1 (i) On the axes below, sketch the graph of $y=2 \cos 3 x-1$ for $-90^{\circ} \leqslant x \leqslant 90^{\circ}$.
(ii) Write down the amplitude of $2 \cos 3 x-1$.[1]
(iii) Write down the period of $2 \cos 3 x-1$.[1]
2 When $\lg y^2$ is plotted against $x,$ a straight line is obtained passing through the points (5, 12) and (3, 20).
Find $y$ in terms of $x$, giving your answer in the form $y = 10^{ ax + b}$ , where $a$ and $b$ are integers.[5]
3 The first three terms in the expansion of $\left(1-\frac{x}{7}\right)^{14}(1-2 x)^{4}$ can be written as $1+a x+b x^{2}$. Find the value of each of the constants $a$ and $b .$
4 (i) On the axes below, sketch the graph of $y=\left|2 x^{2}-9 x-5\right|$ showing the coordinates of the points where the graph meets the axes.[4]
(ii) Find the values of $k$ for which $\left|2 x^{2}-9 x-5\right|=k$ has exactly 2 solutions.$[3]$
5 $\begin{array}[t]{ll}\text { (a) } \text { It is given that } & \mathrm{f}: x \mapsto \sqrt{x} & \text { for } x \geqslant 0,\end{array}$$\mathrm{g}: x \mapsto x+$for $x \geqslant 0 .$
Identify each of the following functions with one of $\mathrm{f}^{-1}, \mathrm{~g}^{-1}, \mathrm{fg}, \mathrm{gf}, \mathrm{f}^{2}, \mathrm{~g}^{2}$
(i) $\sqrt{x+5}$$[1]$
(ii) $x-5$$[1]$
(iii) $x^{2}$[1
(iv) $x+10$[1)
(b) It is given that $h(x)=a+\frac{b}{x^{2}}$ where $a$ and $b$ are constants.
(i) Why is $-2 \leqslant x \leqslant 2$ not a suitable domain for $h(x) ?$$[1]$
(ii) Given that $h(1)=4$ and $h^{\prime}(1)=16$, find the value of $a$ and of $b$.[2]
6 (a) Write $\dfrac{\sqrt{p}\left(\dfrac{q p}{r}\right)^{2}}{p^{-1} \sqrt[3]{q r}}$ in the form $p^{a} q^{b} r^{c}$, where $a, b$ and $c$ are constants.$[3]$
(b) Solve $\log _{7} x+2 \log _{x} 7=3$.$[4]$
7 It is given that $y=\left(1+\mathrm{e}^{x^{2}}\right)(x+5)$.
(i) Find $\dfrac{\mathrm{d} y}{\mathrm{~d} x}$$[3]$
(ii) Find the approximate change in $y$ as $x$ increases from $0.5$ to $0.5+p$, where $p$ is small.[2]
(iii) Given that $y$ is increasing at a rate of 2 units per second when $x=0.5$, find the corresponding rate of change in $x .$
8 (a) Five teams took part in a competition in which each team played each of the other 4 teams. The following table represents the results after all the matches had been played.
$\begin{array}{|c|c|c|c|}\hline \text{Team} & \text{Won} & \text{Drawn} & \text{Lost} \\\hline A & 2 & 1 & 1 \\\hline B & 1 & 3 & 0 \\\hline C & 1 & 1 & 2 \\\hline D & 0 & 1 & 3 \\\hline E & 3 & 0 & 1 \\\hline\end{array}$
Points in the competition were awarded to the teams as follows
4 for each match won, $\quad 2$ for each match drawn, $\quad 0$ for each match lost.
(i) Write down two matrices whose product under matrix multiplication will give the total number of points awarded to each team. $\quad[2]$
(ii) Evaluate the matrix product from part (i) and hence state which team was awarded the most points. $\quad[2]$
(b) It is given that $\quad \mathbf{A}=\left(\begin{array}{rr}1 & -1 \\ 2 & 4\end{array}\right)$ and $\quad \mathbf{B}=\left(\begin{array}{rr}5 & 0 \\ 1 & -2\end{array}\right)$.
(i) Find $\mathbf{A}^{-1}$.
(ii) Hence find the matrix $\mathbf{C}$ such that $\mathbf{A C}=\mathbf{B}$.[3]
9 A solid circular cylinder has a base radius of $r \mathrm{~cm}$ and a height of $h \mathrm{~cm} .$ The cylinder has a volume of $1200 \pi \mathrm{cm}^{3}$ and a total surface area of $S \mathrm{~cm}^{2}$.
(i) Show that $S=2 \pi r^{2}+\dfrac{2400 \pi}{r}$.
(ii) Given that $h$ and $r$ can vary, find the stationary value of $S$ and determine its nature.
A curve is such that $\dfrac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}=2(3 x-1)^{-\frac{2}{3}}$. Given that the curve has a gradient of 6 at the point $(3,11)$, find the equation of the curve. $[8]$
10 The diagram shows a circle centre $O$, radius $10 \mathrm{~cm}$. The points $A, B$ and $C$ lie on the circumference of the circle such that $A B=B C=18 \mathrm{~cm}$.
(i) Show that angle $A O B=2.24$ radians correct to 2 decimal places.
(ii) Find the perimeter of the shaded region.
(iii) Find the area of the shaded region.[3]
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