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</div><h2>HL Paper 3</h2><div class="specification">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Let <em>c </em>be a positive, real constant. Let <em>G </em>be the set \(\{ \left. {x \in \mathbb{R}} \right| - c < x < c\} \) . The binary </span><span style="font-family: 'times new roman', times; font-size: medium;">operation \( * \) is defined on the set <em>G </em>by \(x * y = \frac{{x + y}}{{1 + \frac{{xy}}{{{c^2}}}}}\).</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 44.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Simplify \(\frac{c}{2} * \frac{{3c}}{4}\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 20.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">State the identity element for <em>G </em>under \( * \).</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 28.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">For \(x \in G\) find an expression for \({x^{ - 1}}\) (the inverse of <em>x </em>under \( * \)).</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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 27.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that the binary operation \( * \) is commutative on <em>G </em>.</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 style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that the binary operation \( * \) is associative on <em>G </em>.</span></p>
<div class="marks">[4]</div>
<div class="question_part_label">e.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(i) If \(x,{\text{ }}y \in G\) explain why \((c - x)(c - y) > 0\) .</span></p>
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 30.0px Helvetica;"><span style="font-family: 'times new roman', times; font-size: medium;">(ii) Hence show that \(x + y < c + \frac{{xy}}{c}\) .</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">f.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 32.0px Times;"><span style="font-family: 'times new roman', times; font-size: medium;">Show that <em>G </em>is closed under \( * \).</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">g.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p style="margin: 0.0px 0.0px 0.0px 0.0px; font: 31.0px Times; color: #3f3f3f;"><span style="font-family: 'times new roman', times; font-size: medium;">Explain why \(\{ G, * \} \) is an Abelian group.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">h.</div>
</div>
<br><hr><br><div class="specification">
<p class="p1">A random variable \(X\) has probability density function</p>
<p class="p1">\(f(x) = \left\{ {\begin{array}{*{20}{c}} 0&{x < 0} \\ {\frac{1}{2}}&{0 \le x < 1} \\ {\frac{1}{4}}&{1 \le x < 3} \\ 0&{x \ge 3} \end{array}} \right.\)</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Sketch the graph of \(y = f(x)\).</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the cumulative distribution function for \(X\).</p>
<div class="marks">[5]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p class="p1">Find the interquartile range for \(X\).</p>
<div class="marks">[3]</div>
<div class="question_part_label">c.</div>
</div>
<br><hr><br>