<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find $\int{\left( {{\cot }^{6}}2x+{{\cot }^{4}}2x-\sin \frac{x}{2}\sin \frac{3x}{2} \right)}\mathrm{ d}x$.</span></div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">The diagram shows the graphs of $y={{\mathrm{e}}^{2x}}$ and $y={{\left( x+1 \right)}^{2}}$.</div><div class="list-item-paragraph">$R$ is the finite region bounded by the two curves $y={{\mathrm{e}}^{2x}}$ and $y={{\left( x+1 \right)}^{2}}$.</div><div class="list-item-paragraph">Find the volume of the solid formed when $R$ is rotated through $2\pi $ radians about the $y$-axis, giving your answer correct to $4$ decimal places.</div><p style="text-align: right;">[3]</p><div class="list-item-image"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/public/question-images/converted/1761540959364-qtzjyukhxf.png" alt="List item image" /></div></li><li><div class="list-item-paragraph">A curve $C$ has parametric equations</div><div class="list-item-paragraph" style="text-align: center">$x=2\left( \sin \frac{t}{2}-\cos \frac{t}{2} \right)$, $y=\sin \frac{t}{2}+\cos \frac{t}{2}$, for $-\frac{3\pi }{2}\le t\le -\frac{\pi }{2}$.</div><div class="list-item-paragraph">The line $L$ with equation $y=-2x-5$ meets the curve $C$ at the point $\left( -2,\,-1 \right)$. $S$ is the region enclosed by line $L$, curve $C$and the $x$-axis as shown in the diagram below.</div><div class="list-item-image"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/public/question-images/converted/1761541142288-0319z6id82yx.png" alt="List item image" /></div><ol class="i-list"><li><div class="list-item-paragraph">Show that the x-intercept of curve $C$ is $-2\sqrt{2}$.</div><p style="text-align: right;">[2]</p></li><li><div class="list-item-paragraph">Find the exact area of $S$.</div><p style="text-align: right;">[6]</p></li></ol></li></ol>

Find $\displaystyle\displaystyle \int{\left( {{\cot }^{6}}2x+{{\cot }^{4}}2x-\sin \frac{x}{2}\sin \frac{3x}{2} \right)}\mathrm{ d}x$.

The diagram shows the graphs of $y={{\mathrm{e}}^{2x}}$ and $y={{\left( x+1 \right)}^{2}}$.

$R$ is the finite region bounded by the two curves $y={{\mathrm{e}}^{2x}}$ and $y={{\left( x+1 \right)}^{2}}$.

Find the volume of the solid formed when $R$ is rotated through $2\pi $ radians about the $y$-axis, giving your answer correct to $4$ decimal places.

$x=2\left( \sin \frac{t}{2}-\cos \frac{t}{2} \right)$, $y=\sin \frac{t}{2}+\cos \frac{t}{2}$, for $-\frac{3\pi }{2}\le t\le -\frac{\pi }{2}$.

The line $L$ with equation $y=-2x-5$ meets the curve $C$ at the point $\left( -2,\,-1 \right)$. $S$ is the region enclosed by line $L$, curve $C$and the $x$-axis as shown in the diagram below.

2020 ASRJC P2 Q4

2020 ASRJC P2 Q4

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