1yd5S0qjTnjDRuCcN/dmLrsmzezpITxKextkjduYvruWVH8gPnemHzktd/KSXP2WtcX7729/eB3w6rO/+7u/u+ss7WvbT7tCBHenF7vLDH+jIV/xpkd0nfYuTukNndYA8dk9+6EGGODqwsT8z0uHbRVM69yHHY3HUiF5gAA0U2N6T8RmEcgzKEIwjjnGdc2SZUgEYSwNlNOsPuNFrHByl49QCXuLRo8NPQ6DXUzltaYreC5k4RQpGg8R5AK1zBuVYKhcaBWIkwej+/OE5z3lO/8OC5IfuWUdP98SpGBoOjZvCp49CN09KfzaiDz7psckXxwGc04kT6iyNQozKyUEjXZwi9sQXHbvQgV00lP4hig433XRTnxsOb0f82AINXujIRysNGTor5aDjIUta99GqnNKhIx+t++Lw1cBycPp7okIjHXp5lja6oxHcS3nTUSeq833Ws57VywadOHqlwRZUKJVFo6BDUKHx0DDQ0+BAY4+Hxkb5CM51InQjjw/QPXYQdNZs6SlBGuXs6B7dXdOdfegef5Q/fDzmyyfdpOEfrvEA+pEvLVuyuyCdOH+2QYbtrvmENMoraejBLrGpa/xTFnTyZGxw8dznPrenowfdpcMvthcX/fkpedJo3H/yJ3+y72fPr1PW9JCe/niiE8SRG1vR0+AJH52euPiNtHhJSwd0eNETHXn8T2OKh3I3+MhMgYBGuadeAH9Q5v4sSOejkdTQOnqCswW4zc/kkR7yi57fyh8+fFcz6GnANCLd3eNP6j7d6clG6on41GnppDEAMWDlwy95yUt6XNKwARryxZEZOjyBPXVYOgR/jSmf0qE/CVrGN6P13j004x7a6Orwmte85vDzP//zV3FCTdMqVD/PsVXefmyj1UMbdR1+4Ad+4NBGEFd00klTaRMvoBNaT31oj7qdRxsZXKWrfJznegyJd2yjpkMriEMz6KE1+JM0Nb1jdKOLOEf5aE8Lh/bI3uPkQ7yj9Ml78uE8PFpPfmiFe2idRY9DHxlVduLrfflnj9tvv/3QGrz70cSeuQ6dEF5t1Hz4kR/5kUMb0d0nTeiWrvFvI7RDe9Q8NCftcclflSWAuOgUm7QRz+GOO+44fPazn72SITiv1+HXBhk9vlXsw5133nn4wR/8wS6/jco6L+lCFx6VjxDZ4umLng7oK23SQ+UzlmFrqLsu/Elc0tKl2iHX7tEhoTVUhxe84AXdpyPDEe/wCo/ogg4c6d0GGoc2ULkqx/Co9khcPc/9NtjpfvDOd76z80x8TT/FS5DekezW4B3+1b/6V1fpIOlDg4/zKkOQjzYaP3z4wx8+tIFc51vThYf4BPHf933fd2idyKE92V/Fh6bKcC6Ep3RtkNTrb+ugeprqv0mbUPnlHh58nw8og9BLM9JFr8rXPUH70wZ9PY14aU6FXfvRB02BHpoyvdfRS8Oe3qdlrPeuaEK/BWhCl+MeuYCO/uBNvb/90msbFetZj+FXsYc++YlNMwrYguTfsTlHL4+MIGGNT5Vdy3KPfOWIxqM43UMf3bZAWvRoBXrMga5Ge6bejH6e9KQn9X8SAvKFvcDTEyp9M4LdCroD++ODHp+teYeUQXRQhnvppVeG7OjpQdzeESEaPISMPLcCbfIhoD2mLh0D8qy8MoVMb0/WZMNSHuhbj3vyW4Fe+TvyX4jtxa3ZQJrwUJ+UH5zSdkc19HMkWxUb6fdkCK1QafYaJPI5yItf/OI+t+l/Hv3nJF57+UF4HqtLxSl4wBqf0DnWtHvks6H04RHakecSqv7OlyqcitBGfX0+2SO/x3mVmh6pZHv0BzKj75r8EZU22Csf6F+xV4fonmv88NjLp+IYO8Ix+b8O5FUjaTrG9I2p2PwlJKzpc129q92c47OHV+hDGxyrzxSO6sKSkShSM7oF18kAWs476rAXnIPenAIPDf1o6DVIn7xXXWr8FObo9siG8AmPyjfHJYRur9yADSuPymcPT2nT0K3RadSNeMzdmuPMqLE2aFvyXoFe2EsXkB3aNf2ngDZ0zvc0zhDaKvuY0XTV4RhU+hrODXI9SVk5xnZe0gdL8qMf+r22qkA78sr1FoRmb7nvwVGcayaioONebDXEOUC2qRpv551bmndMHk6B2PM69kBby2FvXo6RHxmhO4YHoMmj9haQm/TJ8978VpCPHq+9lS1yr6sDRIdjbXgM3YjocGxeYodj6Y+FEb3FAGB1nYGDKahL6RHfIa+Ww54yOVUZTuGohv4UBYn+Og51HUS2np9DeBtuxcVezNlhLj4Y7zuPLZboRtT09Vjjt0Da6qhbUWXVsBeh2UKfNEK1WT0X9mBv+ilch0fVO2EvKu0xvnQOXFIHHb9VdBp33zAAOyw1nOfSL3z38pf2XGV3vmeFr2Kk8H0sxaDWrFoCBefqUR/AA3gA50HqrGWR1u+r15maPXWD+bWKP5YNfeBDG85Q/2jkgYb+ATyArz0YCRus+Q7CmnZP6xr5B+rz7+OPZUMfB/CFnpUaPrDgKA/gATyArz3UUbtRvbrsvyV+7/d+74F6fS92fxn7RwUaeo5gft6XhEb0nOIBx3gAD+BrE+qwqRvz9PYd8rWt+vzA9E3rDFuD98fu2UaWOYNjGnZH15ziAcd4AA/gawfqbQZqmZt3rB8v/XHHV19DX7W5am/vH7lX7bHx5giJq7ycV8fY0+gv6XRM5zHF76utEzplnud4zfHZm/4SWLLHFE7lF2vYJCdsr5JOyRn4SHJic+/Nn/TylzrtOnX4/vle4n3ijHwV4Wuoq9tX+GvgCJxgKhxTkUaExyl4fbVDHv845HMr/mjZQl5q+OqDOgu1Ts+XwZifGv6o4kEP+v8Be55aDBkRAkUAAAAASUVORK5CYII="></p>
<p>correct shape with a local maximum and minimum, passing through <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>0</mn><mo>,</mo><mo> </mo><mo>−</mo><mn>2</mn><mo>)</mo></math> <em><strong>A1</strong></em></p>
<p>local maximum and minimum on the graph of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>-</mo><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup></math> <em><strong>A1</strong></em></p>
<p><em><strong><br>[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>This question was very poorly done and frequently left blank. Few candidates understood the connection between the differential equation and maximum and minimum points. Even when the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mtext>d</mtext><mi>y</mi></mrow><mrow><mtext>d</mtext><mi>x</mi></mrow></mfrac><mo>=</mo><mn>0</mn></math> was correctly solved, it was rare to see the curve correctly drawn on the slope field. Some were able to draw a solution to the differential equation on the slope field though often not through the given initial condition.</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">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The sides of a bowl are formed by rotating the curve <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>6</mn><mo> </mo><mi>ln</mi><mo> </mo><mi>x</mi><mo>,</mo><mo> </mo><mn>0</mn><mo>≤</mo><mi>y</mi><mo>≤</mo><mn>9</mn></math>, about the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-axis, where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> are measured in centimetres. The bowl contains water to a height of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo> </mo><mtext>cm</mtext></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the volume of water, <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi></math>, in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math> is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mo>=</mo><mn>3</mn><mi mathvariant="normal">π</mi><mfenced><mrow><msup><mi mathvariant="normal">e</mi><mstyle displaystyle="false"><mfrac><mi>h</mi><mn>3</mn></mfrac></mstyle></msup><mo>-</mo><mn>1</mn></mrow></mfenced></math>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence find the maximum capacity of the bowl in <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mtext>cm</mtext><mn>3</mn></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>attempt to use <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mo>=</mo><mi mathvariant="normal">π</mi><msubsup><mo>∫</mo><mi>a</mi><mi>b</mi></msubsup><msup><mi>x</mi><mn>2</mn></msup><mo> </mo><mo>d</mo><mi>y</mi></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><msup><mtext>e</mtext><mfrac><mi>y</mi><mn>6</mn></mfrac></msup></math> or any reasonable attempt to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mo>=</mo><mi mathvariant="normal">π</mi><msubsup><mo>∫</mo><mn>0</mn><mi>h</mi></msubsup><msup><mtext>e</mtext><mfrac><mi>y</mi><mn>3</mn></mfrac></msup><mo> </mo><mo>d</mo><mi>y</mi></math> <em><strong>A1</strong></em></p>
<p><br><strong>Note:</strong> Correct limits must be seen for the <em><strong>A1</strong></em> to be awarded.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mi mathvariant="normal">π</mi><msubsup><mfenced open="[" close="]"><mrow><mn>3</mn><msup><mtext>e</mtext><mfrac><mi>y</mi><mn>3</mn></mfrac></msup></mrow></mfenced><mn>0</mn><mi>h</mi></msubsup></math> <em><strong>(A1)</strong></em></p>
<p><br><strong>Note:</strong> Condone the absence of limits for this <em><strong>A1</strong></em> mark.</p>
<p><br><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>3</mn><mi mathvariant="normal">π</mi><mfenced open="[" close="]"><mrow><msup><mtext>e</mtext><mfrac><mi>h</mi><mn>3</mn></mfrac></msup><mo>-</mo><msup><mtext>e</mtext><mn>0</mn></msup></mrow></mfenced></math> <em><strong>A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>3</mn><mi mathvariant="normal">π</mi><mfenced open="[" close="]"><mrow><msup><mtext>e</mtext><mfrac><mi>h</mi><mn>3</mn></mfrac></msup><mo>-</mo><mn>1</mn></mrow></mfenced></math> <em><strong>AG</strong></em></p>
<p><br><strong>Note:</strong> If the variable used in the integral is <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> instead of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> (i.e. <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mo>=</mo><mi mathvariant="normal">π</mi><msubsup><mo>∫</mo><mn>0</mn><mi>h</mi></msubsup><msup><mtext>e</mtext><mfrac><mi>x</mi><mn>3</mn></mfrac></msup><mo> </mo><mo>d</mo><mi>x</mi></math>) and the candidate has not stated that they are interchanging <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> then award at most <em><strong>M1M1A0A1A1AG</strong></em>.</p>
<p> </p>
<p><em><strong>[5 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>maximum volume when <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>=</mo><mn>9</mn><mo> </mo><mtext>cm</mtext></math> <em><strong>(M1)</strong></em></p>
<p>max volume <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>180</mn><mo> </mo><msup><mtext>cm</mtext><mn>3</mn></msup></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>A number of candidates switched variables so that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>6</mn><mi>ln</mi><mi>x</mi></math> and then used <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">π</mi><mo>∫</mo><msup><mi>y</mi><mn>2</mn></msup><mi mathvariant="normal">d</mi><mi>x</mi></math>. Other candidates who correctly found <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> failed to use the limits <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math>, using <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9</mn></math> instead. As part (a) was to <em><strong>show that</strong> </em>the volume was equal to the final expression it was necessary for examiners to see steps in obtaining the result. It was common to miss out any expression involving <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi mathvariant="normal">e</mi><mn>0</mn></msup></math>. Since the value <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math> could be written from the answer given, where this value came from needed to be shown. It was encouraging to see correct answers to (b), even when candidates had failed to gain marks for (a). Some candidates successfully used their GDC to calculate the value of the definite integral numerically.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>A number of candidates switched variables so that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>6</mn><mi>ln</mi><mi>x</mi></math> and then used <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">π</mi><mo>∫</mo><msup><mi>y</mi><mn>2</mn></msup><mi mathvariant="normal">d</mi><mi>x</mi></math>. Other candidates who correctly found <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> in terms of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> failed to use the limits <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math>, using <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>9</mn></math> instead. As part (a) was to <em><strong>show that</strong> </em>the volume was equal to the final expression it was necessary for examiners to see steps in obtaining the result. It was common to miss out any expression involving <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi mathvariant="normal">e</mi><mn>0</mn></msup></math>. Since the value <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math> could be written from the answer given, where this value came from needed to be shown. It was encouraging to see correct answers to (b), even when candidates had failed to gain marks for (a). Some candidates successfully used their GDC to calculate the value of the definite integral numerically.</p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>The shape of a vase is formed by rotating a curve about the <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math>-axis.</p>
<p>The vase is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>10</mn><mo> </mo><mtext>cm</mtext></math> high. The internal radius of the vase is measured at <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo> </mo><mtext>cm</mtext></math> intervals along the height:</p>
<p><img style="display:block;margin-left:auto;margin-right:auto;" src="data:image/png;base64,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"></p>
<p>Use the trapezoidal rule to estimate the volume of water that the vase can hold.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>V</mi><mo>=</mo><mi mathvariant="normal">π</mi><munderover><mo>∫</mo><mn>0</mn><mn>10</mn></munderover><msup><mi>y</mi><mn>2</mn></msup><mo> </mo><mo>d</mo><mi>x</mi></math> <strong>OR </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">π</mi><munderover><mo>∫</mo><mn>0</mn><mn>10</mn></munderover><msup><mi>x</mi><mn>2</mn></msup><mo> </mo><mo>d</mo><mi>y</mi></math> <strong><em>(M1)</em></strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>=</mo><mn>2</mn></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>≈</mo><mi mathvariant="normal">π</mi><mo>×</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>×</mo><mn>2</mn><mo>×</mo><mfenced><mrow><mfenced><mrow><msup><mn>4</mn><mn>2</mn></msup><mo>+</mo><msup><mn>5</mn><mn>2</mn></msup></mrow></mfenced><mo>+</mo><mn>2</mn><mo>×</mo><mfenced><mrow><msup><mn>6</mn><mn>2</mn></msup><mo>+</mo><msup><mn>8</mn><mn>2</mn></msup><mo>+</mo><msup><mn>7</mn><mn>2</mn></msup><mo>+</mo><msup><mn>3</mn><mn>2</mn></msup></mrow></mfenced></mrow></mfenced></math> <em><strong>M1A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mn>1120</mn><mo> </mo><msup><mtext>cm</mtext><mn>3</mn></msup><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mfenced><mrow><mn>1121</mn><mo>.</mo><mn>548</mn><mo>…</mo></mrow></mfenced></math> <strong><em>A1</em></strong></p>
<p><br><strong>Note:</strong> Do not award the second <em><strong>M1</strong> </em>If the terms are not squared.</p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
<p>This was a straightforward question on the trapezoidal rule, presented in an unfamiliar way, but only a tiny minority answered it correctly. It may be that candidates were introduced to the trapezium rule as an approximation to the area under a curve and here they were being asked to find an approximation to a volume and they were unable to see how that could be done.</p>
