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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>
<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>
<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>
<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,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"></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>
<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>
<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>
<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>
<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>
<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>
<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 src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAu8AAAE3CAYAAAAaBq+OAAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAADsMAAA7DAcdvqGQAAF/ASURBVHhe7d0LeBRVmjfwVwWRi4xyUQeTDBdFdFAiH2Fc0YERBgEv5GHFUZaPUQmMqxl24rcgxgugBBEUVkCXBRQSGWHEhKgjN5HLkjgOAgkzqAREMQFBMYAJJHKtr/+HU6GJ6aQvVae74P97njbdp1vOpc459Vb1qerzLB8hIiIiIqKYd77+S0REREREMY7BOxERERGRRzB4JyIiIiLyiJ+seU9IiNPPiIiIiIgoWoqLd+lnp9UYvNf0QSIiIiIiMiNQTM5lM0REREREHsHgnYiIiIjIIxi8ExERERF5BIN3IiIiIiKPYPBOREREROQRDN6JiM4yH330kXoQEdHZh8E7EdFZZP/+/XLfffeqx+7du3UqERGdLRi8ExGdRV588UV59NE/quevvPKK+ktERGcPBu9ERGeJlSs/kPz8PBk2bJh6jedIIyKisweDdyKis0BlZaU899xz8vTTT0uzZs1UGp4jDUtpiIjo7MDgnYjoLDBt2jTp1u0W6dXrtzpF1HOkzZ49W6cQEZHXMXgnIvK4zZs3yyuvTJf//M//1CmnIQ3v4TNEROR9DN6JiDyuRYsW8t5771ctl/GHNLzXqFEjnUJERF7G4J2IyOOuvPJK6dSpk371U3jv6quv1q+IiMjLGLwTEREREXkEg3ciIiIiIo9g8E5ERERE5BEM3omIiIiIPILBOxERERGRRzB4JyIiIiLyCAbvFJPwc+67du3Sr8xZsWKFfmZOYWGh+ml7k9C+pn8yH3Xkz/STl0VjTkKe0Rg3+fn5+pk527Zti8pciHyJvITBO8Wk77//XlJTU/Urc1599VXjE3leXp689tpr+pUZn3/+uTz00EP6lRmlpaXSp08foztnbMthw4bpV2Yg2MIBWeiOyt5PZkrKtXGSkIDHXfLkG5/I3pP67VqgTaMRWCLAi0a+o0aNCrONI9O6TVvj9f3ss89k4sSJ+pU5aWlpxudCnDwxPRdWVFTI4MGDjR80EEWCwTvFpPbt26uH6bM/I0eOlDlz5uhXZgwaNEj+lPaY0bNr3bp1k5YtWxpt37i4OElOTpZFixbpFPehD0FWVpb6a8rw4cNDbNuTcmjTLHkq9wp5/JNiKd65QbL/8zLJeXKwPPDyJ3JIfyoQBB4DBgyIytlS5Gs6oMWYQRubDrje+stCmTZtmn5lRu/evVUQbfpgZfTo0efEXIh5qUePHrJ06VKdQuQBVjXx8VfqZ0TRVVRUZCUlJVkVFRU6xYz+/furvE2aPn26epgUjfYtLS1VeZaUlOgU99l5mtymqN8F9eqHkOc+a/WUOdbG8hP6tc+Jrdbcu9pb8Z1esjYe02m1QJ6oZ15ennptai7Pzs5WY8b0OMV48R1s61dmoI6oq93GpiA/5GsS6mp63MC5MhcSBSPQPM4z7xSzcNY0GmdEzpWz79Fo32bNmqkzeibPXiLPqVOnGv1qHGfz1q5ZrfIM7qx0C+mRNlQ6N/Gbks9vJJdc1kAa39FZrqqn02qBPHNyctRyB5Nn4HHmvVevXjJixAijZ8KHDh2qzkibvE6lYcOGMnbsWNXGJuuKb8rA5HZFXaNx9h3bFd+Une1zIVEkGLxTTEtJSVHrPU3vKBEUmFzviQATX8m/+eabOsWMaLRv3759VduaDESwTbFkB0G8KcgT+SG4DSsQKdshn+Q3lQG/7ShNdZLt1Jr4nz5uvvkm2b27RHr27CVHjh7Vn3Yfrk9p06aNjBkzRqe4D8HljBkzpN8ddxpdtpOYmBiVQA8HDZMnT9avzEDfNT0X2gcN58JcSBSu83D6XT9XsAMoLjZ/ARJRIPfdd59ce+21VYHBqg8/VH9tt/Xsqf46mb7l00/lo48+kuHDhtX6eSfzPHLkiKQ99pjagSDQtNP9RfLv+0O6nYadZHxCgtH2/WLHDpkyZYpM9T369utXle4vlH8/mM+ifTMmTJC777pLRj3+eFW6v1DyhGDSc995R3b46pv66KM11rWmf+OaDu3kvE+mSr9lv5HJw34hFxw8HfwHk+e+ffvk9w88KJMmvaDOiNvp/oL5d/zVlO6fhvad8cor0q5du6pvVtzIE/zT1+Xlyfr16wO2L9T274STZ1l5udwz8F55e9FbRsYqIH3W7NlqLrz1lltUuok8g50LIZT02j5rei4EpKN97733XnXQQhQLAsbkavGMH655p1iTOW9eVNYjhrL2/cOVK/WzyGD9cEZGhn7lrEBlxJpwrM/GX5OwXjkzM1O/OsWpdgzEXhceyZr7cMqINbwpKSlB9+ETuxdbf7jr5TPXwIeoVasrzlgDbwLqh3oGu2bZqe1dU19ySqAyYqyaXnMfaG222+MGc2FBQYF+FZ5Qy4j2Nb32PVr7GqJAAsXkDN7JE7CTxGRuEoIeBCImYaeBnce5cJGYfdBg8uJVQD8yvV0BfTioA7PyLdbc9Jes1XuO6ITwYC6vfhGrCfYFwtHIM9IAMxQYq04EtaGK1lwYrQtmo3FSwXT7EgXC4J08LVp3A8AOy2QQAthxmD6jZwc/pneU0agrIE+3ztQGgr5b51npE7ut1ePGWnO3/qATwmfP5XYAbzLIjMZBA+pneo5AnpgjTOYZrbEarbnQ9EkF3nmGYkmgmJwXrFLMw1pE+24A69at06lm+IK8oC4Sq76OMhK4oHPNmjWOX4RXWxntu8DMnDlTp5hhX7xq38PayXasDeqKix3DuZA03DLiQjysBS8rK9Mp1Zz8RtaMny07f5cmD1xjX6J6VPaumSZT13yvX4cuTt+FplGjRjrFfchz/vz5aq1/bZzc3riQdMiQIY5fSFpbGZEn5iaT8xLGKurpf0GniXET7FwYSDhlxPxg8s4zKCPvPENewOCdPAMX3rVu3Vq/MgN3DHnkkUf0KzMQ5M2aNctosAXYUXbv3l2/MgN1ff311+Waa67RKWYgAMJFjvhrEuqbnp6uX/lB4D72DzJkzmx55rfX+d1Bpq10/ffv5cbOkZUTwTSCEpOQH4JMk3DXG9MXG+JAsEuXLvqVGbid4i233KJfmYG50PT2tOdC07Cvadu2rX5FFHsYvJNnRCMAAfzCoWk4oxeNwBI7aNNQT+R9Tju/lfR49j11V4GfPD5/Vno05VQdq9B/ozFWMUeYFo27sERjLsS+JhrtSxQs7hGIiIiIiDyC93knIjoLcS4nIvK2QPM4z7wTEREREXkEg3eKeSbupBApltEZLOO5hdvbGSyjMzi2ySsYvBMREREReQSDdyIiIiIij2DwTkRERETkEbzbDBHRWYhzORGRt/FuM+QpK1asqPrJfFN27dqlforbNPxMf2VlpX5lBtoX9TVp27ZtxvMkcgr6LvqwaTk5ObJ//379ygzUFfmaNmHCBONzYX5+vvF9DVGkGLxTTGrcuLGMHTtWPTd1B4DmzZurQDqcHXQkZSwrK5NFixbpV+7xL+OhQ4dk3Lhx+pUZ+/btU3nWtnN2elsjGMA2dVJdZUTQYzoAQbBlOk8ElDgIjEQ42xt17dq1q7EDQbuMFRUVMnjwYOOBNMbqxIkT9auaOT1uGjVqpPJ08mAlmDI2bdpUXnvtNf3KjJYtW8rw4cPV+HG6HYncwuCdYhJ+pr99+/ZGz/7gJ8fHjx8vkydP1ilmPPzww/LQ0BRjwQjYP3MeafAVCmzTSy+91MiBii0tLU19m2Kyngi2unfvbnR75ubmyogRI4zmifGCbTls2DCjBw746frU1NQ6DwSdhvkI+dYVSDtt4MCBKog2eXa4WbNmMnr0aJkzZ45OMWPQoEFqvJrsx9iuPXr0MDovEUWKwTvFrJSUFLWjPHLkiE5xX+/evdUZYnyVagp2lG/9ZaFMmzZNp5gxcuRIeeqpp4yeSURAEO63G+FAgIkDQNTT1DYdMmSIOgjEAZKpgAtBZd++fVWepuqJtkWf7dSpkzpwMNmP0MZ2viYhkD5w4IDRg0G0M76FtM8Om4K+hHFqei7EHGF6LkSeOIFSVl6uU4hiG4N3ilk4I4Kd9NJly3SKGZMmTVJnbE3uKBF4mT67ZrfvzJkzdYr7sHOeNWuWWn5gqn1xpnb+/PnSvcdvjJ3Rw0Hg1KlTVcBlKvhBsIV6ou/iAKnavQhcgcASBw4IpPv06WP0jCnyBaeXRdUG9R0zZoz0u+NOo3VNTEyMytlhHDScC3Mh5qXXX5sjCxcu1ClEMQ53m/EXH3+lfkYUfaWlpVZSUpJVUlKiU8wYOXKklZmZqV+ZUVBQYPXv39+qqKjQKe5DXmjfoqIinWJGRkaGepiUl5dnvC8hL+Rpsi9hm6L/NmnSxGhd7fZFPzYFdU1JSTE+VrOzs1W+JkVzLkR9TUJfOlfmQqLaBIrJGbxTzMOOAzsQk7CjvKBeffXXpHNtR4m8TULbmq4rgi0EetOnT9cpZrRo0dx4GyNwN52nfYBkui+dSwcN58pcuHz5cjU/EMWKQDE5l81QzGt40UVR+RoV69CDvTjNqbsUYA0v8nRj/XCgMtoXBy9dulSnuA/LD+wlHv51dftuD1ha0qtXL9XO4S4FCLWMWLaDNbybN282enFnw4YXnbGMxkS+WN6BawzsPOvixPZG+9p5ujFHBCrjCy+8YPT6DUD/hepr7t0cN6HOhYGEWkbfQYprc2Eg9S64QN19xuQ1DUThYPBOMa9Bgwbq4kqsvzS99nLNmjVGd84IRLCW1+Q6dMAFW/f+7j7jd3mIxt07kCfuemPylnQ4WJk9e3bVRZam2hltvHbtWikuLjZ2cSf68LJly9TBiqmDBuRpX2Ngqm0R1OLC5FGjRhmdlzAXmr7Q3J4Lo3FNzptvvqlTzLDb1+Q2JQoVg3fyhGidHUZAgJ2zSbijBW79Z/KgwT67ZvouD/bdO0zeEhRwm0EEl6bPsOHA4dZbb5Wvv/5ap7gP/RgXYT/wwAM6xX3oT+hL+A2D0tJSneouzBEYr/bZaRNwYTLuOY98TbGDWpMH+PZcaP/2hilDhw5Vt440OReifZOTk3nrSIpp52HtjH6u8Ce1KVZhAn/77bclPT1dp5iB4B1nLXF2zxTcoWTHjh1qJ20KzjRhB226fXGmdP369UaDLkB98UCgeTY6V+dyzBMIwExBH8JJBZP9F3nirjc4KDMJP3p2zz33GG1fHGDjtxNMti++1XjooYfUSRSiaAo0jzN4JyI6C3EuJyLytkDzOJfNUMxz82Isp7CMzmAZzy3c3s5gGZ3BsU1eweCdiIiIiMgjGLwTEREREXkEg3ciIiIiIo/gBatERGchzuVERN7GC1aJiIiIiDyOwTvFPN6lwBksozN4RwrncHs7g2V0Bsc2eQWDdyIiIiIij2DwTkRERETkEQzeKSbVq3+h+tl8k3JyctTDJPyU+7Bhw/Qrc5AnfmLdJLSt6Tyjq0y2r8mVeU/eJQkpubJXp9buqHyT+x9ybUKculBJPe6eJ9tP6rcpavLz843PSfiZ/lGjRhkfNzNmzIjKXJicnGy8rpgLkTeRlzB4p5j01l8Wyrhx49Tz23r2VH/d1qNHD5k4cWJYE3m4ZWzfvr36a2JH6V/GTp06yZgxY/QrM7Zu3Vpnnk5uawRaXbt2dXzHXFsZEWzhIXJSyta8KHcPSZVn3ig49WYwDm2Wv8zcK8Nzt6g7DKjHuw/I1TE4U6N9Iw20wtneEyZMUMGlKXYZd+zYoeYkk8Fls2bN5JJLLpHXXntNp9TMyXEDCKLv/d19jh6s1FVGzIV4LFq0SKeYMXDgwKoDJKfbkcgtDN4pJg0YMED9NXn2BzvK8ePHGz/ThYA23IOGcA0dOlQOHDhgtH3T0tJUnqYCr7i4OJk6dapc98uO6qypCcXFxdKnTx/Jypov9X/1hHy+80N5NrGxfrcuR+WblfMls/X98rvES3Ra3RBgIdgyVUfbqlWrpHv37rJixQqdYgb60ebNm41/ezRkyJCoHPSivllZWUa3L8aO/wkUU0aPHq3mB5NzYe/evaNy0EAUEdzn3V98/JX6GVF0lZaWWklJSdbCBQt0ihkjR460MjMz9avgfLhypX4WnuXLl1v9+/e3KioqdIrzqpexpKTEuqBeffXXFHub5uXl6ZQzRdqONUH9kCfa2Al1lRF1RB9Cnnv3brIWD73eih+62Nqj3w/oh9VWeocr1RzcYehLVvbqbVa5fqsuaE/0HzzstjUxlxcVFak8Ud9w+lG42xvjJCMjw0pJSXF1zIB/GZEX8pw+fbpOMQPtjP4UqI3dGDeAumZnZ+tXkQm2jCbmwurseSlz3jydQhQbAs3jDN4ppmEi79evn35lhj2RY4cZLCd2nuEcNISipjJGY0dpB9MFBQU65TS3ghA7TyeCrmDLiPotX/7n4IN37cSejda7c9Ot21UQn2VtLT+h36mbfxB/2eUtdar7EODhQBB/Q+lLkW5vbE/UNZwDh2BVLyPqh77k1MFgsNC2CKZr4va4CWUuDCSUMro9F9YE27Nbt25G50KiugSKyfkLqxTz7rvvPrn22murvq6ufi9ee52ik+kbNmyQd997T55MT5e+/fpVpfvD553Ms6y8XH1tnP7EEzLk97+vSvcXyb/vD+l22ptvvqn+ztHrat3ME5C+5dNP5dVXX5VxY8fK73zb1073F8q/H8xn0b6zZ8+Wdu3aybRp06rS/YWSJ9SV/uWX+fKPCf8uI+RZmTns59LvV79S6f6fr+nfaNq0iXRouk3+lDxedg2fL7OGtJUvCgvVe/g85unaHDl6VA6VH5Ybb7xRXnjhBfEFJY7VKVDZ0b4LFy5Uy3iw9CExMdGVPME/fV1enixYsCDsvhROnvv27ZMxvvwee+wxGT58eFW6v2D+HX/BpL80ZYpcf/310vu3v1XpJvIMdi6EUNJr+6zpuRCQjrnw1927qyVSRLEgYEyuQng/PPNOsWbx4sXq7I+bZ9dqEsrZH6fOfNlnTt04+xOojMgLeQZayuIWnEmsXle3ziDakBfOXkay3CK0MhZXnXnPXr1a1Tf4s5iV1ra5g6z4Ti9ZG4/ppBBgLq9pOY3bkA/GK5a14Fus2jg5bpBnTd/mRCpQGZGX6XnJ/lawej3dHjdOnAkPtYzR+FZwyfvvq/Z14psGIidw2Qx5GibyQF8ZuyWc5TNOQNCDh0moo+lABLDsIZJAOhzIy9R6af/gHctm0I/RzsEt3zlh/bD6aatDBMG7zQ7iTa3VRrsi2ENdTR80mMoP7O1psv/aBw0m84zWXBiN5TNoX9MHDUSBMHgnz8NE7tTFU8GKxtkf5IUdpRtnEWtjr6k1vdNCnqaCSn/2eml3nRm8A9o3uH6MM++/t26fsj7oC1f9xcJcjmDPZDBtr9E2OXaicQCKgBbzoUn2AaDJekbzBIrpgwaimgSax3mrSIp59hpFrIHELRVP3UfbDPs2YuvWrdMpNau+jjISDRs2lFmzZlWtpXVKXWXE7TkvvfTSOuvqNKw/LysrU8+dbMe6pKamyqRJk/Sr4IVUxpMVcvC7IyLfHZRy/UNL2L72rVBPOyp7Ny2T3Pc2yV71uTLZnjtB/riws0wY9n+kifqM92DsYL19IE5v77i4OFm2bJkkJCTolMjVVUb0I9xC0hdo6hT34d7kuO2qfUtFE+MG2xG/m7B06VKdEppwyuh/+14T7DLi9py4bsPJ+9wTOYnBO3kGJnLclxx/TcK9jrt06aJfmYGL/Uz/wiEgmMUBi0kIZtPT0/UrsxBcuuX4pimS2LqnPFN4WKTwKenZ+naZuumQ/a7szU31BZk3SEpuyamksg0y89G7pWtr/LLqv8m8g7fJ9LdTpXMTTtOhwPxgeo5AAI8DB1MwZnDxtZv9tyYIavFjdiZhPjL5o1yA9sVBYPPmzXUKUWzhXoE8xeQO0oaJ3HQwANGoKzmnXufHpND+lVT1WC5pne1z6PXkiuQZvrR/yJzkeN/rC+WKHk/JsqrPvicZD/SQqxm4Uww5l+ZC1BP1JYpF3DMQEREREXkE7/NORHQW4lxORORtgeZxnnknIiIiIvIIBu8U80zcSSFSLKMzWMZzC7e3M1hGZ3Bsk1cweCciIiIi8ggG70REREREHsHgnYiIiIjII3i3GSKisxDnciIib+PdZsgzKisrZf/+/fqVOdH4Kexo1JOIqDaYg03DXGg632jta4gixeCdYk5paan06dOnaiI3dQeAAQMGSH5+vn4VmnDLOHPmTGM//W2XMScnR0aNGqWem7Jt2zYpLCzUrwLjHSnOLV7Z3gjwVqxYoVPMwVgNZn5wuh27d++uxqyT6irjX//6V5k6dap+ZYa9r7FP3HBsk1cweKeYg5/CHjJkiIwYMUKnmDF//nxJS0szegYe+a1cudJoYNC3b185cOCAsYMGaNSokQwfPlwFI6agfllZWVE5i0juwNjEgafTgWVd8DP5ixYtMjpmAGN18+bNqh+bhCB68ODBRufCgQMHyldffWV0jsC+JjU1VcaNG8d5gjyFwTvFJEyoYHJn2b59exk/frzK29REjqAAdXzqqaeCOjPtBOQ5bdo0o0EBdpLYKU+cONHYNh00aJBs2bJF7r//fmPBHgJLtKnJr+KjsdzADqLxTZXJvNGPevXqpQJLkwdm/mNmwoQJRvMdM2aMGjPhfisYjm7dusno0aONBrV2XTFHmJoLASeKLr30UuNn/YkiweCdYhZ2lthBb/n0U53ivt69e6vgADsRUxCQYMeBM9OmznRFIyhAPdeuXasCIBMBfLNmzWTSpEnyyCOPqGAPebodiKSkpEhZWZn6Kn7YsGFGglssN7i46c9UfhgvCHzczhPbsn///mp7Im8EtKYCLozRZcuWGT8wswN4nB3Gt4Kmglr7wBff0pkM4LGMsE2bNsbnwlmzZhmdCwEHKevXr5d1eXk6hSjG4W4z/uLjr9TPiKKvpKTEuqBeffXXlIqKCssXhFnTp0/XKWZkZ2dbvoBI5W8K2jUpKcnKy8vTKe7zb19TdUU+GRkZqq4FBQU61T3ID22KPNF/8dftfIuKilQfGjlypMqzSZMmVmZmpsrXzXYuLS21li9frvou2hd5mhqvyNfO01RfAvRd9GGTeWL7nktzoen2tedCE/MDUbACxeQM3inmIQjCpGpyIkdAgjxNBrWAnSSCL5Ows4pWUGB6B426YrsimDaVb7SC25///PIzgnm0NfJ2E+qFPFBP1Bf1Rv3dhO1o8sDMZgfwbtfPnz0Xmhyr0ZoLsU2jMReabl+i2jB4J8/6cOXKqh2lSaGc6UIZnWAHtW6c6aqtjNEICuygyz+Ad6oda+Mf7IUTkERSRv/gFoGtm6rP5ejPbufpD4EQ2rm2MeTk9nbrwKy2MmKcmh439rcN/nV0e9ygfpEe4IdaRtTPxAGnP5QxGmf9iQIJFJNzzTt5QrQuYF27ZrVa+2nyoi17rb/pC9Rwsa7puqanp0unTp2MriG288XdhSZPnmz0Tj9xcafupIT1tVi7bRL6s8k8ExMTVTsfP3ZU1dttyA9r8AG3OjQB8xIu7MS4MbVGG9vQvhuXyXX39lxo6mJsjNMXXnjB+MW69lp/XsBKsYzBO3mGHdSavBMBglrTt63ETsu+QM3kRVt2UGB6p4UACAH8unXrdIoZCGYXLFggt956q04hr7MPzHBBqykI9jBmMDeZYo8Zk2MVcyHyffzxx3WK+3DRuX0LX5N3cEJ+pm9bSRQKBu/kGXZQi3uGm4QdFu63bBLOdGGnFY26IoA3DfmaPhMN6FN40NkFQZ9JCGxx0GBSNMYq8nvggQf0KzNwkI070JiEOQF32enYsaNOIYot52HtjH6uJCTESXGxubN9RETkPM7lRETeFmge55l3IiIiIiKPYPBOMW/Vhx/qZ7GLZXQGy3hu4fZ2BsvoDI5t8goG70REREREHsHgnYiIiIjIIxi8ExERERF5BO82Q0R0FuJcTkTkbbzbDHkWL3RyBsvoDF7U5hxub2ewjM7g2CavYPBOREREROQRDN6JiIiIiDyCwTvFnF27zK/T3b9/v3qYVFlZGZW6Uiwrk+1rcmXek3dJQkqu7NWpRKZgHsTcZFo05kLkGY26EkWKwTvFnKysLJkxY4Z+JXJbz576mXs2bNggffr0CXsHEk4ZS0tLpXWbtpKfn69T3GWXsV79C43laVuxYkVQO0kT2zpS7pXxpJSteVHuHpIqz7xRoNPObl7Z3hgvpg/uoWvXrrJt2zb9KjAn2xFz4YgRIxwPausq44ABA4zPS9jX+NfVC/2RCBi8U8xJS0uTzZs3nxHAu613794yZMgQtQMxdQYoLi5OPvt0i6qvyZ2WnScCalNWrlwp999/f1CBiBOwM0abeuus2vnStMez8vnOD+XZxMY6jfwVFhZGJYj+9ttv1cG96eBy6tSpMnjwYKP5Yi7s1KmTKwF8bXJycozPhcgPTNeVKGK4VaS/+Pgr9TOi6KmoqLBSUlKs6dOnWx+uXKlT3ZeXl2clJSVZJSUlOiU4kZQReSFP5O0m/zIiz/79+6v2NWX58uWqnpmZmWr71sSpbV1aWmqNHDlS5Yc6hro9a+N+fyy2Fg+93oofutjao1PC4YW5PNS2zM7OVtsU29bt8WKzy1hUVKTGTEZGRsD+6wZ7fsC4CcSNPolxg3ydGjvBlNHUXOjP3tfgseT993UqUWwINI/zzDvFpIYNG8q0adPUGfjcd97Rqe7r1q2bOttl+gy86bNOyHPBggWqfUeNGmXkrBPO6C1btky2bNmizsK72b7NmjWTSZMmqfyaNm2qtuewYcNcPRuPM8JYkpScnKzaFN8cYbviGw584+BWffHv4t+v/jh+/HiN6U7UH+2IuqJNUVcsP0BdcWYcebh1dhzbce3ateILomXevHlqWQnyNnE2vn379mrMQPfu3VVdTbDnh3Xr1hn9NjI1NTUq