# 2013 A Level H2 Math

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Paper 1
Paper 2
##### 2013 A Level H2 Math Paper 1 Question 2

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]

##### 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]

##### 2013 A Level H2 Math Paper 1 Question 6

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 $\mu$.

[1]

(i) Explain why $\mathbf{c}$ can be expressed as $\mathbf{c}=\lambda \mathbf{a}+\mu \mathbf{b}$, for constants $\lambda$ and $\mu$.

[1]

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 $\mathbf{c}$.

[1]

(ii) Write down the position vector of $N$ in terms of $\mathbf{a}$ and $\mathbf{c}$.

[1]

(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]

(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]

##### 2013 A Level H2 Math Paper 1 Question 8

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 $\theta$.

[2]

(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 $\theta$.

[2]

(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]

##### 2013 A Level H2 Math Paper 1 Question 10

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 $x$.

[4]

(i) Given that $z<\frac{3}{2}$, solve equation $\text{(A)}$ to find $z$ in terms of $x$.

[4]

(ii)

Hence find $y$ in terms of $x$.

[2]

(ii) Hence find $y$ in terms of $x$.

[2]

(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.

[3]

(iv)

###### * Not in the latest 9758 H2 Math Syllabus

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]

(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]

##### 2013 A Level H2 Math Paper 1 Question 11

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 $t$.

[3]

(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 $x={{y}^{2}}+1$.

[5]

(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 $x={{y}^{2}}+1$.

[5]

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 $M$.

[3]

Show that $4{{t}^{6}}-3{{t}^{2}}+1=0$ at $M$. Hence, or otherwise, find the exact coordinates of $M$.

[3]

(iv)

Find the exact value of the area of the shaded region bounded by $C$ and $L$ for which $y\ge 0$.

[6]

(iv) TFind the exact value of the area of the shaded region bounded by $C$ and $L$ for which $y\ge 0$.

[6]

##### 2013 A Level H2 Math Paper 2 Question 1

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]

##### 2013 A Level H2 Math Paper 2 Question 4

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 ${{\text{p}}_{2}}$.

[3]

(i) Find the acute angle between ${{\text{p}}_{1}}$ and ${{\text{p}}_{2}}$.

[3]

(ii)

Find a vector equation for $l$.

[4]

(ii) Find a vector equation for $l$.

[4]

(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]

(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]

##### 2013 A Level H2 Math Paper 2 Question 6

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]