</div>
<br><hr><br><div class="specification">
<p>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = {\text{si}}{{\text{n}}^2}\theta ,\,\,0 \leqslant \theta \leqslant \pi ">
<mi>y</mi>
<mo>=</mo>
<mrow>
<mtext>si</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mi>θ<!-- θ --></mi>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<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>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}\theta }}">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>θ</mi>
</mrow>
</mfrac>
</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>Hence find the values of <em>θ</em> for which <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}\theta }} = 2y">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>θ</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mn>2</mn>
<mi>y</mi>
</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>attempt at chain rule or product rule <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}\theta }} = 2\,{\text{sin}}\,\theta \,{\text{cos}}\,\theta ">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>θ</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
</math></span> <em><strong> 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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2\,{\text{sin}}\,\theta \,{\text{cos}}\,\theta = 2{\text{si}}{{\text{n}}^2}\theta ">
<mn>2</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
<mo>=</mo>
<mn>2</mn>
<mrow>
<mtext>si</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>n</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mi>θ</mi>
</math></span></p>
<p>sin <em>θ </em>= 0 <strong><em>(A1)</em></strong></p>
<p><em>θ </em>= 0, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\pi ">
<mi>π</mi>
</math></span> <em><strong>A1</strong></em></p>
<p>obtaining cos <em>θ </em>= sin <em>θ<strong> (M1)</strong></em></p>
<p>tan <em>θ</em> = 1 <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\theta = \frac{\pi }{4}">
<mi>θ</mi>
<mo>=</mo>
<mfrac>
<mi>π</mi>
<mn>4</mn>
</mfrac>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[5 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="specification">
<p>A window is made in the shape of a rectangle with a semicircle of radius <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r">
<mi>r</mi>
</math></span> metres on top, as shown in the diagram. The perimeter of the window is a constant P metres.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2017-08-08_om_17.46.34.png" alt="M17/5/MATHL/HP1/ENG/TZ2/10"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the area of the window in terms of P and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r">
<mi>r</mi>
</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>Find the width of the window in terms of P when the area is a maximum, justifying that this is a maximum.</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that in this case the height of the rectangle is equal to the radius of the semicircle.</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>the width of the rectangle is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2r">
<mn>2</mn>
<mi>r</mi>
</math></span> and let the height of the rectangle be <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="h">
<mi>h</mi>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P = 2r + 2h + \pi r">
<mi>P</mi>
<mo>=</mo>
<mn>2</mn>
<mi>r</mi>
<mo>+</mo>
<mn>2</mn>
<mi>h</mi>
<mo>+</mo>
<mi>π</mi>
<mi>r</mi>
</math></span> <strong><em>(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A = 2rh + \frac{{\pi {r^2}}}{2}">
<mi>A</mi>
<mo>=</mo>
<mn>2</mn>
<mi>r</mi>
<mi>h</mi>
<mo>+</mo>
<mfrac>
<mrow>
<mi>π</mi>
<mrow>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<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="h = \frac{{{\text{P}} - 2r - \pi r}}{2}">
<mi>h</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<mtext>P</mtext>
</mrow>
<mo>−</mo>
<mn>2</mn>
<mi>r</mi>
<mo>−</mo>
<mi>π</mi>
<mi>r</mi>
</mrow>
<mn>2</mn>
</mfrac>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A = 2r\left( {\frac{{{\text{P}} - 2r - \pi r}}{2}} \right) + \frac{{\pi {r^2}}}{2}\,\,\,\left( { = \operatorname{P} r - 2{r^2} - \frac{{\pi {r^2}}}{2}} \right)">
<mi>A</mi>
<mo>=</mo>
<mn>2</mn>
<mi>r</mi>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mrow>
<mtext>P</mtext>
</mrow>
<mo>−</mo>
<mn>2</mn>
<mi>r</mi>
<mo>−</mo>
<mi>π</mi>
<mi>r</mi>
</mrow>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mfrac>
<mrow>
<mi>π</mi>
<mrow>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mn>2</mn>
</mfrac>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mi mathvariant="normal">P</mi>
<mo></mo>
<mi>r</mi>
<mo>−</mo>
<mn>2</mn>
<mrow>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mfrac>
<mrow>
<mi>π</mi>
<mrow>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>M1A1</em></strong></p>
<p><strong><em>[4 marks]</em></strong></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="\frac{{{\text{d}}A}}{{{\text{d}}r}} = {\text{P}} - 4r - \pi r">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>A</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>r</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mrow>
<mtext>P</mtext>
</mrow>
<mo>−</mo>
<mn>4</mn>
<mi>r</mi>
<mo>−</mo>
<mi>π</mi>
<mi>r</mi>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}A}}{{{\text{d}}r}} = 0">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>A</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>r</mi>
</mrow>
</mfrac>
<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 r = \frac{{\text{P}}}{{4 + \pi }}">
<mo stretchy="false">⇒</mo>
<mi>r</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mn>4</mn>
<mo>+</mo>
<mi>π</mi>
</mrow>
</mfrac>
</math></span> <strong><em>(A1)</em></strong></p>
<p>hence the width is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{2{\text{P}}}}{{4 + \pi }}">
<mfrac>
<mrow>
<mn>2</mn>
<mrow>
<mtext>P</mtext>
</mrow>
</mrow>
<mrow>
<mn>4</mn>
<mo>+</mo>
<mi>π</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="\frac{{{{\text{d}}^2}A}}{{{\text{d}}{r^2}}} = - 4 - \pi < 0">
<mfrac>
<mrow>
<mrow>
<msup>
<mrow>
<mtext>d</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mi>A</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mrow>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
<mo>=</mo>
<mo>−</mo>
<mn>4</mn>
<mo>−</mo>
<mi>π</mi>
<mo><</mo>
<mn>0</mn>
</math></span> <strong><em>R1</em></strong></p>
<p>hence maximum <strong><em>AG</em></strong></p>
<p><strong><em>[5 marks]</em></strong></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="h = \frac{{{\text{P}} - 2r - \pi r}}{2}">
<mi>h</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<mtext>P</mtext>
</mrow>
<mo>−</mo>
<mn>2</mn>
<mi>r</mi>
<mo>−</mo>
<mi>π</mi>
<mi>r</mi>
</mrow>
<mn>2</mn>
</mfrac>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="h = \frac{{{\text{P}} - \frac{{2{\text{P}}}}{{4 + \pi }} - \frac{{{\text{P}}\pi }}{{4 + \pi }}}}{2}">
<mi>h</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<mtext>P</mtext>
</mrow>
<mo>−</mo>
<mfrac>
<mrow>
<mn>2</mn>
<mrow>
<mtext>P</mtext>
</mrow>
</mrow>
<mrow>
<mn>4</mn>
<mo>+</mo>
<mi>π</mi>
</mrow>
</mfrac>
<mo>−</mo>
<mfrac>
<mrow>
<mrow>
<mtext>P</mtext>
</mrow>
<mi>π</mi>
</mrow>
<mrow>
<mn>4</mn>
<mo>+</mo>
<mi>π</mi>
</mrow>
</mfrac>
</mrow>
<mn>2</mn>
</mfrac>
</math></span> <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="h = \frac{{4{\text{P}} + \pi {\text{P}} - 2{\text{P}} - \pi {\text{P}}}}{{2(4 + \pi )}}">
<mi>h</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mn>4</mn>
<mrow>
<mtext>P</mtext>
</mrow>
<mo>+</mo>
<mi>π</mi>
<mrow>
<mtext>P</mtext>
</mrow>
<mo>−</mo>
<mn>2</mn>
<mrow>
<mtext>P</mtext>
</mrow>
<mo>−</mo>
<mi>π</mi>
<mrow>
<mtext>P</mtext>
</mrow>
</mrow>
<mrow>
<mn>2</mn>
<mo stretchy="false">(</mo>
<mn>4</mn>
<mo>+</mo>
<mi>π</mi>
<mo stretchy="false">)</mo>
</mrow>
</mfrac>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="h = \frac{{\text{P}}}{{(4 + \pi )}} = r">
<mi>h</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mtext>P</mtext>
</mrow>
<mrow>
<mo stretchy="false">(</mo>
<mn>4</mn>
<mo>+</mo>
<mi>π</mi>
<mo stretchy="false">)</mo>
</mrow>
</mfrac>
<mo>=</mo>
<mi>r</mi>
</math></span> <strong><em>AG</em></strong></p>
<p><strong>OR</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="h = \frac{{{\text{P}} - 2r - \pi r}}{2}">
<mi>h</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<mtext>P</mtext>
</mrow>
<mo>−</mo>
<mn>2</mn>
<mi>r</mi>
<mo>−</mo>
<mi>π</mi>
<mi>r</mi>
</mrow>
<mn>2</mn>
</mfrac>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="P = r(4 + \pi )">
<mi>P</mi>
<mo>=</mo>
<mi>r</mi>
<mo stretchy="false">(</mo>
<mn>4</mn>
<mo>+</mo>
<mi>π</mi>
<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="h = \frac{{r(4 + \pi ) - 2r - \pi r}}{2}">
<mi>h</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mi>r</mi>
<mo stretchy="false">(</mo>
<mn>4</mn>
<mo>+</mo>
<mi>π</mi>
<mo stretchy="false">)</mo>
<mo>−</mo>
<mn>2</mn>
<mi>r</mi>
<mo>−</mo>
<mi>π</mi>
<mi>r</mi>
</mrow>
<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="h = \frac{{4r + \pi r - 2r - \pi r}}{2} = r">
<mi>h</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mn>4</mn>
<mi>r</mi>
<mo>+</mo>
<mi>π</mi>
<mi>r</mi>
<mo>−</mo>
<mn>2</mn>
<mi>r</mi>
<mo>−</mo>
<mi>π</mi>
<mi>r</mi>
</mrow>
<mn>2</mn>
</mfrac>
<mo>=</mo>
<mi>r</mi>
</math></span> <strong><em>AG</em></strong></p>
<p><strong><em>[2 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.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>A particle moves along a straight line. Its displacement, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="s">
<mi>s</mi>
</math></span> metres, at time <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> seconds is given by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="s = t + \cos 2t,{\text{ }}t \geqslant 0">
<mi>s</mi>
<mo>=</mo>
<mi>t</mi>
<mo>+</mo>
<mi>cos</mi>
<mo><!-- --></mo>
<mn>2</mn>
<mi>t</mi>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>t</mi>
<mo>⩾<!-- ⩾ --></mo>
<mn>0</mn>
</math></span>. The first two times when the particle is at rest are denoted by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{t_1}">
<mrow>
<msub>
<mi>t</mi>
<mn>1</mn>
</msub>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{t_2}">
<mrow>
<msub>
<mi>t</mi>
<mn>2</mn>
</msub>
</mrow>
</math></span>, where <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{t_1} < {t_2}">
<mrow>
<msub>
<mi>t</mi>
<mn>1</mn>
</msub>
</mrow>
<mo><</mo>
<mrow>
<msub>
<mi>t</mi>
<mn>2</mn>
</msub>
</mrow>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{t_1}">
<mrow>
<msub>
<mi>t</mi>
<mn>1</mn>
</msub>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{t_2}">
<mrow>
<msub>
<mi>t</mi>
<mn>2</mn>
</msub>
</mrow>
</math></span>.</p>
<div class="marks">[5]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the displacement of the particle when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t = {t_1}">
<mi>t</mi>
<mo>=</mo>
<mrow>
<msub>
<mi>t</mi>
<mn>1</mn>
</msub>
</mrow>
</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 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="s = t + \cos 2t">
<mi>s</mi>
<mo>=</mo>
<mi>t</mi>
<mo>+</mo>
<mi>cos</mi>
<mo></mo>
<mn>2</mn>
<mi>t</mi>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}s}}{{{\text{d}}t}} = 1 - 2\sin 2t">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>s</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mn>1</mn>
<mo>−</mo>
<mn>2</mn>
<mi>sin</mi>
<mo></mo>
<mn>2</mn>
<mi>t</mi>
</math></span> <strong><em>M1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = 0">
<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 \sin 2t = \frac{1}{2}">
<mo stretchy="false">⇒</mo>
<mi>sin</mi>
<mo></mo>
<mn>2</mn>
<mi>t</mi>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{t_1} = \frac{\pi }{{12}}(s),{\text{ }}{t_2} = \frac{{5\pi }}{{12}}(s)">
<mrow>
<msub>
<mi>t</mi>
<mn>1</mn>
</msub>
</mrow>
<mo>=</mo>
<mfrac>
<mi>π</mi>
<mrow>
<mn>12</mn>
</mrow>
</mfrac>
<mo stretchy="false">(</mo>
<mi>s</mi>
<mo stretchy="false">)</mo>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<msub>
<mi>t</mi>
<mn>2</mn>
</msub>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mn>5</mn>
<mi>π</mi>
</mrow>
<mrow>
<mn>12</mn>
</mrow>
</mfrac>
<mo stretchy="false">(</mo>
<mi>s</mi>
<mo stretchy="false">)</mo>
</math></span> <strong><em>A1A1</em></strong></p>
<p> </p>
<p><strong>Note:</strong> Award <strong><em>A0A0 </em></strong>if answers are given in degrees.</p>
<p> </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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="s = \frac{\pi }{{12}} + \cos \frac{\pi }{6}\,\,\,\left( {s = \frac{\pi }{{12}} + \frac{{\sqrt 3 }}{2}(m)} \right)">
<mi>s</mi>
<mo>=</mo>
<mfrac>
<mi>π</mi>
<mrow>
<mn>12</mn>
</mrow>
</mfrac>
<mo>+</mo>
<mi>cos</mi>
<mo></mo>
<mfrac>
<mi>π</mi>
<mn>6</mn>
</mfrac>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mrow>
<mo>(</mo>
<mrow>
<mi>s</mi>
<mo>=</mo>
<mfrac>
<mi>π</mi>
<mrow>
<mn>12</mn>
</mrow>
</mfrac>
<mo>+</mo>
<mfrac>
<mrow>
