# 2007 A Level H2 Math

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Paper 1
Paper 2

##### 2007 A Level H2 Math Paper 1 Question 11

A curve has parametric equations $x={{\cos }^{2}}t$, $y={{\sin }^{3}}t$, for $0\le t\le \frac{\pi }{2}$.

(i)

Sketch the curve.

[2]

(i) Sketch the curve.

[2]

(ii)

The tangent to the curve at the point $\left( {{\cos }^{2}}\theta ,\,\,{{\sin }^{3}}\theta \right)$ where $0<\theta <\frac{\pi }{2}$, meets the $x$- and $y$-axes at $Q$ and $R$ respectively. The origin is denoted by $O$. Show that the area of triangle $OQR$ is $\frac{1}{12}\sin \theta {{\left( 3{{\cos }^{2}}\theta +2{{\sin }^{2}}\theta \right)}^{2}}$.

[6]

(ii) The tangent to the curve at the point $\left( {{\cos }^{2}}\theta ,\,\,{{\sin }^{3}}\theta \right)$ where $0<\theta <\frac{\pi }{2}$, meets the $x$- and $y$-axes at $Q$ and $R$ respectively. The origin is denoted by $O$. Show that the area of triangle $OQR$ is $\frac{1}{12}\sin \theta {{\left( 3{{\cos }^{2}}\theta +2{{\sin }^{2}}\theta \right)}^{2}}$.

[6]

(iii)

Show that the area under the curve for $0\le t\le \frac{\pi }{2}$ is $2\int_{0}^{\frac{\pi }{2}}{\cos t{{\sin }^{4}}\text{t}\,\,\text{d}t}$, and use the substituition $\sin t=u$ to find this area.

[5]

(iii)

Show that the area under the curve for $0\le t\le \frac{\pi }{2}$ is $2\int_{0}^{\frac{\pi }{2}}{\cos t{{\sin }^{4}}\text{t}\,\,\text{d}t}$, and use the substituition $\sin t=u$ to find this area.

[5]

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