These Ten-Year-Series (TYS) worked solutions with video explanations for 2013 A Level H2 Mathematics are suggested by Mr Gan. For any comments or suggestions please contact us at support@timganmath.edu.sg.
It is given that
$y=\frac{{{x}^{2}}+x+1}{x-1}$, $x\in \mathbb{R}$, $x\ne 1$.
Without using a calculator, find the set of values that $y$ can take.
[5]
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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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Functions $\text{f}$ and $\text{g}$ are defined by
$\text{f}:x\mapsto \frac{2+x}{1-x}$, $x\in \mathbb{R}$, $x\ne 1$,
$\text{g}:x\mapsto 1-2x$, $x\in \mathbb{R}$.
(i)
Explain why the composite function $\text{fg}$ does not exist.
[2]
(i) Explain why the composite function $\text{fg}$ does not exist.
[2]
(ii)
Find an expression for $\text{gf}\left( x \right)$ and hence, or otherwise, find ${{\left( \text{gf} \right)}^{-1}}\left( 5 \right)$.
[4]
(ii) Find an expression for $\text{gf}\left( x \right)$ and hence, or otherwise, find ${{\left( \text{gf} \right)}^{-1}}\left( 5 \right)$.
[4]
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