<msqrt>
<mn>3</mn>
</msqrt>
</mrow>
<mn>2</mn>
</mfrac>
<mo stretchy="false">(</mo>
<mi>m</mi>
<mo stretchy="false">)</mo>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>A1A1</em></strong></p>
<p><strong><em>[2 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>A particle moves in a straight line such that at time <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> seconds <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(t \geqslant 0)">
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo>⩾</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</math></span>, its velocity <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v">
<mi>v</mi>
</math></span>, in <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{m}}{{\text{s}}^{ - 1}}">
<mrow>
<mtext>m</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>s</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</math></span>, is given by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v = 10t{{\text{e}}^{ - 2t}}">
<mi>v</mi>
<mo>=</mo>
<mn>10</mn>
<mi>t</mi>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mn>2</mn>
<mi>t</mi>
</mrow>
</msup>
</mrow>
</math></span>. Find the exact distance travelled by the particle in the first half-second.</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="s = \int\limits_0^{\frac{1}{2}} {10t{{\text{e}}^{ - 2t}}{\text{d}}t} ">
<mi>s</mi>
<mo>=</mo>
<munderover>
<mo>∫</mo>
<mn>0</mn>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
</munderover>
<mrow>
<mn>10</mn>
<mi>t</mi>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mn>2</mn>
<mi>t</mi>
</mrow>
</msup>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</math></span></p>
<p>attempt at integration by parts <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left[ { - 5t{{\text{e}}^{ - 2t}}} \right]_0^{\frac{1}{2}} - \int\limits_0^{\frac{1}{2}} { - 5{{\text{e}}^{ - 2t}}{\text{d}}t} ">
<mo>=</mo>
<msubsup>
<mrow>
<mo>[</mo>
<mrow>
<mo>−</mo>
<mn>5</mn>
<mi>t</mi>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mn>2</mn>
<mi>t</mi>
</mrow>
</msup>
</mrow>
</mrow>
<mo>]</mo>
</mrow>
<mn>0</mn>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
</msubsup>
<mo>−</mo>
<munderover>
<mo>∫</mo>
<mn>0</mn>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
</munderover>
<mrow>
<mo>−</mo>
<mn>5</mn>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mn>2</mn>
<mi>t</mi>
</mrow>
</msup>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \left[ { - 5t{{\text{e}}^{ - 2t}} - \frac{5}{2}{{\text{e}}^{ - 2t}}} \right]_0^{\frac{1}{2}}">
<mo>=</mo>
<msubsup>
<mrow>
<mo>[</mo>
<mrow>
<mo>−</mo>
<mn>5</mn>
<mi>t</mi>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mn>2</mn>
<mi>t</mi>
</mrow>
</msup>
</mrow>
<mo>−</mo>
<mfrac>
<mn>5</mn>
<mn>2</mn>
</mfrac>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mn>2</mn>
<mi>t</mi>
</mrow>
</msup>
</mrow>
</mrow>
<mo>]</mo>
</mrow>
<mn>0</mn>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
</msubsup>
</math></span> <strong><em>(A1)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Condone absence of limits (or incorrect limits) and missing factor of 10 up to this point.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="s = \int\limits_0^{\frac{1}{2}} {10t{{\text{e}}^{ - 2t}}{\text{d}}t} ">
<mi>s</mi>
<mo>=</mo>
<munderover>
<mo>∫</mo>
<mn>0</mn>
<mrow>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
</munderover>
<mrow>
<mn>10</mn>
<mi>t</mi>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mn>2</mn>
<mi>t</mi>
</mrow>
</msup>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</math></span> <strong><em>(M1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = - 5{{\text{e}}^{ - 1}} + \frac{5}{2}{\text{ }}\left( { = \frac{{ - 5}}{{\text{e}}} + \frac{5}{2}} \right){\text{ }}\left( { = \frac{{5{\text{e}} - 10}}{{2{\text{e}}}}} \right)">
<mo>=</mo>
<mo>−</mo>
<mn>5</mn>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mo>+</mo>
<mfrac>
<mn>5</mn>
<mn>2</mn>
</mfrac>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mo>−</mo>
<mn>5</mn>
</mrow>
<mrow>
<mtext>e</mtext>
</mrow>
</mfrac>
<mo>+</mo>
<mfrac>
<mn>5</mn>
<mn>2</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mn>5</mn>
<mrow>
<mtext>e</mtext>
</mrow>
<mo>−</mo>
<mn>10</mn>
</mrow>
<mrow>
<mn>2</mn>
<mrow>
<mtext>e</mtext>
</mrow>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</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>Consider the curve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = \frac{1}{{1 - x}} + \frac{4}{{x - 4}}">
<mi>y</mi>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>x</mi>
</mrow>
</mfrac>
<mo>+</mo>
<mfrac>
<mn>4</mn>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mn>4</mn>
</mrow>
</mfrac>
</math></span>.</p>
<p>Find the <em>x</em>-coordinates of the points on the curve where the gradient is zero.</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>valid attempt to find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}}">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</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{{{\text{d}}y}}{{{\text{d}}x}} = \frac{1}{{{{\left( {1 - x} \right)}^2}}} - \frac{4}{{{{\left( {x - 4} \right)}^2}}}">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>x</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
<mo>−</mo>
<mfrac>
<mn>4</mn>
<mrow>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mn>4</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</math></span> <em><strong>A1A1</strong></em></p>
<p>attempt to solve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}} = 0">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mn>0</mn>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 2,\,\,x = - 2">
<mi>x</mi>
<mo>=</mo>
<mn>2</mn>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
<mo>=</mo>
<mo>−</mo>
<mn>2</mn>
</math></span> <em><strong>A1A1</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>The cross-section of a beach is modelled by the equation <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>02</mn><msup><mi>x</mi><mn>2</mn></msup></math> for <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mn>10</mn></math> where <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi></math> is the height of the beach (in metres) at a horizontal distance <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math> metres from an origin. <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math> is the time in hours after low tide.</p>
<p>At <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn></math> the water is at the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>0</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>)</mo></math>. The height of the water rises at a rate of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>2</mn></math> metres per hour. The point <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>W</mtext><mo>(</mo><mi>x</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>,</mo><mo> </mo><mi>y</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>)</mo></math> indicates where the water level meets the beach at time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>.</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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"></p>
<p> </p>
</div>
<div class="specification">
<p>A snail is modelled as a single point. At <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>0</mn></math> it is positioned at <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>1</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>.</mo><mn>02</mn><mo>)</mo></math>. The snail travels away from the incoming water at a speed of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math> metre per hour in the direction along the curve of the cross-section of the beach. The following diagram shows this for a value of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math>, such that <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>></mo><mn>0</mn></math>.</p>
<p style="text-align: center;"><img 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"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>When <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>W</mtext></math> has an <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi></math>-coordinate equal to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math>, find the horizontal component of the velocity of <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>W</mtext></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>Find the time taken for the snail to reach the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>10</mn><mo>,</mo><mo> </mo><mn>2</mn><mo>)</mo></math>.</p>
<div class="marks">[4]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Hence show that the snail reaches the point <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>10</mn><mo>,</mo><mo> </mo><mn>2</mn><mo>)</mo></math> before the water does.</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>use of chain rule <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></math></p>
<p>attempt to find <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac></math> at <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>1</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>2</mn><mo>=</mo><mn>0</mn><mo>.</mo><mn>04</mn><mo>×</mo><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mfrac><mrow><mo>d</mo><mi>x</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo></mrow></mfenced><mo> </mo><mo> </mo><mn>5</mn><mo> </mo><msup><mtext>m h</mtext><mrow><mo>-</mo><mn>1</mn></mrow></msup></math> <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>if the position of the snail is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>X</mi><mo>,</mo><mo> </mo><mi>Y</mi></mrow></mfenced></math></p>
<p>from part (a) <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>X</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><mrow><mn>0</mn><mo>.</mo><mn>04</mn><mi>X</mi></mrow></mfrac><mfrac><mrow><mo>d</mo><mi>Y</mi></mrow><mrow><mo>d</mo><mi>t</mi></mrow></mfrac></math></p>
<p>since speed is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math>:</p>
<p>finding modulus of velocity vector and equating to <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>=</mo><msqrt><msup><mfenced><mfrac><mover><mi>Y</mi><mo>˙</mo></mover><mrow><mn>0</mn><mo>.</mo><mn>04</mn><mi>X</mi></mrow></mfrac></mfenced><mn>2</mn></msup><mo>+</mo><msup><mover><mi>Y</mi><mo>˙</mo></mover><mn>2</mn></msup></msqrt></math> <strong>OR </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>=</mo><msqrt><msup><mover><mi>X</mi><mo>˙</mo></mover><mn>2</mn></msup><mo>+</mo><mn>0</mn><mo>.</mo><mn>0016</mn><msup><mi>X</mi><mn>2</mn></msup><msup><mover><mi>X</mi><mo>˙</mo></mover><mn>2</mn></msup></msqrt></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>=</mo><msup><mover><mi>Y</mi><mo>˙</mo></mover><mn>2</mn></msup><mfenced><mrow><mfrac><mn>1</mn><mrow><mn>0</mn><mo>.</mo><mn>0016</mn><msup><mi>X</mi><mn>2</mn></msup></mrow></mfrac><mo>+</mo><mn>1</mn></mrow></mfenced></math> <strong>OR </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo>=</mo><msup><mover><mi>X</mi><mo>˙</mo></mover><mn>2</mn></msup><mfenced><mrow><mn>1</mn><mo>+</mo><mn>0</mn><mo>.</mo><mn>0016</mn><msup><mi>X</mi><mn>2</mn></msup></mrow></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>Y</mi><mo>˙</mo></mover><mo>=</mo><msqrt><mfrac><mn>1</mn><mstyle displaystyle="true"><mfrac><mn>1</mn><mrow><mn>0</mn><mo>.</mo><mn>08</mn><mi>Y</mi></mrow></mfrac><mo>+</mo><mn>1</mn></mstyle></mfrac></msqrt></math> <strong>OR </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mi>X</mi><mo>˙</mo></mover><mo>=</mo><msqrt><mfrac><mn>1</mn><mrow><mn>1</mn><mo>+</mo><mn>0</mn><mo>.</mo><mn>0016</mn><msup><mi>X</mi><mn>2</mn></msup></mrow></mfrac></msqrt></math> <em><strong>(A1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∫</mo><mrow><mn>0</mn><mo>.</mo><mn>02</mn></mrow><mn>2</mn></munderover><msqrt><mfrac><mn>1</mn><mrow><mn>0</mn><mo>.</mo><mn>08</mn><mi>Y</mi></mrow></mfrac><mo>+</mo><mn>1</mn></msqrt><mo>d</mo><mi>Y</mi><mo>=</mo><munderover><mo>∫</mo><mn>0</mn><mi>T</mi></munderover><mo>d</mo><mi>t</mi></math> <strong>OR </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><munderover><mo>∫</mo><mn>1</mn><mn>10</mn></munderover><msqrt><mn>1</mn><mo>+</mo><mn>0</mn><mo>.</mo><mn>0016</mn><msup><mi>X</mi><mn>2</mn></msup></msqrt><mo>d</mo><mi>X</mi><mo>=</mo><munderover><mo>∫</mo><mn>0</mn><mi>T</mi></munderover><mo>d</mo><mi>t</mi></math> <em><strong>(M1)</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>T</mi><mo>=</mo><mn>9</mn><mo>.</mo><mn>26</mn></math> hours <em><strong>A1</strong></em></p>
<p> </p>
<p><em><strong>[4 marks]</strong></em></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><strong>EITHER</strong></p>
<p>time for water to reach top is <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>2</mn><mrow><mn>0</mn><mo>.</mo><mn>2</mn></mrow></mfrac><mo>=</mo><mn>10</mn></math> hours (seen anywhere) <em><strong>A1</strong></em></p>
<p><strong><br>OR</strong></p>
