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

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2. On a sketch of an Argand diagram, shade the region whose points represent complex numbers $z$ satisfying the inequalities $ |z \ + \ 2 \ – \ 3\mathrm{i}| \le 2 \enspace \mathrm{and} \enspace \mathrm{arg} \ z \le {\large \frac{3}{4}\pi} . $

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3.

The variables $x$ and $y$ satisfy the equation ${x}^{n}{y}^{2} = C$, where $n$ and $C$ are constants. The graph of $ \ln y$ against $\ln x$ is a straight line passing through the points $(0.31, 1.21)$ and $(1.06, 0.91)$, as shown in the diagram.

Find the value of $n$ and find the value of $C$ correct to 2 decimal places.

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4. The parametric equations of a curve are

$$ x = 1 \ – \cos \theta, \enspace y = \cos \theta \ – \ \frac{1}{4} \cos 2\theta.$$

Show that $\displaystyle \frac{\mathrm{d}y}{\mathrm{d}x}= -2 \ {\sin}^{2} \Big(\frac{1}{2} \theta \Big)$.

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5. The angles $\alpha$ and $\beta$ lie between ${0}^{\circ}$ and ${180}^{\circ}$ and are such that

$$ \tan(\alpha \ + \ \beta) = 2 \enspace \mathrm{and} \enspace \tan \alpha = 3 \tan \beta.$$

Find the possible values of $\alpha$ and $\beta$ .

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6. (a) Prove that $ \mathrm{cosec} \ 2\theta \ − \ \cot 2 \theta \equiv \tan \theta$.

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(b) Hence show that $\displaystyle \int_{\frac{1}{4} \pi}^{\frac{1}{3} \pi} (\mathrm{cosec} \ 2\theta \ − \ \cot 2 \theta) \ \mathrm{d}\theta = \frac{1}{2} \ln 2$.

7. (a) By sketching a suitable pair of graphs, show that the equation $ 4 \ – \ x^2 = \sec \frac{1}{2}x$ has exactly one root in the interval $ 0 \le x \lt \pi$.

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(b) Verify by calculation that this root lies between 1 and 2.

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(c) Use the iterative formula $ \displaystyle x_{n+1} = \sqrt{4 \ − \ \sec \frac{1}{2} x_{n}}$ to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

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8. (a) Find the quotient and remainder when $8x^3 + 4x^2 + 2x + 7$ is divided by $4x^2 + 1$.

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(b) Hence find the exact value of $ \displaystyle \int_{0}^{\frac{1}{2}} \frac{8x^3 + 4x^2 + 2x + 7}{4x^2 + 1} \ \mathrm{d}x$.

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9. The variables $x$ and $y$ satisfy the differential equation

$$ (x \ + \ 1)(3x \ + \ 1) \frac{ \mathrm{d}y }{ \mathrm{d}x } = y,$$

and it is given that $y = 1$ when $x = 1$.

Solve the differential equation and find the exact value of $y$ when $x = 3$, giving your answer in a simplified form.

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10. The points $A$ and $B$ have position vectors $2\mathbf{i} + \mathbf{j} + \mathbf{k}$ and $\mathbf{i} \ − \ 2\mathbf{j} + 2\mathbf{k}$ respectively. The line $l$ has vector equation $ \mathbf{r} = \mathbf{i} + 2\mathbf{j} \ – \ 3\mathbf{k} + \mu (\mathbf{i} \ – \ 3\mathbf{j} \ – \ 2\mathbf{k})$.

(a) Find a vector equation for the line through $A$ and $B$.

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(b) Find the acute angle between the directions of $AB$ and $l$, giving your answer in degrees.

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(c) Show that the line through $A$ and $B$ does not intersect the line $l$.

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11.

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

(a) Find the $x$-coordinate of $M$, giving your answer correct to 3 significant figures.

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(b) Using the substitution $u = \cos x$, find the area of the shaded region enclosed by the curve and the $x$-axis in the first quadrant, giving your answer in a simplified exact form.

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