These Ten-Year-Series (TYS) worked solutions with video explanations for 2013 A Level H2 Mathematics Paper 1 Question 5 are suggested by Mr Gan. For any comments or suggestions please contact us at support@timganmath.edu.sg.
2013 A Level H2 Math Paper 1 Question 5
It is given that
$\text{f}\left( x \right)=\left\{ \begin{matrix}
& \sqrt{1-\frac{{{x}^{2}}}{{{a}^{2}}}}\,\,\,\,\text{for}\,\,-a\le x\le a,\\&0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{for}\,\,a<x\le2a, \\\end{matrix} \right.$
and that $\text{f}\left( x+3a \right)=\text{f}\left( x \right)$ for all real values of $x$, where $a$ is a real constant.
(i)
Sketch the graph of $y=\text{f}\left( x \right)$ for $-4a\le x\le 6a$.
[3]
(i) Sketch the graph of $y=\text{f}\left( x \right)$ for $-4a\le x\le 6a$.
[3]
(ii)
Use the substitution $x=a\sin \theta $ to find the exact area of $\int_{\frac{1}{2}a}^{\frac{\sqrt{3}}{2}a}{\text{f}\left( x \right)\,\text{d}x}$ in terms of $a$ and $\pi $.
[5]
(ii) Use the substitution $x=a\sin \theta $ to find the exact area of $\int_{\frac{1}{2}a}^{\frac{\sqrt{3}}{2}a}{\text{f}\left( x \right)\,\text{d}x}$ in terms of $a$ and $\pi $.
[5]
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