<p>or at time <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><mn>9</mn><mo>.</mo><mn>26</mn></math>, height of water is <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo>.</mo><mn>2</mn><mo>×</mo><mn>9</mn><mo>.</mo><mn>26</mn><mo>=</mo><mn>1</mn><mo>.</mo><mn>852</mn></math> <em><strong>A1</strong></em></p>
<p><strong><br>THEN</strong></p>
<p>so the water will not reach the snail <em><strong>AG</strong></em></p>
<p> </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;">
<p>In part (a), a small minority of candidates found the horizontal component of velocity correctly. Few candidates made any significant progress in part (b).</p>
<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>A curve has equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3x - 2{y^2}{{\text{e}}^{x - 1}} = 2">
<mn>3</mn>
<mi>x</mi>
<mo>−<!-- − --></mo>
<mn>2</mn>
<mrow>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mi>x</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mo>=</mo>
<mn>2</mn>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an expression for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}}">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
</math></span> in terms 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>.</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>Find the equations of the tangents to this curve at the points where the curve intersects the line <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 1">
<mi>x</mi>
<mo>=</mo>
<mn>1</mn>
</math></span>.</p>
<div 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>attempt to differentiate implicitly <strong><em>M1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3 - \left( {4y\frac{{{\text{d}}y}}{{{\text{d}}x}} + 2{y^2}} \right){{\text{e}}^{x - 1}} = 0">
<mn>3</mn>
<mo>−</mo>
<mrow>
<mo>(</mo>
<mrow>
<mn>4</mn>
<mi>y</mi>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>+</mo>
<mn>2</mn>
<mrow>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mi>x</mi>
<mo>−</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span> <strong><em>A1A1A1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>A1 </em></strong>for correctly differentiating each term.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}} = \frac{{3 \bullet {{\text{e}}^{1 - x}} - 2{y^2}}}{{4y}}">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mn>3</mn>
<mo>∙</mo>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mn>1</mn>
<mo>−</mo>
<mi>x</mi>
</mrow>
</msup>
</mrow>
<mo>−</mo>
<mn>2</mn>
<mrow>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mn>4</mn>
<mi>y</mi>
</mrow>
</mfrac>
</math></span> <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>This final answer may be expressed in a number of different ways.</p>
<p> </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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3 - 2{y^2} = 2 \Rightarrow {y^2} = \frac{1}{2} \Rightarrow y = \pm \sqrt {\frac{1}{2}} ">
<mn>3</mn>
<mo>−</mo>
<mn>2</mn>
<mrow>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>2</mn>
<mo stretchy="false">⇒</mo>
<mrow>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
<mo stretchy="false">⇒</mo>
<mi>y</mi>
<mo>=</mo>
<mo>±</mo>
<msqrt>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</msqrt>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}} = \frac{{3 - 2 \bullet \frac{1}{2}}}{{ \pm 4\sqrt {\frac{1}{2}} }} = \pm \frac{{\sqrt 2 }}{2}">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mn>3</mn>
<mo>−</mo>
<mn>2</mn>
<mo>∙</mo>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</mrow>
<mrow>
<mo>±</mo>
<mn>4</mn>
<msqrt>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</msqrt>
</mrow>
</mfrac>
<mo>=</mo>
<mo>±</mo>
<mfrac>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
<mn>2</mn>
</mfrac>
</math></span> <strong><em>M1</em></strong></p>
<p>at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {1,{\text{ }}\sqrt {\frac{1}{2}} } \right)">
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<msqrt>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</msqrt>
</mrow>
<mo>)</mo>
</mrow>
</math></span> the tangent is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y - \sqrt {\frac{1}{2}} = \frac{{\sqrt 2 }}{2}(x - 1)">
<mi>y</mi>
<mo>−</mo>
<msqrt>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</msqrt>
<mo>=</mo>
<mfrac>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
<mn>2</mn>
</mfrac>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>−</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</math></span> and <strong><em>A1</em></strong></p>
<p>at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {1,{\text{ }} - \sqrt {\frac{1}{2}} } \right)">
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mo>−</mo>
<msqrt>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</msqrt>
</mrow>
<mo>)</mo>
</mrow>
</math></span> the tangent is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y + \sqrt {\frac{1}{2}} = - \frac{{\sqrt 2 }}{2}(x - 1)">
<mi>y</mi>
<mo>+</mo>
<msqrt>
<mfrac>
<mn>1</mn>
<mn>2</mn>
</mfrac>
</msqrt>
<mo>=</mo>
<mo>−</mo>
<mfrac>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
<mn>2</mn>
</mfrac>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>−</mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
</math></span> <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>These equations simplify to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = \pm \frac{{\sqrt 2 }}{2}x">
<mi>y</mi>
<mo>=</mo>
<mo>±</mo>
<mfrac>
<mrow>
<msqrt>
<mn>2</mn>
</msqrt>
</mrow>
<mn>2</mn>
</mfrac>
<mi>x</mi>
</math></span>.</p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>A0M1A1A0 </em></strong>if just the positive value of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span> is considered and just one tangent is found.</p>
<p> </p>
<p><strong><em>[4 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>Find the coordinates of the points on the curve <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{y^3} + 3x{y^2} - {x^3} = 27">
<mrow>
<msup>
<mi>y</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>3</mn>
<mi>x</mi>
<mrow>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>27</mn>
</math></span> at which <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}} = 0">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mn>0</mn>
</math></span>.</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 at implicit differentiation <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3{y^2}\frac{{{\text{d}}y}}{{{\text{d}}x}} + 3{y^2} + 6xy\frac{{{\text{d}}y}}{{{\text{d}}x}} - 3{x^2} = 0">
<mn>3</mn>
<mrow>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
</mrow>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>+</mo>
<mn>3</mn>
<mrow>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>6</mn>
<mi>x</mi>
<mi>y</mi>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>−</mo>
<mn>3</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span> <em><strong>A1</strong></em><em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Award <em><strong>A1</strong></em> for the second & third terms, <em><strong>A1</strong></em> for the first term, fourth term & RHS equal to zero.</p>
<p>substitution of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}} = 0">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mn>0</mn>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3{y^2} - 3{x^2} = 0">
<mn>3</mn>
<mrow>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>3</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow y = \pm x">
<mo stretchy="false">⇒</mo>
<mi>y</mi>
<mo>=</mo>
<mo>±</mo>
<mi>x</mi>
</math></span> <em><strong>A1</strong></em></p>
<p>substitute either variable into original equation <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = x \Rightarrow {x^3} = 9 \Rightarrow x = \sqrt[3]{9}">
<mi>y</mi>
<mo>=</mo>
<mi>x</mi>
<mo stretchy="false">⇒</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>9</mn>
<mo stretchy="false">⇒</mo>
<mi>x</mi>
<mo>=</mo>
<mroot>
<mn>9</mn>
<mn>3</mn>
</mroot>
</math></span> (or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{y^3} = 9 \Rightarrow y = \sqrt[3]{9}">
<mrow>
<msup>
<mi>y</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>9</mn>
<mo stretchy="false">⇒</mo>
<mi>y</mi>
<mo>=</mo>
<mroot>
<mn>9</mn>
<mn>3</mn>
</mroot>
</math></span>) <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = - x \Rightarrow {x^3} = 27 \Rightarrow x = 3">
<mi>y</mi>
<mo>=</mo>
<mo>−</mo>
<mi>x</mi>
<mo stretchy="false">⇒</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>27</mn>
<mo stretchy="false">⇒</mo>
<mi>x</mi>
<mo>=</mo>
<mn>3</mn>
</math></span> (or <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{y^3} = - 27 \Rightarrow y = - 3">
<mrow>
<msup>
<mi>y</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>=</mo>
<mo>−</mo>
<mn>27</mn>
<mo stretchy="false">⇒</mo>
<mi>y</mi>
<mo>=</mo>
<mo>−</mo>
<mn>3</mn>
</math></span>) <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\left( {\sqrt[3]{9}{\text{,}}\,\,\sqrt[3]{9}} \right)">
<mrow>
<mo>(</mo>
<mrow>
<mroot>
<mn>9</mn>
<mn>3</mn>
</mroot>
<mrow>
<mtext>,</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mroot>
<mn>9</mn>
<mn>3</mn>
</mroot>
</mrow>
<mo>)</mo>
</mrow>
</math></span> , (3, −3) <em><strong>A1</strong></em></p>
<p><em><strong>[9 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>A camera at point C is 3 m from the edge of a straight section of road as shown in the following diagram. The camera detects a car travelling along the road at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> = 0. It then rotates, always pointing at the car, until the car passes O, the point on the edge of the road closest to the camera.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
<p style="text-align: left;">A car travels along the road at a speed of 24 ms<sup>−1</sup>. Let the position of the car be X and let OĈX = <em>θ</em>.</p>
<p style="text-align: left;">Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}\theta }}{{{\text{d}}t}}">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>θ</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
</math></span>, the rate of rotation of the camera, in radians per second, at the instant the car passes the point O .</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>let OX = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x">
<mi>x</mi>
</math></span></p>
<p><strong>METHOD 1</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}x}}{{{\text{d}}t}} = 24">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mn>24</mn>
</math></span> (or −24) <em><strong>(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}\theta }}{{{\text{d}}t}} = \frac{{{\text{d}}x}}{{{\text{d}}t}} \times \frac{{{\text{d}}\theta }}{{{\text{d}}x}}">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>θ</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
<mo>×</mo>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>θ</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
</math></span> <em><strong>(M1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3\,{\text{tan}}\,\theta = x">
<mn>3</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>tan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
<mo>=</mo>
<mi>x</mi>
</math></span> <em><strong>A1</strong></em></p>
<p><strong>EITHER</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3\,{\text{se}}{{\text{c}}^2}\,\theta = \frac{{{\text{d}}x}}{{{\text{d}}\theta }}">
<mn>3</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>se</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>c</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>θ</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{d}}\theta }}{{{\text{d}}t}} = \frac{{24}}{{3\,{\text{se}}{{\text{c}}^2}\,\theta }}">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>θ</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mn>24</mn>
</mrow>
<mrow>
<mn>3</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>se</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>c</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
</mrow>
</mfrac>
</math></span></p>
<p>attempt to substitute for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\theta = 0">
<mi>θ</mi>
<mo>=</mo>
<mn>0</mn>
</math></span> into their differential equation<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><em style="color: #000000;font-family: Verdana,Arial,Helvetica,sans-serif;font-size: 14px;font-style: italic;font-variant: normal;font-weight: 400;letter-spacing: normal;text-align: left;text-decoration: none;text-indent: 0px;white-space: normal;"><strong>M1</strong></em></p>
<p><strong>OR</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\theta = {\text{arctan}}\left( {\frac{x}{3}} \right)">
<mi>θ</mi>
<mo>=</mo>
<mrow>
<mtext>arctan</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mi>x</mi>
<mn>3</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}\theta }}{{{\text{d}}x}} = \frac{1}{3} \times \frac{1}{{1 + \frac{{{x^2}}}{9}}}">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>θ</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mn>3</mn>
</mfrac>
<mo>×</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mn>9</mn>
</mfrac>
