# 2016 A Level H2 Math

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Paper 1
Paper 2
##### 2016 A Level H2 Math Paper 1 Question 3

The curve $y={{x}^{4}}$ is transformed onto the curves with equation $y=\text{f}\left( x \right)$. The turning point on $y={{x}^{4}}$ corresponds to the point with coordinates $\left( a,b \right)$ on $y=\text{f}\left( x \right)$. The curve $y=\text{f}\left( x \right)$ also passes through the point with coordinate $\left( 0,c \right)$. Given that $\text{f}\left( x \right)$ has the form $k{{\left( x-l \right)}^{4}}+m$ and that $a$, $b$ and $c$ are positive constants with $c>b$, express $k$, $l$ and $m$ in terms of $a$, $b$ and $c$.

[2]

By sketching the curve $y=\text{f}\left( x \right)$, or otherwise, sketch the curve $y=\frac{1}{\text{f}\left( x \right)}$. State, in terms of $a$, $b$ and $c$, the coordinates of any points where $y=\frac{1}{\text{f}\left( x \right)}$ crosses the axes and of any turning points.

[4]

##### 2016 A Level H2 Math Paper 1 Question 7

(a)

Verify that $-1+5\mathbf{i}$ is a root of the equation ${{w}^{2}}+\left( -1-8\mathbf{i} \right)w+\left( -17+7\mathbf{i} \right)=0$. Hence, or otherwise, find the second root of the equation in Cartesian form, $p+\mathbf{i}q$, showing your working.

(a) Verify that $-1+5\mathbf{i}$ is a root of the equation ${{w}^{2}}+\left( -1-8\mathbf{i} \right)w+\left( -17+7\mathbf{i} \right)=0$. Hence, or otherwise, find the second root of the equation in Cartesian form, $p+\mathbf{i}q$, showing your working.Â

##### 2016 A Level H2 Math Paper 1 Question 8

It is given that $y=\text{f}\left( x \right)$, where $\text{f}\left( x \right)=\tan \left( ax+b \right)$ for constants $a$ and $b$.

(i)

Show that $\text{{f}’}\left( x \right)=a+a{{y}^{2}}$. Use this result to find $\text{{f}”}\left( x \right)$ and $\text{{f}”’}\left( x \right)$ in terms of $a$ and $y$.

[5]

(i) Show that $\text{{f}’}\left( x \right)=a+a{{y}^{2}}$. Use this result to find $\text{{f}”}\left( x \right)$ and $\text{{f}”’}\left( x \right)$ in terms of $a$ and $y$.

[5]

(ii)

In the case where $b=\frac{1}{4}\pi$, use your results from part (i) to find the Maclaurin series for $\text{f}\left( x \right)$ in terms of $a$, up to and including the term in ${{x}^{3}}$.

[3]

(ii) In the case where $b=\frac{1}{4}\pi$, use your results from part (i) to find the Maclaurin series for $\text{f}\left( x \right)$ in terms of $a$, up to and including the term in ${{x}^{3}}$.

[3]

(iii)

Find the first two non-zero terms in the Maclaurin series for $\tan 2x$.

[3]

(iii) Find the first two non-zero terms in the Maclaurin series for $\tan 2x$.

[3]

##### 2016 A Level H2 Math Paper 1 Question 10

(a)

The function $\text{f}$ is given by $\text{f}:x\mapsto 1+\sqrt{x}$, for $x\in \mathbb{R}$, $x\ge 0$.

(a) The function $\text{f}$ is given by $\text{f}:x\mapsto 1+\sqrt{x}$, for $x\in \mathbb{R}$, $x\ge 0$.

(i) Find ${{\text{f}}^{-1}}\left( x \right)$ and state the domain of ${{\text{f}}^{-1}}$.

[3]

(i) Find ${{\text{f}}^{-1}}\left( x \right)$ and state the domain of ${{\text{f}}^{-1}}$.

[3]

(ii) Show that if $\text{ff}\left( x \right)=x$ then ${{x}^{3}}-4{{x}^{2}}+4x-1=0$. Hence find the value of $x$ for which $\text{ff}\left( x \right)=x$. Explain why this value of $x$ satisfies the equation $\text{f}\left( x \right)={{\text{f}}^{-1}}\left( x \right)$.

[5]

(ii) Show that if $\text{ff}\left( x \right)=x$ then ${{x}^{3}}-4{{x}^{2}}+4x-1=0$. Hence find the value of $x$ for which $\text{ff}\left( x \right)=x$. Explain why this value of $x$ satisfies the equation $\text{f}\left( x \right)={{\text{f}}^{-1}}\left( x \right)$.

[5]

(b)

The function $\text{g}$, with domain the set of non-negative integers, is given by

$g\left( n \right)=\left\{ \begin{matrix} 1\text{for}\,n=0, \\ 2+\text{g}\left( \frac{1}{2}n \right)\text{for }n\text{ even,} \\ 1+\text{g}\left( n-1 \right)\text{for }n\text{ odd}. \\ \end{matrix} \right.$

(b) The function $\text{g}$, with domain the set of non-negative integers, is given by

$g\left( n \right)=\left\{ \begin{matrix} 1\text{for}\,n=0, \\ 2+\text{g}\left( \frac{1}{2}n \right)\text{for }n\text{ even,} \\ 1+\text{g}\left( n-1 \right)\text{for }n\text{ odd}. \\ \end{matrix} \right.$

(i) Find $\text{g}\left( 4 \right)$, $\text{g}\left( 7 \right)$and $\text{g}\left( 12 \right)$.

[3]

(i) Find $\text{g}\left( 4 \right)$, $\text{g}\left( 7 \right)$and $\text{g}\left( 12 \right)$.

[3]

(ii) Does $\text{g}$ have an inverse? Justify your answer.

[2]

(ii) Does $\text{g}$ have an inverse? Justify your answer.

[2]

##### 2016 A Level H2 Math Paper 1 Question 11

The plane $p$ has equation $\mathbf{r}=\left( \begin{matrix} 1 \\ -3 \\ 2 \\ \end{matrix} \right)+\lambda \left( \begin{matrix} 1 \\ 2 \\ 0 \\ \end{matrix} \right)+\mu \left( \begin{matrix} a \\ 4 \\ -2 \\ \end{matrix} \right)$ , and the line $l$ has equation $\mathbf{r}=\left( \begin{matrix} a-1 \\ a \\ a+1 \\ \end{matrix} \right)+t\left( \begin{matrix} -2 \\ 1 \\ 2 \\ \end{matrix} \right)$, where $a$ is a constant and $\lambda$, $\mu$ and $t$ are parameters.

(i)

In the case where $a=0$,

(i) In the case where $a=0$,

(a) show that $l$ is perpendicular to $p$ and find the values of $\lambda$, $\mu$ and $t$ which give the coordinates of the point at which $l$ and $p$ intersect,

[5]

(a) show that $l$ is perpendicular to $p$ and find the values of $\lambda$, $\mu$ and $t$ which give the coordinates of the point at which $l$ and $p$ intersect,

[5]

(b) find the Cartesian equations of the planes such that the perpendicular distance from each plane to $p$ is $12$.

[5]

(b) find the Cartesian equations of the planes such that the perpendicular distance from each plane to $p$ is $12$.

[5]

(ii)

Find the value of $a$ such that $l$ and $p$ do not meet in a unique point.

[3]

(ii) Find the value of $a$ such that $l$ and $p$ do not meet in a unique point.

[3]