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1.     Solve the equation $\ln (x^3-3)=3\ln x-\ln 3$. Give your answer correct to 3 significant figures.
$$\begin{array}{}\text{[3]}& \end{array}$$

2.     The polynomial $a{x}^3+5x^2-4x+b$, where $a$ and $b$ are constants, is denoted by $\mathrm{p}(x)$. It is given that $(x+2)$ is a factor of $\mathrm{p}(x)$ and that when $\mathrm{p}(x)$ is divided by $(x+1)$ the remainder is 2.

    Find the values of $a$ and $b$.
$$\begin{array}{}\text{[5]}& \end{array}$$

3.     By first expressing the equation $\tan (x+{45}^{\circ })=2\cot x+1$ as a quadratic equation in $\tan x$, solve the equation for $0^{\circ }<x<{180}^{\circ }$.
$$\begin{array}{}\text{[6]}& \end{array}$$

4.     The variables x and y satisfy the differential equation
$$(1–\cos x)\frac{\mathrm{d}y}{\mathrm{d}x}=y\sin x.$$
        It is given that $y=4$ when $x=\pi$.

        (a)     Solve the differential equation, obtaining an expression for $y$ in terms of $x$.
$$\begin{array}{}\text{[6]}& \end{array}$$
        (b)     Sketch the graph of $y$ against $x$ for $0<x<2\pi$.
$$\begin{array}{}\text{[1]}& \end{array}$$

5.     (a)     Express $\sqrt{7}\sin x+2\cos x$ in the form $R\sin (x+\alpha )$, where $R>0$ and $0^{\circ }<\alpha <{90}^{\circ }$. State the exact value of $R$ and give $\alpha$ correct to 2 decimal places.
$$\begin{array}{}\text{[3]}& \end{array}$$
        (b)    Hence solve the equation $\sqrt{7}\sin 2\theta +2\cos 2\theta =1$, for $0^{\circ }<\theta <{180}^{\circ }$.
$$\begin{array}{}\text{[5]}& \end{array}$$

6.    Let ${\displaystyle \mathrm{f}(x)=\frac{5a}{(2x–a)(3a–x)}}$ where $a$ is a positive constant.

        (a)     Express $\mathrm{f}(x)$ in partial fractions.
$$\begin{array}{}\text{[3]}& \end{array}$$
        (b)     Hence show that ${\displaystyle {\int }_a^{2a}\mathrm{f}(x)\mathrm{d}x=\ln 6}$.
$$\begin{array}{}\text{[4]}& \end{array}$$

7.     Two lines have equations $\mathbf{r}=\left(\begin{array}{c}1\\ 3\\ 2\end{array}\right)+s\left(\begin{array}{c}2\\ -1\\ 3\end{array}\right)$ and $\mathbf{r}=\left(\begin{array}{c}2\\ 1\\ 4\end{array}\right)+t\left(\begin{array}{c}1\\ -1\\ 4\end{array}\right)$.

    (a)     Show that the lines are skew.
$$\begin{array}{}\text{[5]}& \end{array}$$
    (b)     Find the acute angle between the directions of the two lines.
$$\begin{array}{}\text{[3]}& \end{array}$$

8.     The complex numbers $u$ and $v$ are defined by $u=-4+2\mathrm{i}$ and $v=3+\mathrm{i}$.

    (a)     Find $\frac{u}{v}$ in the form $x+\mathrm{i}y$, where $x$ and $y$ are real.
$$\begin{array}{}\text{[3]}& \end{array}$$
    (b)     Hence express $\frac{u}{v}$ in the form $r{\mathrm{e}}^{\mathrm{i}\theta }$, where $r$ and $\theta$ are exact.
$$\begin{array}{}\text{[2]}& \end{array}$$
In an Argand diagram, with origin $O$, the points $A$, $B$ and $C$ represent the complex numbers $u$, $v$ and $2u+v$ respectively.

    (c)     State fully the geometrical relationship between $OA$ and $BC$.
$$\begin{array}{}\text{[2]}& \end{array}$$
    (d)     Prove that angle $AOB=\frac{3}{4}\pi$.
$$\begin{array}{}\text{[2]}& \end{array}$$

9.     Let ${\displaystyle \mathrm{f}(x)=\frac{{\mathrm{e}}^{2x}+1}{{\mathrm{e}}^{2x}–1}}$, for $x>0$.

        (a)     The equation $x=\mathrm{f}(x)$ has one root, denoted by $a$.

        Verify by calculation that $a$ lies between 1 and 1.5.
$$\begin{array}{}\text{[2]}& \end{array}$$
        (b)     Use an iterative formula based on the equation in part (a) to determine $a$ correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
$$\begin{array}{}\text{[3]}& \end{array}$$
        (c)     Find ${\mathrm{f}}^{‘}(x)$. Hence find the exact value of $x$ for which ${\mathrm{f}}^{‘}(x)=-8$.
$$\begin{array}{}\text{[6]}& \end{array}$$

10.    

The diagram shows the curve $y=\sin 2x{\cos }^{2}x$ for $0\le x\le \frac{1}{2}\pi$, and its maximum point $M$.

        (a)     Using the substitution $u=\sin x$, find the exact area of the region bounded by the curve and the $x$-axis.
$$\begin{array}{}\text{[5]}& \end{array}$$
        (b)     Find the exact $x$-coordinate of $M$.
$$\begin{array}{}\text{[6]}& \end{array}$$