</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{d}}\theta }}{{{\text{d}}t}} = 24 \times \frac{1}{{3\left( {1 + \frac{{{x^2}}}{9}} \right)}}">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>θ</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mn>24</mn>
<mo>×</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mn>3</mn>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mn>9</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
</mfrac>
</math></span></p>
<p>attempt to substitute for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 0">
<mi>x</mi>
<mo>=</mo>
<mn>0</mn>
</math></span> into their differential equation <em><strong>M1</strong></em></p>
<p><strong>THEN</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}\theta }}{{{\text{d}}t}} = \frac{{24}}{3} = 8">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>θ</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mn>24</mn>
</mrow>
<mn>3</mn>
</mfrac>
<mo>=</mo>
<mn>8</mn>
</math></span> (rad s<sup>−1</sup>) <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Accept −8 rad s<sup>−1</sup>.</p>
<p> </p>
<p><strong>METHOD 2</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}x}}{{{\text{d}}t}} = 24">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mn>24</mn>
</math></span> (or −24) <em><strong>(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3\,{\text{tan}}\,\theta = x">
<mn>3</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>tan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
<mo>=</mo>
<mi>x</mi>
</math></span> <em><strong>A1</strong></em></p>
<p>attempt to differentiate implicitly with respect to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3\,{\text{se}}{{\text{c}}^2}\,\theta \times \frac{{{\text{d}}\theta }}{{{\text{d}}t}} = \frac{{{\text{d}}x}}{{{\text{d}}t}}">
<mn>3</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>se</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>c</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
<mo>×</mo>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>θ</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</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{d}}\theta }}{{{\text{d}}t}} = \frac{{24}}{{3\,{\text{se}}{{\text{c}}^2}\,\theta }}">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>θ</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mn>24</mn>
</mrow>
<mrow>
<mn>3</mn>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>se</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>c</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
</mrow>
</mfrac>
</math></span></p>
<p>attempt to substitute for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\theta = 0">
<mi>θ</mi>
<mo>=</mo>
<mn>0</mn>
</math></span> into their differential equation <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}\theta }}{{{\text{d}}t}} = \frac{{24}}{3} = 8">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>θ</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mn>24</mn>
</mrow>
<mn>3</mn>
</mfrac>
<mo>=</mo>
<mn>8</mn>
</math></span> (rad s<sup>−1</sup>) <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> Accept −8 rad s<sup>−1</sup>.</p>
<p><strong>Note:</strong> Can be done by consideration of CX, use of Pythagoras.</p>
<p> </p>
<p><strong>METHOD 3</strong></p>
<p>let the position of the car be at time <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> be <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="d - 24t">
<mi>d</mi>
<mo>−</mo>
<mn>24</mn>
<mi>t</mi>
</math></span> from O <em><strong>(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{tan}}\,\theta = \frac{{d - 24t}}{3}\left( { = \frac{d}{3} - 8t} \right)">
<mrow>
<mtext>tan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mi>d</mi>
<mo>−</mo>
<mn>24</mn>
<mi>t</mi>
</mrow>
<mn>3</mn>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mfrac>
<mi>d</mi>
<mn>3</mn>
</mfrac>
<mo>−</mo>
<mn>8</mn>
<mi>t</mi>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>M1</strong></em></p>
<p><strong>Note:</strong> For <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{tan}}\,\theta = \frac{{24t}}{3}">
<mrow>
<mtext>tan</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mn>24</mn>
<mi>t</mi>
</mrow>
<mn>3</mn>
</mfrac>
</math></span> award <em><strong>A0M1</strong></em> and follow through.</p>
<p><strong>EITHER</strong></p>
<p>attempt to differentiate implicitly with respect to <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t">
<mi>t</mi>
</math></span> <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\text{se}}{{\text{c}}^2}\,\theta \frac{{{\text{d}}\theta }}{{{\text{d}}t}} = - 8">
<mrow>
<mtext>se</mtext>
</mrow>
<mrow>
<msup>
<mrow>
<mtext>c</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>θ</mi>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>θ</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mo>−</mo>
<mn>8</mn>
</math></span> <em><strong>A1</strong></em></p>
<p>attempt to substitute for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\theta = 0">
<mi>θ</mi>
<mo>=</mo>
<mn>0</mn>
</math></span> into their differential equation <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="\theta = {\text{arctan}}\left( {\frac{d}{3} - 8t} \right)">
<mi>θ</mi>
<mo>=</mo>
<mrow>
<mtext>arctan</mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mi>d</mi>
<mn>3</mn>
</mfrac>
<mo>−</mo>
<mn>8</mn>
<mi>t</mi>
</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{{{\text{d}}\theta }}{{{\text{d}}t}} = \frac{8}{{1 + {{\left( {\frac{d}{3} - 8t} \right)}^2}}}">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>θ</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mn>8</mn>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mi>d</mi>
<mn>3</mn>
</mfrac>
<mo>−</mo>
<mn>8</mn>
<mi>t</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</math></span> <em><strong>A1</strong></em></p>
<p>at O, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="t = \frac{d}{{24}}">
<mi>t</mi>
<mo>=</mo>
<mfrac>
<mi>d</mi>
<mrow>
<mn>24</mn>
</mrow>
</mfrac>
</math></span> <em><strong>A1</strong></em></p>
<p><strong>THEN</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}\theta }}{{{\text{d}}t}} = - 8">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>θ</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>t</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mo>−</mo>
<mn>8</mn>
</math></span> <em><strong>A1</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>Given that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_{ - 2}^2 {f\left( x \right){\text{d}}x = 10} ">
<msubsup>
<mo>∫<!-- ∫ --></mo>
<mrow>
<mo>−<!-- − --></mo>
<mn>2</mn>
</mrow>
<mn>2</mn>
</msubsup>
<mrow>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
<mo>=</mo>
<mn>10</mn>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_0^2 {f\left( x \right){\text{d}}x = 12} ">
<msubsup>
<mo>∫<!-- ∫ --></mo>
<mn>0</mn>
<mn>2</mn>
</msubsup>
<mrow>
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
<mo>=</mo>
<mn>12</mn>
</mrow>
</math></span>, find</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_{ - 2}^0 {\left( {f\left( x \right){\text{ + 2}}} \right){\text{d}}x} "> <msubsup> <mo>∫</mo> <mrow> <mo>−</mo> <mn>2</mn> </mrow> <mn>0</mn> </msubsup> <mrow> <mrow> <mo>(</mo> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mrow> <mtext> + 2</mtext> </mrow> </mrow> <mo>)</mo> </mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_{ - 2}^0 {f\left( {x{\text{ + 2}}} \right){\text{d}}x} "> <msubsup> <mo>∫</mo> <mrow> <mo>−</mo> <mn>2</mn> </mrow> <mn>0</mn> </msubsup> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mrow> <mtext> + 2</mtext> </mrow> </mrow> <mo>)</mo> </mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </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 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="\int_{ - 2}^0 {f\left( x \right){\text{d}}x = 10} - 12 = - 2"> <msubsup> <mo>∫</mo> <mrow> <mo>−</mo> <mn>2</mn> </mrow> <mn>0</mn> </msubsup> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> <mo>=</mo> <mn>10</mn> </mrow> <mo>−</mo> <mn>12</mn> <mo>=</mo> <mo>−</mo> <mn>2</mn> </math></span> <em><strong>(M1)(A1)</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_{ - 2}^0 {2{\text{d}}x = \left[ {2x} \right]} _{ - 2}^0 = 4"> <msubsup> <mo>∫</mo> <mrow> <mo>−</mo> <mn>2</mn> </mrow> <mn>0</mn> </msubsup> <msubsup> <mrow> <mn>2</mn> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> <mo>=</mo> <mrow> <mo>[</mo> <mrow> <mn>2</mn> <mi>x</mi> </mrow> <mo>]</mo> </mrow> </mrow> <mrow> <mo>−</mo> <mn>2</mn> </mrow> <mn>0</mn> </msubsup> <mo>=</mo> <mn>4</mn> </math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_{ - 2}^0 {\left( {f\left( x \right){\text{ + 2}}} \right){\text{d}}x} = 2"> <msubsup> <mo>∫</mo> <mrow> <mo>−</mo> <mn>2</mn> </mrow> <mn>0</mn> </msubsup> <mrow> <mrow> <mo>(</mo> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mrow> <mtext> + 2</mtext> </mrow> </mrow> <mo>)</mo> </mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> <mo>=</mo> <mn>2</mn> </math></span> <em><strong> A1</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="\int_{ - 2}^0 {f\left( {x{\text{ + 2}}} \right){\text{d}}x} = \int_0^2 {f\left( x \right){\text{d}}x} "> <msubsup> <mo>∫</mo> <mrow> <mo>−</mo> <mn>2</mn> </mrow> <mn>0</mn> </msubsup> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mrow> <mtext> + 2</mtext> </mrow> </mrow> <mo>)</mo> </mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> <mo>=</mo> <msubsup> <mo>∫</mo> <mn>0</mn> <mn>2</mn> </msubsup> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </math></span> <em><strong>(M1)</strong></em></p>
<p>= 12 <em><strong>A</strong><strong>1</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>A right circular cone of radius <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="r">
<mi>r</mi>
</math></span> is inscribed in a sphere with centre O and radius <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="R">
<mi>R</mi>
</math></span> as shown in the following diagram. The perpendicular height of the cone is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="h">
<mi>h</mi>
</math></span>, X denotes the centre of its base and B a point where the cone touches the sphere.</p>
<p style="text-align: center;"><img 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"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the volume of the cone may be expressed by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="V = \frac{\pi }{3}\left( {2R{h^2} - {h^3}} \right)">
<mi>V</mi>
<mo>=</mo>
<mfrac>
<mi>π</mi>
<mn>3</mn>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mi>R</mi>
<mrow>
<msup>
<mi>h</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mrow>
<msup>
<mi>h</mi>
<mn>3</mn>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</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>Given that there is one inscribed cone having a maximum volume, show that the volume of this cone is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{32\pi {R^3}}}{{81}}">
<mfrac>
<mrow>
<mn>32</mn>
<mi>π</mi>
<mrow>
<msup>
<mi>R</mi>
<mn>3</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mn>81</mn>
</mrow>
</mfrac>
</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 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 use Pythagoras in triangle OXB <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow {r^2} = {R^2} - {\left( {h - R} \right)^2}">
<mo stretchy="false">⇒</mo>
<mrow>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mi>R</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mi>h</mi>
<mo>−</mo>
<mi>R</mi>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p>substitution of their <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{r^2}">
<mrow>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
</math></span> into formula for volume of cone <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="V = \frac{{\pi {r^2}h}}{3}">
<mi>V</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mi>π</mi>
<mrow>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
<mi>h</mi>
</mrow>
<mn>3</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{{\pi h}}{3}\left( {{R^2} - {{\left( {h - R} \right)}^2}} \right)">
<mo>=</mo>
<mfrac>
<mrow>
<mi>π</mi>
<mi>h</mi>
</mrow>
<mn>3</mn>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msup>
<mi>R</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>h</mi>
<mo>−</mo>
<mi>R</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{\pi h}}{3}\left( {{R^2} - \left( {{h^2} + {R^2} - 2hR} \right)} \right)">
<mo>=</mo>
<mfrac>
<mrow>
<mi>π</mi>
<mi>h</mi>
</mrow>
<mn>3</mn>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msup>
<mi>R</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mrow>
<mo>(</mo>
<mrow>
<mrow>
<msup>
<mi>h</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mi>R</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>2</mn>
<mi>h</mi>
<mi>R</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p><strong>Note:</strong> This <strong>A</strong> mark is independent and may be seen anywhere for the correct expansion of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{{{\left( {h - R} \right)}^2}}">
<mrow>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>h</mi>
<mo>−</mo>
