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.
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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 $-4a\le x\le 6a$.
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(i) Sketch the graph of $y=\text{f}\left( x \right)$ for $-4a\le x\le 6a$.
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(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 $.
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(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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The origin $O$ and the points $A$, $B$ and $C$ lies in the same plane, where $\overrightarrow{OA}=\mathbf{a}$, $\overrightarrow{OB}=\mathbf{b}$ and $\overrightarrow{OC}=\mathbf{c}$ (see diagram).
(i)
Explain why $\mathbf{c}$ can be expressed as $\mathbf{c}=\lambda \mathbf{a}+\mu \mathbf{b}$, for constants $\lambda $ and $\mu $.
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(i) Explain why $\mathbf{c}$ can be expressed as $\mathbf{c}=\lambda \mathbf{a}+\mu \mathbf{b}$, for constants $\lambda $ and $\mu $.
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The point $N$ is on $AC$ such that $AN:NC=3:4$.
(ii)
Write down the position vector of $N$ in terms of $\mathbf{a}$ and $\mathbf{c}$.
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(ii) Write down the position vector of $N$ in terms of $\mathbf{a}$ and $\mathbf{c}$.
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(iii)
It is given that the area of $ONC$ is equal to the area of triangle $OMC$, where $M$ is the mid-point $OB$. By finding the areas of these triangles in terms of $\mathbf{a}$ and $\mathbf{b}$, find $\lambda $ in terms of $\mu $ in the case where $\lambda $ and $\mu $ are both positive.
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(iii) It is given that the area of $ONC$ is equal to the area of triangle $OMC$, where $M$ is the mid-point $OB$. By finding the areas of these triangles in terms of $\mathbf{a}$ and $\mathbf{b}$, find $\lambda $ in terms of $\mu $ in the case where $\lambda $ and $\mu $ are both positive.
[5]
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The complex number $z$ is given by $z=r{{e}^{\text{i}\theta }}$, where $r>0$ and $0\le \theta \le \frac{1}{2}\pi $.
(i)
Given that $w=\left( 1-\text{i}\sqrt{3} \right)z$, find $\left| w \right|$ in terms of $r$ and $\arg w$ in terms of $\theta $.
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(i) Given that $w=\left( 1-\text{i}\sqrt{3} \right)z$, find $\left| w \right|$ in terms of $r$ and $\arg w$ in terms of $\theta $.
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(iii)
Given that $\arg \left( \frac{{{z}^{10}}}{{{w}^{2}}} \right)=\pi $, find $\theta $.
[3]
(ii) Given that $\arg \left( \frac{{{z}^{10}}}{{{w}^{2}}} \right)=\pi $, find $\theta $.
[3]
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The variables $x,y$ and $z$ are connected by the following differential equations.
$\frac{\text{d}z}{\text{d}x}=3-2z$Â $\text{(A)}$
$\frac{\text{d}y}{\text{d}x}=z$Â Â Â Â Â Â $\text{(B)}$
(i)
Given that $z<\frac{3}{2}$, solve equation $\text{(A)}$ to find $z$ in terms of $x$.
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(i) Given that $z<\frac{3}{2}$, solve equation $\text{(A)}$ to find $z$ in terms of $x$.
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(ii)
Hence find $y$ in terms of $x$.
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(ii) Hence find $y$ in terms of $x$.
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(iii)
Use the result in part (ii) to show that
$\frac{{{\text{d}}^{2}}y}{\text{d}{{x}^{2}}}=a\frac{\text{d}y}{\text{d}x}+b$,
for constants $a$ and $b$ to be determined.
[3]
(iii) Use the result in part (ii) to show that
$\frac{{{\text{d}}^{2}}y}{\text{d}{{x}^{2}}}=a\frac{\text{d}y}{\text{d}x}+b$,
for constants $a$ and $b$ to be determined.
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(iv)
The result in part (ii) represents a family of curves. Some members of the family are straight lines. Write down the equations of two of these lines. On a single diagram, sketch one of your lines together with a non-linear member of the family of curves that has your line as an asymptote.
