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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.

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2. The polynomial $ax^3 + 5x^2 − 4x + b$, 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 $b$.

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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 ${0}^{\circ} \lt x \lt {180}^{\circ}$.

$$\tag*{[6]} $$

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 $x$.

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(b) Sketch the graph of $y$ against $x$ for $0 \lt x \lt 2\pi$.

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5. (a) Express $ \sqrt{7} \sin x + 2 \cos x $ in the form $R \sin(x + \alpha)$, where $ R \gt 0 $ and ${0}^{\circ} \lt \alpha \lt {90}^{\circ}$. State the exact value of $R$ and give $\alpha$ correct to 2 decimal places.

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(b) Hence solve the equation $ \sqrt{7} \sin 2\theta + 2 \cos 2\theta = 1 $, for ${0}^{\circ} \lt \theta \lt {180}^{\circ}$.

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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.

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(b) Hence show that $\displaystyle \int_{a}^{2a} \mathrm{f}(x) \ \mathrm{d}x = \ln 6$.

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7. Two lines have equations $ \ \mathbf{r} = \begin{pmatrix}

1 \\[1pt]

3 \\[1pt]

2

\end{pmatrix} + s\begin{pmatrix}

2 \\[1pt]

-1 \\[1pt]

3

\end{pmatrix} \ $ and $ \ \mathbf{r} = \begin{pmatrix}

2 \\[1pt]

1 \\[1pt]

4

\end{pmatrix} + t \begin{pmatrix}

1 \\[1pt]

-1 \\[1pt]

4

\end{pmatrix} $.

(a) Show that the lines are skew.

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(b) Find the acute angle between the directions of the two lines.

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

(a) Find $ {\large\frac{u}{v}} $ in the form $x + \mathrm{i}y$, where $x$ and $y$ are real.

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(b) Hence express $ {\large\frac{u}{v}} $ in the form $ r { \mathrm{e} }^{ \mathrm{i}\theta}$, where $r$ and $\theta$ are exact.

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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 $BC$.

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(d) Prove that angle $ AOB = \frac{3}{4} \pi $.

$$\tag*{[2]} $$

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

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(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.

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(c) Find $ { \mathrm{f} }^{‘}(x) $. Hence find the exact value of $x$ for which $ { \mathrm{f} }^{‘}(x) = −8$.

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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.

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(b) Find the exact $x$-coordinate of $M$.

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