<mi>R</mi>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</math></span>.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{\pi h}}{3}\left( {2hR - {h^2}} \right)">
<mo>=</mo>
<mfrac>
<mrow>
<mi>π</mi>
<mi>h</mi>
</mrow>
<mn>3</mn>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mi>h</mi>
<mi>R</mi>
<mo>−</mo>
<mrow>
<msup>
<mi>h</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{\pi }{3}\left( {2R{h^2} - {h^3}} \right)">
<mo>=</mo>
<mfrac>
<mi>π</mi>
<mn>3</mn>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mi>R</mi>
<mrow>
<msup>
<mi>h</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mrow>
<msup>
<mi>h</mi>
<mn>3</mn>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>AG</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>at max, <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}V}}{{{\text{d}}h}} = 0">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>V</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>h</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mn>0</mn>
</math></span> <em><strong>R1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}V}}{{{\text{d}}h}} = \frac{\pi }{3}\left( {4Rh - 3{h^2}} \right)">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>V</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>h</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mi>π</mi>
<mn>3</mn>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mn>4</mn>
<mi>R</mi>
<mi>h</mi>
<mo>−</mo>
<mn>3</mn>
<mrow>
<msup>
<mi>h</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow 4Rh = 3{h^2}">
<mo stretchy="false">⇒</mo>
<mn>4</mn>
<mi>R</mi>
<mi>h</mi>
<mo>=</mo>
<mn>3</mn>
<mrow>
<msup>
<mi>h</mi>
<mn>2</mn>
</msup>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow h = \frac{{4R}}{3}">
<mo stretchy="false">⇒</mo>
<mi>h</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mn>4</mn>
<mi>R</mi>
</mrow>
<mn>3</mn>
</mfrac>
</math></span> (since <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="h \ne 0">
<mi>h</mi>
<mo>≠</mo>
<mn>0</mn>
</math></span>) <em><strong>A1</strong></em></p>
<p><strong>EITHER</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{V_{{\text{max}}}} = \frac{\pi }{3}\left( {2R{h^2} - {h^3}} \right)">
<mrow>
<msub>
<mi>V</mi>
<mrow>
<mrow>
<mtext>max</mtext>
</mrow>
</mrow>
</msub>
</mrow>
<mo>=</mo>
<mfrac>
<mi>π</mi>
<mn>3</mn>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mi>R</mi>
<mrow>
<msup>
<mi>h</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mrow>
<msup>
<mi>h</mi>
<mn>3</mn>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</math></span> from part (a)</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{\pi }{3}\left( {2R{{\left( {\frac{{4R}}{3}} \right)}^2} - {{\left( {\frac{{4R}}{3}} \right)}^3}} \right)">
<mo>=</mo>
<mfrac>
<mi>π</mi>
<mn>3</mn>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mi>R</mi>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mn>4</mn>
<mi>R</mi>
</mrow>
<mn>3</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mn>4</mn>
<mi>R</mi>
</mrow>
<mn>3</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>3</mn>
</msup>
</mrow>
</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=" = \frac{\pi }{3}\left( {2R\frac{{16{R^2}}}{9} - \left( {\frac{{64{R^3}}}{{27}}} \right)} \right)">
<mo>=</mo>
<mfrac>
<mi>π</mi>
<mn>3</mn>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mn>2</mn>
<mi>R</mi>
<mfrac>
<mrow>
<mn>16</mn>
<mrow>
<msup>
<mi>R</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mn>9</mn>
</mfrac>
<mo>−</mo>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mn>64</mn>
<mrow>
<msup>
<mi>R</mi>
<mn>3</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mn>27</mn>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</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="{r^2} = {R^2} - {\left( {\frac{{4R}}{3} - R} \right)^2}">
<mrow>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mi>R</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mn>4</mn>
<mi>R</mi>
</mrow>
<mn>3</mn>
</mfrac>
<mo>−</mo>
<mi>R</mi>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{r^2} = {R^2} - \frac{{{R^2}}}{9} = \frac{{8{R^2}}}{9}">
<mrow>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mi>R</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>R</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mn>9</mn>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mn>8</mn>
<mrow>
<msup>
<mi>R</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mn>9</mn>
</mfrac>
</math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \Rightarrow {V_{{\text{max}}}} = \frac{{\pi {r^2}}}{3}\left( {\frac{{4R}}{3}} \right)">
<mo stretchy="false">⇒</mo>
<mrow>
<msub>
<mi>V</mi>
<mrow>
<mrow>
<mtext>max</mtext>
</mrow>
</mrow>
</msub>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mi>π</mi>
<mrow>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mn>3</mn>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mn>4</mn>
<mi>R</mi>
</mrow>
<mn>3</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{4\pi R}}{9}\left( {\frac{{8{R^2}}}{9}} \right)">
<mo>=</mo>
<mfrac>
<mrow>
<mn>4</mn>
<mi>π</mi>
<mi>R</mi>
</mrow>
<mn>9</mn>
</mfrac>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mn>8</mn>
<mrow>
<msup>
<mi>R</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mn>9</mn>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <em><strong>A1</strong></em></p>
<p><strong>THEN</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{{32\pi {R^3}}}{{81}}">
<mo>=</mo>
<mfrac>
<mrow>
<mn>32</mn>
<mi>π</mi>
<mrow>
<msup>
<mi>R</mi>
<mn>3</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mn>81</mn>
</mrow>
</mfrac>
</math></span> <em><strong>AG</strong></em></p>
<p><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>Let <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = {{\text{e}}^x}\sin x">
<mi>y</mi>
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mi>x</mi>
</msup>
</mrow>
<mi>sin</mi>
<mo><!-- --></mo>
<mi>x</mi>
</math></span>.</p>
</div>
<div class="specification">
<p>Consider the function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f">
<mi>f</mi>
</math></span> defined by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f(x) = {{\text{e}}^x}\sin x,{\text{ }}0 \leqslant x \leqslant \pi ">
<mi>f</mi>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mi>x</mi>
</msup>
</mrow>
<mi>sin</mi>
<mo><!-- --></mo>
<mi>x</mi>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mn>0</mn>
<mo>⩽<!-- ⩽ --></mo>
<mi>x</mi>
<mo>⩽<!-- ⩽ --></mo>
<mi>π<!-- π --></mi>
</math></span>.</p>
</div>
<div class="specification">
<p>The curvature at any point <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(x,{\text{ }}y)">
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>,</mo>
<mrow>
<mtext> </mtext>
</mrow>
<mi>y</mi>
<mo stretchy="false">)</mo>
</math></span> on a graph is defined as <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\kappa = \frac{{\left| {\frac{{{{\text{d}}^2}y}}{{{\text{d}}{x^2}}}} \right|}}{{{{\left( {1 + {{\left( {\frac{{{\text{d}}y}}{{{\text{d}}x}}} \right)}^2}} \right)}^{\frac{3}{2}}}}}">
<mi>κ<!-- κ --></mi>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<mo>|</mo>
<mrow>
<mfrac>
<mrow>
<mrow>
<msup>
<mrow>
<mtext>d</mtext>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</mrow>
<mo>|</mo>
</mrow>
</mrow>
<mrow>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>+</mo>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mfrac>
<mn>3</mn>
<mn>2</mn>
</mfrac>
</mrow>
</msup>
</mrow>
</mrow>
</mfrac>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find an expression for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}}"> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>y</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mfrac> </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>Show that <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{{\text{d}}^2}y}}{{{\text{d}}{x^2}}} = 2{{\text{e}}^x}\cos x"> <mfrac> <mrow> <mrow> <msup> <mrow> <mtext>d</mtext> </mrow> <mn>2</mn> </msup> </mrow> <mi>y</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mrow> </mfrac> <mo>=</mo> <mn>2</mn> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mi>cos</mi> <mo></mo> <mi>x</mi> </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>Show that the function <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f"> <mi>f</mi> </math></span> has a local maximum value when <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \frac{{3\pi }}{4}"> <mi>x</mi> <mo>=</mo> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>4</mn> </mfrac> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>-coordinate of the point of inflexion of the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f"> <mi>f</mi> </math></span>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f"> <mi>f</mi> </math></span>, clearly indicating the position of the local maximum point, the point of inflexion and the axes intercepts.</p>
<div class="marks">[3]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the area of the region enclosed by the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f"> <mi>f</mi> </math></span> and the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x"> <mi>x</mi> </math></span>-axis.</p>
<p> </p>
<div class="marks">[6]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value of the curvature of the graph of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f"> <mi>f</mi> </math></span> at the local maximum point.</p>
<div class="marks">[3]</div>
<div class="question_part_label">g.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the value <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\kappa "> <mi>κ</mi> </math></span> for <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \frac{\pi }{2}"> <mi>x</mi> <mo>=</mo> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </math></span> and comment on its meaning with respect to the shape of the graph.</p>
<div class="marks">[2]</div>
<div class="question_part_label">h.</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="\frac{{{\text{d}}y}}{{{\text{d}}x}} = {{\text{e}}^x}\sin x + {{\text{e}}^x}\cos x{\text{ }}\left( { = {{\text{e}}^x}(\sin x + \cos x)} \right)"> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>y</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mfrac> <mo>=</mo> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mi>sin</mi> <mo></mo> <mi>x</mi> <mo>+</mo> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mi>cos</mi> <mo></mo> <mi>x</mi> <mrow> <mtext> </mtext> </mrow> <mrow> <mo>(</mo> <mrow> <mo>=</mo> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mo stretchy="false">(</mo> <mi>sin</mi> <mo></mo> <mi>x</mi> <mo>+</mo> <mi>cos</mi> <mo></mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> </math></span> <strong><em>M1A1</em></strong></p>
<p><strong><em>[2 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="\frac{{{{\text{d}}^2}y}}{{{\text{d}}{x^2}}} = {{\text{e}}^x}(\sin x + \cos x) + {{\text{e}}^x}(\cos x - \sin x)"> <mfrac> <mrow> <mrow> <msup> <mrow> <mtext>d</mtext> </mrow> <mn>2</mn> </msup> </mrow> <mi>y</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mrow> </mfrac> <mo>=</mo> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mo stretchy="false">(</mo> <mi>sin</mi> <mo></mo> <mi>x</mi> <mo>+</mo> <mi>cos</mi> <mo></mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mo stretchy="false">(</mo> <mi>cos</mi> <mo></mo> <mi>x</mi> <mo>−</mo> <mi>sin</mi> <mo></mo> <mi>x</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{{\text{e}}^x}\cos x"> <mo>=</mo> <mn>2</mn> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mi>cos</mi> <mo></mo> <mi>x</mi> </math></span> <strong><em>AG</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="\frac{{{\text{d}}y}}{{{\text{d}}x}} = {{\text{e}}^{\frac{{3\pi }}{4}}}\left( {\sin \frac{{3\pi }}{4} + \cos \frac{{3\pi }}{4}} \right) = 0"> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>y</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mfrac> <mo>=</mo> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>4</mn> </mfrac> </mrow> </msup> </mrow> <mrow> <mo>(</mo> <mrow> <mi>sin</mi> <mo></mo> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>4</mn> </mfrac> <mo>+</mo> <mi>cos</mi> <mo></mo> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>4</mn> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </math></span> <strong><em>R1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{{\text{d}}^2}y}}{{{\text{d}}{x^2}}} = 2{{\text{e}}^{\frac{{3\pi }}{4}}}\cos \frac{{3\pi }}{4} < 0"> <mfrac> <mrow> <mrow> <msup> <mrow> <mtext>d</mtext> </mrow> <mn>2</mn> </msup> </mrow> <mi>y</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mrow> </mfrac> <mo>=</mo> <mn>2</mn> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>4</mn> </mfrac> </mrow> </msup> </mrow> <mi>cos</mi> <mo></mo> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>4</mn> </mfrac> <mo><</mo> <mn>0</mn> </math></span> <strong><em>R1</em></strong></p>