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(iv) The result in part (ii) represents a family of curves. Some members of the family are straight lines. Write down the equations of two of these lines. On a single diagram, sketch one of your lines together with a non-linear member of the family of curves that has your line as an asymptote.
[4]
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A curve $C$ has parametric equations
$x=3{{t}^{2}}$, $y=2{{t}^{3}}$.
(i)
Find the equation of the tangent to $C$ at the point with parameter $t$.
[3]
(i) Find the equation of the tangent to $C$ at the point with parameter $t$.
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(ii)
Points $P$ and $Q$ on $C$ have parameters $p$ and $q$ respectively. The tangent at $P$ meets the tangent at $Q$ and some point $R$. Show that the $x$-coordinates of $R$ is ${{p}^{2}}+pq+{{q}^{2}}$, and find the $y$-coordinates of $R$ in terms of $p$ and $q$. Given that $pq=-1$, show that $R$ lies on the curve with equation $x={{y}^{2}}+1$.
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(ii) Points $P$ and $Q$ on $C$ have parameters $p$ and $q$ respectively. The tangent at $P$ meets the tangent at $Q$ and some point $R$. Show that the $x$-coordinates of $R$ is ${{p}^{2}}+pq+{{q}^{2}}$, and find the $y$-coordinates of $R$ in terms of $p$ and $q$. Given that $pq=-1$, show that $R$ lies on the curve with equation $x={{y}^{2}}+1$.
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A curve $L$ has equation $x={{y}^{2}}+1$. The diagram shows parts of $C$ and $L$ for which $y\ge 0$. The curves $C$ and $L$ touch at the point $M$.
(iii)
Show that $4{{t}^{6}}-3{{t}^{2}}+1=0$ at $M$. Hence, or otherwise, find the exact coordinates of $M$.
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Show that $4{{t}^{6}}-3{{t}^{2}}+1=0$ at $M$. Hence, or otherwise, find the exact coordinates of $M$.
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(iv)
Find the exact value of the area of the shaded region bounded by $C$ and $L$ for which $y\ge 0$.
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(iv) TFind the exact value of the area of the shaded region bounded by $C$ and $L$ for which $y\ge 0$.
[6]
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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.
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(ii)
Find an expression for $\text{gf}\left( x \right)$ and hence, or otherwise, find ${{\left( \text{gf} \right)}^{-1}}\left( 5 \right)$.
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(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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The planes ${{\text{p}}_{1}}$ and ${{\text{p}}_{2}}$ have equations $\mathbf{r}\cdot \left( \begin{matrix}2 \\-2 \\1 \\\end{matrix} \right)=1$ and $\mathbf{r}\cdot \left( \begin{matrix}-6 \\3 \\2 \\\end{matrix} \right)=-1$ respectively, and meet in the line $l$.
(i)
Find the acute angle between ${{\text{p}}_{1}}$ and ${{\text{p}}_{2}}$.
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(i) Find the acute angle between ${{\text{p}}_{1}}$ and ${{\text{p}}_{2}}$.
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(ii)
Find a vector equation for $l$.
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(ii) Find a vector equation for $l$.
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(iii)
The point $A\left( 4,3,c \right)$ is equidistant from the planes ${{\text{p}}_{1}}$ and ${{\text{p}}_{2}}$. Calculate the two possible values of $c$.
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(iii) The point $A\left( 4,3,c \right)$ is equidistant from the planes ${{\text{p}}_{1}}$ and ${{\text{p}}_{2}}$. Calculate the two possible values of $c$.
[6]
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The continuous random variable $Y$ has the distribution $N\left( \mu ,{{\sigma }^{2}} \right)$. It is known that $\text{P}\left( Y<2a \right)=0.95$ and $\text{P}\left( Y<a \right)=0.25$. Express $\mu $ in the form $ka$, where $k$ is a constant to be determined.
[4]
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