30aangvtfQ3MeOUVnoEnT+CPNFFMw0R6zz33SLt27aom2Or34rXXKTqZvuXTT+XVV1+VcWPHyu/uu68q3R8+72Se+/btkzG+/B568EEZ9fjjVen+Ivn3/SEdaUeOHJHs7GzZs3evvP3222pH5maegHSsq3197lwVxI8ZM6Yq3V8o/34wn8U2/eijj1TQhcAEQfa2oiL1vi2UPKGmdGzHGzt3lp07d8rf/vY3KSkulkOHD6sA6OO/r5fONyZK27ZtpXXr1tL7t7/9yb/R7qrGUvDccBkhz8r6OcnyWQ15Yp62lZUfEuvkSf3qTH379vN9NkE9L/aVA+7z9eemF1/sSF3j4uPVXzuo/fzzz9XfL7/8UjYVFMpNv+qqArLbbrtNrvKNYSfyBDv9LwsXSr5vm+IAKTExUf74xz+qA/BQ/p1wyvLFjh0yZcoUucWX13RfQG1i3CxdskQFl5D66KPSt18/1/NEerBzIYSSXttn7bnwkUceUUta7HR/4eQJgdLt9m3VqpXa12CbEkVbwJhcnX/3w2UzFGvwVaa9hMakUJbQOPW1tf21sRt1ra2MTn9FHgzkZX9dbefrxtf/1WFJDZYgoL7IO9Sv6CMtI/LHEozAbX32LJtBHVFX1LkmTm1vLH3AdsT2xHbF9g2UZ6gClRF5YgkNltKgjiYgT4xV/zEDbo+bUObCQEItI/JCniaX0GBfg22K9jW5NIookEDzOIN38gR7XWIsB/BOsXdaputqr0k3FYjYsI75gnr11V/TsH1Nr2Gu27mz5t0NGD8YOwUFBTrFXcgH46a2azmcZh98mpyXojkXmgzgwT5AYgBP0cbgnTzvXArg7brWdpGaG6JRV0B+CKKdOlvqbQzevQb9Fv3X5LiJxli18zR5gI/6RTOAJ4qmQPM4L1ilmGevUbQvLMKaRJOwhnbWrFniC6h1yk9VX0cZKbuuWOvvlGDKiLriIs/mzZvrFDPi4uIkPT1dCgti//7mTm/rnzhZIQe/OyLy3UEpr3kp+1nD9bZ0QDBlxAXS6L/ox6ZgrOLiToxVU+1oz4XhCLeMaFPU0wT/MuK6mIEDB+pXRLGFwTt5CoJa3P3ANFwMh7tNmIS6YmdpGgIRXqwVHcc3TZHE1j3lmcLDIoVPSc/Wt8vUTYf0u0RnQmBreqxGYy5EPaMxF+IOWUSxiME7EVGMqNf5MSks3qXuLnDqsVzSOjfR7xIRETF4JyIiIiLyDN7nnYjoLMS5nIjI2wLN4zzzTkRERETkEQzeKeadLXekiDaW0RleKKNXcHs7g2V0Bsc2eQWDdyIiIiIij2DwTkRERETkEQzeiYiIiIg8gnebISI6C3EuJyLyNt5thoiIiIjI4xi8U8zjXQqcwTI6g3ekcA63tzNYRmdwbJNXMHinmLJr1y7p2rWrFBYW6hQzkGdWVpZ+ZcaECRPUo7KyUqe4Lz8/X9XTZJ5EFLoVK1ZITk6OfmVOvfoXqrxNwpw0bNgw2b9/v05xn72vwZxI5DUM3immxMXFyaxZs2T48OFGg+lly5bJunXrjO5A0tLS1N/u3bvLtm3b1HO3XXvttbJlyxa5//77jR0gIQBJTk5WAQEPGqLLZHB0tvhixw4V5KEfm2y/6667Tj7++GM1dkzND7Dzqy/l1VdflVGjRhkbr0OGDJFbb71V+vTpY2xewr5m/vz5ah6eMWMG5ybyFAbvFHMSExPPCKbLysv1O+5p1qyZzJ492+gOpGHDhpKenq4OVgYPHmzkjDjqOWnSJBk7dqw6QMKZf7cDkgEDBqj8Vq5cqQ5UsKN0O0/7GwYcMCDwiZWgFWXBA4GgibObOLuIfLCdEYA+9NBD+h13oR+jnhhHqKsTwSe2IbYp/j38u078m8G4ql07FeR98803ctnlV6j+ayJvBJcYq4888oiaH0wFmMh3wYIFkpCQoMarqWAaAbx94sZUXdu3by9r166V4uJiGTFihOzbt0+/QxTbeLcZimnYWWMix6SOoN4E7KywA8HOZOjQoSrIdhsCk4kTJ8qBAwdkzJgxagfqNuwcFy1apNp39OjRKsh2G+r55ptvqu2KM4r33HOP2oE6DUHr+vXrVcCFHfOaNWtkU0GhPPjA76VNmzbSoUMHufzyy6Vly5YSHx/v2DZGvhUVFbJz5045dOiQbN26VQ4ePHhG/pdeeql07NhR2vmCwm7duun/0xnI/7PPPpMNGzbIs88+5zsQvV169eolN954o/rWBQdvTrEDWHyTAzhLjP47d16mev1Y2p/U35tuukkdODjRp3EggnZFXsj/r+8vkTvv6Kf6ENq0SZMm6m+LFi0crasN/RfbEmO1R48e0r9/f8e3YU0wVqdOnSq5ubkyfvx46d27t37HXfZcmJqaKgMHDjQyF6KumANNzoWAg0JsV7SziW1KFIxAMTmDd4p5/jsQBNQmRHMHcu/v7pMl7//V2A4aQdDkyZPV85EjR7oSTFeH9l26dKkK4gH5du7c2dXgAHmWlJSos2vffvutCqy/+uorFWyuXbM64h02AmccACGow1nLVq1aVR0guBVMAsbHkiVLVGCHPO1g/Xe/G+gr0zf6U85CEI2lFf5Bc9u2baVRo0aOHgwFwz5gwkEEAnv8RYCNtkCbuAXf7tjjBvNS3759Xa83xiqWs6DdcabY1EE+5kDkjQP9szmYRh3xLYfJEzdEtQkYkyN49xcff6V+RhQbPly50iotLbVSUlLUA89Nyc7OtpKSkqy8vDydUjOU0Sm+ANPq37+/5QtoLV9QolMjV1cZly9fruqamZnpaL51Qduirsj75Zdf1qmxy8lt7QS0Hx7Vx4UX5vJYa8ua1FXGoqIiKyMjw7qgXn01X5iAfOz8MFZNtCPmh0jqGE4Z0baYF6ZPn25kTkIZkQ/mI+xrMBcTRVOgeZxr3skTqq9JxxkSE3Am1f+iJhNwZgtrTnFGE2tOTa3Xxpl+XGuAM5m4oNUUnFXD2l6cabvkZz/TqRQstB8ebp3Zp9rhLDiuXfnu273qmxYTMC8hPywfwplpEzA/4GJWfGOGs/8moG3tNemYk0zMhTjbjvkI36SgnU2t+ScKBYN38hT7oiaT7B0IvoY3BTuQaNQVASACETeXGwSCgxY8iLwIY8fU8g5AfggyMV5NwficNm2aDBo0SKe4zw6msbTOJPvEzeHDh3UKUexg8E6egwtXEVCbhB0IJnPTUFeeUSWiWIG50NTNA/xF49sl7GdMHpARBYsXrBIRnYU4lxMReVugeZxn3omIiIiIPILBO8W8VR9+qJ/FLpbRGSzjuYXb2xksozM4tskrGLwTEREREXkEg3ciIiIiIo9g8E5ERERE5BG82wwR0VmIczkRkbfxbjNERERERB7H4J1iHu9S4AyW0Rm8I4VzuL2dwTI6g2ObvILBOxERERGRRzB4JyJyw8m98sm0oXJtQpwkJNwlT77xiew9qd8L6Kh8k/sf+v/Rj7vnyfY6/z8iIjpXMHgnInLcQdn08r/Lvy5pI/+9/kvZuf5PcsGf02Tsu8VSaxx+aLP8ZeZeGZ67RV2kpB7vPiBXc6YmIiKNd5uhmDJjxgy58cYbpVu3bjrFfYWFhZKXlyeDBg2SZs2a6VR37d+/X9asWSM9evQwlifs2rVL4uLi9Ctyy8nt8yS55xy5PitXMnq08KX8KNvnDZWeL7STrL+PlcSTZdKgQQNp2LDhqf9BwVn3kdJv2W9kyavJ0irCgN1/Lkd/e/PNN6V9+/bSu3dvlWYK+1x4otFu+fn56m/nzp2r9U33oG/OnDlT7rnnHtU/TcnKypImTZpI3759jdWVKFS82wx5AgKLd955R5KTk2XFihVSWVnp+kVECQkJ6u9ll1+hDh6wMwlVqGXEzuKbb76pynPbtm36HfegjKmpqaptseMKp57hQBASbF5nx0VtP8qO/A+kUBIlqcMlOu0iafMvvSTx8Cr5YNN+mTZtmhQVFen3tLKP5JUnsmX/klTpOXyK5KzZLof0W+FCu6N/oZ9Bly5d1F+3Id+cnBzV19DnAm1/L2zv3Nxc1YdNwrxnj1W0Y13jx6l2/MUvfqHm3+7du6t8UQ6nBCojTl506NBBBg8eLKNGjTIyF8JNN90kH3/88Rl19UJ/JAIG7xRTcOZl0qRJ6rFy5Uo1sa7Ly3N0J1Iddh7YUX737V71OpIgPlgI3pFnedkP0qpVq6odF74FcBMCEbRtWVmZ9OnTR4YNG1Z1kOQWHCggr3r1L1R1xGvkaWonDagf8sMDeWNnjXKgPHjYZxydcUh2b//a17HaSkKLeipP9KevDxyRn8l38s+dpTI6PV0SExP157WmPSTj812yc/27MumWUpk15DeSlPKGFB0KfcE7+u6hQ4ergnb0bfQ3t7/lQTuiPZEvDk7Hjh2r+pxb+SKoRvti3GCb4jFhwgRVBvx1wubNm1Xbof9ivGBbIh/k69a4wfyAdkP7oR0xfux+6uZYxZl+zA/z58+XrVu3ysVNf+b6XAgDBgyQtWvXSq9evarGJNrXTfa+BnW1g3i39zVETuGyGYppmMCfeeYZtXPGzhhfcf7to4/0u6fc1rOn+lv9rEm46dhprlq1Sl7975nyyL8/rHagCD5q+rxTeS5dskS2f/GFvPXWW+r1Cy+8oJYOOfXv15SOtC2ffir/8AUn87LekGfHjZV+/frJ/tJS/alTnMrzX26+WUpKSmTBggVS6stjx44dsnzFB3LnHf2ka9eucuzYMfn5z38uV7Vrpz4fyr9fPQ074c8//1y+/PJL2VRw6oDonn8doP5edtllarkSHCovV39btmyplrGEWqea0ttd1VgKnhsuIzY8IC9l9paXH31YbrnlFhn1+44y6vbRsiHtXckZ0Fx2+rY31PRv/OIX8XL5ib/Ln5LHy67h8+XVQQmyc8sW9R4+j3m6JidPnpSKikopKz8sTS9uLK+/PledvYVI6gQ1pSMNATSCIBwUtfNtu9tvv10FuQhAncyzzLetFi5cKOW+v+g3cNOvulYtLbn22mvl+uuvl8svv1x2+NoW2xPbFWr794Mti91/cxcvll27d8t3330nb2fnSOcbE1V/aty4cdj9F2pLR0A5e/Zs+cg39yFPzEsPPfSQa2PVTq8+F+IgBkGvE/9+bZ/FvIS58Gc/+5mk+w503Z4LISszU5YsXarmjNGjR3M5DcWEgDE5gnd/8fFX6mdEseHDlSutoqIia+TIkVZSUpKVnZ1tVVRU6Hfd4wswrenTp6s88RevA0EZnZKXl2elpKRY/fv3d7SutZURdVu+fLnK166viTYGX0Ck6oy6Dn3oIZ0aGdQHfaa2bRau2tqxoKDAWr/+r9bioddb8R3HWXkHD+l3fPYstobGt7fumrvVOqGTaldpbZs7yIrv9JK18ZhOCsDuqxfUq1/VV92ey9Ff0EfRXzIzM9V2DFWo4wbbFA+T6ioj6o3tjv6LOcqNPucP/z7yQtvjgbZf8v77+l13YC5AntjWqCPqG6pw5kjMC3Y98dxtKGM09jVEgQSaxxm8k6f4T6ymJlX/HZeJHYjNrit2XCYhGEFAYDpIOhtkZGRYW7duPhV0d3jaWv3D6TD92MaXrE7x3az01ft0Sl1OWD+sftrqUEfwju3lH7Tb3JzLkQ/qGk4QR87Btsd2D+fAKRyYC+2DNsyJpthBPPqcKf77GrcPyIgCCTSPc9kMeRLWYLq9frc6fHXNr1EpGHXdbaZH02AuN8L/87D88cCj8nZakjTRqTWpaTxwLqezTTTm/Wjc9YfIFmge5wWrFPOqr1EE0xM41Ba411TGWMMyOiOYMp7f7jZ5qN9RyZm/VIoOHZdD25fJvIVfS7/nU+TXNQbuR2XvpmWS+94m/UNOZbI9d4L8cWFnmTDs/9QauEM0xoMT2Cedca6U0e1+XlMZGbhTLGLwTkTktPMTJPnFN+T5y9+S317XWq7r+ZrIw6/Ji8kJetI9LntzUyUh4QZJyS1RKVK2QWY+erd0bY1fVv03mXfwNpn+dqp0bsJpmoiITuNegYjIDU2ukeSM99RXnsXF70lG8jV+Z9DryRXJM3zp/5A5yfG+1xfKFT2ekmXqs/rzD/SQqxm4ExFRNdwzEBERERF5BC9YJSI6C3EuJyLyNl6wSkRERETkcQzeKebxbg/OYBmd4YUyegW3tzNYRmdwbJNXMHgnIiIiIvIIBu9ERERERB7B4J2IiIiIyCN4txkiorMQ53IiIm/j3WaIiIiIiDyOwTvFPN6lwBksozN4RwrncHs7g2V0Bsc2eQWDdyIig7Zt2yYTJkzQr8xYsWKF1Kt/oX5lDvLNycnRr4iIyAkM3ononIVAGsHl/v37dYp7KisrZcaMGTJ48GDp0qWLTnUX8sSBwquvvio7v/pSp7pv165dMmrUKJVvx44ddaoZ2JY4aMjPz9cpRERnFwbvFDOww8fZwWHDhklWVpYUFhaq4MNtycnJ6oHACjt8E4Ec6mfXE3mi7m4rKy9XQQ0CVhP5hePIkSP6mXtQd7Q7tjkCzEOHDkmzZs30u+5AX+7evbuUlZXJsmXLpHfv3vod95w4cULuv/9+9XzBggUSFxennrsJ4xUHQ63btJWbbrpJcnNzpX379vpd9yBfjCMcqFx2+RWyYcMGadmypX7XHKfmDvRR1Mf0WLXHBLahqfkXY7Fr165q2yFf1NltaNNo7GuInMK7zVDMweS9ZcsW+fjjj2XK1P+SBx/4vdx6661yww03yDXXXCMNGzbUn3QOJvPPPvtM7fQRcAB2ZDhDet111zke+GBHUVJSouq5detWWb9+vfz1/SXyWNqfVNDTtm1b31hMcDSoRB1Rt+LiYtXGyK/zjYnSo0cP9T7yBftMqVNBFw4YECDbsF1tBw4ckLnzMvUrUeVZu3at49sYQdWaNWvUjvqbb76RIUOGqO3rdkCLfCdOnKjae+zYsZKYmKjfcRfavN8dd8qS9/9q5EABUEcEfug3I0aMMHKwgAAX/eWZMWPl2XFj1QFS586dHe0/COy+/PL0txYYrwcPHtSvRM1R/j77dEvEYwdjFX0V+aDfbioorBqrl1xyiXTo0EF9zumxiv6K+eEf//iHmptQtzvv6KeCa+SJ/OLj410Zn59//rkUFBTIypUrz5gL3coTqu9rUNdevXrJjTfeKNdee63rB/VEdQkUkzN4p5gXKJi/7bbbXAsQ7GAeeWNngoAPwR6Cg27duulPOcvecSJQQF2x0wbssEePHu3KjgR5fv/991JRUaHyRZCNtkZQ3aZNG0lPT9efDF/1Nc84MGnUqJF+5VzgUR3qhoOxRYsWyebNm1XAfssttxgLoFFvBO6pqakycOBAV4KP6nBQOHXqVHUw+Mknf5fdu/fod9yDPF977TUVbI4fP971gwUE03l5eSq/Tp06qbbFQbZbgVb1g8/LL7/8jLP6bgWW1dljdd++ffLtt9+qOQnzBcYq5kP0bzdgLvz6669VYI1xhINtew52K0/0qaKiInUQsW7dujPydHMsBQrm0afdmqeIahMwJkfw7i8+/kr9jCg2fLhypX52im9St7Kzs9VfU0pLSy1fwGAtX75cp5ypehmdYufrC651SvjcKqOTnCqjL9CwLqhX38rIyFDt56S6yoh+mZKSoh4lJSU61X3Iq3///qrO6C8m5nK0bVJSUlWeoQp1e6N+eGAcYmyYcC6Nm2Chj2OMhSKSMqJvIT/M+25u9+pltPc1odaVyCmB5nEG7xTzuPN0BsvojNrKiJ08glns8E1CEI2DFf+DS7fn8szMTBVIR3IQzT7pDJbRGV4oI51bAs3jvGCViMghuCYDF6QOGDBAp5iDu8mYWt8Od955p7oQlssJiIjMYvBOROQQrMWNxkVuuA4jzsAFov5QTxNrvYmI6Ey8YJWI6CzEuZyIyNsCzeM8805ERERE5BEM3inmrfrwQ/0sdrGMzmAZzy3c3s5gGZ3BsU1eweCdiIiIiMgjGLwTEREREXkEg3ciIiIiIo/g3WaIiM5CnMuJiLyNd5shIiIiIvI4Bu8U83iXAmewjM7gHSmcw+3tDJbRGRzb5BUM3omIiIiIPILBOxERERGRRzB4JyIypLKyUmbMmCH5+fk6xX3Ic9iwYbJixQqdYsauXbtk1KhR6i/R2Wz+/PnqgbFGZALvNkNE56Rt27bJ5MmTpU2bNpKenq5T3YMd+4gRI+TSSy+VcePGScOGDfU77sBcXlS0XeXZqVMnSU1N1e+4DwH7gAEDZPTo0eqvKWjjpUuXyr2/u0+++3avNGvWTL9D5J7t27fLk08+qZ5nZGTI1VdfrZ4TRYp3myGimISAC2eiJ0yYoM4Quw354ez34MGDpW/fvkYCdwSz999/vwqiJ02a5HrgDjgvE43AHduydZu2MnXqVKOBe2FhoWrjjz/+2HjgjrzRp5KTk3UKnUsQrL/11lsyaNAg6dnzN/LCCy/wLDy5imfeKWZgsispKdGvTvv4b3+Tm/7lX/Sr01q0aOHIDhqBVUVFhX5Vt/bt2+tnp+EuBbf17KlfhSZQvVGmL7/8Uj3v2rWrxMXFqefhqqmMdt52Xt98841v/Bers9K9evVyLejbv3+/bNiwQT2eGTNWnh03Vrp06aLK4WYAhKUjTz31lMrj4YcfDqv/hLqto3EWGtv1sssuk+eff9544J6WlqaWENQ0TqqLZNzY0JcmTpyo+uzYsWMlMTFRv+OMQONm06ZNsnbtWtV/H3zg9zJw4EDVh906aLDH6r59++Tbb789Y6wmJCTItGnT9CedEWhetPNv0qSJ9O7dW6fWLdC2xvb7/vvv9au6xcfHR3zwG2jOdWJfs3v3bhk/frx89tmnvvE3UW6++Wb9DlHoAsXkDN4pZuDs1ZtvvqlfnfY//zNTPzvThRde6JvEL9KvwoPuX1ZWrl/V7bzzz5emFzfRr8J36NBhKSs/rF+JNG5Ucz1Qx1N/68sFF1ygnofr5MmTcuTIUTl69KjvuSWVPx5R6Xbedl7169eT8847L+L8qkP+R48e8+04f1R5N2nSSBo0uFAurF9f5eem48ePS7mvvVGGn/3sYqlXr55+x11HfG1dWnpQmje/RBro9nUb+vTBg2VqOzZp0linug99uqLiR1VXp/tOTVDPH3396ICvrpde0lQuuqiBq/3I7r9Hjx3z1bVCjZsGDRqosXm+b15w0okTJ+RH31g97ssr0FhFXevVu8CxsYr67f32dBDdwFcv/PvVYQ6sr8dPpPMvHK6oVPUMVqNGDX19u75+FZ5jvvwqfPkGK5J9zXvvva++/SIKR8CYHMG7v/j4K/UzotjghT7phTK2anWFlZ2dbRUUFFhFRUVWRUWFfsddy5cvt/r3728lJSVZ06dPV/kHytvpdkQ+yBN5o+5OCLaMeXl5Kl+0tSmob0pKiqqzqT5ptzHyLS0t1anBCbeM6EPoUyNHjgw5z1D9/OeXq7pdUK++lZGRobar22OnpKRE9Vf0nWD6jxfmn3OljOgbEydOVP/W4sWLdSpReAL1SZ55p5jnhT7JMgaGb1QgmOUMTpbRXsIRyRKZmgRTRqx/zsrKkpycnIiXOwULSwH817ib2N52noBlG6EuZwi1jMhvzJgxri2RqYnvoFeWLFkm11xzTcTLNdwSrbEdinOhjB999JE88cRo6dbtFnn00Uflyiuv1O8QhSdQn+QFq0TkKgRYJoKs6g4fPqzWXuOCVLfWIdcEQfvmzZvVeuhoBe6m2HnOnj3bSGCLdco33XSTLFiwwFifwhIr5BWrgTtFH9btY55B4P7000+ri+8ZuJObGLwT0VkJF9MFc9Gk09q1axfWWehIlJaWyq233mo0cEfAgos0TeaJ7YmLfhlIUyx58cUX1d/Fi3OlV6/fqudEbmLwTkTkoG7durkWXJ7c+5FMS+mqvkpN6POkvLFpr5z0peMM/5AhQ059yBB8mxHK3UaIzlY4046HyW/46NzG4J2IyAsOfSIvP/CwLGn1vKzf+aWsH91Q/vxvz8u73xzVHyAionMBg3ciopj3o2x/+79k6tfJMvo/fyNXnH+hXPHrgXLf1cvkiVc+kjL9KSIiOvsxeCciinUnv5b8nA0i3TpLh6Z62j7/F9JtQBc5nLNKNpVh8QwREZ0LGLxTzIv124sBy+gMljGAQ3tk+/bD0uyXCdJCJ4nUk4svuVTk8A7Z+a03l85wezuDZXSGF8pIBAzeKebhZ7VjHcvoDJYxgIqD8u3pH+St5ns5UH5cP/cWbm9nsIzO8EIZiYDBOxFRCCL9aXbntZBLLz71c/XkvKZNm+pnFAneiYXIOQzeKeb9smNH/YwiwZ2nM26++Wb9zKDLrpVbEhvLke/LpEIn4SLWb77aLtK4nbS+/EKdRk7rkpSkn1EkEm+8UT8jokgxeKfYcnKvfDJtqFyL+1jre1mv2G2pe1nX6mSx5D6s73+tHtfI3fOK6v7/HHbGfbh9Zejz5J9l094orUc+tF3W5L4mT/bpKim5JTqxLkflm9z/ON3+eNw9T7Ybvx7ypK/4/yu5856UPgmpkrs3istCzuiTd8kzCwpkb53t4XA71nRx6snd8o/VO6XxgNuks30RawwJdE/6WsXIOD6z/z0a3f7n4XEcO/PhUdn7yUxJudZuj7vkyTc+MT+OiZxiVRMff6V+RmTaAWvjlBFW+uKtVrnv1Yk9+dbLQ5N8fTLZmrLxwKmP1OiEVb7xZev221+2Npaf0GlRUL7emjJ0rLV42w++F0esPev/2xra4UorPirl2metTu+mxnN8/PXW0MXFOr0OqMPt99bR3gb8sNpKR9up8j9qLd5zTL9hGvpksm8bPmet3nPE1ydXWk/f3s36w+Kvfb2uFi6044ndi60/dEiyhs7d4hsfP1jbFj9t3d5hhLV49xH9iTNFdS5X9b/euv3pldaeE76xsPq5Wst6SoyMY4iZ/ufhcRwz8yH61XRraPpiaxvyPbHHWv/yQ1aH+PbW7VPWq31NQLEyH9I5K9A8zuCdYscPa60pL585mZ7YNte6y9cnO03ZaAXcfZ742lr8h951B1SuOuHb3/+39fIZk3yltW3uIN+Y6u2b/GvdRbjmVPsFu9M/Yu1ePMLq9IfF1u4ox06n2O0XveDpVPt1s9JX79MpukwdnrZW/xCokdxqR18Qsm2xlX57ezVPx9/+tA6Maha9ubyGNjqx1Zp7V3urQ/pq32FHADExjv1Fv//ZvDeOY2k+9B0ATZlz5gGD7o/xnV6yNgbctLE2H9K5KNA8zmUzFDua/lrSRiRJE/0Szr/4ErlMWssdNyZIzZfknZSy/50jTyz5VJaMuEeGT10ka7ab/8maVR+ulqY9HpYRnS/RKaBv5df4V3LjVY10mlmn2u+UwoIC/SyAso/klSeyZf+SVOk5fIrkrNkuh/Rbppx5twfdfi7atWuXVFZW6lfV/Sg78j+QQkmUpA72dr1IWtxwiyQeXiUfbNqv06qppR2R36hRo/SrUJ0vTa5OloxlReqWdsXLnpXkq+u+mBJ5du3aVb8ywO+e9OeVfH4qLcA96U9vb+fGMbbn/v0Btk1I3O9/wfIfxzU5Y9zEwDhGX60+H6768H+jNB+2kB5pQ6VzE79w5/xGcsllDaTxHZ3lKr8dS+y1I1HNGLxTDPPt0LdukvzGt8lvOwe62BI7iWfl8+IvZX3uWLnl+ywZ0rOXpMz7NAYm2oOy9ZPCmF2T/BNNe0jG57tk5/p3ZdItpTJryG8kKeUNKTpkZoFnfn6+7Nu3T79yX1ZWlgwYMEC