<p>hence maximum at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \frac{{3\pi }}{4}"> <mi>x</mi> <mo>=</mo> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>4</mn> </mfrac> </math></span> <strong><em>AG</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">c.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{{\text{d}}^2}y}}{{{\text{d}}{x^2}}} = 0 \Rightarrow 2{{\text{e}}^x}\cos x = 0"> <mfrac> <mrow> <mrow> <msup> <mrow> <mtext>d</mtext> </mrow> <mn>2</mn> </msup> </mrow> <mi>y</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mrow> </mfrac> <mo>=</mo> <mn>0</mn> <mo stretchy="false">⇒</mo> <mn>2</mn> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mi>cos</mi> <mo></mo> <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=" \Rightarrow x = \frac{\pi }{2}"> <mo stretchy="false">⇒</mo> <mi>x</mi> <mo>=</mo> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </math></span> <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Award <strong><em>M1A0 </em></strong>if extra zeros are seen.</p>
<p> </p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><img src="images/Schermafbeelding_2017-02-28_om_14.29.02.png" alt="N16/5/MATHL/HP1/ENG/TZ0/11.e/M"></p>
<p>correct shape and correct domain <strong><em>A1</em></strong></p>
<p>max at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \frac{{3\pi }}{4}"> <mi>x</mi> <mo>=</mo> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>4</mn> </mfrac> </math></span>, point of inflexion at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \frac{\pi }{2}"> <mi>x</mi> <mo>=</mo> <mfrac> <mi>π</mi> <mn>2</mn> </mfrac> </math></span> <strong><em>A1</em></strong></p>
<p>zeros at <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = 0"> <mi>x</mi> <mo>=</mo> <mn>0</mn> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \pi "> <mi>x</mi> <mo>=</mo> <mi>π</mi> </math></span> <strong><em>A1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Penalize incorrect domain with first <strong><em>A </em></strong>mark; allow <strong><em>FT </em></strong>from (d) on extra points of inflexion.</p>
<p> </p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">e.</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="\int_0^x {{{\text{e}}^x}\sin x{\text{d}}x = [{{\text{e}}^x}\sin x]_0^\pi - \int_0^\pi {{{\text{e}}^x}\cos x{\text{d}}x} } "> <msubsup> <mo>∫</mo> <mn>0</mn> <mi>x</mi> </msubsup> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mi>sin</mi> <mo></mo> <mi>x</mi> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> <mo>=</mo> <mo stretchy="false">[</mo> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mi>sin</mi> <mo></mo> <mi>x</mi> <msubsup> <mo stretchy="false">]</mo> <mn>0</mn> <mi>π</mi> </msubsup> <mo>−</mo> <msubsup> <mo>∫</mo> <mn>0</mn> <mi>π</mi> </msubsup> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mi>cos</mi> <mo></mo> <mi>x</mi> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mrow> </math></span> <strong><em>M1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_0^\pi {{{\text{e}}^x}\sin x{\text{d}}x = [{{\text{e}}^x}\sin x]_0^\pi - \left( {[{{\text{e}}^x}\cos x]_0^x + \int_0^\pi {{{\text{e}}^x}\sin x{\text{d}}x} } \right)} "> <msubsup> <mo>∫</mo> <mn>0</mn> <mi>π</mi> </msubsup> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mi>sin</mi> <mo></mo> <mi>x</mi> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> <mo>=</mo> <mo stretchy="false">[</mo> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mi>sin</mi> <mo></mo> <mi>x</mi> <msubsup> <mo stretchy="false">]</mo> <mn>0</mn> <mi>π</mi> </msubsup> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mo stretchy="false">[</mo> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mi>cos</mi> <mo></mo> <mi>x</mi> <msubsup> <mo stretchy="false">]</mo> <mn>0</mn> <mi>x</mi> </msubsup> <mo>+</mo> <msubsup> <mo>∫</mo> <mn>0</mn> <mi>π</mi> </msubsup> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mi>sin</mi> <mo></mo> <mi>x</mi> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> </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="\int_0^\pi {{{\text{e}}^x}\sin x{\text{d}}x = [ - {{\text{e}}^x}\cos x]_0^\pi + \int_0^\pi {{{\text{e}}^x}\cos x{\text{d}}x} } "> <msubsup> <mo>∫</mo> <mn>0</mn> <mi>π</mi> </msubsup> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mi>sin</mi> <mo></mo> <mi>x</mi> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> <mo>=</mo> <mo stretchy="false">[</mo> <mo>−</mo> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mi>cos</mi> <mo></mo> <mi>x</mi> <msubsup> <mo stretchy="false">]</mo> <mn>0</mn> <mi>π</mi> </msubsup> <mo>+</mo> <msubsup> <mo>∫</mo> <mn>0</mn> <mi>π</mi> </msubsup> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mi>cos</mi> <mo></mo> <mi>x</mi> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mrow> </math></span> <strong><em>M1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_0^\pi {{{\text{e}}^x}\sin x{\text{d}}x = [ - {{\text{e}}^x}\cos x]} _0^\pi + \left( {[{{\text{e}}^x}\sin x]_0^\pi - \int_0^\pi {{{\text{e}}^x}\sin x{\text{d}}x} } \right)"> <msubsup> <mo>∫</mo> <mn>0</mn> <mi>π</mi> </msubsup> <msubsup> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mi>sin</mi> <mo></mo> <mi>x</mi> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> <mo>=</mo> <mo stretchy="false">[</mo> <mo>−</mo> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mi>cos</mi> <mo></mo> <mi>x</mi> <mo stretchy="false">]</mo> </mrow> <mn>0</mn> <mi>π</mi> </msubsup> <mo>+</mo> <mrow> <mo>(</mo> <mrow> <mo stretchy="false">[</mo> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mi>sin</mi> <mo></mo> <mi>x</mi> <msubsup> <mo stretchy="false">]</mo> <mn>0</mn> <mi>π</mi> </msubsup> <mo>−</mo> <msubsup> <mo>∫</mo> <mn>0</mn> <mi>π</mi> </msubsup> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mi>sin</mi> <mo></mo> <mi>x</mi> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mrow> <mo>)</mo> </mrow> </math></span> <strong><em>A1</em></strong></p>
<p><strong>THEN</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_0^\pi {{{\text{e}}^x}\sin x{\text{d}}x = \frac{1}{2}\left( {[{{\text{e}}^x}\sin x]_0^x - [{{\text{e}}^x}\cos x]_0^x} \right)} "> <msubsup> <mo>∫</mo> <mn>0</mn> <mi>π</mi> </msubsup> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mi>sin</mi> <mo></mo> <mi>x</mi> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mrow> <mo>(</mo> <mrow> <mo stretchy="false">[</mo> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mi>sin</mi> <mo></mo> <mi>x</mi> <msubsup> <mo stretchy="false">]</mo> <mn>0</mn> <mi>x</mi> </msubsup> <mo>−</mo> <mo stretchy="false">[</mo> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mi>cos</mi> <mo></mo> <mi>x</mi> <msubsup> <mo stretchy="false">]</mo> <mn>0</mn> <mi>x</mi> </msubsup> </mrow> <mo>)</mo> </mrow> </mrow> </math></span> <strong><em>M1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int_0^\pi {{{\text{e}}^x}\sin x{\text{d}}x = \frac{1}{2}({{\text{e}}^x} + 1)} "> <msubsup> <mo>∫</mo> <mn>0</mn> <mi>π</mi> </msubsup> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mi>sin</mi> <mo></mo> <mi>x</mi> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mo stretchy="false">(</mo> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mi>x</mi> </msup> </mrow> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[6 marks]</em></strong></p>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}} = 0"> <mfrac> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>y</mi> </mrow> <mrow> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </mfrac> <mo>=</mo> <mn>0</mn> </math></span> <strong><em>(A1)</em></strong></p>
<p> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{d^2}y}}{{d{x^2}}} = 2{e^{\frac{{3\pi }}{4}}}\cos \frac{{3\pi }}{4} = - \sqrt 2 {e^{\frac{{3\pi }}{4}}}"> <mfrac> <mrow> <mrow> <msup> <mi>d</mi> <mn>2</mn> </msup> </mrow> <mi>y</mi> </mrow> <mrow> <mi>d</mi> <mrow> <msup> <mi>x</mi> <mn>2</mn> </msup> </mrow> </mrow> </mfrac> <mo>=</mo> <mn>2</mn> <mrow> <msup> <mi>e</mi> <mrow> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>4</mn> </mfrac> </mrow> </msup> </mrow> <mi>cos</mi> <mo></mo> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>4</mn> </mfrac> <mo>=</mo> <mo>−</mo> <msqrt> <mn>2</mn> </msqrt> <mrow> <msup> <mi>e</mi> <mrow> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>4</mn> </mfrac> </mrow> </msup> </mrow> </math></span> <strong><em>(A1)</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\kappa = \frac{{\left| { - \sqrt 2 {{\text{e}}^{\frac{{3\pi }}{4}}}} \right|}}{1} = \sqrt 2 {{\text{e}}^{\frac{{3\pi }}{4}}}"> <mi>κ</mi> <mo>=</mo> <mfrac> <mrow> <mrow> <mo>|</mo> <mrow> <mo>−</mo> <msqrt> <mn>2</mn> </msqrt> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>4</mn> </mfrac> </mrow> </msup> </mrow> </mrow> <mo>|</mo> </mrow> </mrow> <mn>1</mn> </mfrac> <mo>=</mo> <msqrt> <mn>2</mn> </msqrt> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mn>4</mn> </mfrac> </mrow> </msup> </mrow> </math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">g.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\kappa = 0"> <mi>κ</mi> <mo>=</mo> <mn>0</mn> </math></span> <strong><em>A1</em></strong></p>
<p>the graph is approximated by a straight line <strong><em>R1</em></strong></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">h.</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>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">d.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">g.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">h.</div>
</div>
<br><hr><br><div class="specification">
<p>The diagram shows the slope field for the differential equation</p>
<p style="text-align: center;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mrow><mo>d</mo><mi>y</mi></mrow><mrow><mo>d</mo><mi>x</mi></mrow></mfrac><mo>=</mo><mi>sin</mi><mfenced><mrow><mi>x</mi><mo>+</mo><mi>y</mi></mrow></mfenced><mo>,</mo><mo> </mo><mo>-</mo><mn>4</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mn>5</mn><mo>,</mo><mo> </mo><mn>0</mn><mo>≤</mo><mi>y</mi><mo>≤</mo><mn>5</mn></math>.</p>
<p>The graphs of the two solutions to the differential equation that pass through points <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>0</mn><mo>,</mo><mo> </mo><mn>1</mn><mo>)</mo></math> and <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mn>0</mn><mo>,</mo><mo> </mo><mn>3</mn><mo>)</mo></math> are shown.</p>
<p><img style="display: block; margin-left: auto; margin-right: auto;" src="data:image/png;base64,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"></p>
<p>For the two solutions given, the local minimum points lie on the straight line <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mn>1</mn></msub></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find the equation of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mn>1</mn></msub></math>, giving your answer in the form <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>.</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>For the two solutions given, the local maximum points lie on the straight line <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mn>2</mn></msub></math>.</p>
<p>Find the equation of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mn>2</mn></msub></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>sin</mi><mfenced><mrow><mi>x</mi><mo>+</mo><mi>y</mi></mrow></mfenced><mo>=</mo><mn>0</mn></math> <em><strong> A1</strong></em></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo>=</mo><mn>0</mn></math> <em><strong> (M1)</strong></em></p>
<p>(the equation of <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>L</mi><mn>1</mn></msub></math> is) <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>-</mo><mi>x</mi></math> <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><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>+</mo><mi>y</mi><mo>=</mo><mi mathvariant="normal">π</mi></math> <strong>OR </strong><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mo>-</mo><mi>x</mi><mo>+</mo><mi mathvariant="normal">π</mi></math> <em><strong> (M1)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;">
[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 the functions <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f,\,\,g,">
<mi>f</mi>
<mo>,</mo>
<mspace width="thinmathspace"></mspace>
<mspace width="thinmathspace"></mspace>
<mi>g</mi>
<mo>,</mo>
</math></span> defined for <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>, given by <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f\left( x \right) = {{\text{e}}^{ - x}}\,{\text{sin}}\,x">
<mi>f</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−<!-- − --></mo>
<mi>x</mi>
</mrow>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>sin</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g\left( x \right) = {{\text{e}}^{ - x}}\,{\text{cos}}\,x">
<mi>g</mi>
<mrow>
<mo>(</mo>
<mi>x</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mtext>e</mtext>
</mrow>
<mrow>
<mo>−<!-- − --></mo>
<mi>x</mi>
</mrow>
</msup>
</mrow>
<mspace width="thinmathspace"></mspace>
<mrow>
<mtext>cos</mtext>
</mrow>
<mspace width="thinmathspace"></mspace>
<mi>x</mi>
