$\require{\cancel}$ $\require{\stix[upint]}$

Qtn No.	1	2	3	4	5	6	7	8	9	10	11	12	Total
Marks	4	5	4	5	5	7	7	10	9	8	11	11	86
Score

Question 1

$\text{(i)}$ Express $4 x^{2}-12 x$ in the form $(2 x+a)^{2}+b$. $[2]$

$\text{(ii)}$ Hence, or otherwise, find the set of values of $x$ satisfying $4 x^{2}-12 x>7$. $[2]$

Question 2

$\text{(i)}$ In the binomial expansion of $\displaystyle \left(2 x-\frac{1}{2 x}\right)^{5}$, the first three terms are $32 x^{5}-40 x^{3}+20 x$. Find the remaining three terms of the expansion. $[3]$

$\text{(ii)}$ Hence find the coefficient of $x$ in the expansion of $\displaystyle \left(1+4 x^{2}\right)\left(2 x-\frac{1}{2 x}\right)^{5}$. $[2]$

Question 3

A sector of a circle of radius $r \mathrm{~cm}$ has an area of $A \mathrm{~cm}^{2}$. Express the perimeter of the sector in terms of $r$ and $A$. $[4]$

Question 4

Angle $x$ is such that $\sin x=a+b$ and $\cos x=a-b$, where $a$ and $b$ are constants.

$\text{(i)}$ Show that $a^{2}+b^{2}$ has a constant value for all values of $x$. $[3]$

$\text{(ii)}$ In the case where $\tan x=2$, express $a$ in terms of $b$. $[2]$

Question 5

 

The diagram shows the graph of $y=\mathrm{f}(x)$, where $\displaystyle \mathrm{f}(x)=\frac{3}{2} \cos 2 x+\frac{1}{2}$ for $0 \leqslant x \leqslant \pi$.

$\text{(a)}$ State the range of f. $[2]$

A function $\mathrm{g}$ is such that $g(x)=f(x)+k$, where $k$ is a positive constant. The $x$-axis is a tangent to the curve $y=\mathrm{g}(x)$.

$\text{(b)}$ State the value of $k$ and hence describe fully the transformation that maps the curve $y=\mathrm{f}(x)$ on to $y=g(x)$. $[2]$

$\text{(c)}$ State the equation of the curve which is the reflection of $y=\mathrm{f}(x)$ in the $x$-axis. Give your answer in the form $y=a \cos 2 x+b$, where $a$ and $b$ are constants. $[1]$

Question 6

 

The diagram shows the graph of $y=\mathrm{f}^{-1}(x)$, where $\mathrm{f}^{-1}$ is defined by $\displaystyle\mathrm{f}^{-1}(x)=\frac{1-5 x}{2 x}$ for $0 < x \leqslant 2$

$\text{(i)}$ Find an expression for $\mathrm{f}(x)$ and state the domain of $\mathrm{f}$. $[5]$

$\text{(ii)}$ The function $\mathrm{g}$ is defined by $\mathrm{g}(x)=\frac{1}{x}$ for $x \geqslant 1$. Find an expression for $\mathrm{f}^{-1} \mathrm{~g}(x)$, giving your answer in the form $a x+b$, where $a$ and $b$ are constants to be found. $[2]$

Question 7

A point $P$ is moving along a curve in such a way that the $x$-coordinate of $P$ is increasing at a constant rate of 2 units per minute. The equation of the curve is $y=(5 x-1)^{\frac{1}{2}}$.

$\text{(a)}$ Find the rate at which the $y$-coordinate is increasing when $x=1$. $[4]$

$\text{(b)}$ Find the value of $x$ when the $y$-coordinate is increasing at $\frac{5}{8}$ units per minute. $[3]$

Question 8

 

The diagram shows a triangle $A B C$ in which $A$ is $(3,-2)$ and $B$ is $(15,22)$. The gradients of $A B, A C$ and $B C$ are $2 m,-2 m$ and $m$ respectively, where $m$ is a positive constant.

$\text{(i)}$ Find the gradient of $A B$ and deduce the value of $m$. $[2]$

$\text{(ii)}$ Find the coordinates of $C$. $[4]$

The perpendicular bisector of $A B$ meets $B C$ at $D$.

$\text{(iii)}$ Find the coordinates of $D$. $[4]$

Question 9

The function $\mathrm{f}: x \mapsto 5+3 \cos \left(\frac{1}{2} x\right)$ is defined for $0 \leqslant x \leqslant 2 \pi$.

$\text{(i)}$ Solve the equation $\mathrm{f}(x)=7$, giving your answer correct to 2 decimal places. $[3]$

$\text{(ii)}$ Sketch the graph of $y=\mathrm{f}(x)$. $[2]$

$\text{(iii)}$ Explain why $\mathrm{f}$ has an inverse. $[1]$

$\text{(iv)}$ Obtain an expression for $\mathrm{f}^{-1}(x)$. $[3]$

Question 10

The first, second and third terms of an arithmetic progression are $a, \frac{3}{2} a$ and $b$ respectively, where $a$ and $b$ are positive constants. The first, second and third terms of a geometric progression are $a, 18$ and $b+3$ respectively.

$\text{(a)}$ Find the values of $a$ and $b$. $[5]$

$\text{(b)}$ Find the sum of the first 20 terms of the arithmetic progression. $[3]$

Question 11

A curve has equation $y=\mathrm{f}(x)$ and is such that $\mathrm{f}^{\prime}(x)=3 x^{\frac{1}{2}}+3 x^{-\frac{1}{2}}-10$.

$\text{(i)}$ By using the substitution $u=x^{\frac{1}{2}}$, or otherwise, find the values of $x$ for which the curve $y=\mathrm{f}(x)$ has stationary points. $[4]$

$\text{(ii)}$ Find $\mathrm{f}^{\prime \prime}(x)$ and hence, or otherwise, determine the nature of each stationary point. $[3]$

$\text{(iii)}$ It is given that the curve $y=\mathrm{f}(x)$ passes through the point $(4,-7)$. Find $\mathrm{f}(x)$. $[4]$

Question 12

Functions $\mathrm{f}$ and $\mathrm{g}$ are defined for $x \in \mathbb{R}$ by

$$ \begin{aligned} &\mathrm{f}: x \mapsto \frac{1}{2} x-2 \\ &\mathrm{~g}: x \mapsto 4+x-\frac{1}{2} x^{2} \end{aligned} $$

$\text{(i)}$ Find the points of intersection of the graphs of $y=\mathrm{f}(x)$ and $y=\mathrm{g}(x)$. $[3]$

$\text{(ii)}$ Find the set of values of $x$ for which $\mathrm{f}(x)>\mathrm{g}(x)$. $[2]$

$\text{(iii)}$ Find an expression for $\mathrm{fg}(x)$ and deduce the range of $\mathrm{fg}$. $[4]$

The function h is defined by h : $x \mapsto 4+x-\frac{1}{2} x^{2}$ for $x \geqslant k$.

$\text{(iv)}$ Find the smallest value of $k$ for which $\mathrm{h}$ has an inverse. $[2]$