/qskh2b39a5FmbSWhxek9/MmGTXyB1D75587SmtdiB2jHreUnZMyYMdK/f3/9QfchkB03bpyMHj1apxhwaI9s3XbqnvTtL7NDzrruSe/MOMaBSvfu3WXDBt/Bg2Hbtm1T+UddlMdxIA0bnoid+bBsh3yS31QG/LajBDz8jdF2JAIG7xS7Tu6SVQv/Lt2fT5Ff1znZXyhXdO4nD2T8WT54trPkP/OUzN50UL8XHSe/yZOF7/8fef7RmwPvIIKAoMCk0gvi5Zb+j8nbHzwv3fLHy3/M3ujqgRDqh7PRkydP1inuQkA7Y8YMWbduneTk5PiCiob6neoq5eC3Pz37e/z4Cd9/D0vxgcM1B+/a+Vd0lrseeK6qHf80Z5P8KT1dunXrpj/hvqlTp0qbNm3qOEhxWMVB2alvidOyZctTT6rUdU/68MdxYWGhquf48eOld+/eOtV9OMs/YcIEGTx4sHz//fc61QyMHfTlsvJynXJa9f7n5DjGGAr1QKV1428dmQ8jd1S+WbVY3u/+n/Lor0//5FkgbrYjUbgYvJP79uZKiv/V+jU9UnJlr/74Kb4J9t1p8vo142XUvzRQHRU7CwRbCPTq1b9QVqxYceqjZ2gq1wxJk8cTP5PMtV9KpPeI2L//O1/xU2suc9XjBnmzvL3+P7STxfLus7lyzevjJLlVaLfxQzCAs9AICLDcAfU9tQygRHJTbqghf/9HeHdmwb+P9kxOTpY77rhD3nvvfWl41QB5/PEu8lnmOtnmxM02AvSDXr1uk4UL35RNmzbIyJUXVesHzkLQMWJEisQVzpLly5fKzTff9JPynHqgHRGk/9RFDbA9G0vCpY2DmEDPlwZtk6va8dgv2ul091VW/ihfffWVpKWl6RT3YYz++ONRCXQ4FPw96UMbx+i7w4cPl1mzZhkL3NGXMB9ddvkVEhcXJ2vXrpXExET9rnvKfYE68sVYxdzQqlUraXBhoDnmfGlyzUBHxrE9/w4bNkwubvozWbVqlX4nCL758O//vSbk+RBtjIMyfEuGfDHvR/rtxslvlsizr7eT11+8W1oFGMDIA33KnoOzsuY71o5EjtBr36vwglWKvhNW+db5Vvq4ldYXP5RbU6dOtZKSkqz+/ftb06dPt/Ly8qzS0lL92ZqcukNDrRe5BoB/F/8+8kF+yDd0P1hb5461xq3eHfSFdyUlJVZ2draVkpJiXVCvvpWRkWEtX768jnoGYc9ia2gtF7oVFBSovOw88fo0XHT2tNWh1ou63HTMV/xHHbtgEG2M7YltG5yaL049tvElq9MZF7HWbvr0adZ3K54w2o7Yjg0aXKTqbEpFRYUaM6X7/l7Dxanl1sYpveu40Le64MYxtie2q/N1Ddz/MEcgz5EjR0Y+RoPhN459Abuq85ljtTbhjWNsT+Rhz4X2/Is0vBe80ObDoqIiKzMzs2ouxF/MjUiPWPkWa276S+rOUf7QdzDfYg4MvK+J9nxI56JAMTmDd4oZ2CF8/PHHVuXXS61x6fOtvC92W7Nnz1YTaGVlpf5UENSdBIK/vVf1nQV2yJjIwwsGcFu8l6x0dTu/wFBX+yDBf2cR/A45SDUE76gX6ot8H3zwQVXXmnfGCF5/X/ft1EKAOqOuaOO629e54B3tivqirqEI724zp2GbpqQ8aBW9PsTRdqwN2hV1veKKy3SK+9B/MH5Q3xrbKJi7zVRXxzhGnuhHyLfm/hupn/Y/zBV2IOtIMFkHbEu06XtZT1WN4x9//FG/G6zgxzHywxhBu/oHznWP1UDqng8RHCMP/zwxP4V+kFCHE7ut1ePGWnO3nuqBqGdWVpb1xRdfWGlpaaqdsf8JfDDm/HxIVBcG7xTzDh48aP1YvEwF7lv9buv1Y8VXvkl3es3B0ok91sZ3F1vvbtxz6qxO+VZrcXpySBMszrZEembnw5Urff+teUeFe4OPm7L2jKAFQWxkBwnBORV8trfumrvV2rhpk8oTgR0Oio4fP64/Bb6yb1xqLX53o7VHNaS+j7hD92RG22KnjKCntgOUU+1o00Fg/CBr7rYQDt6qQVujzvgbMnX7wiSrw9AsX588pm/VWNt93k+34+bde6xx49Ktz95ON3bvcjuIRn82OZdjDKFv2YK9J/3xY/qgLMRxjDGDeiLgcidwh9P977XPy6vGTlj9KESoE8YK8kMge+gf/1M1jmvvRZGNY2xHtCnqGHm7njkfHjp0SKVWnw+RF/J0PFj3Vy1wP+brd3v3lvjK97I1pcZv0NydD4mCxeCdYt6pH8DR97Cu9jh9xq7YWjz0el+aPhuGSfnpO09/9vZ0a+7qbUEH7k755z83+XYEz1m3+5X59CP4JRbO0csU/MtxwwvWit179fvV2lHtaP3Lf6dvp7v61I+aRAiBO4KQYM56VwXvxzZaUzr5ld33CGcZFIIC5B3RAZIKJO0+5msX/SNip9hnZ+1vN9xrx2Ag+MIDkL8JaGME0mcGXnXdk97uf4+EPI6xLe2g1jU19L+O41ZY5aF8AxihUycTahjHZyzbqNaOUe5/p1Uvh//D8HxYvW/5P6q+HYqtcUxkQ/+ryXn4j17+ruBiLdxDmIiCt+OLL6TdVVfpVxQu3Gf5tp499avI4KIzXOiGCzYD31EmdKWlpdK8eXP96qdwkd15550nF110kU4xAxcTLl26VKZNm6bqa2IuR55o4wULFjjaxrVBfu3atTN61x4iomgINI/zbjMU88784Z7Y9PXXX+tnscsL7eikuLg4SU9Pdzyo3FxYqJ/VDMH71q1b9StzmjRpou4jbyqIhrZt26pbFYab598++kg/C96QIUOMBu5eGDcbPvlEP4tdbEci5/DMOxHRWYhzORGRt/HMOxERERGRxzF4p5jnha9bWUZnsIznFm5vZ7CMzuDYJq9g8E5ERERE5BEM3omIiIiIPILBOxERERGRR/BuM0REZyHO5URE3sa7zRAREREReRyDd4p5vEuBM1hGZ/COFM7h9nYGy+gMjm3yCgbvREREREQeweCdiM5JlZWVsn//fv3KBSf3yifThsq1CXGSkHCXPPnGJ7L72AnZuHGj/kBNjso3uf+h/x/9uHuebD+p3w7Dtm3bJCsrS78yJycnR3btMr/mPhp5EhGZxOCdiKIOAZfJABOB+4gRI2TNmjU6xWkHZdPL/y7/uqSN/Pf6L2Xn+j/JBX9Ok+fe3y3trr5af6YGhzbLX2buleG5W9RFSurx7gNydZgzNeo5atQoueGGG3SKGfn5+TJx4kRp3ry5TjED+Q4YMEC/MgcHgTNmzNCviIjcxeCdYt5tPXvqZ6fgTKLpHaW9cy4sLNQpZ6peRicg8MLZS6eC2mDKiHoi2Bs2bJhOcR8CrtZt2kq7du1caceajBkzRi699NKwAr1gynhye66MnbpP/u/oh6XHFRfK+Vd0k/973y9kyRNzpFCa6E9Vd1S+WTlfMlvfL79LvESnRWbRokXStWtXSUxM1Cnuw4FYWlqazJ8/Xxo2bKhTa+bk9kY/Qr4YM06qq4yYj/r06SNNmzbVKWagnTFWJ0yYYGzcAOqLPAPNhYFEWkZsX6e3bXXVy2jva1z9ho4oDAzeyTMwkWJnNXjwYGnVqpVOdRcCaEzel11+hXp9zTXXqL9uw06qe/fu8vHHH8udd96pU921YsUKFYR07NhRpk2bplPdhQMTBFyffbpFunXrplPdhe154MABGTdunE5x2o+yI/8DX5CeKEkdTgXhWVlvycUdfiWJh1fJB5sCBAJlH8krT2TL/iWp0nP4FMlZs10O6bfCcfz4cVVXtK9JaNfRo0dL+/btdYr77AOGWbNmSVxcnE51HwLK637ZUaZOnSpDhgzRqe7D/IAD3ptuusnY9rVPYGD+7dChg7G5EG2cnJws77zzjpqbTPCvK/Y1zZo10+8QxQjc591ffPyV+hlRbMicN88aOXKklZSUZGVnZ1sVFRX6HfcgD+SFPKdPn26Vlpbqd2r24cqV+llk8vLyrP79+6v6FhUV6VRnBCpjSUmJlZKSoh5O5xkI2hd1RJ7+betUOwaCbYk8I+lDdZdxn7U6vZsV3+kla+MxyyooKFD96FjJ29bQ+PbWXXO3Wif0J2tyYs9G69256dbtvrm4w9Asa2t5bZ+uGerXoMFFKm+TMjMzVfsGy4ntjf6L9sXYcUOgMqIvIV/kbwrGpz0/+Ofr5rhBX0JdL6hXX23fcMdOqGW050I83Nq21S1evLhqu5ra1xDVJlBMzh9popiFM+1z5syRd999V+6//3511h1fw1e/nZf9VacT6TjTPmnSJFmwYIH07t1bbrvtNnXWB2r6vFNlycrMlFmzZ6vnOIM3fPhw9dypf7+m9KVLlsja//1fyc3NVe2LpSTgZp7wl4ULJeuNN9Qymb59+kjffv1UeiT/fjCf3bBhg7w+d65ai13bNoVI0ttd1VgKnhsuIzY8IG988KD84dZ/kccee0x+16OJ/L9eo2RD2ruSM6C57PziC/X56v8GzvK1adtaLtj7V/lT8njZNXy+TL7rUtn39dfqfXwe83RtyspPnbPPeTvbkTpBTen+aV/s2CFTpkxxvX3BTkffxVn+Rx55RDr+8pdh/Tuh5olxM+OVV9Tz1EcfdaT/Qm3pmJcef/xx9e3YQw8+KF26dFHpbucZylwIoaQH+iy+9XvrrbfU83vvvVddmwJO5Ak1peNM+9ixY1X7YklQ375961zyRWRCwJhchfB+eOadog1nl/zPtC95/339jntwhsU+056RkVHnmfbqwj3zZdfVxNkl/zLijKx9Bi/UukbCPguNtq6JW2cQ0baRnyUtthYPvV7NkYEfj1qL93x16nOdXrKW7CxRZ/KUPYuDOvN+WqW1be6gqjP4wbLbOC6ulU5xH8YP8gz1TH8k2xt54ix/Vfu6xL+M6D8YN8jT1FlZu+9iXgqUp9Pjxs4T84NT38bVVUbkafpMO+Y+/zPtJvY1RKHAfqUmDN4ppmAHZU+k9o7KrYDOZgc7keyowiljTXV1E8qInZWdb6iBVqSWL19eZ75ubGtnAvdQ6KC7w5PWXz79qmrbHtv4ktUpvpuVvnqfel23E9YPq5+2OoQQvCMvu41NzuUYO+EE0eFub9TTROAOdhntfoR+bALGqn1gX9e85NS4sQ9O0LZOBe22QGXEtjQdtIN9ssbkvoYoVAzeyROww7AnUlOwkzQX2J2GIMB0XbFzNHnW0Ia1stg5m25n5FfXAYMbTmyba93lC9SfWPmtTrED+qet1T8Eu4Yd/8/vrdunrLfKdUpdcGCGB5iayxH8RHodQagQ1Nr1NME+wDfZjzA/RLLGPBwYLyYDaED9nD5QCAbqanoeJApVoHmca96JyHW4Gwju+W16HSmuk/AdNBi7k41t55cfSeGkP8kTx/8ouf91v1y556/y/B9flNKH35RXkxNquM3XUdm7aZV8vPsyuemOznLF+WWyPfdF+ePMFjLh7VTp3KTuG4Ph1n24VmLt2rWqnU3M5VgrjDsx7fzqS2N3ecG1MJMnT1Zro031J/tWgbzrCBGZFGge560iKeZVv8AoFrGMtUNgF0yg5XQZccGd04F7MGWsd2GCJL/4hjx/+Vvy2+tay3U9XxN5+DV5sSpwPy57c1N9E/MNkpJbolKkbIPMfPRu6doav6z6bzLv4G0yPcjAHVq0aKFuIWjyAAl5RRK4h7O9cQvK2bNnG6snyoigPZYD92iO7WCxjETOYfBOROQwFcw2uUaSM95TZ02Ki9+TjORr/H6eqZ5ckTzDl/4PmZMc73t9oVzR4ylZpj6rP/9AD7k6yMAdkKeps982BNCm8yQiOtcxeCciIiIi8ggG70REREREHsELVomIzkKcy4mIvI0XrJJn8UInZ7CMzuBFbc7h9nYGy+gMjm3yCgbvREREREQeweCdiIiIiMgjGLwTEREREXkEL1glIjoLcS4nIvI2XrBKRBRl+Gn/yspK/cqMwsLCqp/3Nwl1JSIi5zF4p5jHuxQ4I9gyItAzHWBmZWXJrl27jLUj6te1a1eVZ6jCLSPyGjx4sNG2RV7Dhw+X77//XqeYkZOTI5MnT9avAnN6e6MfzZgxQ79yRjBlzM/PVw/T0KewjU2Nm0iwjETOYfBOnoId1ahRo1RwYBLyS05O1q/MQDAQboAZLuR52eVXSGlpqU5xH4KtdevWSfPmzXWK+8aMGSOpqalGf9p/2rRpMnr0aGnWrJlOcd+iRYukR48e0r59e53iPhz8TZw4UUaOHKlTzMA3DOhLgwYN0ilmYMykpaXJL37xC51iBvIdMGCA8YNBtPGECRN0ihn4Fgdzoclvc1BX1NPpg0EiJzB4J8/AxN29e3dJSEiQvn376lR32QcLS5cuNTqJ4wwiAoL58+cbCzBXrFih8vzs0y1G80RdEUw3bNhQp7oLB2IHDhyQgQMH6hT3oZ7ovwi2TDl58qQ8NDRFHTCYNHPmTHVgZPqAAd8wzJo1y+jBkR24o0+ZPBDEXGTna6q+aOMRI0ZIcXGxytsU1BHfWE2dOtVYn8IJk/vvv189Hzp0qPpLFFNwwaq/+Pgr9TOi2PDhypVWdna2lZSUZOXl5elU9xUVFak8p0+fblVUVOjUmqGMTkA+I0eOtFJSUqzS0lKd6ozayog6oq4lJSU6xX3YltXzdKodA7G3aST1DLWM2KbIE3mb1LTpxWrcmGS3b13jxebU9sZ4yczM1K+cFaiMqOsF9epbBQUFOsUMjNXq84OpcRNJG4czbuy50NS8hDIuX75cbVf8JYq2QDE5g3eKadGYwCEaBwuoH/IM5mDBKcgnIyNDta+pPKGmwN1tqJ/pbQrYnmhjkxBsNWhwkdFtirz69+9vvH0RUKL/mmSPVZN1RfuinqbHarTmQvSlaMyFyNfkvERUGwbv5Dn22Z5oBbMmJ3DsGE2f7UFdUU8cHJlqX0BdTQfugHqiL5kU6plopyAAuezylvqVGQjy0MYm4aw32tfpb6lqE43AHfXDWD0XgtlonPlG/VBP1Nf0WCWqDYN38hT/sz1ufyVsi2QCj6SM2CGbCGb9y4i87GDAJOSLHXOgwMetbY3+5NQZy1DKiP5k+ut35Ie6mpzLEVxiu4YaREeyvbEtMW7cXrZSfdyYDtztPGsbq06PG3t+cDKYrauMyCfaBwtOtyNRpALN47xglWKK/wWiuFCpW7du+h134YLC1m3ayiOPPCLp6elGLp5EXYcNGyabN2+WtWvXGrvgDRdj4cLJW2+9VV1YaIqd79o1q41tV8CForj7icmLYgH9t2XLltK7d2+d4j70qaeeesr4nV7Qvm/9ZaHRi0WxPYcMGSKJiYk6xV1oW/RfjBlT/Rd30EGeuFjT1Fi188RNAUzNhf4XiC5YsMDIXIjtibvJvPrqq7Lzqy+NjlOiSDF4p5hSWloqHTt2VLfVMxXMwsqVK41P4Js2bVIB9OzZs40GlXYwgMDHFNypws7XZOBuHwyOHz/eaH+yb5eIANMk3BoStzQ1eacX3HEFB0im7gAFODDCHYNM3gkEd1rBmDE5bnAHHdNjZsmSJequPRivpnz99ddGT5wA5oamTZsaO1ggcpQ+A1+Fy2Yo1njhq0yWsXb4SjyYr8KdLiPydPouJMGUEct03Lr7SW38l66YmsuxzCHcpSvhbm8srzC1tMIuI65fiFXRHNvBYhmJQhdoHj8P/9FxvJKQECfFxeZ+FIaIiJyBM/720hXO5URE3hZoHueyGSKis4TJNedERBQdDN4p5q368EP9LHaxjM5gGc8t3N7OYBmdwbFNXsHgnYiIiIjIIxi8ExERERF5BIN3IiIiIiKP4N1miIjOQpzLiYi8jXebISIiIiLyOAbvFPN4lwJnsIzO4B0pnMPt7QyW0Rkc2+QVDN6JiIiIiDyCwTsRnWHXLrPrpCsrK2XFihX6lRmoY1ZWln5lxrZt22TYsGH6VSBlsn1Nrsx78i5JSMmVvTo1XGjbrl27ql9eNSknJ0c9TJsxY4bx/gum+y9g20ajrkQUfQzeyVPy8/OlXv0L9SszsJNE0IXgyyQEIhMmTNCvzECepoPaqVOnyoYNG/QrM6ZNmyZNmjTRr8yYM2eO9O3bV7+qyUkpW/Oi3D0kVZ55o0CnRWbp0qXSo0cPo7+8ivEyceJE6dixo04xo7CwUFauXCnNmzfXKWbgIGXRokX6lTmvvfaa8bGKgwUcDJo+aIjGXGjXlSgm4W4z/uLjr9TPiGJLXl6edUG9+uqvKRUVFVZKSoo1ffp0nWIG8kO+yN+U5cuXW0lJScbz7N+/f1TyNKmoqCj4ep7Yas29q70VP3SxtUcnhSMurpXansjbpMzMTOPjBe2KuhYUFOgUM9C2yLekpESnmBGN+QF1RF2zs7N1ihnRrKvJfQ1RTQLF5AzeyRMYuLsL7Wo6CEFe2KYm87SDPNMBLQL34PtusbV46PURB+8tWjS3Ro4cqV+ZUVpaqrYp/pqEA4aMjAz9ygy7L5kO8KJxkI280IfPhbmQgTvFkkAxOZfNUMzDEofuPX4ja9eslm7duulUd+Gr/xEjRkinTp0kNTVVpwbm1F0K8PXw5s2bVZ0bNmyoU50RqIxYDoT2xdf/cXFxOtVdaF+065L3/3pGnm7f7QFLdIYMGSLt27fXKaELtYxY6gWm+i6gfcvLD4sv8NEpZrz55pvyX1OnBL1Mx4ntjf6LcfPwww/rFGcFKiPmB/Qlk9sVfempp55SY9V/fnBz3NhzYa9evYKaCwMJtYxuzoWB/GXhQhkwYICaJ0xuV6JQ8UeaKKZhZ4XAcspLL6odCFTfCdzWs6f661T60iVLZMYrr0i7du0kuX//Wj/vZFly33lHNm3aJE+mp0vffv2q0v1F8u/7QzrS9u3bJ2PGjpVHHnnEWPsiHUEeDBo0yJF/P5jPYg3rhOefl6lTphhpX0D6M2PGyL333isdf/nLGj9fPa3dVY2l4LnhMkKelfVzkuWzGv5tzNO1qaz8UY4ePSqLcxYHVUZ/oaT7p5WVl8vo0aPVevfk5OSqdH9ulOUl3/bs/utfS5cuXcL6d8LJc8UHH8g///lP+X+PPRZWnhBqelZmpjyUMkz+/EaWtGzZUqW7nWcocyGEkl7bZzEX7tixQ1IffdTYWEXgjrkQ1xIwcKdYETAmV+ff/XDZDMUKfG15riyVwTpS08tWsLQhGl8Po66mvwq3v/Y3v8Thz6eWwPjm1cCPR63Fe47p/wMiWzaDumK7ml4ahCU6ptdD233JJKyrR/uaXBoUjaUc0ZoLuVSG6DTsI2rC4J1iEibRcyVwRx2x4zC99jsadY3WBX4I8kyv/7aD6NBFFrxHo672djUZcNkHn9E44DV5Yazdj7DW3ZRozQ/Iz/Q2ZeBOsYzBO3kGA3d3RauudhBieieJto3GRZThB9HhB+92G58LZ92RJy5UNQnjxmSe0Rir0ZofGLgT/VSgmJwXrFLMwVpHkxenAtaa33rrrRFdkBWOd955x+iForBu3Tr113Rd7YtFTa8nHTdunLz1l4VRude5LwjSKSE4WSEHvzsi8t1BKT+p04Jk39c9kgtyQ4ULRvGo/R72zsI93ZHnwIEDdYr77B9iQh82Bdsz2IvmneI7yJU2bdrI0KFDdYoZxcXFxufCVatW8eJU8iResEpERuCCUfyAjqk7R9gQEOAOEiahrggMQg30jm+aIl2Sp8jp30P9paTlZkta5+B+UAo/ZHPPPfcYDd5xUXnjxo0lMTFRp7gPgTuYrCd+pRZ913T/JaJzV6CYnME7EREREVGMCRSTc9kMEREREZFHMHgnIiIiIvIIBu9ERERERB7B4J2IiIiIyCMYvBMREREReQSDdyIiIiIij2DwTkRERETkEQzeiYiIiIg8gsE7EREREZFHMHgnIiIiIvIIBu9ERERERB7B4J2IiIiIyCMYvBMREREReQSDdyIiIiIij2DwTkRERETkEQzeiYiIiIg8gsE7EREREZFHMHgnIiIiIvIIBu9ERERERB7B4J2IiIiIyCMYvBMRERERecR5lo9+riQkxOlnREREREQULcXFu/Sz034SvBMRERERUWzishkiIiIiIo9g8E5ERERE5BEM3omIiIiIPEHk/wOAOpSgVA3rcQAAAABJRU5ErkJggg=="></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>
<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>
<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>
<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>
<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>
<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>
<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 src="data:image/png;base64,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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>
<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>
<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>
<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>