</math></span>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f'\left( x \right)"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </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>Find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g'\left( x \right)"> <msup> <mi>g</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </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>Hence, or otherwise, find <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int\limits_0^\pi {{{\text{e}}^{ - x}}\,{\text{sin}}\,x\,{\text{d}}x} "> <munderover> <mo>∫</mo> <mn>0</mn> <mi>π</mi> </munderover> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mo>−</mo> <mi>x</mi> </mrow> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> </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>attempt at product rule <em><strong>M1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f'\left( x \right) = - {{\text{e}}^{ - x}}\,{\text{sin}}\,x + {{\text{e}}^{ - x}}\,{\text{cos}}\,x"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mo>−</mo> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mo>−</mo> <mi>x</mi> </mrow> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>+</mo> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mo>−</mo> <mi>x</mi> </mrow> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </math></span> <em><strong> 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><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g'\left( x \right) = - {{\text{e}}^{ - x}}\,{\text{cos}}\,x - {{\text{e}}^{ - x}}\,{\text{sin}}\,x"> <msup> <mi>g</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mo>−</mo> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mo>−</mo> <mi>x</mi> </mrow> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>−</mo> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mo>−</mo> <mi>x</mi> </mrow> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </math></span> <em><strong> A1</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>METHOD 1</strong></p>
<p>Attempt to add <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="f'\left( x \right)"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="g'\left( x \right)"> <msup> <mi>g</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mi>x</mi> <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="f'\left( x \right) + g'\left( x \right) = - 2{{\text{e}}^{ - x}}\,{\text{sin}}\,x"> <msup> <mi>f</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>+</mo> <msup> <mi>g</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mo>−</mo> <mn>2</mn> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mo>−</mo> <mi>x</mi> </mrow> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </math></span> <em><strong>A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\int\limits_0^\pi {{{\text{e}}^{ - x}}\,{\text{sin}}\,x\,{\text{d}}x} = \left[ { - \frac{{{{\text{e}}^{ - x}}}}{2}\left( {{\text{sin}}\,x + {\text{cos}}\,x} \right)} \right]_0^\pi "> <munderover> <mo>∫</mo> <mn>0</mn> <mi>π</mi> </munderover> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mo>−</mo> <mi>x</mi> </mrow> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </mrow> <mo>=</mo> <msubsup> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <mfrac> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mo>−</mo> <mi>x</mi> </mrow> </msup> </mrow> </mrow> <mn>2</mn> </mfrac> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>+</mo> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mn>0</mn> <mi>π</mi> </msubsup> </math></span> (or equivalent) <em><strong>A1</strong></em></p>
<p><strong>Note</strong>: Condone absence of limits.</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = \frac{1}{2}\left( {1 + {{\text{e}}^{ - \pi }}} \right)"> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mo>−</mo> <mi>π</mi> </mrow> </msup> </mrow> </mrow> <mo>)</mo> </mrow> </math></span> <em><strong>A1</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="I = \int {{{\text{e}}^{ - x}}} \,{\text{sin}}\,x\,{\text{d}}x"> <mi>I</mi> <mo>=</mo> <mo>∫</mo> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mo>−</mo> <mi>x</mi> </mrow> </msup> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = - {{\text{e}}^{ - x}}\,{\text{cos}}\,x - \int {{{\text{e}}^{ - x}}} \,{\text{cos}}\,x\,{\text{d}}x"> <mo>=</mo> <mo>−</mo> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mo>−</mo> <mi>x</mi> </mrow> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>−</mo> <mo>∫</mo> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mo>−</mo> <mi>x</mi> </mrow> </msup> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </math></span> <strong>OR </strong><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = - {{\text{e}}^{ - x}}\,{\text{sin}}\,x + \int {{{\text{e}}^{ - x}}} \,{\text{cos}}\,x\,{\text{d}}x"> <mo>=</mo> <mo>−</mo> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mo>−</mo> <mi>x</mi> </mrow> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>+</mo> <mo>∫</mo> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mo>−</mo> <mi>x</mi> </mrow> </msup> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </math></span> <em><strong>M1A1</strong></em></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" = - {{\text{e}}^{ - x}}\,{\text{sin}}\,x - {{\text{e}}^{ - x}}\,{\text{cos}}\,x - \int {{{\text{e}}^{ - x}}} \,{\text{sin}}\,x\,{\text{d}}x"> <mo>=</mo> <mo>−</mo> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mo>−</mo> <mi>x</mi> </mrow> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>−</mo> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mo>−</mo> <mi>x</mi> </mrow> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>−</mo> <mo>∫</mo> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mo>−</mo> <mi>x</mi> </mrow> </msup> </mrow> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> </math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="I = \frac{1}{2}{{\text{e}}^{ - x}}\left( {{\text{sin}}\,x + {\text{cos}}\,x} \right)"> <mi>I</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mo>−</mo> <mi>x</mi> </mrow> </msup> </mrow> <mrow> <mo>(</mo> <mrow> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mo>+</mo> <mrow> <mtext>cos</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> </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="\int_0^\pi {{{\text{e}}^{ - x}}\,{\text{sin}}\,x\,{\text{d}}x = \frac{1}{2}\left( {1 + {{\text{e}}^{ - \pi }}} \right)} "> <msubsup> <mo>∫</mo> <mn>0</mn> <mi>π</mi> </msubsup> <mrow> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mo>−</mo> <mi>x</mi> </mrow> </msup> </mrow> <mspace width="thinmathspace"></mspace> <mrow> <mtext>sin</mtext> </mrow> <mspace width="thinmathspace"></mspace> <mi>x</mi> <mspace width="thinmathspace"></mspace> <mrow> <mtext>d</mtext> </mrow> <mi>x</mi> <mo>=</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow> <msup> <mrow> <mtext>e</mtext> </mrow> <mrow> <mo>−</mo> <mi>π</mi> </mrow> </msup> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> </math></span> <em><strong> A1</strong></em></p>
<p><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.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>The folium of Descartes is a curve defined by the equation <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{x^3} + {y^3} - 3xy = 0">
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mi>y</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>3</mn>
<mi>x</mi>
<mi>y</mi>
<mo>=</mo>
<mn>0</mn>
</math></span>, shown in the following diagram.</p>
<p><img src="images/Schermafbeelding_2018-02-07_om_18.23.15.png" alt="N17/5/MATHL/HP1/ENG/TZ0/07"></p>
<p>Determine the exact coordinates of the point P on the curve where the tangent line is parallel to the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span>-axis.</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="{x^3} + {y^3} - 3xy = 0">
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mi>y</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>3</mn>
<mi>x</mi>
<mi>y</mi>
<mo>=</mo>
<mn>0</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="3{x^2} + 3{y^2}\frac{{{\text{d}}y}}{{{\text{d}}x}} - 3x\frac{{{\text{d}}y}}{{{\text{d}}x}} - 3y = 0">
<mn>3</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mn>3</mn>
<mrow>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
</mrow>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>−</mo>
<mn>3</mn>
<mi>x</mi>
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>−</mo>
<mn>3</mn>
<mi>y</mi>
<mo>=</mo>
<mn>0</mn>
</math></span> <strong><em>M1A1</em></strong></p>
<p> </p>
<p><strong>Note: </strong>Differentiation wrt <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y">
<mi>y</mi>
</math></span> is also acceptable.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{{\text{d}}y}}{{{\text{d}}x}} = \frac{{3y - 3{x^2}}}{{3{y^2} - 3x}}{\text{ }}\left( { = \frac{{y - {x^2}}}{{{y^2} - x}}} \right)">
<mfrac>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>y</mi>
</mrow>
<mrow>
<mrow>
<mtext>d</mtext>
</mrow>
<mi>x</mi>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mn>3</mn>
<mi>y</mi>
<mo>−</mo>
<mn>3</mn>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mn>3</mn>
<mrow>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>3</mn>
<mi>x</mi>
</mrow>
</mfrac>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mi>y</mi>
<mo>−</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mrow>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mi>x</mi>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</math></span> <strong><em>(A1)</em></strong></p>
<p> </p>
<p><strong>Note: </strong>All following marks may be awarded if the denominator is correct, but the numerator incorrect.</p>
<p> </p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{y^2} - x = 0">
<mrow>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mi>x</mi>
<mo>=</mo>
<mn>0</mn>
</math></span> <strong><em>M1</em></strong></p>
<p><strong>EITHER</strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = {y^2}">
<mi>x</mi>
<mo>=</mo>
<mrow>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{y^6} + {y^3} - 3{y^3} = 0">
<mrow>
<msup>
<mi>y</mi>
<mn>6</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mi>y</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>3</mn>
<mrow>
<msup>
<mi>y</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span> <strong><em>M1A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{y^6} - 2{y^3} = 0">
<mrow>
<msup>
<mi>y</mi>
<mn>6</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>2</mn>
<mrow>
<msup>
<mi>y</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>=</mo>
<mn>0</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{y^3}({y^3} - 2) = 0">
<mrow>
<msup>
<mi>y</mi>
<mn>3</mn>
</msup>
</mrow>
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mi>y</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>2</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(y \ne 0)\therefore y = \sqrt[3]{2}">
<mo stretchy="false">(</mo>
<mi>y</mi>
<mo>≠</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
<mo>∴</mo>
<mi>y</mi>
<mo>=</mo>
<mroot>
<mn>2</mn>
<mn>3</mn>
</mroot>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = {\left( {\sqrt[3]{2}} \right)^2}{\text{ }}\left( { = \sqrt[3]{4}} \right)">
<mi>x</mi>
<mo>=</mo>
<mrow>
<msup>
<mrow>
<mo>(</mo>
<mrow>
<mroot>
<mn>2</mn>
<mn>3</mn>
</mroot>
</mrow>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mrow>
<mtext> </mtext>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mo>=</mo>
<mroot>
<mn>4</mn>
<mn>3</mn>
</mroot>
</mrow>
<mo>)</mo>
</mrow>
</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="{x^3} + xy - 3xy = 0">
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>+</mo>
<mi>x</mi>
<mi>y</mi>
<mo>−</mo>
<mn>3</mn>
<mi>x</mi>
<mi>y</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="x({x^2} - 2y) = 0">
<mi>x</mi>
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>2</mn>
<mi>y</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x \ne 0 \Rightarrow y = \frac{{{x^2}}}{2}">
<mi>x</mi>
<mo>≠</mo>
<mn>0</mn>
<mo stretchy="false">⇒</mo>
<mi>y</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<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="{y^2} = \frac{{{x^4}}}{4}">
<mrow>
<msup>
<mi>y</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>4</mn>
</msup>
</mrow>
</mrow>
<mn>4</mn>
</mfrac>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x = \frac{{{x^4}}}{4}">
<mi>x</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mi>x</mi>
<mn>4</mn>
</msup>
</mrow>
</mrow>
<mn>4</mn>
</mfrac>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="x({x^3} - 4) = 0">
<mi>x</mi>
<mo stretchy="false">(</mo>
<mrow>
<msup>
<mi>x</mi>
<mn>3</mn>
</msup>
</mrow>
<mo>−</mo>
<mn>4</mn>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mn>0</mn>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="(x \ne 0)\therefore x = \sqrt[3]{4}">
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>≠</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
<mo>∴</mo>
<mi>x</mi>
<mo>=</mo>
<mroot>
<mn>4</mn>
<mn>3</mn>
</mroot>
</math></span> <strong><em>A1</em></strong></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="y = \frac{{{{\left( {\sqrt[3]{4}} \right)}^2}}}{2} = \sqrt[3]{2}">
<mi>y</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mroot>
<mn>4</mn>
<mn>3</mn>
</mroot>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mn>2</mn>
</mfrac>
<mo>=</mo>
<mroot>
<mn>2</mn>
<mn>3</mn>
</mroot>
</math></span> <strong><em>A1</em></strong></p>
<p><strong><em>[8 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br>