<p>A curve $C$ has parametric equations</p>
<p style="text-align: center">$x={{\sin }^{3}}\theta $, $y=3{{\sin }^{2}}\theta \cos \theta $, $0\le \theta \le \frac{\pi }{2}$.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Show that $\frac{\mathrm{d}y}{\mathrm{d}x}=2\cot \theta -\tan \theta $.</div><p style="text-align: right;">[3]</p></li><li><div class="list-item-paragraph">Show that $C$ has a turning point when $\tan \theta =\sqrt{k}$, where $k$ is an integer to be determined.</div><div class="list-item-paragraph">Find, in non-trigonometric form, the exact coordinates of the turning point and explain why it is a maximum.</div><p style="text-align: right;">[6]</p></li><li><div class="list-item-paragraph">Show that the area of the region bounded by $C$ and the $x$-axis is given by</div><div class="list-item-paragraph" style="text-align: center">$\int_{0}^{\frac{1}{2}\pi }{9{{\sin }^{4}}\theta {{\cos }^{2}}\theta \mathrm{ d}\theta }$.</div><div class="list-item-paragraph">Use your calculator to find the area, giving your answer correct to $3$ decimal places.</div><p style="text-align: right;">[3]</p></li></ol>
<p>The line with equation $y=ax$, where $a$ is a positive constant, meets $C$ at the origin and at the point $P$.</p>
<ol class="i-list" start="4"><li><div class="list-item-paragraph">Show that $\tan \theta =\frac{3}{a}$ at $P$. Find the exact value of $a$ such that the line passes through the maximum point of $C$.</div><p style="text-align: right;">[3]</p></li></ol>

$x={{\sin }^{3}}\theta $, $y=3{{\sin }^{2}}\theta \cos \theta $, $0\le \theta \le \frac{\pi }{2}$.

Show that $\frac{\mathrm{d}y}{\mathrm{d}x}=2\cot \theta -\tan \theta $.

Show that $C$ has a turning point when $\tan \theta =\sqrt{k}$, where $k$ is an integer to be determined.

Find, in non-trigonometric form, the exact coordinates of the turning point and explain why it is a maximum.

Show that the area of the region bounded by $C$ and the $x$-axis is given by

$\displaystyle\displaystyle \int_{0}^{\frac{1}{2}\pi }{9{{\sin }^{4}}\theta {{\cos }^{2}}\theta \mathrm{ d}\theta }$.

Use your calculator to find the area, giving your answer correct to $3$ decimal places.

The line with equation $y=ax$, where $a$ is a positive constant, meets $C$ at the origin and at the point $P$.

Show that $\tan \theta =\frac{3}{a}$ at $P$. Find the exact value of $a$ such that the line passes through the maximum point of $C$.

N2015 P1 Q11

N2015 P1 Q11

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