<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 src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAusAAAIdCAYAAACEMjM5AAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAADsMAAA7DAcdvqGQAAK1iSURBVHhe7Z0LnBV1+f8fAy2QSvGaXLxQoESJxNKFDIRERGyJUtBMRRbyQiT8AtNMwFxCMFAgM8XLon9BE8RCaVdAkKULKrsaqUtekN3ESwumuJuKnf9+hu93HY7nnJ0583xhzjmf9+s1nDPfYZ955vN9vs/3mTln5uyXaEIIIYQQQgghseMT5pUQQgghhBASM1isE0IIIYQQElNYrBNCCCGEEBJTWKwTQgghhBASU1LeYNq5c0fzjhBCCCGEEOKarVvrzLs9SVusp/sDQgghhBBCiB6Zam9+DYYQQgghhJCYwmKdEEIIIYSQmMJinRBCCCGEkJjCYp0QQvKEp59+Wv75z3+aNUIIIfkAi3VCCMkDtm/fLmeeeYaUlIyWxsZG00oIISTXYbFOCCF5wA033OC99u37TZk7d673nhBCSO7DYp0QQnIcfP1l/fpK7/1ll10mv/nNPH4dhhBC8gQW64QQksPgKy/jx/9YfvGLX3jrHTp0kLlz58vPf/5zfh2GEELyABbrhBCSw9x1113SvfsX5dvfPtW0iAwbNsx7XbJkifdKCCEkd+EvmBJCSI6Cr7oMHHiK/OUvf/OuqPtzd/I2Qggh8YW/YEoIIXlI27Zt5Y9/fDhlMf6FL3xBFi++XxoaGkwLIYSQXITFOiGE5Cgo0k888USz9nG+8Y1veEU7IYSQ3IXFOiGEEEIIITGFxTohhBBCCCExhcU6IYQQQgghMYXFOiGEEEIIITGFxTohhBBCCCExhcU6IYQQQgghMYXFOiGExIJ35bUnbpGSEzp6P47Rues5ckNFjbxpthJCCClMWKwTQsg+Z5e8/ZcFMu8vX5afb6qTrS+vkduHvyVzSy6XhdXvmv9DCCGkEGGxXoBs375dBp12mrz44oumRZeKigoZ+6MfmTV9YBv7cAE0Ofvss82aPvfdd59MvuIKs6YPbGMfLkDcwL6ruLH28eoCa98V0CX7uGwtn/n6T6R0/Dfk2FZNq60+L6f+eJR8Td6Ul7e+7f2PJ554wltcAG3guyvtYdtV3PzqV79ylg9c2r7llluc2XaZZ+Czq/yO+EP+dRWH6E9X+RG6uJw7YNvVGCI5QCIFnTp1MO9IPlJeXp5o1Xp/79UFffr0SXzuc58za7q88MILnu/Tp083LbqMGTvWs4/9uKBdu3aJUwcNMmu62H51rc2GDRtMiy7duh3vzH59fX2zfRfAZ/Stnv0PEm+tvCpx4o8eTGzalUhMmjzZs91S7GSTu+G71UY77qE7fIbt3/72t6ZVj7POOsuz7SLmYdOVbTtWMaa0geYu8wzyO+y7ADECXVzMTdDFleYAsehKFxsvixcvNi0kH8mUv1msFyB24ndZkCJxucBOoK4KRpxkoHBxgZ2IXBQtwE6imJS0gd7w3VUB4LLo8hfqLrS3/QrtdeLytcQ/lv8qMfrEXyT+8MZ/vX6Ffby21Ldhczd8h99YtLVBfrG6uyiQbMy4yDW2OHKRC9CHyDOuxqrN7y4KXnuxxFXBi/wCXVyAQhe+uyp40acYoy6w856rOZvEg0z5ez/8Yy6yN4Obm7ZurTNrJN/Ax2krVqyQBbfdZlpERowc6b3et3ix9+onzLaXXnpJfjFlqvzy2mly5ZVXem3Jf2f/BqTblm5f+PrOusfXyV133mFad5Pu78Lsy/p+9ve/J/cuWuS1pfMDhN02e84cefbZZ/fQHQTxEaTbhvZnnnlGrp91gww5fbD84Nxzs/Ix076mTJ0m//pXnZRed50cccQRpnU3UfdVUlIid5UtlKLeX5HLf/KTjH4E2eZv37lzp1wzZYq8/sabMn/eXDn4oIPMlo9I5yNIt8224+sAd9xxpxxx+GFyxeTJnjbZ2APetveWyZyvjpM59lsAbU6Xy2adJ/XrH5GZ11/v5eaWmDXzBu/V7gsk76+pKJKLL75Ylj64zPN93GWXyXHHHedty8b/5H1hLF37y+vkg127PN2xL7vNT7b7Qg6D7zZmLMl++Am6r9dff12u+Nnu3AXfRzfFJwjroyV521NVVTJ79hz5f/fcjY+2Tetuovp/+4IFMu7H46Vjxw5y3S9/GcgeCLqv25py15qm/Fvz/HPyZNJXsjLZA+ls2nZ89eXwI47co0+z8RGk2mZ9f+P116R9+/Zp7YGw2/D1lG7Hn9Ccfy1hfQSp9nXjTTfJpk2b5J133vHWSX6Ssfb2SvYkeGU9v8HVIlwBWLxokWnRw15l1LryvWrlSvNuN7jq4uqqvf2aR9irF0F0dH1FqqWr6sk6hsFeqXNx1dtesUdMNp3EmFYdXFxRtzrCtv16B161r5CW3zM7cf/1FyS6NOXi4696LFFn2lsiSO6G5lYXjCVt3/1X61Nd3Y0Si8DlFXV/zGjlMD823jWuwKbS0eVVdYA+DeN7mDnGzh17+8p31HgE9qq9K92POuoolZgh8SZT/uYNpgXICy++KP369zdruqxevdp7LSoq8l41wQ08jf99TwYMGGBadFn+xz/K57t0kS5NizbXz5zpveLqqzbQ5amNVXLJpZd6V4w0wRWj3958s3yl10nNn5RogStp6Ms2n/qkPPzwcmkqBMyW6MD2N77R14t1/5VdDazt1asfk4suGiUV5eWquuPToyEXXiGv7HekTOjeRt6te0N232IaHcQKNMenJNDl/vvvV/UdN/Dhym6HDh29XDBo0CCzRQd8koEr6sO/O8zzXRvECWLmwgvOd5LDfvjD8714v/fee02LHohLO1a1dQe4IRb5d9SoUaZFl6UPPuhpM2LECNOiB/LYG2/+29m8t8rMey50B/hk8MSePc0aKUhM0b4HvLKev9grO66uXuAqIxYXuP7eHmy7uHoMcFXE1ZURXE3LdFU9CtqflPix3wt2cTXK2ta6ou4HWqAvXdgGsL37E5jNiceuGpA4bfaGxBu7N7VIS7kbPsO+i/4E8NvFJw0W+O7qkzWAmHFlH5q4zDEuxxNA3yLPuAK2XWlvtXEV94h5fCLjAjtnu8o3JD5kyt/8znoBgqsMLq4eA9g++OCD1a/wWlz7nou2cUUNuNActl31JcnETnlh0UQZV9ZZ/u/2SXJqh0+a9swwd8cb1+MpV/MMwGNJXXyaYXFpP5fnPRIfMuVvfg2mAHE56GFbM2GtXrXKvNuNa99d4dI29G5J82Qdg+Jq8slVstWxZerkiRuGecl693KW3PneMPnlnZMDF+q5gjsN44/L3Aj2dZ6JgstCHaSzrxGP2vNeMi77leQGLNYLmFR3qJPwUEcdClvHjlL002XeVZXdywopvXCIFH3uALOd7E04pnWgjjpQR8JinRBCCCGEkJjCYp0QQgghhJCYwmKdxJoBAweadyQK1FEH6hgdaqgDddSBOpJcgE+DIYSQPIG5mxBCchM+DYbkLIX85AhNqKMO1DE61FAH6qgDdSS5AIv1AoZ3mOtAHXWgjiQuMBZ1oI46UEfCYp0QQgghhJCYwmKdEEIIIYSQmMJincQa3qmvA3XUgTpGhxrqQB11oI4kF2CxTgghhBBCSExhsV7AjBg50ryLL7lwpz511IE6FgaMRR2oow7UkeQCLNYJIYQQQgiJKSzWCxg+DkoH6qgDdSRxgbGoA3XUgToSFusFyn333Wfe6bN9+3apqKgwa9FIdfMPbL/44otmTR/X2jzxxBNmTR/okkobrZuoYBvHUKjwZrToUMPdRM0FmXR0nWdc2taaO1KB/JXsu1Y8as57qXA5L5H4w2K9AEGy+sF5P5Q//+UvpkWXiy++WL73ve+ZNV2QEIecMVR+d+utpkWXW265xdPG1WR0+umnyw9/eL5Z0wXa9OzZU66fOdO06AJNuh1/gsy4/nrTogvisfX+B8ivfvUr06ILJrtPf/rTMvmKK0yLLrB/1FFHeTHkAujy1a9+1UlsoogZdNppMvZHPzIteiAu4TfsawPbxzfFJOxr49p2jx49ZNy4caZFF5d5BuPn69/oa9Z0QWwjv7sqepEfZ82aZdZ0+dmVV3q+u+Cll17y5iVXuYXEHxbrBchTTz3lvR55xBHeqzaVlZXSvXt3sxaN5Jt/Hn30Ue/1K716ea/a3HnnnXL4YYdKUVGRadEDBdFTG6vkW/2+ZVp0QRHd+N/35PspTpSi3kSF4qK4uFjafOqT8qOxY02rHpikb731Nk/7HzkoGO1JGIr1s77/fdManlQ6Qpuzzz7bsw9OPfVU71ULFC4oGn9xzRT5z3/eli5dupgt0YHvKL5wErZ69WPS66STzBYdcAJz9NFHe3F/Sv/+XpvWDX2IGRS8LzSNq1GjRplWHXbu3CnfaCpIXdgGKOzeePPfMnHiRNMSnnQ6Il6g93eKv2NadPntzTc35V/dOLHYQrp3797eqyaIF+THk5JiXCsen66uls8rjk0/LzYV6+C4447zXkkBkkhBp04dzDuSj4wZOzbRqvX+icWLFpkWPerr6z3bkyZPNi3RWLVypXm3m7POOsuzj/1o88ILL3i2oY8LrO/YjzbW9z59+piWPUnWMSzNMbN4sWnRY8OGDYl27dp5C95rY3Xv1u34yHGTrGN5eXnic5/7nGcf+9GMS9iyukOb6dOnmy3pCZO70ZfWd8SNZlzCd6s79gGdLFFjEcCejRkXMQk94Ptvf/tb06IH/IXtUwcNMi3ZkU5H6A1dXORI6OFKFwC/o+qSDowf+J4c5xrxCGDb1dwxcMAAzz7JbzLlbxbrBQgmonRFXVTsROSfnKOQnEgxEbny3SZzFwUjJk6XE5EtjNL5HmVC0iouUmF1waKtO2zDZ/iuVUj7dfQX0toFI4oh2LW6B/U9SO5GsWJ1ceF7SycwUYsjWzC6iBlgx5LWBQc/Nt6hT9R4TKWj1SbIiV02wG8sLrB5xtWJAOaNVL5HjUeAmIfv2mPJAt9xsYHkNyzWyR4gqWBCcoEtYDQKI+BPpPbqsauJCMnQ1URkTwS0TmL8oGBpqU+znZA0i4tkYA+au9DFb1vzahd0hN7WdphCOgiwhYkZtqF5WF1ayt0oQO1JAHTR7tMgJzBRiiNrH/pr+w5soe4qP9qTJI14T9YReiBmXOUwW5C6OIkBVnsX/QqbsJ0qF0SJRws0gX3NT6f8YDy5ikkSH1isk2ZsYYfi0cXXYFxeAXB55dueCGRT2AXREZq4mkRtAeBiorCTkIuTDGvbXknTjEdb1Lm4Sge9MXm6OGm0mmRbELWUu2EbY9TFGLLFnPYJjGVv2UdR5CI3WvvZ5Jgg2Pzo6sq0jfsw2ofREbaxDxdY7V1e+XaV311fpCLxgcU62QNMFki4LiYkJBQkRhcgaWVbxATB6hKWIDpiAnVRIAHo7WoSgm2X/enXRDMeYdeV3sDFiZEG+zp3u9QFY9NVLAK/fRe5Edq4KqQB7Lss6LLJM2F0RG53NWbRty7nDvSrqxwMvva1r2U1N5HcgsU6SYmLCUkbjY8oXUMddaCO0cmF3M1Y1IE66kAdSVzIlL/56EZCCCGEEEJiCov1AmbEyJHmHYkCddSBOpK4wFjUgTrqQB0Ji3VCCCGEEEJiCot1EmsGDBxo3sWX+xYvNu/iC3XUIRd0jDuMRR2oow4c0yQXYLFewORCIiWEEEIKGc7VhMU6iTWrV60y70gUqKMO1DE61FAH6qgDdSS5AIt1QgghhBBCYgqLdUIIiQH/e+3PMrekj3Tu3LFp6SL9xv1GKre+Z7YSQggpVFisFzB8HJQO1FGHgtbxP4/JTVc/Ku3GrZCtW/8pf1k8Xj77p1/JuaPnydI3d5n/RPYWHNM6UEcdqCPZD7+MZN43gys7W7fWmTVCCCHu2CVvr1kgN37i+3LNtw41bW/L87eWyKDrauXs2x+RG0492LRnhrmbEEJyk0z5m1fWSazJhZt/cuFOfeqogxsdW8tn+l/sK9TBAXLQ4YeLtPmm9DzxM6YtP2As6kAddcgFHQlhsV7A8HFQhMSVf8nzT1TLgd8/QwYe3sq0EUIKEc7VhF+DKWD8CcD/nbjkxBBkW6pkkm7b3twXoB97Qj/2JJttQfwA6ba1tK+Tv/CWXDv6Oel608XSYdvTzduQm1ti1swbvFcNP5JJty3IvgD9iL4tjB8g3Tb6se/8yMae/29IfpKx9kaxnkynTh3MO5LPLF60yLyLL6tWrjTv4gt11IE6GnY9nXjwRxckrvtzvWkITi7kbsaiDtRRB+pI4kKm/M2vwRBCSGx4TZ67a4lUnVUqV3y9vWkjhBBSyLBYL0AaGxulT58+0u34402LLuvXr5dhw4aZtWgMGDjQvPuIMWPGSEVFhVnTpa6uztNm+/btpqVlwnw8Cb+1tEnFwoULZf78+WbtI1LpmA2w7Up76Aj/sbiiurpali5datbC05KOiBvsIzvelW2r7pYy+b6MG9hBWptWP7AfJjbDsnnzZvNOH+QF+K4VixbYRJ9q6mLHNHyePHmy914b2J4+fbpZC08mHWEbedIFiBHkSMwjLRH2qxuwifzoKg6hd3J+1IpHzXkvme8UF3uaZ59bSK7DYr0AqampkY1V1dK2bVvTostDDz0khx12mFnTBcX0nXeVmTV9li1b5mnTpk0b06IHJqKrr75aunbtalp0wQR30egS+cxn3Dw9BAXR5RMmys6dO02LLphE4b8LoD0m6t5FfeT55583rXrAPvw//IgjZezYsaY1DCjUb5b5tUPkytFfEjt6/vevP8m1cx6XmrfearZ/0UUXma162EKj+xd7eO81sbb79T9FnnvuOdOqA2wPHjxYzh4xUhoaGkyrDrANn3fs2GFa9MBYhe2XX37ZtOiBWJwwYYJZ02fBggXmnT7r1q2T5Q8/4mRugi7XTJnqLD+uXbtWXn31VbOmS21trdM5m+QA5uswe8DvrOc3S5YsSbRqvX9iYVmZadEFtufNm2fWopH8fcLy8nLPfn19+O/zBqGoqChRWlpq1nSprKz0fK+qqjItuhQXF3v+NxUtpuUjon4vEz7D90mTJpkWXRAvmnHjp+nk1NMG9suaYj6VPkFJ1hG2MJ6gu/U/fGzuTLy6cmpiQFPeRe7dc/lK4idLn/dsB7EfJncn+w6NEKNaQPeSkpKP2db4jjB8xziFbeyjqZgxW3SwYxW2o8RLKmAPmqcbq0FJp6MdS9BfG9iEbcSNC6A3YsUFtk+TddGIR4D+dDV3TJwwwfNdOxZJvMiUv1msFyAouJAQXdy0YpO51qSfnEhdJnNbkIb1PaiO8NuV7/YELJ3vUSYkFIeYiOC7i8LFFl2jRo0yrXpYXeC/xkmSX0doDU1gH2NKsziCLihAN2/enJgzZ07gk4AguRu2/UU6xpTWeAXwE3pY3ZMLu6jFEXS2ukc9+UoFtLC6aNsGsAv7UeMllY42/7o46QX2RCCoLmHmGMQNbLs6EUCeQTwmEzUeAcYqfMfFJBeMHDHC2dxB4gOLdbIHSFhIXC6KdVscBSksguBPpJggYBsTtAvSJfOWCKKjLQA0iyKLLaYzXfXOdkKC5ra40L566beNIkAzHv0FI14149FfLOJV85MSxIfVxBa6YQrGTLkbGthiy+qifYLht5+ukI5SHNn8Am1cfEJlx+mAUwaE0j0o0AT2NYq6VDoidqCNC98RP/A9zIlAmDFttdEaq8lAl1RXvqPEo8V+4qudIy29e/dO6TvJL1isk2ZswsWk56JYRwGgeQXAn0jtRKpZYFgwuYWdiCxBdLSTqAtsgZRJl2wnJDuBujjJwOTjt60Zj9AatrWv0sFH2IV97atodmzCdtgi3ZIpd1u98epiDNmTI8RjpoIr21i0cY79uChG7SdrGKsuviJo7WsVXck62vyoHfMWq3+YYjrMmMa8Ae1dgHiH76nymEaxDm1c5Xc7N7nqVxIfMuVv3mBagAw9Y4h3Z7kLDjroIDn//PPNWnSS79QfdeEFzm7Q7HVSTxk0aJBZ0+e6664z7/S5/77FGXWJ8sSDO25fIH379jVrevTu3VvWrnnMie2f/exnsuXll2T48OGmRQc8leHZf2zybibTjpX27dt7Pm/YsMHzW/smZ9x0+Mbrr8lVV13lZAyNHz/e83/cuHHesaQj21g86aSTvHiZOXOmkxvAcfPejXNmy9y5c+WAAw4wrbpcO22q2s2fyTq+++67nn3tmPeDXJCpb6OAhxJcdtllZk0fzHu9evUyax+h9TQYxL0LcGPsST1PdDZnk9yAv2BKCCF5AnM3IYTkJpnyN6+sk1izetUq845EgTrqQB2jQw11oI46UEeSC7BYL2DuW7zYvCNRoI46UEcSFxiLOlBHHagjYbFOCCGEEEJITGGxTgghhBBCSExhsU5ijdad+oUOddSBOkaHGupAHXWgjiQXYLFOYg1v/tGBOupAHaNDDXWgjjpQR5ILsFgvYEaMHGnekShQRx2oI4kLjEUdqKMO1JGwWCeEEEIIISSmsFgvYPg4KB2oow7UkcQFxqIO1FEH6khYrJNYw5t/dKCOOlDH6FBDHaijDtSR5AIs1gkhhBBCCIkpLNZJrOGd+jpQRx2oY3SooQ7UUQfqSHIBFusFDO8w14E66kAdSVxgLOpAHXWgjoTFOiGEEEIIITGFxTohhBBCCCExhcV6gbJw4UK5fcECs6ZLXV2dLF261KxFI9Wd+hUVFVJdXW3WdGlsbJT58+d7r0EJ81gtaAP/XQH769evN2sfofXEg+3bt6e0rwF0hO6bN282LfGDT46ITi5o6B/TGFOIe20Q61HGUiYdYVsrBydjbQfJkdk8chC2XegNMG8ka64Vj4gTzKsugI6w7UoXEn9YrBcgKIYuGl0i/3r1VdOiy7Jly2TGjBlmTRdMEEPOGCrPPPOMadFl3bp1cvmEiVJfX29adJk7d65cffXVZk0XaDN8+HC56667TIsumIwGDx4sDz30kGnRBRPR+PHjpfsXe5gWXaAPJrzW+x/g7IQJxcD06dOdFUroA9h2cUIDfVDIQJsghVgY0LfQHtpoFxzwediwYTJmzBjTogdO3I859jh58sknTYseiPV+/U/x+lSbKVOmyNkjRqr3I1ixYoVn20WORFzD9nPPPWdadBk7dqysXbvWrOmyevVqb151AcYMbLvShcQfFusFyJYtW7zXww491HvVZuXKld7kqUHynfobN270Xr/85S97r9r8/ve/l6FnDJGOHTuaFj0wEc2ec6P87Gc/My26wPeNVdUyadIk0/IRUZ94gEl/2rRp3nsUGdpAm1/Pni1PP/20rF3zmGnVAwVov379vAnv2mlT5eSTTzZbwpFKR2iDAhox37uoj3ey2q5dO7M1OvaKHeyjcEQxs2nTJrM1GrZARxH96c981iseNU8m7cnL4Ucc6Wn/1ltvSXVVldkaDcTM5MmTPZ/BZZdd5r1q8P7773vFP07cb5wzO+t4SQdOAu68q0weeXh51rkm3ZiG5sgz99+3WNq0aWNadUC84ELMxAmXO8mR9iS6V69e3qsmiBfkR+QBP1pPg8GFHujigjf//W/v9eijj/ZeSQGSSEGnTh3MO5KPzJs3L1FUVJRYvGiRadGjvr4+0ar1/ony8nLTEo1VK1ead7spLS31fHdBbW2t5/uSJUtMSzCC6thURHu+NzQ0mBY9ampqPN+hTyqSdQwD/C0pKfHsV1ZWmlY9ECuw3aVLF68PNIEuxcXFnn0cA9aj4NcRtjCWYNvahz4a/QsdysrKmn239qEVxlg6guRu+Ac/ESvWNuISx1JVVWX+V/bAPsaQ9d3atn5HiUUAO1Z32Ma+NMcUtEcswr5WHvODfoVtHEMUUukIHaAJtNfUxAKt4XvQcRRmjoG/Grqkw+qerEvUeATW97BzR1BGjRrl2Sf5Tab8zWK9AMGkj8LRBSgCkFS0iq7kRGonfhfYZJ6pGMoWW0y7SOaYKDA5Q5t0E3SUCckWRi4KdWsbMZnO92xAHyLGYRu6aBVdjzz8sKeDv4jGMUQ9CQDZFuh+0uVuaOuyQAdBT16ixCLGD/yGfexLe6zCX9jHotGnycA+fE93Uh2GVDra/nXhO/oRurieO1z4DjCuEJPJaBTrGEMufXc5Z5P4wGKdNIOEi6Ti6goAJlAkdC38idQmRCR1F8BvjUk0FbZw1CxILfYkI1PRle2EZIsvvGoCHTABubBtJ30s0EZT8759+3p2MfGjiNa0jfiD7bAFup90uRv2YBv7gN5aBbrF2scC+5mKlmxiETrbkxiMJRdFkb1qDP21P+EBNn/BvkbcJOto7SPmXWD1cVmQoo9dgP6E74jTZDSKdZuDXeGyX0l8YLFOmrFJS3uytmAidVXw2slCs0CyWF2yOREI8lEvbGsXpRaXJxmu/E41eWp9LQs2oYeLT0gQH67GDvyN6nOm3O2qyALwG7q7GJsA9pFbXJ2oAxSLiBscg4uvCMI2ilFXGrm2D/2xhCGMjsgHri4i2RN4FzkBZKNNUFzP2SQ+ZMrf++Ef8/X1Zjp37ihbt+rfoU7iAW5A6tmzp/c4KO1fRsPNcIcccoj6jU0ANzdhad++vWnRBb53zOKmqSA6Zms7CC5t701cxGOhwdytQy7GIp4YgrzrIveCbOyH0dFlHsO80VSoO83BruY9cP3118sVV1xh1ki+kil/82kwBQgKdVcgGWomLP+d+rDrqlAHLgvefW1b64kHhQ51jA411CFZR+RGV8UicG3fZY6E3+nsa8QjbLvU5hg+BabgYbFOCCGEEEJITGGxTgghceKdGlmz7HYZ840vyTn3vGgaCSGEFCr8zjohhMSGf8qi84fLFWt2NL0/UPpOf0QWnddl96YAMHcTQkhuwu+sE0JITvAFOWfh3+XlpZfK8dLatBFCCClkWKwXMLhTP+7kws1o1FEH6vgRrQ4+WA4x7/MNxqIO1FEH6khyARbrhBBCCCGExBR+Z72A8Z+t+5+Fm3wWH2RbqjP/dNv25r4A/dgT+rEn2WwL4gdIt62lfZ142BaZOmqhyJR75bsHPt+8Dbm5JWbNvMF71fAjmXTbguwL0I/o28L4AdJtox/7zo9s7Pn/huQnGWtvFOvJ8BdMCwMXv9KnjcZPQbuGOupAHX28cHNiZKfuiZF3v2AagpELuZuxqAN11IE6kriQKX/zazCEEEIIIYTEFH4NhhBC4saLv5VzTpkvMv0PfHQjIYQUAJnyN6+sE0JIzPjwrbflNXlPdrzTILtMGyGEkMKExXoBk+qGmLiRC4/VygWoow7udXxdKq/tL8d+d568KP+VZ391uhz3pVJZ+p7ZnAcwFnWgjjrkgo65MFcTt7BYJyQiTKQ6UEdwhHzzmjXeR6HNy99/LsM/aTaTvQJjUQfqSIgOLNYJIYQQQgiJKSzWC5S6Orc3obm0v337dmlsbDRr+uSyNgD6EEIKG+RIV3nSZY5xmd9h16XvLnO763mDxBsW6wUIktUxxx4nzzzzjGnRZeHChTJ8+HCzFo0BAweadx9x0UUXye9//3uzpsvmzZs9bfDqgqVLl3r2XU1G0P7wI440ax+RSsdsWL9+vbTe/wCzpg8mpOnTp3v7cQX6tqKiIqs+0NKxkHGhIfoSfapdiMEuxtTkyZNNix6I9T59+ng5IRsy6VhdXS2f/sxnpba21rTogfGDHOMiR6L/YHvdunWmRZfx48fLLbfcYtZ2oxWPiD/kdhfs3LnTs419kMKExXoBsnXrVu/1ggsv9F61QaI98cQTzZoumOCWP/yIdOkS/HF2YbDJsFOnTt5rEIL+shwm/hkzZsjECZdLmzZtTKsemKAvGl0i106balp0QQHdr/8pMurCC0yLLvsfcIA3IS1btkwOPPBA06oH/EfR1f2LPWTIGUNNqx6ITcQPTjZwQqN9woH4QYGE4m7+/PnqV9pgDz6jOM22gEwF/EZswudhw4Z5ixbwGXZRmKJPn3zySbMlGhjT0Pqcc87xxlSPHj3MFh3gt72ggYJdExS8Y8eOlaFnDAmVx4KyYMEC6XVSz0C2w/7q5po1a7zX7t27e6+aQJc77yqTjh1b/hXgbFi5cqWnuQu+0ru393rMMcd4r6QA8X4aKQn+gml+U1ZWlmjVen+zpktDQ4NnG/vQIPnX5azv9fX1pkUP6/u8efNMiy5Llizx7NfU1JgWPeB7UVFRori42HufTNRf6ausrPR8LykpSWk/CrW1tZ5d2J80aZJq38JX6A5dYB8aYT3bY/DrCD+hC+IFdmEfC/aFtijHgb+tqqryfIUm1n+7YH+p4ihI7sax42/Ly8s9P632/qW0tNT87+yAfYzVZNs4lptuusn8r+yB7rBl7eI4tMYV9LF5BjqjHzRBvMMuFrzPlnRj2uoSxXY6bB5AXLoAcY6YcQHiPZUuWr9gauPQBXbu0M69JF5kyt8s1gsQJHMkRBc/YYyJDUlFa4JLTqR7I5mHnfSD6IjiC5MztHeBnaDT+R5lQnJZqNtJCNpcdeWVpjU6mJAxccK29R3HEQUc++9+97uPFaHwHcUt4ifbAgl/B7v+AtQuiHnYh1bo30wnAelyN/4WNpKLfizYJ/YNfbIteOE/jh/78NuGTrDtzwfZxiL0xz7sMUB3HFeUk6JkcPzWPo5FO96hE/zGkm2sWFLpaMcTXl0AbbAE1SXMHGPnjqjjNB2IRfiejEaxjrhx6fuZZ57p+U/yGxbrZA+QVFDIuCjW7RUprUnOn0gxucE2JmwX2IkoLEF0tIVjtsVQJuxJRqYJOtsJyVWhjgLLFrwoFrGuEY+w4y94Na+4+gtR+A69tWzb+IDveI8+zcZ2qtxtC0TENuzDbxRGWkWuLVSwYB/wH3GTLl6yiUX4jGOw+8hkPxtgy+Yu7GfGjBlmix6ahTpI1tHmR/SxC2wuCFOQhhnTGF/QRrNfLYh1+I4+TkajWEd8wr7WmEoGtjGuSH7DYp00YxM6Eq6LYh1FjKvJwmVCtLpkcyLQko6YfGAbk5ELYNvFVReriXahDmzhiz61aMQjCkdbSGvHCfRAkeuimNBiX+Ru6IF8An1cgZyCmNE6OUrGXtXFPnA8LnKjZqGeCmgE+64KxmwuZgTV0RbTrgpSe6LhUnsXORjYPOzqIhWJDyzWSTOYiJAQXU1IKJJcTaiw6y/uNPHrEpYgOsK2q0nURWEKoAU0z0aTloDNZJ9dxGOhwdydPf5CzkUsYiy5ygEABamr3Auy+bQnjI646u1KH9h1NXcAaOP/qpcmyJWjRo1ykodJvMiUv/fDP+Ze02Y6d+7o/XIeIfsa/BR03B+Xh1/pC/vUg70NddQh7jrmQu5mLOpAHXXIBR1JYZApf/PRjYREJO6TUa5AHUlcYCzqQB0J0YHFOiGEEEIIITGFxXoBg48oSXSoow7UkcQFxqIO1FEH6khYrJNYw+8S6kAddaCO0aGGOlBHHagjyQVYrBNCCCGEEBJTWKyTWIM79Ul0qKMO1DE61FAH6qgDdSS5AIv1AoZ36utAHXWgjiQuMBZ1oI46UEfCYp0QQgghhJCYwmKdEEIIIYSQmMJivYDJhcdB5cKd+tRRB+pYGDAWdaCOOlBHkguwWCeEEEIIISSmsFgnsYZ36utAHXWgjtGhhjpQRx2oI8kFWKwTQgghhBASU1isFzB8HJQO1FEH6kjiAmNRB+qoA3UkLNYJIYQQQgiJKSzWC5QxY8bI5s2bzZou1dXVnn0NUt2pP336dFm6dKlZ06WxsTG0NmHu1K+rq5PJkyfL9u3bTYsu6exrPfEA+lRUVDiJHasj9uEqNqOSC0+OiDu5+PQNxCPiUhvkyvXr15u1cGTSEb4iT8K+NrCNHBNkjGbzFBPYzlaTlsC8MX/+fLO2G614hNbDhg0za/pgXnLRnyQ3YLFegCDJ3nlXmTz44IOmRZdHHnlEnn76abMWjeSbf1CEXjNlquzcudO06LJu3TpPG1fMnTtX1qxZY9Z0wSQ6bty4lPY1bqKC/fHjx8uQM4bKli1bTKsumEz79esn3b/Yw9kJDSa8hQsXfmzSDgJvRouOCw1xkoqTSBSoeK8FciWKR8QjcoMmiL/eRX1k7dq1piUcmXScM2eOlyddsGLFCpk950azpgvGpivbYMaMGfL222+btd1oxWNlZaUsf/gRs6bLjU39iXnp3XffNS2k0GCxXoBs2rTJez30kEO8V22WLVsm559/vlnT5cknn/ReBwwY4L1qc/PNN8vQM4ZI165dTYsediL62c9+Ju3btzetemCCxmSBV237KIDOOeccb8JYu+YxGTRokNmiA66k3fDrX8vZI0bKiSeeKM/+Y5PaMaDot4Vc6/0P8Aqki0aXmK3RwAkMCjr4b6/aobjDFbY+ffpEOuGA5rDtX+x+sES5yga/km1jsbax4NiikGwTumAZNPh072QpCvANxw+9ofUxxx7nnURu2LBB2rZta/5X9mzbtq25SMfJ7/33LZaTTz7ZbI0GfEcsXj5hYlMRNlsmTJhgtuiAWEehDts9e/Y0rTogbjBGJ0643EmOvPfee6XXST2lV69epkUPxMvGqmrvYoALVq5cKddOc3OC9Oabb3qvJ5xwgvdKCpBECjp16mDekXyktLQ0UVxcnFi8aJFp0aO2tjbRqvX+icrKStMSjVUrV5p3uykpKfF8d0FVVVVWvgfVEX5jaWhoMC16LFmyxPO9rKzMtOxJso5hqKmpSRQVFXkLNNIE9qAJfO/du7da3MAutLC2seD9vHnzvG1h+wD/HxpfccUViUmTJnlxaO0mL9iOBf8/CPX19Z62qWxlWuBDMqlyN3RI9fctLUH7Gv8Px4q8guNOZQsL/MV2aIg8ERb8TXl5+cf2gf2iHTpGBTYQI7CLPsFxaY5X2LKxk26sBiXVmMZYhe1UsaEBtIb9oFqHmWNgU0OXdKBf0afJ/RklN1rsvIc4dMHIESM830l+k6n2ZrFegGDQI+m6KNZt0agxcQJ/InWdzFEEpErmLRFER6uLVjHqB8USbMP/dGQ7IcFf2IYu2RRY6UBRYYsu2MZ+NOLRFhN2QdGCCTSq77BhbcJvLOhTLDgWLNliTwSSF2hibfuXTKTK3baYTl5S2Q4b+/j/6D/ogpMh6IKiCPat/8m5IGwswk66ky4t4KMt0rGMGjVKLYdZEINWKxxTVJJ1hL+wjyVsPwYBfQnfw+TfMGPa6q+tuwW6YB/JaBTrNj9o5kg/uJCB3EbyGxbrpBkkQiQVV1cAMFljMnWBy4RodXFxImALGmijjZ2gobn2BI2CApqg4NW0jf6DXfiNok7LNrTAhIYJGb67KFjiTj7mbsQIxo7GSVc67MkAYsdFsWjHKRYUvS6wV+xdaQT78N/FuIJN2HZVkNoLGponeH7gN/x3gZ2bMA7CsH79+sSVV17pJJ6JGzLlb35nvQDBdwK7d+9u1nTZsWOHs++rg1EXXiAdO3Y0a3r8+9//9r6rPnToUNOiB76jetRRR0nTZGda9Gia5KR///7ed3fbtGljWnU48MADve+94qZYTduHHHKI97133Fg3fPhwNdv4jvtVV13l3WTbt29fdT3IvgExMnPmTO8+CRdjH0ydOtW7TwKx4+J+EoBxiu/uu/iuNzj44IO9ceVKI3xvGvfDuBhX9v6I73//+96rNrgxE/ld+zv8fhA7rsCcjXtgwnD88cfLZz/7Wfnud4d595GRHMcU7XvAK+uFgYuvwWij8RGla6ijDtQxOrmQuxmLOlBHHfJdx+rq6sS3vnVy4uKLL07U1dWZVhJHeGWdEEIIIaTAwNO1/vSncu/Txq9//atyzz33RH7SE9n77IeK3bxvpnPnjrJ1q96zakk8QT8TQgghpHC48sqfyyWXXGLWSFzIVHuzWC9gcqGf6aMO9FGHuPtIDXWgjzrQRx2i+ogr6eXl5TJ+/DivUL/wwgt5T08MydTP/BoMIYQQQkge8s9//lMuuOAC7wenVq16zLuizkI992CxTgghhBCSR+Bq+m9/+1sZOPAUOffcc6WsrEy+8IUvmK0k12CxTgghhBCSR+BrL88884z85S9/k2HDhvFqeo7DYp3EmrvuLDPvSBSoow77Ssf//etxmVvSx/tOY+fBP5e7N74mH5ptuQZjUQfqqEO+6ogCHVfWO3ToYFpILsNinRBC4sx/HpPZoy6Thw6eIstfel4eu2iHLDz3Wln2r/fNfyCEEJLPsFgnhJDY8rY8f99vZe4rg2XUpMHy5dbtpMvw4TLy86vkypkV8ob5X4QQQvIXFuuEEBJXPnxOHvp/T4l8rUi+cnir3W2ti+Rb3+8t/13+gKyvz9UvwxBCCAkKi/UCZtbMG8y7+DJg4EDzLr5QRx2oYwp21ErNv96T9j0/L8eZJpFPyqcPOljkgxqpe/U905Y7MBZ1oI46UEeSC7BYL2BGjBxp3sWX1atWmXfxhTrqQB1T0PBfecl8Nf1jv14nO2XHfz4w73MHxqIO1FEH6khyARbrhBASVz5skM+Ytx+nnRz82f3Ne0IIIfkKi/UC5r7Fi807EgXqqAN1TEHnE6XvsZ+U/76+Q3aYJpH/yKsv/1PkgC/KUZ/7pGkjJH7gh3lIdHbt2mXekUKFxToR+d9WeWLuaDkBz3BuWj7/3Z/Lqn++bTaSoPzvtT9/9Czszp+Xr100S57YxsfrhWeH/HPNMrlx3EDpfM498pxpLUhanSDFP/iKNCx9WCpeNzeT7npWnnz0ZWn73bPl5EPNTadEh53/lDXLbpfvfeV4KVlWaxpJGPx5sFu3LzEPZkWDvPbELVJywu45+bjuP5ApDz7TdJpOChUW6wXK0qVLZefOnU3vtsnG+XfKsuOvlr9vrZOXV14n33nrfhl18R2yIcsHTWzfvt2zr0Gqm382b97sLa5Yv369dwyh+M9jctPVj8r/Ri+XrVufl7ULfijHVt4k3zv7Onl0x55CZmU/BNAm2b7mTVS4Wubuitkuee3WUTL4/HEy+w81pi0+7P2b0T4jXU/7rpz6icfl4UVPyRvyqlQvfECWvny6TBnXTw43/yuXiO8NfZul7KJiOX/8FHnizf+ZNvdkmwta0rGurs5bXJA2B/9npZcHPzH24aY8WCO3/mKIdPtzUx783jRZ9VawCaW6utpZfocesO9HKx6RE3XmvV3yzt/ukF+v7S7/t6lOtr60XOaf+Zbc+ZOfyAPP85OKQoXFegGChHX2iJHy0ksvNa19Tno1TU6lg46V1k1rrbqOlMk/OU3kX29K9ZvZffS2fPlyz74rzjvvPHnggQfMmi6YJPr1P0WefPJJ0xKE/8lbG56RHT+4VC7/OsqndnLsoMtl6hXfFHnlUVlTjZOi3VRUVHj2//3vf5sWXXAi0P2LPUL6HxzEzjnnnCP9+vUzLdq0lgO+f4fMv3mCXNWjrWkrbD5x9Fly0x+vli/86YfSu3Mf+c79h8iYZdfKOUfzKzC6dJUL7n9WXl56qXQyLQBFGMbV/PnzVU+yYXf69Oly+BFHqhenGKfDhw+XhQsXmhY94CtyzKZNm0yLZZfsqKySd86/TMb1Oaxp/UD5z6d7ypUTv9Hk0EpZv+mjPJgO6Nu7qI/89a9/NS26TJs2Te69916zpsuKFSu8eS96jLSWT391nMz66bfkBHxw1rqndPvOqTJQ3pOnnt6y+7+QwiORgk6dOph3JB+prKxMtGq9f6K2tta0+PlX4rGrhibOvv25xAemJSzFxcWJkpISsxaNVStXmne7qaqq8nzHMbigtLQ0UVRUlGhoaDAt2dKYqL13dKJTtysS976xy2upr6/3bGtpk4ztV9hP9j9Zx2xYsmSJZx/HgH7QBv5PmjTJ20er1gclHhz9pUSnkXcnnjXbowJNampqvP3gWLCvsHGkoWMq4AcW+IclW1LlbsQdjtfaxvq+RFtDHBPiEcc4b948b4k0fl+4OXFmpy8kRj+41bOFeEdMIq9Fzwu7gc/WbllZmWkNRzodkddhG0vqHB8N6ADbwbRoTLxy94WJTl1/mrjP5MFM2Bzjwm/YhO3y8nLTshuteETedZPbP0jUP3pl4ssl9yUqXnzFtJF8JFPtvR/+MXV7M/iO1Natbj4+I/seXCG6fMJE2fVB0vcI33lOHp57nfys/jx5fPbpcrBpDgOuKuBK0f33Lfau7EQFj9Xyf0xpfX/n7f9ImzZtTKsO1vcb58yWcePGmdZseVHW/PwCuUSuk9Wl/eVzTS24inbNlKmy5eWXpGPHjrv/mxK4knbMscfJqAsvkLlz535Mm2Qdw4ArgFOmTJHZc2707F9//fXSvn17szUa8Hv16tVev26sqpZeJ/X0tB86tIM8PnmCjH93spQvOk9OMP8/CLjy9+abb8rrr78uzz//vLz11lue78kMPWOINBXs0rdvX9OSHlyhvGh0iVnbDbQ4+ODMo2T8+PEt9jXibvDgwd7xJwM9+vfv773v0aOHtGvXzluOOeYYOfTQQz/WD6lydyrfLfYYDjroIDn++OO9tuOOO07atm0rXbt29dYzYa/g+n2Hrpn+duvWrTJkyBA5//zzTUvLYD9NxaF3NRdf38PrmjVrPqYZ9Bo2bJhcddVVpiUciPXXnpovl557m3SYu0ruOquLXDttqudvz549zf+Khu0P+HrPPfcE0jkVqcY0/McnX6+++qr3lQztPAObuHr8yMPLZdCgQaY1Ex/lwcea8uCRpjUV8B2f2CHeZ86caVr1sLq/8fpre4ybKLnRAt8//ZnPyh23LwgV1y2zXf656h6Z/ONNcsnaG2XQYfy0MZ/JVHuzWCce7z8yTnpdvEzsbaWf+mapPHL3BfL5kPev4eNifM1DqyD1J1KbzJEMoxfTH8eeCCQn82z43yv3yKXFT8iJ98+QS7q28b4niY939ZP5RwUT+NOf/pTS92wnJPg9duxYryjSOgELxgZZVlISqlifPHnyx4pyW+zaQhevKETDxiZ0wNfG/v73v8uXvvQlry3IR/VBinWLLUix7P6KmjSfbIDkY0Ohfdttt5m13aTL3Rg7tbW7b5jcsmWLV/DaohfgBGf5w4947y3P/mNTi4Uk7OLjfz9+n1MRpli34yaZiRMubz7BOOKII+Swww7LuugFOH58RQ3j//e3/lRunfKAV6zP/NaBaiemOCm74oor5M67yrwTgAkTJkS64JA8ptEXiDfYf/KJDWonFxZ7UhmmmP7w5TK57Lsb5SsPzJAxn898rPZEIEjcZQNO4hAnyWNGo1i3856q7v+9V67tPVkW2El5/5Nk7rolMuyoA0wDyTcy1t4o1pPh12DygG0PJkY39SP6Mu0y+sHENvPfd/NOYtPvpyVmnP+Vpu09Ez+teN20B8d+jUQL/0eU+JoAPsZ08RUMfDUAtvGx90d80CTjZR/XbY/lS97H5Xuw6++JB390QWLaun97q/i4GB8da36MbrFfrWnpI+9sPurFx/PQBH5H+WpGdjr+LfTXYOxXIfAazd/0aH+FIxvQ5+m+zgItowLbrvQDYTTEMaJP7VeEtMcPwJhHnNvxX//UnMQQ8zUYLeA/xij2kfw1jGxJ1hFfwYB97MsF9itqgb+isutpLw9e9+dgX7uyOdIF6b4CAzTGtI0hF7y94Y7ElaMHJbo1je0vXPZQIvysTHKFTPmbxXqBgQkPSQUFzeJFi0zrnuz627TEaZ26ZzVZIdnuWfDqAbuaJwJ+7IlA4InIx546vp54/s6piZ+t2NL8nX+/5trY73i6KK6gNU6+tAskaJ1Kiz11DF+sE+bubMDYQRHXHOcv3KxerGMcoZjOJr8EweYY5AMXQJtw9l/z8uDkh4Ld+2T9d3WiYfOki5M9gL5FH7vA0+aA9onbzuia6PSDexMvmHaSf2TK33waTIGBjxfXrnks40d1rY7tJl85sKsce8SBpiU4+Hh09OjRZk0XfIyJr3m44IQTTlD46k6DvLbmHvld43dkwuCjvafrgM9//vPy5huvq38sDfCRNO49cPGx8YYNG7zv/mrfG4DviLesBVMT2Ttg7OD719px7gfjCF+/iJZf0tOpUyfvPh5XX1ODNvh6SjD778q2VQvlzg+Hy9fa1DbnwUygD/AVkiD3j2QD8iT8d9XHl112mVx88cVmTRdoM+eGH8uxPY6S7l/5vBxk2klhwRmxAMmYED98XlbduEieOmucfL9P+O9qIrFoJkR8n9CCiU7r+6PJwG70Qv0WmbflNLn2kpOan3/9v9dWyXU3Vcp/W7gRMVuC6uHXMSf48F2p3dooUr9Dkh5Tv0/JOR1jSNw1/HDHDnlLdskbbzU0/Rtf/Doi57o82QDBLgg0NBXqN8v82iFy9egvyXdPPdVrRR68ds7j8rq3lhoXFzMsyJPp/NeIR/juam6Cpn0P/Y/Mqh4ol5/1ZTnEtJLCgsV6ATNi5EiR7X+Q60/t6t3Y4C1n3C21X7tG5k4ZKN3444iBGDHyzKZCfbb88PzZcvc1p0l3q2XTckyfn8tbJ3TzngZDMuPFYxPvP/pT6XnsD2QWfgDk+etl5LFNJ0CV/O0+4prXZeOc0+TY4TdLrbwv1dcMlON6zpZH+eObAdnp5cELRt2UIg9eKe/0OF6OMP+TZOIV+dv1ZzZr17nzCFny3qkyZcEkGdyRv61QqPBpMCTW4KpH1Dv1CXXUIu465kLuZizqQB11oI4kLmTK37yyTkhE7lu82LwjUaCOJC4wFnWgjoTowGK9gMmFRMorHjpQRx2oY3SooQ7UUYdc0JEnPYTFOiGEEEIIITGFxTqJNfg+IYkOddSBOkaHGupAHXWgjiQXYLFOCCGEEEJITGGxXsDYR+WRaFBHHagjiQuMRR2oow7UkfDRjYQQkicwdxNCSG7CRzeSlPAOcx2oow7UkcQFxqIO1FEH6khYrJNYw5t/dKCOOlDH6FBDHaijDtSR5AIs1gkhhBBCCIkpLNYJIYQQQgiJKSzWCYkI79TXgToadv5T1iy7Xc7v013OuedF00j2JoxFHagjITrwaTAFyvbt26V9+/ZmTZ9ctt/Y2Oi9tmnTxnslZO+xWX475HT51ab3mt63k77TH5ZF53XZvSkAzN0kH3CZ33PZNuYkzkv5C58GQ/YAxejhRxwpFRUVpkUX2IV9V0yePFluueUWs6YLtOnXr5+sWLHCtOiChNunTx9n2sP/MWPGeBq5AsewcOFC2bx5s2khenSVSx55UV5eeqkcL61MG8lXMF6XLl0q69evNy16zJ8/X1rvf4BZ08PmGNh3wbBhw5zld/h80UUXmTVd0IeY95AfXTB48GBZt26dWSOFBov1AqSmpsZ7raqq8l61efLJJ2XoGUPMWjSS79RHIpw950bp2LGjadEFRfrGqmrp0aOHaWmZMI/VmjFjhme/e/fupkUPTKLjx4+XO+8qk+LiYtO6G40nHtjCAhPSRaNLpKGhwWzRATpiH9XV1d7JAE44sL84AL9wgrWwrMzJZIwTHxwvjhvHn/j0QXKI2abB9OnTvQXHUFene+UdvsPnoCdvufD0Df+Yhl7QzWqoBYq7c845R84eMVJefDH8150y6Yii9PIJE+XGObNNix6///3vvRxz0kknmZb0hH3kIDRZ/vAj3gUTbTCGocm3v/1t07IbrXjEfNrrpJ5OrqzfOGeON28cfvjhpoUUGizWC5CXXnrJez3qc5/zXjVBQrxmytSPJUQt1qxZ470OHTrUe9UEvqOYnjjhcunatatp1QMTPk407r9vsfrJBny3hfraNY9J3759zRYdMIliAkVhAX3eeP016dmzp9maPfDbFue33nqbfPozn5XeRX28k4EdO3bIcccdZ/5ny0BfFPdhlqBXNHGCO+SMoXJRyRjvZAVXLFFco3iDHRxDlCK4bdu23vHiuHH8f/xznfxPEpL471uej1FPEBBvy5Yt847hmGOPa/Yfvkf9hGTWrFmez92/2MOziyuj1jb6BPbRz9kATW1fhVmy3R/APp955hmvb/EpGPSCbhs2bFAZt/bErF//U7z1J5/YIOeff773XgN/oT5u3DjTqgN8R4xeO22qeo4BiCUUvC5sb9y40Xv95je/6b1qgxyG2HdBbW2t99qtWzfvlRQe/M56AYKJAoXBtwcONC173giUfDUkyDbbvuWVV+TnV/9CSq/7pVxxxRVeWzb2QKptSIbvvP2OjB07xrTuJtkPP+m2Je8Lk/FN8+bLDTOvb5rsJpgt6f8uzL5QbN15111y4okn7qE7SPbDT5BtdzdNEnehSF+3Tq75xdXSrelEIxsfU+1r27Zt8sc/Lvds49OSmTNnSpWZ9Cxh9vX+++/Lq002tzbFyfsffOCdXFj6nXyyHH98Nxkzdqw3KeG7mUF8BDt37pTf3Xqrd/UpDNjnqtW7r6y15D/28c4773gT54lNJyqbNm3yTh6T9wmdcLKHE9Ydpsj2+90SuIrW+5i3Zd5P7hO58neyZPzueIHdjRuf9N5nYtbMG7zXVFrhGP716qve1b+VK1d6VzEtQ04fLF/4/Oflh02FI/T/w0MPmS0fkdyflpN69ZI333xTXn/9dfn9/ffLG2+86cWMHxRhRx55pBx66KFS1Lu350M6ewDbkKtwghsWFMA1zz9v1naTSg+AGO989NHyfNP/v+OOO5py2Fav/aSeJ8qXvvQlGT16tJxwwgnyaNOJRzLp/E+1L2i//s9/loV33+Np8bOf/Uw+aBoPftL5CNLtC9ht+IoH7J//w/PktEGDvLYwPlpSbcPYnTtvnuy3336ydu3aUPEBWvLDuyDQdALzkx+P806SQBAfg+4LFwPefudteeKJJ7z1dPZA2G04icHJqs2/lnQ+BtmXv/3/3XuvfOITn/BOuEn+krH2RrGeTKdOHcw7ko+0ar1/oqysLLF40SLTose8efM8+w0NDaYlGqtWrjTvEomamhrPdnl5uWnRA/4WFRUlJk2aZFqCE0THkpISz35ToWdadIDfsA1dKisrTevH8esYhPr6+kRpaalnF35raW59xYL3iMOqqirvOFzEozbpdERsQv8lS5Z4MYRjKy4u9nRMPH1lYnRTTkVeTbuMfjDxtLHl8cLNiZGduidG3v2CZxv6oz/QF1Y/z3YSsBUG2ID+6Ad/32Aca4B4h//QBTahDey7GMPZAD/sMaO/4CP6ccFtt5n/oQOO3/Yd9qGRH5Nj0eZerb5LBjEC+4iXoIQZ04gNaKQ1d/hBnMN3HEMyYXNjKqw2LnwHLvuVxIdM+ZvFeoFhC15MSC6KI0x4KCq08CdSOxm5SIh20oY+YWlJR2vbRYFiC+pMhToIOyHZyUersLCgeLPFeTK5XKyr4yvWk7EFcCqi5m70C2y7KjrAXtMwADhOxGPyiY9mLNqci2JU82Tdr6PNMa4KOvgN+2Fze1AdrX2c1LjA6pPqBFcjHnGii8UF/jmb5DeZ8je/BlNg4Lucc+bMkQkTJjh5BBS+Lzlo0CBn3/kGsK8NPsb861//qvrdUQu+z/zGG2848RsfHR944IEq3x9PBl/dcfUYMtICL/5Wzjllvsj0P/DRjXmA67GE/LVlyxYnOQbAf3zl6/TTT3cyb8D+vffe633lyIV95GDcqzV8+HDTogvuk8BDCVzMe3bOvvjii5mP85xM+ZvFOiGExIwPn7lR+g+dK5++ern8YWx3aW3aW4K5mxBCcpNM+ZtPgylgkm9siSO59pi3uEIddXCv4+uycc5pcuzQG+QVeV82XTdIjus5W5bhN5LyBMaiDtRRB+pIcgEW64QQEhuOkF4Tyr2rK81L9UQZ9kmzmRBCSMHBYp0QQgghhJCYwmKdxJoBSc8kJ9lBHXWgjtGhhjpQRx2oI8kFWKwXMP4fZiCEEEJI/OBcTVisk1iTCzf/5EIipY465IKOcYexqAN11IFjmuQCLNYJIYQQQgiJKSzWCxg+DkoH6qgDdSRxgbGoA3XUgToSFuuEEEIIIYTEFBbrJNbwTn0dqKMO1DE61FAH6qgDdSS5AIt1QgghhBBCYgqL9QKGd+rrQB11oI6FAWNRB+qoA3UkuQCLdUIIIYQQQmIKi3VCCCGEEEJiCov1AoaPg9KBOupAHUlcYCzqQB11oI6ExXqBMnnyZNnyyitmTZe6ujoZM2aMNDY2mpbsSXWn/vr162Xp0qVmTZ+FCxd6+3AFfHdpH/on68MnHuhAHaNDDXVoSUfkX+SB7du3mxY9qqurvTzpgoqKCmf5HX7Pnz/frO1GKx6hs9a8lwrM2fCfFCYs1gsQFHOz59wob//nP6ZFl2XLlsmdd5WZNX0mTJggzz//vFnTZfPmzXLR6BJ5/fXXTYsuKNLPHjFSXnzxRdOiC5L58OHDZcaMGaZFHxwDJo7W+x/gpBCIK3aix/G7mJBRoAwbNsxJgYUxD9vTp09Xn/ChBXRBkYX9kOBAL2iHsQT9tIDd8ePHe7mmoaHBtOoA272L+jjpa8T9kDOGOsvvU6dOlZUrV5o1XZYvX+7Ney5yA3TBnP3GG2+YFlJosFgvQJ599lnvtWPHjt6rNkiG106bKm3atDEt2ZN8pz4KpY1V1dKvXz/TosuCBQuk10k95fTTTzctemByw4nGqAsvkLPOOsu06oHJHpPoUUcd9bErU1GfeGCv0qHg69f/FFmzZo3cf99iad++vfkf0YA227Zt8wpJ7AcLTgiwBC0KcFUL/uFvUADBBuzhBEwL2MXxf/ozn/X2A821CusePXrIYYcd5hVYhx9xpFdYa30Cg7H+7W9/2zuRRoxAJ1wZ1fIdYx5F1jHHHtdsO91JQZhYhA0UstAaeqBPoQn6VPuExs/777/v7QOLjUfsH35o9AlswBb0unzCRLlxzmzp3bu32RqMdDrCZ5ywP/3007J2zWOqeR55YNy4cV6OvPjii02rHvfee6/36sI2dFn+8CNy6aWXmpbdaD0NBjGC3K6VE/3YHNi9e3fvlRQgiRR06tTBvCP5SGlpaaKoqMis6VJbW5to1Xr/RHl5uWmJxqqVK8273UyaNMmZ71VVVZ7vS5YsMS16NDQ0JIqLiz3foZE28+bN83wvKSnx9pVMso5Bqa+vb7Zt7VdWVpqt4YBf0Bb20I9YrN1UC/aF/wMfgoD+Q2xnsott+D/wA8cR1LYFOmI/ZWVlXn9au3iPtpqaGvM/swfxAVuIFdjGKzQLEjct5W70AY4b2lrfbZ+mipswQEuMe7/+8B16++2HiUWrBWz69fYvNk7Qp1jCjC/4i7/B32OxmqdasA3/J2r82+OAPayHjUFLKh3hm7XtIs+gL2EfY0Ab6ADbiHUX2DyWHOfZ5kY/1neteS8Zl3M2iQ+Z8jeL9QIEgx6D3wWYWJG0sp2AkvEnUpsQsQ8X2GI6atGSCleTHHxFAQHbmIzS+R52QoKf1i4W+B+1ELWFBBZbHKEvUbBgG+xrFLt+YA/Hgn3YkwT0sfUjbGGQrCMKIkzQ/uIX9jP1RRigi40dLIhR7C+d7TC5G+MJ+vuLR/itUeTBP/gOe369odPvfvc787+yA/6hX9GnWNCnfv2xBMk/+D/WN1vww19r18Zj1H6Ev7BrfcO+oE1UkmPR5l7Y18q/fhB3sO8q/9o4d+E7+hC20Q/JaBTrVhsXJ0gAY9TVnE3iA4t10gySCZIKksviRYtMqx6YKLBo4U+kdjJykcxtIQldwtKSji4nOWgN2yguMhFmQrIxgkIGPrvQOxUu4jETOM6whVgmHW2BikkVk6vmiQf6AHFkC+t0J33Z5m7Ys8WS5vi1QAvEEvz/8Y9/bFr1gU4a8aoZi/aEAIWiZkykyo3ow6gnF6mwOSFsbATVEbpYjVxgc3Aq/TWKdZzoIbZdgHiG79nMTSS3YLFOmkHSxeSBVxfFEWy7SiqYiJAUXYBiN9sipSUdMZG68ht2Na7SJYNJzcWkn4m9XaznKpmK0ai5G32+t/s9jmjGInKtxglEJlDkurgYYLEnimGPI6iOyGGYO1zpBG1cnIRaoE1LF0yyxZ4o4ZXkNyzWyR7YhOiiONKe6P1XPeJaRORCkalx9cg11DE6uZC7GYs6UMfgZJo7NHR0PTctuO02847kM5nyN58GU4C4uFvdovEEmHS4tE0IISQ/cT13uLbfrl07844UKizWCSGEEEIIiSks1guYESNHmnckCtRRB+pI4gJjUQfqqAN1JPvhuzDmfTOdO3eUrVv5S3SEEJJLMHcTQkhukil/88p6AXPf4sXmHYkCddSBOpK4wFjUgTrqQB0Ji3USa7R+CrrQoY46UMfoUEMdqKMO1JHkAizWCSGEEEIIiSks1gkhhBBCCIkpLNYJiQjv1Neh0HX832t/ljmjirybjDp37ip9zp8hT2x732wlexOOaR2oIyE68GkwhBCyr3nnz3L9uEfk3fMvlWsHfkZeeLhUfjHhbll/xEVy60NTZHD7VuY/Zoa5mxBCchM+DYYQQmLLLnlzxQr5Z/HFTYX6UU3r7eTzZ/xMrr2qn8grj8ufNvxn938jhBBSkLBYL2By4XFQuXCnPnXUoXB1bC2Hnf1LWTC8o1kHn5RPH3SwSNuvyZdP+qxpyw8YizpQRx2oI8kFWKwTQkjseEOef6Ja2gzpJ6cdEewrMIQQQvITfme9gPGfrftvBEo+iw+yLdWZf7pte3NfgH7sCf3Yk2y2BfEjDLBh/+6Rhx+Wb/dolPHF6+QT40+Vfm13Nm9Dbm6JWTNv8F6D+BjmmEG6bUH2BehH9G1h/ADpttGPfedHNvb8f0Pyk4y1N4r1ZDp16mDekXxm8aJF5l18WbVypXkXX6ijDvmnY2Ni24OXefk0/fKlxOgHt5r/b/jwhcSDP7ogMe7hV01DcGAz7jAWdaCOOlBHEhcy5W9+DaZAWb9+vZw6aJBZ06WxsdGz74rt27dLdXW1WdMH9jdv3mzW9IH9ujp+cpX/fEqOHDbfu1KSfnlGFgzrZP4/2CE1C++RZd/8mcwZ8jnTRki8QI53lSNd5nfYdjU3QZOKigqzpguuqsNv+E8KFFO070EuXJ0h2dPQ0JBo1Xr/xJIlS0yLLuXl5Z79+vp606LLpEmTEsXFxWZNF2hTVFTk7cMFsA/fXdkH0B/H4Kp/LehfV32cLbW1tZ7GroC28+bNc3Tc/01se+zXiZ/ctC7x8ttvm7ZwZMrdlZWVXtzhVZuamhpPFxe20Z/o13wGxwgNNYE9V7kGNpHjXVBSUuIsvyNGXfmN2Idt7X4EiA+XczaJB5nyN4v1AqSqqspZUgGayTb5I0pM2i6TFuy61gb20QfaIKGXlpZ69rEff4Gj9VEv/C4rK/P6F/vBSUFcsLFhjx99qV3kTZ06tXkf0FovTv6bqKv4VWLcr9cmqjdvTmxuWl577bXEe1v+kPj5jEcTdeZ/tUSm3I0TDNtveNUsrKEzYsHaxklNupOmsLFoxwzsothCDKaznUtAM+hkjw9LmHjNpKPNY+gT7VxjC14XOdgWvNBFG8Q/bMN/P1q5EfnAVT60c7aLeYPEBxbrZA9QbGHgu/genE2I2IcGyYnUThQurmxa38NeiQqqo/XdxUSEJG6LpVTaZzshQRNMoPYkwC4oMHAcYYoLG3d2QfEFre1y5plnNr+HVigGwhbcKJ79JxNYoAvs4Tiixg10hA3Y8xen0QrfdxPbHvtl4rSmvIvcu+fSN3HBLY8HLk6D5G74avXBq+aJqd+21T1Z87CxiL9HrCEubJ9iwXqUEzJbANkF/tr488ciFsS/jcdsCyb0If4WmvjjE+8Rs2HtptIRWtjiH35rn9Sgf2EbxxCEMHMMfEUfQA8XoO/ge3K8ZJsb/cD3MLqE5dJLLvHsa/cniReZ8jeL9QIEyRyJ3EWxjkk1VULMFn8itckcE6cLbEEatqALoqOdKLSTOTSxRXCmwivMhIS+g03/FT9beGGyznbCgF1b8GCB3v6CyF8gYX9239kWR+hH+Iv9+O3huHB82cRocjziOPzFKda1JlT4Dy3C2A6Tu2HP6oL9aBbtsGV9x4I+sPajFkeIh0wnZEGx/WcX/L2Nv+Ri3b8vxE9Q0p1oQA+0h801fpJ1hD3ogAXvtYHu9viDxniYOQZ9APuacegHfZiq76LGI7AnMdnmqpYYcMqAUHFHchMW66QZTA5IKkiMLop1TEJIilr4E6k9EXCRzGETtlEEhKUlHW0ix4StCYpNW0RA90wTaNAJCTZgDwsmB8SJ1olXS7iIRz/oYxwPjgvHh6ImLOl0RB/7CzLNSRt+W9vwOZPtsLnbFqywC/vaMYp8gyLY2kecahRHFlsMw67dh4b2WrEIv+ATxil00IwLqyP60MYHYtvFeIVN6IvjCFqog6A62nlJO/4sNgenOpnTiEcbfy6wORnjlOQ3LNZJM7YoxauL4gi2syl4g4BErnki4Mcm2zATkaUlHeE3JtFsbGfCFlnaV3MwMWv7GgTXxbofHF+Uq5rpwLgKe4U3KLCNOM00aWebu6EH7CJWXfS9tQ9tXAKNNNCKRRy365Ndm9NdFnP2hCvssQTVEb7jGFyMSWD9dwVsu4pt279asU3iS6b8zR9FKjDweKmNGzdK3759TYsueORWt27dpE2bNqZFD9ju3LmztG/f3rToYR+l2LFjyz86kwx+vCLTD1bAdtu2bZ34nU+0pCNpGeZuHRiLe2IfGRg2hwXVMUr+DQIeM4kc7Mo+Hqt4wgknOMvxeCTkIEePWibxIVP+ZrFOYs3qVatkwMCBZo1kC3XUIe465kLuZizqQB11oI4kLmTK3/xRJEIIIYQQQmIKi/UCBh9RkuhQRx2oI4kLjEUdqKMO1JGwWCeEEEIIISSmsFgnsYbfJdSBOupAHaNDDXWgjjpQR5ILsFgnhBBCCCEkprBYL2By4dFkuFM/7lBHHahjYcBY1IE66kAdSS7AYp0QQgghhJCYwmK9gOEd5jpQRx2oI4kLjEUdqKMO1JGwWCeEEEIIISSmsFgnsYZ36utAHXWgjtGhhjpQRx2oI8kFWKwTQgghhBASU1isk1jDp2/oQB11oI7RoYY6UEcdqCPJBVisFzB8HJQO1FEH6kjiAmNRB+qoA3UkLNYJIYQQQgiJKSzWC5TGxka5e+FCs6YP7GuQ7uYfLfsa5MJjtXLhJirqWBgwFnWgjtmRPHdo6rh9+3bzThfoCL/jNO+RvQuL9QKlX79+8vzzz5s1XTZv3iyf/sxnnSWupUuXyjnnnGPW9Jk/f75MnjzZrOmzfv16p/YB+gA6FWJyx7G7AnpCV1f7wJhB/NXV1ZkWXSoqKjz/XYC4hn0XMQc9XOWTfAaaob+14wn2kMNcxCn87dOnj1nTBfGJuc8F0OLwI450ogmA3+vWrTNrpNBgsV6AoNDYWFUtn/zUp0yLLkiIoE2bNt5rFJJv/kEhMGPGDOnatatp0QXaXD5hovTo0cO06IKCpl//U2THjh2mRRdMztOnT5fuX+zh6WT7wMVNVOgL6BWmOENsVFdXOyu8oC+OHZM9Jn3t/fzlz3/2dMU+oLO2/YaGBlm4cKEcc+xxTopq9NfZI0bKmDFj1IuK119/XYacMdQ7kUY/pCObWBw+fLhXCEFzzRMl2IKvmjazBT6E6ZNMOtqTSmiG/tYEPqI/1qxZI23btjWtOmA8wd/+/fubFl1uvvlmOeqoo8zabrRy4+rVq73XQw45xHvVBLpgzj7wwANNCyk4Eino1KmDeUfykSVLliRatd4/seC220yLLkVFRYnS0lKzFo1VK1ead7spLy/3fK+pqTEtejQVSoni4mLPf7wPyuJFi8y7zFRWVnq+l5SUhLIfBNgrKyvz7GPBe/8+knUMCmxAa/iOuEG/Tpo0qXk/WIL2dX19/R5/hwW25s2b5/XrnNmzVXSBr34f4Z9WvEBHv9aIFeii4bcFOln/ESu1tbVmS8sEyd3QGn7DPnzXpKqqyhtDsI1jSOV7NrEIO4gT6zf2gX6Oqjv0hT27YB3xMnHCBC9m0Bea4DigEXTH8UAje0xY8D4o6XT09y/sax4D/IdtLEHiMmhutNi4D2I7LDb/4tVPtrkxGcSk1ryXzFVXXun5rh2PJF5kyt8s1gsQJEQklrCJNAiYiFIlxGxJTqTwG4sL7EmMlu9+rC7ahTpswW87OWOySJXQg05I8DNVEWEXaI9tKFaxXxQ0YY4H/xd/g4LC7ifdPmAf/mSLiwLPryPsW/9hO4qvqcimqA6au6OcELREckyiD/yaRymOYAe6QG/Yxj5gP0oRg2NHXMBnaJIq7tGOmMe+g8YPfMLf4G+tv6nswn8b62H6IVlHHIPdD/pU6wTVAt+gDRbNeLHYHBk01sNi+zaZKPFogR7w3cXcAc4880yvb0l+w2Kd7AGSCiYRF2Digf0oBZEffyJFInSVEDGxwjYSujb+SU5LF+CfnOF3psk56ISEYh9+JhcR2hN/MtAf+8H+4IO94glfohRiIF2Blw2pdITf/n7QLGRw7LAJ20GK6rC5O5sTgqDAd/Sl1Rz7AhrFEYDu1r629oiZdCeVQfOPzVf4W/gJfdEGuxp5wOoIe9Y/xKGL/OjPYZrxbYEesA3/NXOkBRpBn1QxrhGP9pM2F74D2M42Z5HcgcU6acZevcCrC5BwMTG5ABMS7LvATvpRC8NUuJjk4Cf8dTU5xwXt/rAFnnZ/YJJGIQC76BNt/EV1JrLJ3f4TAu2CHaBQsidfeK8N/EexBH1cXYSwuCrGooB4w7Gj71z5Z/vPRaEObLHrIj6A9kWkZNAHruY9e6KRz3me7IbFOmkGyRYTM5KWi6/BIGG5SrhI6K4SFmxnewLTko6YRF1Mcq4mzn2Fi3jc26BwdNUvsN3S2IqSu2Fb++TIj6u84IJcikX0meuTCOTdbPJjUB1h38WJogW27Sc7LsDJgKv4Rv/iazBxPFEkurBYJynJhQlJ6yNzl1BHHahjdHIhdzMWdaCOOlBHEhcy5W8+upEQQgghhJCYwmKdEEJiwbuy7W+/kZITOkrnzk1L13PkhooaedNsJYQQUpiwWC9gRozU/bGMQoU66lDYOu6St/+yQOb/7ST5xaY62fry43L78LdkbsnlcsdT75r/Q/YWHNM6UEcdqCPZD9+FMe+bwVWdrVvd/GQuIYSQAPzrfvnu138lHeY/IvO/8znTmBnmbkIIyU0y5W9eWSckIvctXmzekShQRz+75D/PPy0vD5kil54RrFAnejAWdaCOhOjAYr2AyYVEunrVKvOORIE66rB3dHxNnn34Bvm//2slM371HeneyjTnCYxFHaijDrmgI096CIt1QgiJC+8tkzk9e8vgS+ZLxfY7ZOzXx8ltLzSajYQQQgoRfme9gPGfrftvYEk+iw+yLdWZf7pte3NfgH7sCf3Yk2y2BfED3Ld4unxy5WIZX7HdtKRimMzdcKMMO7K1t/b42rVy6K5aeXbjSrli3krZ/9w7ZdWMU6VD0zbk5paYNfMG7zWIj3tfD/oRdVsYP0C6bfRj3/mRjT3/35D8JGPtjWI9Gf4oUmHAH6zQgTrqQB2T2FWZuPm0zyc6nX9f4lnT1BL8USQdGIs6UEcdckFHEh3+KBJJSS6cqQ8YONC8I1GgjjrsVR1bHS4nfKWzdO95rBxqmvIBxqIO1FGHXNCRV9UJvwZDCCGxY6e8sGiijCvrLP93+yQ5tcMnTXtmmLsJISQ3yZS/eWWdxBo+8UAH6qiDOx3rZOOc73rJevfyPbnzvWEy/a7JgQv1XIGxqAN11IE6klyAxXoBk+qGGBIe6qhDYevYUXpNeNC7qrJ7KZfSC4dIryMPMNvJ3oRjWgfqqAN1JCzWCSGEEEIIiSks1gkhhBBCCIkpLNYLlPnz58uWV14xa7ps375dpk+fLo2N0X/MJdWd+ps3b/bsu6Kurk4WLlxo1txQUVHhHcfeopCeHIG4Qwy6ADrCNmLEFYgLV7EBbdavX2/WdIFt+N3SuM82Fl1qnovs6zFdXV3tJE4xvjA/uQAxlDx3aOmIuIdtV7kHcxI0JwWK9wDHJPic9fymvr4+0ar1/oklS5aYFl3Ky8s9+7W1taZFl0mTJiWKiorMmi4NDQ2J4uJip/ZLS0s9faCTC7CPysrKRElJibevOIGYmDdvnucf/HSB1besrMzJPjBuYB/7cWEf8Q370CgsLeXumpoazzZiQ9t32INtjB3sRxPrN7TRtA2fEY8Yi8iL+UZVVZUXp8hpmv0NzdAfLuYQG/8u+sP67cK2y3nP9ZxN4kGm/M1ivQBBEeAqqQBMDCgGNEj+wQo7abtKWjaZY5LTBpMldIF9FJLaIKHDLool7AP9YAu+uPzwB/rP+ocFhYR2oQRbdsLHvjRPiqAj+hE6W421x5E/ThCPYQiSu21RAW20fff3b7oxmm0swp61DV20YsZqjQX9ib51lRs1Sacj4gd9jGOx/ayZb2yOdHGyij6GbRf53Ra8yWNKKzcijqC5C+ycrXmiSuIHi3WyB0iySOAuwCSHpKJVICUnUntVXXuSACjQUyVzDTBR2KIgmyummYA9W5xiwfvkk40gExJ8tMVQmCWb/oB/0NkWFFigj2bhjonNao79aJyA+XWEPauXdnEBPW1RhGMIqm/Q3O0vqrXjEb7aeITvyf0ZpTjy62J1Dxt7qYCPiD0bL1hs4Z5tgeS3FWYJur9kHfF3fm3gP/pWQx/g71cXOdLOHdiHC6w2mvFogU3Y1jwp8gPfMV5JfsNinewBBj0Kdhc/YWyvOGoVXP5EiskItl1cdcFEBF0wwYWd3FrSEZOQdmEEfaGDtYtXaJ9O96ATEgoW2A2zRC2CoQ987927t3csWFDowLYG0NzqhEIgylXTZB2hty1g8KpVGFnQH7Z/g/gdJnfDd1tQuii+/L77Y0SjOIIWVnfY1zzhgC5XXXnlHsU29gGNwsQOjjl5rARZggIdEW84dv9JL/zM9gQjHdiP1SNMQRpmjoF96Jwuh0UBNq02yWjEo431KLklE126dPHmbJLfsFgnzSCZIKkgwbso1jFpIOlq4U+kSLTw3UUyRyKE7WwmuUw6Qm9MQFg0E7mdnFGwBClUNCYk10BHaISCxRYGWpqh2IBd9APsZnuCkU5H2IZdF8UGYjKo32FzN3Sx40pz3FrQfzZWbZGnGYvQxtrXLGbsmLbFsM0PYQpV10BH23fQAAUj/NUG8WzHY9iToqBzjB0/OAYXZJo7NOIR+qAPXIAx5FIbEh9YrJNmkGxt0tIu1jFRZJPQg4JkmOrKiAYohrKdiDPpCC2iXs1NBexp29zXaMdjKhD3KAy0rzwC2HTR1wB+o2BsaWxlm7tRCMB3FyAvYGy5LHShC/pVi70RixogLlzEsh/Ec9CLAskE1RH2XcUfcG0f84erYhon6OlONEh+kSl/74d/zINhmsHPXeNX9Eh+gkdLtW/f3vtVtBEjR5pWHaxtF7i0HQUXOhYi1DE6zN06MBZ1iIuOeKximzZtzJo+ruem2xcskNElJWaN5CuZ8jeLdRJrVq9aVVDPCHcFddQh7jrmQu5mLOpAHXWgjiQuZMrf/FEkQgghhBBCYgqL9QIGH1GS6FBHHagjiQuMRR2oow7UkbBYJ4QQQgghJKawWCexht8l1IE66kAdo0MNdaCOOlBHkguwWCexBjf/kOhQRx2oY3SooQ7UUQfqSHIBFuuEEEIIIYTEFBbrBQyfI6wDddSBOpK4wFjUgTrqQB0Ji3VCCCGEEEJiCov1AiYXHgeVCzf/UEcdqGNhwFjUgTrqQB1JLsBinRBCCCGEkJjCYp3EGt6prwN11IE6Roca6kAddaCOJBdgsU4IIYQQQkhMYbFOSER4p74O1JHEBcaiDtSREB1YrBcwTKSEEEJIvOFcTVisFyibN2+Wuro6s6bP+vXrzbtopLpTv7GxUaqrq82aG6BPPsGnmOhAHaNDDXXIZx2Rf5HnXZA8d2jqiHnPld+u52wSb1isFyjdv9hD5syebdZ0QTLs1/8UZwXvihUrpHdRH2dJceHChZ4+QRNj2Mdqbd++XcaMGSOTJ082Le7AvuIGJrT58+d/zDetx5MhLlrvf4C3Dxcx4rfvgqVLl3r2tU54k4FtxJ8LbRDTffr0cVJUTJ8+3fNb2zbiEH3p1zvfHpWHvtbsb9hCfyCWMpGNjugH5N/a2lrTokdFRYU3d7iYm2AT815NTY1p0QM6nnfeebJ69WrTQgoNFusFiL2y8NmDDvJetXnkkUek10k9pWvXrqYle5Lv1MckMWPGDJk44XJp06aNadUD2lw0ukSunTZVOnbsaFr1gP3BgwfLnXeVSXFxsWnVBfvACQcKm8OPONIrRII+8QB/i2IRE4+rky1w+YSJnm+Y8LX3g7i4cc5sbx/nnHOOanEHHf32XRS9p59+uoy68AJv4ndRsK9d85gXf+PHj1f3vaSkxHsdPnx4c55JJtunb/Tr10+efvppOebY47wY1QTjBXoPGzZMTXPEHfyEPcS4ttYt6Wj3jxOoT3/ms95Y0AB2YeuaKVPljtsXmFYdYHvChAle/Hfq1Mm06gD9r776ahl6xpA95qZs4zEZnAiAbt26ea+abNu2TTZWVUuXLl1MCyk4Eino1KmDeUfykbKyskSr1vsnFja9atPQ0ODZnjdvnmmJxqqVK8273Vjfa2pqTIse9fX1iaKiokRTEe0dR1AWL1pk3mXG+g77tbW1pjU6sLVkyZLEpEmTPPt2aSqcvHYcS7KO6Ui2gQWaoB19CnuVlZWe/mE0SgZawx5sYx+w33QSZrbqUFVV1Wwffmvg1xE2YRs643g0ee+997zXpiLP0zsoQXM3bFrfo/RjKhCPiHHYT+V70FhMBXS2MYpXLd2hAXy1fjcVRc1jJ1tsfCQv8BsLtpeXl3tjKZvjSNYRNmCvtLS0Oe6x4JiQezRyDuzDJuxjfLVE0NwIoDXiEbY186PF9kdyTEaJRwt8h9/Q3gWXXnKJ57v2WCXxIlP+ZrFegCB5IymGSaRBsclcq5j2J1JMRrCNic4F0AT2w04ULekIv61tJPOoCRd/jwkn1aSMAhiTaPI+wkxI+Fv0H/aBCQ42obl/X/4F/ydbsC/8vbWNY0hV4GULtIfvsI0+yKYo8pOsI3yFbd0CY2fi1ZVTE9/s1Dfx8z8+59nHuApCmNxtfYcuUWMyGdizMY9C0Y9GcWQLL+gepGgMA3QZcMqAZvvwP0rcYCxhgc+whXi0JwXJC7YF7QvoiGPH+PTbg89ow3FEjXcLfIJN2A8zjsLMMda+5vi3wH/oAp2S0YhHO5a0Y9GCeITuJL9hsU6aQZJFUolSYGUCCSVVQswWfyK1yVzrRMCPveodtCgKCpI3JglN2yjSYQ928T7IpKwxIVmSiw+NyRWTKezYogOvmv3svyIYxd9UOqJIh10sGpP1h1vuTvzo+A5NebhPYurjO5qLXsR/S4TN3bbIgN7aVzPRpzZW4bstQrViEf7aeNE4CU4G8WdP9LC4yJk4BuwH8WnHU9DjuH7GjGbf4CdsaPchgM0wMZgNdny6sg9tYT/V2NeIR3vhxAWu52wSH1isk2ZsUnSV1F0mFdjGpKQNJkfYRsLVxl5F09Qb/rrov7iACRW6aV+lskWHi0kVttHPWCKx6++JB380IHHaaX29Yn3aure8/g56oppN7rYnlK7GbVDfswHa2BNtF/YB+hbHgP3EDVfH7AfHjviIcpLbEug/l1eOMS5dzB0W+O8qPuwJdT7nfLIbFuukGZylo2B3BWwHvTIUFtgO+vFrWDDpZet3Sx/1utIj33Dxtay9Cfo5WvG0LfH8nZclzpi1LvGPe0c3F+uWIDGabe7GuHJZDEQZX/uCXI9FbVzlRovr+EDB62ruAC7nPdh1OWeT+JApf/NpMAVG+/btZdCgQd57F48ng23Np7T479SHbfjvAjwdwMXTZYAru2HQeuJBoZNJR/Rz9k9A2iVv/+UumVT5VZl0YS9p9+GHpv0jXMYoxlVHB08/svh9ZyzqsDd1dJ3DXMY26Nu3b9q5Q0NH7XnPD+zuiOEjeMneZT9U7OZ9M507d5StW/nw/XzHX6z7fyEtuYgPsi1V4Z9u297cF6Afe0I/9iSbbUH8AOm2JbcXnZCQh+a9Ip1//hPZ9bcl8sV3H5Uh056TCxfcL1/c/tfmv0NubolZM2/wXrPxA2SzLci+AP2Ivi2MHyDdNvqx7/zIxp7/b0h+krH29q6vJ8GvwRQGufBRr+aNka6gjjrkn46NiW0PXubl0/TLlxKjlz6ReOwX4xOzn9ph/u69RN19Y5u29UnM+Ovbpi0YsBl3GIs6UEcdqCOJC5nyN78GQwghTviUHDlsvnelJP3yjCz4ep3cedcSmTOsh3dlpXPn4+TrP3246e//Jb856wTpPGqJPLfbICGEkAKExXoBkwsfqw0YONC8I1Ggjjo40fHIYVK2RwH/svxtzplNGzrI+CXPydY7vycn7P6feQFjUQfqqEMu6MivwBAW64REhIlUB+po+VA+aPyv967hvf95r2TvwljUgToSogOLdRJr+OQIHaijDntXx/3Na37BWNSBOupAHUkuwGK9gEl19zoJD3XUgTpaPiWdz7tTtm6tlGu++VnTRvYmjEUdqKMO1JGwWCeEEEIIISSmsFgnhBBCCCEkprBYJ7GGTzzQgTrqQB2jQw11oI46UEeSC/AXTAkhJE9g7iaEkNwkU/7mlXUSa3invg7UUQfqGB1qqAN11IE6klyAxXoBwzvMCSGEkHjDuZqwWCckIkykOlBHEhcYizpQR0J0YLFOCCGEEEJITGGxXqCMGTNGNmzYYNZ02b59u/Tp00fq6qLf6JbqTv3Nmzd79rEfF8DvyZMnq/ifDvi+dOlSp/vwE7cnHlRUVHj96IqFCxd6S2Njo2nRwepo7bsC8VddXW3WdIHf8+fPN2u6BBk72caiy3EJrRGTucS+HtMYWy7GMexOnz5d1q9fb1r0QN4dNmzYHjGkpSP8xrzkwm+AOdvVuCXxh8V6AYKEdeddZfKpT33KtOiyfPly2VhVLW3btjUtusyaNct7bdOmjfeqCRLutGnTZM2aNc78xwQ3ePBgOXvESNMSHxAbmCgxKWDS0S52LTfffLN0/2IPb18uTrqOPPJIuWh0iYwfP97ZMcC+i8kT/qIAGjt2rJPCtF27dnL5hIlOfMeYwdgZPny4uu9+29onMi+99JIMOWOoV8hpFp/QGDGOMe8izvcV0P+cc87xNNuyZYtp1WHKlClyzZSpcuCBB5oWPWbMmCHLH37ESW5ft26dN+8dffTRpkWPnTt3enP2UUcdZVpIwYFHNybTqVMH847kI+Xl5YlWrfdP1NbWmhZdiouLE5MmTTJr0Vi1cqV5t5vKykrP9yVLlpgWXeA37GM/2tTU1CRKSko8+3jFugYNDQ2eraqqKk+XsrIy7zjQD9gXlrK77jL/u2VKS0ub/876Crta/gL4DD9hv6ioyEl/2liBDlqx7o/HefPmefbxqg38hS5YwvgeNHfvS9+Tx3QYYM/GNfKYJogX+AzbGAP19fVmS/Ygrv3jEO+hOcYqxkAUUumIMYoF+8WCPGBzDpao4wya2PyAY8FxaGLj0kU+gC6wjbzjJ0o8+oEeWFzges4m8SBT/maxXoDYQs4FSN5IKlrFbnIidZkQXU0UyYVplCLD2kqehJMXbMdiJ+1sJiRMbtiXfz/wH5M1+lejmMHkAz9hG/2qeUIAEI+2cNQoLJJ13FtFb1Ctw+Rul4WR3/fkAiNqcYQxYGNSW3dXJ5HQAOPexrpd7BhN1igI18+Y0TzO/Tb9C3TCdhwT9hNlzOLvoQnswl7Uk41k7Mm1i7EEoAP8T/Zbo1i3JwLaJ5AW5FxX8x6JDyzWSTNIVDbZLl60yLTqgaSSKiFmiz+RYrKA71onAn6iTBSZdIRdO8FBm6i62OLTTtLQBAsmi0yFbtQJCZM8jsX2L44HC4oBDWC7S5cuzTppnAhYUAhhooPtqLGTSkdb9LooMuC71TlI7ITN3bbQi6pLKuA7YgWLvxjVKI6A1R3HoJVvLDf/5jfN2rg6iUQOtnGJBToFjXv8v759+36sGIdd+KqpB+xZP7GvMCcWQecY+A37QeM8LIhv2IdGyWjEo41FF77DJmy7yC8kXrBYJ83YpIXkqF2sYwJxlVSQsDCZaRWHfsIWRMmk09FOQC4m+30NNMPEh77WmqAWNhUcKDpsX2gCH2ETtl1c/bKTtQvbdsziJKYlwuZuvy5hirCgWN8xdl2AGAyqTRjsmIb/8B370DyB9AO7iBvEkIs+iILN6dAgm9gOMsdgH7CPPKmVS5KBbVf2YRMauSqm7TyCV5LfsFgnzaBoxNURoF2sI2khIbqa1MJe1QkKbEa56p1OR+jABBscqyP6w8XJDfoXBYerEyeXthFHQYqlbHI3dMFJkqtCEb6nuqKpBeyjqNbEP6ahD/o12/yQy+CYoW22xx5kjoFt5F+XJyqw7yoXw3+c8LryH7F35plnmjWSz7BYJynRLtZdoPWRuUuoow7UMTq5kLsZizpQRx2oI4kLmfI3H91ICCGEEEJITGGxXsCMGBm/53znItRRB+pI4gJjUQfqqAN1JCzWCSGEEEIIiSks1guY+xYvNu/iS9x+Jj8V1FEH6lgYMBZ1oI46UEeSC7BYJ4QQQgghJKawWCexZvWqVeYdiQJ11IE6Roca6kAddaCOJBdgsU4IIYQQQkhMYbFewPAOcx2oow7UkcQFxqIO1FEH6khYrBNCCCGEEBJT9sMvI5n3zXTu3FG2bq0za4QQQnIB5m5CCMlNMuVvXlkvYPg4KB2oow7UkcQFxqIO1FEH6khYrJNYwzv1daCOOlDH6FBDHaijDtSR5AIs1gkhhBBCCIkpLNYJISQufPiMLLu4j/fdxd1LFxl6+3Oyy2wmhBBSeLBYL2D4OCgdqKMO1HGXvPNkhdz80jly+9OveDcabd36oiwffYK0Nv+D7B0YizpQRx2oI2GxXqBUVFRIdXW1WdNn/vz5sn37drOWPQMGDjTvPqKxsVEWLlyoYj8d0Gbz5s1mLfdJpeO+BH3nUl/YR4xrY3WE7+r2P3xWVt35qBzzo+/Kl3dtl9dee81s0KWurs6JNpalS5dmHJtRYhG24b82yCnoU7zmCnEb05og/65fv96s6ZIcQ5o6Yl5ylddcz9kk3rBYL0AwIQ05Y6hUVlaaFl2QZC+fMFGee+4505I9qW7+WbFihVw0ukQaGhpMiy7wv3dRH9m0aZNpyUw2d+ojoU+ePFmGDRtmWtwS5iYq+DZmzBinJ0RPPvmkdP9iD5k+fXpzgaT5xIN///vfXozjpFETq+OWLVs8+3pF7y55a/W9MumRf8ifJgyRkqm/keffaSvvvvuu2a7Hs88+6/mOosUFZ48YKVdccUXawjfKDX0zZsyQ4cOHqxfsbdq08eLxnHPO8WxrxiLGEuJQu4jb1zdGIjfguPr06ZO2P7LREbaQf9euXWta9MB4RXy+8sorpkVPR/Qv5qU333zTtOjics4m8YfFegFSU1PjvbooBMBDDz0kvU7qKX379jUtemCCQLKdOOFy6dixo2nVA4V6v/6nyKgLL5DTTz/dtOoB+5i8URisWbNGzj//fLMlPhx66KFy8MEHexPP4Ucc6RXU2oXGoEGD5I7bF8g1U6Y2F0iadO3aVW6cM9s7aYTe2ldM4T9iBBOozhXA1nLQqTPkn1tfkL/8frqc/sm/yPkDe8m4ezfLtvffN/9HB/iO8YNxpOP7nqxd85jceVeZjB8/3rToYU8wXBTsTz6xQV599VU55tjj5JlnnjGt0Tn55JO9OMSYRyxC81y6gp+MvdCA3IDjQg7TysXoU/Qt5g/t3AjNr776am/cupibHnjgAc/vXr16mRY9rr/+eu/1m9/8pvdKCg/+KFIBgqshSLL/7+6FpmXP78QlXw0Jss2279y5U350yaXyozEl8pubb/basrEHUm3DJDF7zo3yu9/eLO3atTNbPu6Hn3TbkvdV0zQJXfvL66Rf0+S6/OHl3tU2u81PNvuC7Y0bNzbZfUSOObqzFH/nO9KzZ0854IADPuaHn+Rtjz/+uPzutgWmJTjX/OJq6dZUwIJ0PibvCydGTzz5pLfPLa9slaFnDJFJkybJSy++6PltSWcPtLQNH+t+//vf9+xfMemn8uUvf/ljfvgJss3fXl5RIQvvvseboOfOnSt/aDqRTKYlH0GqfWHyR0GKwnTeTTdK+/btm7dZgvho8W8rKuoq7z72W/netVvkwntul6u+1dnbhtzcErNm3uC9ZvLjO8XFnu+PPfaY/N/Eic2+g6A++kne14YNG+SmefPl/B+eJ3fccYfZkv7vwuwLcXnnXXd570ddeGFK30FLPvqx225fsEDuvXeRrF23Ts763nAZMmSI/NAUjWF8TN7X+00nXIj1x5vsVlU/7RV148aNkw+a2lPlMZDOJtq3vPKK/PzqX3jrYUB/nNZ0sgaC7AvYbdu2bZPVTfHyyIo/ef736NFD+n7jG2p5GIV6//79vfc2JoP4GHRfNhc8+49N3sl8Onsg7DbEJE5e/PqCdD4G2Ze/3fr+ztv/aZ6XSP6RsfZGsZ5Mp04dzDuSjxQVFSVKS0sTixctMi16zJs3L9Gq9f6J+vp60xKNVStXmneJRFVVlWd7yZIlpkWPyspKz3ZJSUmioaHBtAYjnY6wU15eniguLvZs4xX7iUJtba13/GGWsrKySH2dfByIH/SzVh/DDnSHbcRlWP1bAsefbd8m449HgP6AHljw/mM8fWVidFM+RU5Nu4x+MPG0+e97sOuJxK8GdEmcOPupxKKmfgwC7AXF+o5+1dYc2FyAGPSTrGE2tKh7RC695JLmMattH3ms6aTXs29jHm1hQH9NnTr1Y2O9pSWb/IO/8Y992AkaL0Hzjuv+hE2rdTIa8QhNYN+F76BLly4pfSf5Rab8zWK9wLAFLxKwdrGOBI5kq5lU/IkUEwbsaxcWNTU1nibZFnPpdLQTHF6jFulR0ZiQAOIH/YvjwqJVsAN/gaRpF0B/28dRSKWjLTQ0Tgb25MVExU+/5hXrn2zyPUhBFzZ323yA4tEFtij1x79WLLos8DCmoQ1sw/8g2ocFMY4TGrsPnBCHQUvHTNixjjEZpki3BJljXPajBXEI+6nyioaOsO1qDEGT5DFE8hMW66QZTAgY+NoFL7BJxcXE5jJhYRJyMVFAaxdaxAFMejg+7TiCXpj0XEzatmB3gbWNEz81dlUm7jzz7MTUx9/wiiV8QtAS2eRue1XQBYgPnMS4LGQwdnEMLkCco2B1ZR9AI8SP9gmqBvDLRc714yr/WqAv4ttVH7qcm4Cds+MYH0SXTPmb31kvMPA9Wyz+73lqgu8ddnRw4yfA9wJd+U3IPmXXs/K3hzbJK0d9Q7739Y7Savt6uf9XM+SOo66ReyYUyaHmv7UEczchH8f13OFy3nM9Z5P4kCl/82kwBQZuTrGDPvnGFg20E5b/sVpMVtmzrx/zli8403G/hLz9xJ3y0xFfk2ObEnbncx+RhjNmy70hCvVcgbGoA3UMTqa5Q0NHV4U6wJz9qNojYkmuwmKdkIi4OOkpRApax1ZflFNnrPCuqnjLn0rlwv5fEJ6e7hs4pnWgjoTowGKdEEIIIYSQmMJinRBCCCGEkJjCYp3EmgEDB5p3JArUUQfqGB1qqAN11IE6klyAT4MhhJA8gbmbEEJyEz4NhuQsfOKBDtRRB+oYHWqoA3XUgTqSXIDFegHDO/V1oI46UEcSFxiLOlBHHagjYbFOCCGEEEJITGGxTgghhBBCSExhsU5iDe/U14E66kAdo0MNdaCOOlBHkguwWCexhjf/6EAddaCO0aGGOlBHHagjyQVYrBcwI0aONO9IFKijDtSRxAXGog7UUQfqSFisE0IIIYQQElNYrBcwfByUDtRRB+pI4gJjUQfqqAN1JCzWC5Tt27d7iyvq6uqksbHRrGVPupt/4LuG/UKBN1HpYHVE7LkcPy5t72vfGYs6UMfsQHz6Y1RTR9h1NS/BNuZVUpiwWC9QLrroInn88cfNmi5IKscce5ysW7fOtOiChHX4EUc6s49kO336dGm9/wGmxR04Fv/EERcqKio8DVz6Bn3nz59v1vRxab+2ttaLwfXr15sWXWbMmCHDhg1zMvFb39HHLrjiiitkzJgxZk2XhQsXSp8+fZzE5eTJkz2/tQsi2IPt6upq05JfoC+0+wPj1kX+hZ+DBw+WNWvWmBY9YBvjasWKFaZFF8zZiH9SmLBYL0AweSx/+BE57rjjTIsu9957r/fau3dv7zUKqe7UnzZtmvQ6qaeK/WRQHI0fP16umTJV7rh9gWnVBfoj6aIYw0kNCrOgwL+lS5d6hdbmzZsDT5Jhn3hw+OGHexpg8sHE6aJovHHObLl8wkSvkHFpX3OCszp26tRJRl14gUyYMEG9uAPnnnuuN0bnzJljWvTo2rWr5/vVV1/txPezzjpL7ryrLK3uUZ6+MWDAANlYVe2dEGjHTHFxsTz99NMyfPhw1eKzbdu23ljtXdTHi3W812Dm9dd7uQD2XPRjJqAPchBObpAjUABrgXyDcYvxqw1yLeIHJ3wWrafB2BOAHj16eK+aQG/kAxdzHskREino1KmDeUfykbKyskSr1vsn6uvrTYseDQ0NiaKiokRpaalpicaqlSvNu90sWbLE8728vNy06AE9SkpKPPuVlZWmVYfa2lpP96aCwLOPBfvCcYTpB/xf6Gtt2GXSpEmJefPmefpUVVV5+/OTrGMQsC/0I+xjn7Ctje1PaIHY0cb6r9Wffh2hMXRx5bsdp2F8D5q7XfueSfdsYtEPbLqKGasLFowjLeAnYh124TvGatT8O2TIEM+Wf4Em0B77gk41NTXmf0cH/iJfYR92f8hniNPkfJMN0Mjahj7awHfYTs5jUePRgr5FHnaByzmbxIdM+ZvFegGCBIuk6AKbELUmieTiCLZdJEQ7ScO+VmEHmxoFeirw99AYEw/2AU2s//7FTt6PPPyw+cvwYD+wD3s4Fi19LLb4gm2NSd+PvwDQ8Dt5Yre+Q2NtrO/o16C6hMnd1nfEjzbwHf2ZyneN4sj67qKog792zGrEjB+MW/gM29AG4xdaZQN0xN9ifCKnwBbGqT/f2AVt2BZ2fLku0C3+cYr9aQP70Bu+J6MZj9rxYoE2WEh+w2KdNIME6yohAiTDVAkxW/yJ1Bak2lcXoAnspiosssVOyFiQZDUK9CDYyRuThp28UUg++OCD5n9kD2yib3FM2Uz8mcBVTO0+sEAT+K1hO9XEbq96uRhTNjaDTtRhc7e9Aq55Fdni9x19YNEojoAdYy4Kdn/xmHwlVgNogzEE+4jNbPRvSUfkAdiF/+hn7A9tQcHfwT/ro3aBboFNm1dcFbs2zlMdv0Y8wn8sLsC8Ad9dzdkkPrBYJ83YBIwEsHjRItOqAxKhq4RrbbuYOJFkoxRyqXSEBnurQN+boIhBH0Av7cnj5t/8JlRhGgZbOKJgcYEt7FwAnYNO1mFzt/9ExgUYB0F9zwZbsGsXkRjT0MYW1GGK3DCgmIb+LmI+KjhmjPUo2gaZY6Ax4s/FCSOA/+hDF58gAWvfVYzb8a8d4yR+sFgnzdgEDLSLdRSmrhKWS9vQJEpRra1joQId0Q+uCiPYdjXhobBzFZ8ARW+QGM0md8OuqyuaAEWYqz6F7rCNV038Y9qF/UIhSG7EmIySf1vCjk1XfQi7mFNd2Uf8TZwwwayRfCZT/ubTYAoMPAkCTztwQfv27WXQoEFmTQd7p74L2xZoAvv5jNYTD1yDfkB/uAC2O3bsaNayI52Obdq0cRafoG/fvs5iFHZh3xU9e/bco081YxG6wzZeXeHafrbkyphuCYxJl/nXjs10fRhVR9jFnOoqRhB//qfXkMKExTohhBBCCCExhcV6ATNi5EjzjkSBOupAHUlcYCzqQB11oI6ExTohhBBCCCExhcU6iTUDBg407+LLfYsXm3fxhTrqkAs6xh3Gog7UUQeOaZILsFgvYHIhkRJCCCGFDOdqwmKdxJp8eeLBvoY66kAdo0MNdaCOOlBHkguwWCeEEEIIISSmsFgnhJA48U6NrFk6W87r1lE6d+4iva950mwghBBSiOyHX0Yy75vp3LmjbN1aZ9YIIYS4Z1dTnf6QzPjJVXL3u6fI4FO+JeOuOVe+3NpsDgBzNyGE5CaZ8jevrBNCSAz43yu/l8uLfyIPHPp/cvuS+XLrteEKdUIIIfkJi/UCJhfuMM+Fm3+oow4FreOHm+S+KTfIo/8bLFf98nw59fD8rdIZizpQRx2oI8kFWKwTQsg+5n/PPia3rn5dDvzWMfJG6be8j0M7dz1H5vxtm3xo/g8hhJDChN9ZL2D8Z+v+nzNOPosPsi3VmX+6bXtzX4B+7An92JNstgXxA6Tbtmd7g/T67yoZeM0WGTFllnz+sB3SdWeVVD36kNy4qr1cu+ohufALn/L+J3JzS8yaeYP3Gt6P3WSzLci+AP2Ivi2MHyDdNvqx7/zIxp7/b0h+krH2RrGeTKdOHcw7ks8sXrTIvIsvq1auNO/iC3XUIf90bExse/AyL5+mX76UGP3g84mnZg9KdDrme4nl2z80f/tBYsfKqxLf7NQlccaCZxO7TGtLwGbcYSzqQB11oI4kLmTK3/waDCGEOOFTcuSw+d6VkvTLM7Jg2EHy4YcJkU+0M38HWstBnQ6TjvJfqd2xU3aZVkIIIYUHi/UCZf78+fLmv/9t1vSZPHmyVFRUmLXsGTBwoHm3J9OnT5f169ebNX0aGxtl6dKlUlfX8tfBcuHjyXQ6ZgLHv3nzZrOmz8KFC73Foq0jbOMYNPHriBjX0ecQOa7LF+STH/xd1m2s91oQ2+ueqW8q9zvL0N7Hyie9Vj3ge3V1tVnTBZr7+zWZbGLRAr0x9jE+tcFYt7a1YxH9qZ2voui4t8hWR/QB5g8X+Sc5L2jqqDXvJQMdMWdjIYUJi/UCBInw8gkTzZo+mJRmz7lRDjzwQNOiCxLtNVOmmjV9MEH069dPzh7hvgjfvn27t78wxQeKLEwK0CHIyUS2zJgxQ7p/sYd6wevnotElziagt99+2+tDFyd16K81a9bIeeedF6rvUtNaDvnOaJkx+BNy7/U3y/0vNja1bJFX1v5F/jroEhl+0sHm/+mBmBs7dqyC7x9n586dXr+6KFoaGhq8sT9lyhTTogtsjx8/Xl2XtWvXSr/+p3jjVss2cgfsofh0deIVBBwP4gn+aICchj4YcsZQ2bJli2nVATGJ2ESMaoM+wLznAmjscs4mOYD5Oswe5ML3Hkn2VFZWJlq13j/RVIyZFl2Ki4u9RYPk7xPW1NR4vk+aNMm06LJkyRLPflFRkbcvTWAP2peVlXn+Yz92QVtQYKekpKT5b6H1vHnzElVVVeZ/fJxsvpfZVBg1+4n91dfXmy16wG/Yx6s28B9+oy9ra2tNazT8OtpYLC0tNS0Rqa9MPHDFEPNd9q6J0666J7Fi48uJf/3rX+Y/tEzQ3K3uexJW91QxE/U7wnaMhhkzQbG5Ef4jfjTx55ZMYzUojzz88B55BHbRnzgGbd8tGEfwHceCfWOfdv8aORm+wyYWvNcEvsMu+taP1nfWrR4utJ82bZqnsUbckPiSKX+zWC9AkNBR4Lm4aQXJBEmlvLzctETDn0iRBOG3i4QIe3biw2sY+8k64m9RDEEDFKGYHOyEZhe0YRv+D/5vNseDQgh/j/7028Y62v2FUpQJCbZgF7rjvTa2YB81apRp0cNO0IgbjZhJ1hEFI3x3MYnu3LkzZXGRiTC526Xv0B22U/muURzZmHfhu8uCHWMdsQj70D+KfasjbMBnjCPEC2xb/1FUZ3uianMY/LS50b/APvoB+0iXw4LOMfhbmwdgN1ufMwG70CfZtkY84vjhO7RwwcgRIzzfSX7DYp00g6SIpIIE7KJY17664E+kriZoJFo7yWWTbJN1tEUQFtiFJmjDhOpiEgLQG7qkmrCxz6gTEmzAFmyiH7T612InarxqA12s31FJ1hE62BNIF5882BOloHEZJndb37VOZJJJ57tGceTX3cWYslcyEfPa2sCezWU4hmz9T6cj8hnyDWxjH3Y/Ya5U2xMWu9gchj6F/aAEmWP8uQXj30Us2pyc6mKDRjxqz3t+YNNqQ/IbFuukGTuBIkFqF+suri7YRGoLLiRdTaAD7CLRhpmE/CTriOQKWy6Kt6DguKAVJhH4ojEhATvpYfLXBlePYDtMUREU63fU2Eylo40haO0C2IX9IPEUNne7GleWVL5rxSJ0x7hFoacNxrQtWDVO8lIB7eE/9oFjCUsQHaE7cj76IVWhmgnkjWz88tPSHINcCQ2wuBj3wM5L6foxajxCY9h3VUzbOES8kPyGxTppxiZuoF2sI5m4uBIFkLBcXNFFokUBF8Wuto5xB5Of5gmZZWFTwYg+djVpYzJ1ZRvjytVkjRjFmA1SOGWTu1GouyrWw/ieDehPm880sWMa9l1pA2z+wWs+0lJutMfvKj4AbCNGXMxLAPZx8cJVHyIGB5wywKyRfCZT/uYvmBJCSJ7A3E0IIblJpvzNRzeSWLN61SrzjkSBOupAHaNDDXWgjjpQR5ILsFgvYO5bvNi8I1GgjjpQRxIXGIs6UEcdqCNhsU4IIYQQQkhMYbFOYk0u/KR2LkAddaCO0aGGOlBHHagjyQVYrBNCCCGEEBJTWKwXMCNGjjTv4gtv/tGBOupAHaNDDXWgjjrkgo65MFcTt7BYJyQiTKQ6UEcSFxiLOlBHQnRgsU4IIYQQQkhMYbFewPBxUDpQRx2oI4kLjEUdqKMO1JGwWCexhnfq60AddaCO0aGGOlBHHagjyQVYrBNCCCGEEBJTWKyTWMMnHuhAHXWgjtGhhjpQRx2oI8kFWKwXMLxTXwfqqAN1JHGBsagDddSBOhIW64QQQgghhMQUFuuEEEIIIYTEFBbrBUp1dbX8/KqrzJo+sL9582azlj3p7tTfvn27rF+/3qztW3LhsVrZPPEAGtfV1Zk1fWAbcWLR1hHx57evgV/HiooKZ/pAe9h3BcaOK99b0j3K0zcaGxs9XfDqAugC29qxCJsa+dBPLjzFJI65MXnu0NRRa95LBjrCtsucQOINi/UCZerUqfL3TZvMmi5Ihr2L+shf//pX06ILiozBgwfLQw89ZFr0QcIdNmyYt8SRpUuXyvTp052esKxZs0aOOfY4Z/t49tlnvThxNQHBLuy7mDzB1VdfLdOmTTNrujz55JMy5IyhzrSfMGGCjBs3zqzpsqkpr7jSvb6+3tPl97//vWnRpV//U2T8+PHy/vvvmxYdamtrpfsXe8j8+fNVTzRQwE2ePNnpCcy+ADl+zJgx0nr/A0yLDtAIc8esWbNMix523nvggQdMiy6Ys1euXGnWSMGRSEGnTh3MO5KPVFVVJVq13j/RVGiYFl3mzZvn2W+aWE1L9qxaudK8201DQ0OipKQkUVRUlGiaAE2rHvC5tLTU8x/7gFYtsXjRIvMuPDiempqa0FrBL/gHP4uLixPl5eWerXQk6xgEqzX2gT51gbVfWVkZScdUwH9oBH0yaRMGv47QHL4vWbLEtOgCv+F/GN+D5m6Xvrekezax6MeOzyBjMyyIQ9geNWqUadHD5kXEvEY8QkfkDegMu9C8rKxMJe+GBTksVT4OO6ahC47BHg/iVBObb+CvJWo8Wmz/+m1rcfNvfuPZ1taDxItM+ZvFegFik8rCpqSoDSYK2NYq7pITqZ2oMalqg8IFEwTsY8IIOqEmT0hIqJMmTfrYYm2nWrAt7CQL/6BDkMk62wkJ+7DxgmPQKDL8wJ49+cKEpI09MXUVj7ZfXRRImPThO/o0KGFyt9Xdhe9W91QnA1GLI8RMNicyQbHxHkb3oNiTAfge9WKDX0fojViEbSw4hihFo7WXvNhck25JPqYwxTq0gS6wgzyvHZe2X5ML3qjxCOArbEMjF+DkEfZdxDuJDyzWyR4gISIZusAmRK1E60+kmPhhW3sSxaRmr7jgNcokaotPLMkTHTTHMSQv2H/UiRs2sA8cA5bkyTrqhATNrT7akyiOHTGJxUXhaGNS40psso7wHbahvQvsyWnQ+AiTuxEfsO0qF9h4TPZdoziyvrvS3fru4qIA4tDGe5SYTKUjxo+Nd6tPNseAmAiTw3Ac/nwTBvgM2/AXJwPZ2smEPUmCNsloxKPV3IXvwOWcTeIDi3XSDJIqkoqLSQhJN11CzBabSK3fSOqaYKKBXSTDfPiIMXmyxvEBjQnJTnjQKurJRTK26EWBoH31yH8lNurJQCodbQy5GlPwO2jch83d9iTMRZFhfUef+tGIRWB1dzFu7Uk37GvHOoBNe5U627jJpCP8hz7QH/vQzpta2LyOBf5qj30ArW0cprIfNR7tvOdKY5dzNokXLNZJMzaBA+3vCGPSRFKJWhClAlcVMLlpJ3MUtlii2NXWUQMcD/oDiV4TO/EhjrTBPRSIHxeFoz0ZcHVChth0NVnbojTIuAqbuxEn6E/NE2w/Nie4KHiBvfqrDca0Pdlw8XUYAO1d+e8HRd6+KvRayo3QFrndxZxhsXOeqxi0Me7qqjr879Kli/rcR+IHi3XSDAa8TYzaRabfthb2qkecE1Uci/VktK5musS1joihqHGUTkcXse8naKGRTe7e275rx6KLIoxjWoc46NhSbEfVEePH1YkAgP0Ft91m1kg+kyl/89GNBUabNm2kffv2Zk0X17YJiQJiyFUcuYx90LFjR/NOn1z2Hbi2T3Ibl7ENMH5cj8927dqZNVKosFgnhBBCCCEkprBYJ4SQfc5r8rfrz5TOnTt+bDnh52vkVfO/CCGEFB774bsw5n0zmCC2bnX3M+OEEEI+4n+1S+QXt4icMXGYfOOQVrsbP3xS7vruFfJkyT0y/zuf293WAszdhBCSm2TK37yyTggh+5Rd8vYbh8tplxd/VKg38eE/KmXpP/vK1752uGkhhBBSiLBYL2DuW7zYvIsvq1etMu/iC3XUoXB1bC0HfeVk+dZhrc06+I+8tPEJ+efwATLg8I8K+HyAsagDddSBOpJcgMU6IYTEjQ83yfqlr8nAr3aTo0wTIYSQwoTfWS9g/GfrI0aONO8+fhYfZFuqM/902/bmvgD92BP6sSfZbAviRxhgw7+vk496RS7+0bty8vyh0vnNl5q3ITe3xKyZN3ivQXxMbgfZbAuyL0A/om8L4wdIt41+7Ds/srHn/xuSn2SsvVGsJ8MfRSoM+MMfOlBHHfJPx8bEtgcv8/Jp+uVLidEPbjX/3/JWYvOd5yaOv+qxxL9MS1BgM+4wFnWgjjpQRxIXMuVvfg2GEEKc8Ck5cth870pJ+uUZWTCsk/n/Bn4FhhBCiA9+DYYQQmLEh8/cKN+d/Am5+N7LZEj7cDeXMncTQkhukil/88o6IYTEhq3y9Kq/iIw8TQaFLNQJIYTkJyzWC5hUN8TEDT5WSwfqqINzHd9YK0tubS0DenUS/4Mc8wnGog7UUQfqSHIBFuuEEBIXDv+hlD73/+TyL7c1DYQQQgodFuuEEEIIIYTEFBbrBUpjY6Ps3LnTrLlh+/bt5h0hex/EOBZX1NW5vZHT5fhx6btr3ZlXSJxxGZ+M/cKFxXqBMmfOHHnoD38wa/oMGzZMbrnlFrOWPQMGDjTv9mT+/PnSev8DzJobkBgrKiqcF2XZAg3gXxDS6ZiJzZs3exqvX7/etOji2v66deukX79+qhOcX8dp06bJ5MmTzZou0ObwI470Xl0wd+5cZ75v3LhRPv2Zz6YdN9nEoh/osnTpUrOmC/LW9OnTzZouiHWMWS2sjsgBCxcudHqCtC/A8eDYqqurTYsOiB3/3BE1Hv1gTM2YMcOs6YK4dGWb5ADe09aTyIUf1iDZU19fn2jVev9EWVmZadGlsrLSs4/XqKT6wYolS5Z49ufNm2dadIE+sI19YCkvLzdbgoG/3xuUlJR4/hUXFydqampMa2qy+eGPhoaG5n1o9GUy1n5RUVGitrbWtOoBm/B90qRJpiU6fh1tHFZVVZkWPaANdIE+YQiau63vLcVNNtj8kk73qD9CA7uw7yJmkBNhG/poY3OKVt6yOlqfES9hc1U6EH9Y9gXYL/TH8eC4NOcpOzf5YzNqPFqsbRexY8eUqzmPxINM+ZvFegGChI6B72Kys0UGCkgNkhNpqmSrRXKRjvfZFDN2ksGC9/AVCyYdJHLoD7tYohb20MPur7S0NK29bCck9KfLgh0xCP+xDxfFgS1KtYoYv47asZ6MHadhdA+au137bnVPdSITtThCjNtx5QLYhe8uTmQ0C3a/jvDVjlO8RvUduQS2sFitscBv9C0Wm8O05hGbf7E/7Bf70+wDxKLVx59rosajBWPJdS5wMWeT+MBinewBEgoSloufMM40SWeDP5GmS7ZRwYRgJ2gsmDDCFNHJOsKendAw6dmJztpPteCYsgVa2KtrWLDfZH2iTEiwBf9g20XBbk/ARo0aZVp0ge8oADRiJllH6zs0d4EtAIL6HiZ3W99d9Cn8TVe8aBRHNs9onYT5ge9dunRRi5lktAr2VDpCD1vsZjp5bwn8nc1h8NPmMGs71YJtyfsLMsfYIt3awX40i3RgLwqkGksa8ZjNiXUYevfunXIskfyCxTppBkkLSQXJRbtYRxJEQkSy1cIm0kzJNlswSdgiGrbDFumWsDpiH5iMsKAfMCFqJHn/8UAn/wlT1AkJmrss2O1k7aL4gs6wjeIlKql0tEVMNrHTEragDnoyEDZ3I07gu4uiNJ3vGsURsCdhLnSfM3u257tmLvNj4x2v2ZJOR/Sl/6sxQWMnDNhHcg5LNXYz5UbY8F/Bzzb/toSdO7CkujKtkRth21Ux7Z+zSX7DYp00YycJJBjtYh3JBLaRwLXJlGyzBcUEbGKigR7Zoq1jVFCkY+JAX2gCjWzBrtkPlgGnDFD32WKLFxd+25MBze/W+rEnA0EIm7ut764KgTC+h8UWMRonYclgTCMvuNTG5QkqgD725N1FTg5CptyIPIXYcFWkWxAf2nOHH5fzHrBx4lIjEg9YrJNmkCDt5KBdZPpta4PC2lWyjUrcinVLlBOQdMAmJiUXthfcdpuTq4AA/sK2qwkP8elqskbcBz0RyCZ3Y8y69N1VnwJXutsxDW1cFknw3XUR5mKsBiUOuREx4v+UURv0oat5D8D3q6680qyRfCZT/t4P/5gHwzTTuXNH2bo1no+rI4UFfgpa89FahQp11CHuOuZC7mYs6kAddaCOJC5kyt98zjohhBBCCCExhcU6IYQQQgghMYXFegFz3+LF5h2JAnXUgTqSuMBY1IE66kAdCYt1Emv4XUIdqKMO1DE61FAH6qgDdSS5AIt1QgghhBBCYgqLdRJrcKc+iQ511IE6Roca6kAddaCOJBdgsV7AjBg50rwjUaCOOlBHEhcYizpQRx2oI2GxTgghhBBCSExhsU5IRHinvg7UkcQFxqIO1JEQHVisFzC5kEh5p74O1FEH6hgdaqgDddQhF3TkSQ9hsU4IIYQQQkhMYbFOYg3v1NeBOupAHaNDDXWgjjpQR5ILsFgnhBBCCCEkprBYL2D4OCgdqKMO1JHEBcaiDtRRB+pI9ks0Yd4307lzR9m6tc6sEUIIyQWYuwkhJDfJlL95Zb1AqaiokHMcnq0vXbpUFi5caNb0aWxs9Paxfft207Lv2Fd36tfV1Xn96JLJkydLdXW1WdPHb19bR+gD+4gVFyD+sLgAcQ3fXcW3S9+t7q58nz59uqxfv96s6QJNkLdcjGnYdaG5zYWu4jwKcXyKicu5A/nYxbwHHWEbsU8KExbrBcrVV18t7zY0mDVdkATPHjFS3n77bdOSPalu/kGyHT9+vLePBkfHAFB0IPGi8MD7oNjJAEWoywn02WeflSFnDJUxY8a06F+2N1Ht2LFDxo4dG+r4w7BmzRrPviudZs+5UW6//XazFh2/jq+++qoXg660ge+33HKLWdPF+u6qoIbv9957r1nbk6g39G3YsEEmTJjgJGZ27twpF40ukS2vvGJadIHmWgW71bG+vt6z269fP9UTaxSHOClyFSMt4c+/WseVau6IGo8W6IR5ddOmTaZFl5tvvllefvlls0YKDnwNJplOnTqYdyQfqaysTLRqvX9i2rRppkWXSZMmefabJhHTkj2rVq4073ZTW1ubKCoq8uzjOLSB/bKyskRxcbG3Dyw4nqbEbv7Hx1m8aJF5t5uamprmv8VSUlLi2YS/Gpr4KS8v9/TAgvfpSNYxKFZvLHivjdWqtLT0YzpqMG/ePM8+9qOBX0f0JWwjPlzQVCyF9j1o7ra+Qx8XWN1TxXu2sWixMYMxpQ3GOcZ+ly5dMo75bEEugO8aucuvIzSxOQu6RPUdf2/zLBbYRp/C7zB5IOyYTpV/oZlG7sExpdI/ajxakMNg20WenDFjxsf8JvlHpvzNYr0AQXGBROyiOKqqqvKSypIlS0xLNPyJFEnQTiCaSQs+YyLyT05IvNhHkEkvlY74O9iFDvbkxS7+iU+jeIcudhKC36l8jjIhWd3hdxA9woLJGb5jQtIG+sJ3rYI6WUf0L3zXOhlIprGxMXHDDTeYtZYJk7ttQe3Cd3sygHhMRqM4clkY2ZOBVL5HxV8wRtU9WUfYtrpoFbjw0eYwf37Ee+wLFwgy7SfIHIO/T1Wgw7ZGfgR+3ZPnDo14tDGDMeWCkSNGeJq7yL8kPrBYJ83YSRTJ0UWxjoSrWdTZRGoLRixaE3SUAj0b/BOf3ScW+KAxKdmiF/onFwJRJyToAtuY8LT1gT347GoysgV18iSdDakKJPitcTLw4StrEwuvGurlX2/5ekli4VPbEuvWrw9c2IXJ3TYXaJ3IJJPuZECjOILvWrqnwo4ljZhJxsYMlii5LJ2O8Nnaz/RpWzbAX9hEroR9aIQF73FxIgwYly4LdAv0TleoA414hH1o4CJ/2XHq6kSAxAcW66QZW7hoJ0SgWRT5sRMzFq1C3Rb/rgv0TNiJD0lYqz8wYeK40A9BC7ygQCfYhWbawFfYdlV87Y2TgbDFyh588GTigTG9E51OvSpRunZbU0NdYs3kUxLdOo1I3Pr3twLrEjZ326JUO1aALUpd9amrfANsgQf/XeRKm39c2rcFKnRyBXyH/shhYXKzzemuCnQ/mQp1DeA/7OPVBTbOw+hLcpNM+Zs3mBYYuLHs2mlTpX379qZFD2u7b9++pkWHpolT+vfv792Y1bFjR9MaDdjBjWpXXXWV52+bNm3Mlr0HfBg0aJCMGzdOrT969uwpa9eulTtuX2Ba9IBO99+3WK0P/HTt2lUeeXi5WdNn5syZ5p0+p59+ugw9Y4i88cYbpiULdtXI1g3b5ICjT5JLvnVkU0MH6XfpBfJNqZMnqrfK5s2bd/8/Zc466yzPd4wxbTCmrrvuOme+Q/eJEy6Xd99917ToAd+nTJkiRx11lBNtMIaQz5DXXNm/7bbbvPF6xBFHmFZ9kLeQF5DDwuQF/B3yL3xEDnQxH1kOPvhgWbvmMfV5yfLaa695cYjjcIGdV13kXZJDmKJ9D3hlvTBw8TUYbTQ+onQNddShoHX84M+JW7/XI9Gp69jEXS81ek0fbr4zcVrXnybm/esDbz0IuZC7GYs6UEcdqCOJC7yyTgghcab112XMb66TiSc+J4/+6TnZ8eFmefzuRyUxcoicdVRr858IIYQUIvwF0wLG/4MV/p8zTv4hiyDbUv34Rbpte3NfgH7sCf3Yk2y2BfEjDLAxYuSZsu1vC+XG0gfkoeq3peuon8vCXwyV5594Qr729a97/w+5uSVmzbzBew3iY5hjBum2BdkXoB/Rt4XxA6TbRj/2nR/Z2PP/DclPMtbe3vX1JPg1mMKAH1HqQB11yEsdtz2YGN2UT5FT0y6jH0w8nXgnUVfxi8Spp85OrKh/J1Hz0E8TF3Zr2vbVSxIrav9rjLUM7MUdxqIO1FEH6kjiQqb8za/BEEKIK44cJgu21nlXS9IuC4bJl1+7V2788WL51LlD5Nvt20nX7/xKfn3nJfKtHY/K3EdelF3GHCGEkMKDX4MhhJB9zK7HJ8p3zlsucvUy+cPY7rL7W+pPyrKSi2TK8bfL335aJJ/y2jLD3E0IIblJpvzNK+uEELKPaf2NQXJm94Rsmj1Hbvv7jqaWBtny4P+TWyq6yNCvHBOoUCeEEJKfsFgvYFLdEBM3Vq9aZd7FF+qoQ0Hr2HqwjJlTKlPOrJNfnfEl6dy5q3zrd21l+H3zZNoph5n/lB8wFnWgjjpQR5ILsFgnhJAY0PqEs2X0rBUffZf9T6Uy9usdzVdiCCGEFCos1gkhhBBCCIkpLNZJrBkwcKB5R6JAHXWgjtGhhjpQRx2oI8kFWKwXMPyRBUIIISTecK4mLNZJrMmFm39yIZFSRx1yQce4w1jUgTrqwDFNcgEW64QQQgghhMQUFusFDB8HpQN11IE6krjAWNSBOupAHQmLdUIIIYQQQmIKi/UCZfv27VKzebNZ0wf2q6urzVr2ZLpTH/vAUshsburDxsZGs5aebJ94UFdX51Tj9evXO7VfUVERSJ+g+HWE9lhcoe27n33pe9Snb7iMGZe6wGeMJy38Oroep/kGtLJ6RY1HP4h5xKcL4K8r2yT+sFgvUGbMmCELFy40a/pcccUVMnbsWLOmD5LW4MGDZfny5aYlXmDyhL4uCyLQ/Ys9ZPz48c6Kurlz53o6uyoEZs2a5cWKC9AHQ84YKitWrDAtuqAgPe+885xoD5vw/fbbbzcturj03bXud911l1x00UVmTZdNmzZ5Y8pFvK9Zs0aOOfY4JwWXHaeaJwM2h2lcdIkTyMnQCnOgNlOmTJF+/U8xa7rA3wkTJpg1UnAkUtCpUwfzjuQj9fX1iVat90+UlZWZFl2WLFni2S8vLzct2bNq5Urz7iOs/ZKSkkRtba1p1aWmpsbzf968eYlJkyYliouLPd2CUlVV5fmIBX8Ln8P8fVAqKyub95FJi1Q6BgE2rdYNDQ2mVQ9oDPvQxwXou6KiIjXf/ToiRuLme9DcvS99zzYWLTbmNfJLMhij8Bvxrg20gF3Y18hbfh1hD3axQB8NrBbQGq+YL1zl22QQnzgO7NPGkkb+tLEDe/ZYosajxWUuw7HDtqs5m8SDTPmbxXoBggIUA99F8WgTPBKsBv5EisnO+g77GgUYbGBiQIItLS31il7Y9y/YVzYJGLaRwDFBW1vYh9ZkasGJATTHgmNJRZQJyU5w0N4FmpNxMrYo1ZrkknW0vmvEYjLZFNRhcve+8l2jOMI4deW7LbpcnAzYohr+R/U9WUfYtrlG03eMf8QK7GLBPtCmoT1s2PyL/OLPlf79YVvU/dm5A/b8tjTiEfbQr7DtAuQv+O4iR5L4wGKdNGOTCopGF9ikrnUFxiZS+G0TuUbRCBvQAfbsgnX4j4kDBbDmVSTY8u8Tr1jX2gfsWNupTgaiTkh2otM+0QDwHbahvQsQ67CvMdEl65hNQR2GsAV1mNy9r3zXKI5c+45cA99dFEf2U7eoOTiVjtp50g+0gN7+CxrYB44nLCg+ky+MQG/EDLYhz2jlRmgCu9ZfF/FocwziUhv4a7Uh+Q2LddIMkiCSChLs4kWLTKsO1rbmBIpE2lIhGhYkPyRXLPAVCTY5gYchrI44BpvctY4J4BjsRJ3cB1EnJL9trUnUD/yF7Wwm/pZAkQHbGsVLKh3DFtRhCFuUhs3d+8J3jeIIuPT95t/8xvM9akGdDnulNGi/piKTjvbkGv676lu7Dyzpxla63Ijjt/kXY97VFWN/3sI+UxE1Hu3JVzr7UfHP2SS/YbFOmsEEhysaQLtY99vWBAkdk7KLqxYaZKsjJih8XK05UWFyQj8guWsDP9EPLgoY+N27d2/PdxfYwsIFtijVOulKBpqg4AhC2NxtCw1XYyuM72Gxurv4ugrGtC2oXRS7wNU4tVj/XeZNaIO4T7cP7TkmLJg7oIGrsQnQjxpfa0oH7CM3kvwnU/7eD/+Ye02b6dy5o2zdqndXOYkPuBP+0EMPlfbt23s/tKD5c9B4ekDbtm0924WEto5xxj4lw0Uf375ggZw2eLB07NjRtOiBp57U1tZK165dTYsuLmMfmv/73/8O5Hs2uRtP++jWrZu0adPGtOgRxvdsgO6HHHKIuu8Y098pLnYaM4jJphNgJ/EeFwohN7rMiQBz9rrHH5fRJSWmheQrmfI3H91YYGDicZVUMOlo2169apV5R6KgpSP611X8tGvXzlnhgmJOo+hKp6OL2LfArquCEfTs2dNJoQ5S+a45pqG7K9+1YiYdsB8l3pkbdYiqo8ucCBCDyI2ksGGxXsAUytVg11BHHagjiQuMRR2oow7UkbBYJ4QQQgghJKawWC9g8H1CQgghhMQXztWExTqJNQMGDjTv4ksuJFLqqEMu6Bh3GIs6UEcdOKZJLsBincQa3kSlA3XUgTpGhxrqQB11oI4kF2CxTgghhBBCSExhsU4IIYQQQkhMYbFewPBxUDpQRx2oI4kLjEUdqKMO1JHwF0wJISRPYO4mhJDchL9gSlLCx0HpQB11oI4kLjAWdaCOOlBHwmKdxBreqa8DddSBOkaHGupAHXWgjiQXYLFOCCGEEEJITGGxTgghceDtZ2XVnNFyfOeO0rlzNxn88/8nf6l7z2wkhBBSqLBYJ4SQfc2Hm2TJTy+Qy545WX7z1BbZ+tRcGfn6TTLigjny2H8+NP+JEEJIIcKnwRQw27dvl/bt25s1QvY+jY2N0qZNG7Omj8sY1/T9f8/+Vr4z+EH5yj2/l2nf+qzX1vjkTBky/Fb59LUPy9ILu0lrrzUzQXM3fAeutM8V3ZNxHY+ERIFzdn7Dp8GQj7F06VIZPHiwWdOnoqJCWu9/gFlzA/YxefLk5sIjblRXV8v69evNmhtw/GPGjHGmAeJk2LBhzuxPmTLFOwYXYGI7/IgjnfXBihUrpE+fPgra/Ffe2PwPeabp9b0PPrp28qetB0r/YxJS88xLUm/atIDv/fr1M2u6uNZ9zpw5zmJm3bp1Xp/iGLTZvHmzlxPxqg3GKfyuq9O9yIYchjwb1xybLTguxJB2P9t5z0X8uJ6zSbxhsV6AIPHOmDFDjjzySNOiCyaMIWcMlVEXXmBasifVnfrwf/r06d4+duzYsc+vhKV7rFZlZaX063+KV+y6KlyKi4vlzrvKZPz48Rkn1GyfeHDcccfJ8ocf8QokF3zta1+T2XNu9CZP7ceT4QrU0DOGyKxZs0xLdPw6HnHEEbKxqlo2btxoWrKltbT59Gflk/KuvP/BR99R/9QBn5D/NWXoxm31oj31W99dxGVLukd9+sbxxx/vxYyLord79+6eLpddeqlp0aNTp07S66Sect5556kUv34dUaiD4cOHqxbsb7zxhpdncWKHYjGs33F75CD8nz9/vvQu6uPFD+aOqPFoQYF+9dVXe/Oe9pwEHTFn9+/f37SQggNfg0mmU6cO5h3JR5qKyESr1vsnmga/adGlpKQkUVRUlKivrzct2bNq5UrzbjdVVVWebfhfVlZmWsNTU1OzxwJNlixZ8rFl0qRJzQv+TyoWL1pk3n0c/E1TQe35i9d0NqJg+xO6NzQ0mNY9SdYxDNAZ9l34Dn+hC5ZMOmaL1UbL92Qdre+RqX8ocdkXOiQ6nXpV4uYn3mhq2Jn4+8O/TJzWpUOi6+V/SLy2+3+1SJjcreZ7CjLpHiUWAWIGOQBj0gXz5s3zfEde0Ab5C7ZLS0tNS/Yk61hbW+vpggXvtYAO0Bp+wzbyYro8k0y6Me3PvVgQJ8m5F0uQ/BsU7AfxjuNAH9tjiBqPFuRf2NbU3jJt2jTPtlYeI/EkU/7md9YLEFzpBef4fsLY/pxxqishYbY9/vjj8rvbFsgjDy+XQYMGeW3Jf+f/6eR025Lb33//fUGgXjS6RE7qeaL84Nxz5XOf+9zujU2k+7tU+9qwYYPcNG++974l+p18shzY7kDvfe/evaVb165p9wXSbavZvNm7Aour1Mn+p/LREmQb2mH/2l9e5/l74YUXyA/PP795WzLpfEy3L2j/+Lp18vTTT8uPxo6Vdu3amS27SWcPBNnXtdde6/n+kx+P864QpvMDBNmW3H7Dr38tHTp0kGXLloXyEaTbZttxZRqfnlzzi6u92AAf/c090u9Tf5U+45d566lpL4PmPiwjD/yHfLXT23Ltj6+W+2oaRLr/QCZ85VmZc/cr8sOFq6W0/6He/0ZubolZM2/wXq0fIJX/+MgeV01Lr/ulHHP00WZL+mMG6bYl7wsxM62pX3Gl+g9/+IPZkv7vwu4LV3nPHjFSbph5fco8ALLdl/X96M5Hy9ixYwL/nZ9MfrzXZB95bO2ax6Sutta07ibT3wXx47c33yy/nj3be/9/EyfKJb5PCML4mGpf27Ztkz/+cbmsbcoF+OSkV69ezTEPMtkDdrsd70HAVeqDDz7Ye4+8k2p/mfZlt9mcf8zRneWBBx6Qnj17pj1mEHabjccrJv1UvvzlL5st4Xy0pNqXP4eR/CVj7e2V7Enwynr+Yq/suDhDx5UL2Na4auTHf0UEtoNe1UkH/t5e0bGLi6shqcDVIvvJAK4Wae4XfQq7ma6wZwv8tLZdAC2gi7bfwOri6qqUmyvUryYeu6pvolPfKxKL/rLRi1FtoDU0h/YuQKxD91z23UXMwHfNTx+TwViFbSwu8ho0sfkYr5hTwuLPvXZxAY7fXvFGrLjILzY3uopFO2eXl5ebFpKvZKq9WawXGCh2kcRdJC0kbhe2kahgN5tJIY5AH1u0ayd4W5jCvja2gHExaWCyhm18PO0CNwX1bqwueoVRXWLjnHMSXTsNScx+akdi4sSJ6ifAFn3fPwJxjhjPRd+BzWcusAWeK21g30V+8eMv2uOK7UOXha7dh4sTL+Byzibxgl+DIc3g4zrcNIiPAbWB7R49ekhX30eVGuBGINyclW+PVMPNTi6OCTdrtm3bVr0fAL46cfjhhzuJH3uzY9++fb1XTRBD8H3cuHGmRQ/04+233y7nnntuxMeq7ZLXV86Xex5YJjet7SI/XVgqlxUdKX9v6s+XXnrJu3lQG9wUd8stt8iECROcxaIr363u+Fpfx44tfz0oLLhRc/Xq1XK++UqZNi61AfD/lVdecTKe/LjKYxpg3B966KFOH3eIvHLMMcc4ybfA5ZxN4kWm2pvFOiGE7GPeW3qBfOnyVfJf6SY/vPb/5HuDvy29jnT76FNCCCHxgcU6IYQQQgghMSVT7c3nrBNCCCGEEBJTWKwTQgghhBASU1isE0IIIYQQElNYrBNCCCGEEBJTWKwTQgghhBASU1isE0IIIYQQElNYrBNCCCGEEBJTWKwTQgghhBASU9L+KBIhhBBCCCFk7xDqF0wJIYQQQggh+x5+DYYQQgghhJCYwmKdEEIIIYSQmMJinRBCCCGEkJjCYp0QQgghhJBYIvL/AUEWjv0RIfHdAAAAAElFTkSuQmCC"></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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<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>
<br><hr><br>