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1. Solve the inequality $ | 2x \ − \ 1| \gt 3 |x \ + \ 2| $.

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2. Find the exact value of $ \ \displaystyle \int_{0}^{1} (2 \ – \ x) {\mathrm{e}}^{-2x} \ \mathrm{d}x $.

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3. (a) Show that the equation

$$ \ln (1 \ + \ \mathrm{e}^{-x}) \ + \ 2x = 0 $$

can be expressed as a quadratic equation in $ \mathrm{e}^x $.

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(b) Hence solve the equation $ \ln (1 \ + \ \mathrm{e}^{-x}) \ + \ 2x = 0 $, giving your answer correct to 3 decimal places.

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4. The equation of a curve is $ \ y = x \ {\tan}^{-1} \big( \frac{1}{2} x\big) $ .

(a) Find $ \frac{\mathrm{d}y}{\mathrm{d}x} $.

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(b) The tangent to the curve at the point where $x = 2$ meets the $y$-axis at the point with coordinates $(0, p)$. Find $p$.

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5. By first expressing the equation

$$ \tan \theta \ \tan (\theta \ + \ {45}^{\circ}) = 2 \cot 2\theta $$

as a quadratic equation in $ \tan \theta$, solve the equation for $ {0}^{\circ} \lt \theta \lt {90}^{\circ} $.

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6. (a) By sketching a suitable pair of graphs, show that the equation $x^5 = 2 + x$ has exactly one real root.

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(b) Show that if a sequence of values given by the iterative formula

$$ x_{n+1} = \Big(\frac{4{x}_{n}^{5} \ + \ 2}{5{x}_{n}^{4} \ – \ 1}\Big) $$

converges, then it converges to the root of the equation in part (a).

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(c) Use the iterative formula with initial value $x_{1} = 1.5$ to calculate the root correct to 3 decimal places. Give the result of each iteration to 5 decimal places.

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7. Let $ \displaystyle \mathrm{f}(x) = \frac{2}{(2x \ – \ 1)(2x \ + \ 1)} $.

(a) Express $ \mathrm{f}(x)$ in partial fractions.

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(b) Using your answer to part (a), show that

$ {(\mathrm{f}(x))}^{2} = \frac{1}{ {(2x – 1)}^{2}} – \frac{1}{(2x – 1)} + \frac{1}{(2x + 1)} + \frac{1}{ {(2x + 1)}^{2} } $

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(c) Hence show that $ \displaystyle \int_{1}^{2} {(\mathrm{f}(x))}^{2} \mathrm{d}x = \frac{2}{5} + \frac{1}{2} \ln \Big(\frac{5}{9}\Big)$.

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8. Relative to the origin $O$, the points $A$, $B$ and $D$ have position vectors given by

$ \hspace{2em} \ {\small \overrightarrow{OA}} \ = \ \mathbf{i} + + 2 \mathbf{j} + \mathbf{k} $, $ \ {\small \overrightarrow{OB}} \ = \ 2 \mathbf{i} + 5 \mathbf{j} + 3\mathbf{k}$ and $ \ {\small \overrightarrow{OC}} \ = \ 3\mathbf{i} + 2\mathbf{k} $.

A fourth point $C$ is such that $ABCD$ is a parallelogram.

(a) Find the position vector of $C$ and verify that the parallelogram is not a rhombus.

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(b) Find angle $BAD$, giving your answer in degrees.

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(c) Find the area of the parallelogram correct to 3 significant figures.

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9. (a) The complex numbers $u$ and $w$ are such that

$$ \hspace{2em} u \ – \ w = 2\mathrm{i} \enspace \mathrm{and} \enspace uw = 6.$$

Find $u$ and $w$, giving your answers in the form $x + \mathrm{i}y$, where $x$ and $y$ are real and exact.

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(b) On a sketch of an Argand diagram, shade the region whose points represent complex numbers $z$ satisfying the inequalities

$$ |z – 2 – 2\mathrm{i}| \le 2 , \ 0 \le \mathrm{arg} \ z \le \frac{1}{4} \pi \ \mathrm{and} \ \mathrm{Re} \ z \le 3. $$

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10.

A tank containing water is in the form of a hemisphere. The axis is vertical, the lowest point is $A$ and the radius is $r$, as shown in the diagram. The depth of water at time $t$ is $h$. At time $t = 0$ the tank is full and the depth of the water is $r$. At this instant a tap at $A$ is opened and water begins to flow out at a rate proportional to $\sqrt{h}$. The tank becomes empty at time $t = 14$.

The volume of water in the tank is $V$ when the depth is $h$. It is given that $ V = \frac{1}{ 3} \pi (3r{h}^{2} − h^3)$.

(a) Show that $h$ and $t$ satisfy a differential equation of the form

$$ \frac{ \mathrm{d}h }{ \mathrm{d}t } = – \frac{B}{ 2r{h}^{\frac{1}{2} } \ – \ {h}^{\frac{3}{2} } } $$

where $B$ is a positive constant.

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(b) Solve the differential equation and obtain an expression for $t$ in terms of $h$ and $r$.

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