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1. Use the trapezium rule with 3 intervals to estimate the value of

$$ \int_{0}^{3} | {2}^{x} \ – \ 4 | \ \textrm{d}x. $$

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2. Showing all necessary working, solve the equation $ \ \ln (2x \ − \ 3) = 2 \ln x − \ln (x \ − \ 1)$. Give your answer correct to 2 decimal place.

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3. Find the gradient of the curve $ \ x^3 + 3xy^2 − y^3 = 1 \ $ at the point with coordinates $(1, 3)$.

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4. By first expressing the equation $ \cot \theta − \cot(\theta + 45^{\circ}) = 3 \ $ as a quadratic equation in $ \tan \theta$, solve the equation for $ \ 0^{\circ} \lt \theta \lt 180^{\circ}$.

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5. (i) Differentiate $\displaystyle \frac{1}{\sin^{2} \theta}$ with respect to $\theta$.

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(ii) The variables $x$ and $\theta$ satisfy the differential equation

$$ x \tan \theta \frac{\textrm{d}x}{\textrm{d}\theta} + \textrm{cosec}^{2} \theta = 0, $$

for $0 \lt \theta \lt \frac{1}{2} \pi $ and $x \gt 0$. It is given that $x = 4$ when $\theta = \frac{1}{6} \pi$. Solve the differential equation, obtaining an expression for $x$ in terms of $\theta$.

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6. (i) By first expanding $\sin (2x + x)$, show that $\sin 3x \equiv 3 \sin x − 4 \sin^{3} x$.

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(ii) Hence, showing all necessary working, find the exact value of $\displaystyle \int_{0}^{\frac{1}{3} \pi} \sin^{3}x \textrm{d}x$.

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7. The diagram shows the curves $y = 4 \cos \frac{1}{2}x$ and $y = \frac{1}{4 \ – \ x}$, for $0 \leq x \lt 4$. When $x = a$, the tangents to the curves are perpendicular.

(i) Show that $ \ a = 4 \ – \ \sqrt{(2 \sin \frac{1}{2}a)}$.

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(ii) Verify by calculation that a lies between 2 and 3.

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(iii) Use an iterative formula based on the equation in part (i) to determine $a$ correct to 3 decimal places. Give the result of each iteration to 5 decimal places.

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8. Let $ \displaystyle f(x) = \frac{16 \ – \ 17x}{(2+x)(3-x)^{2}} $

(i) Express $f(x)$ in partial fractions.

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(ii) Hence obtain the expansion of $f(x)$ in ascending powers of $x$, up to and including the term in $x^2$.

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9. The diagram shows a set of rectangular axes $Ox$, $Oy$ and $Oz$, and four points $A$, $B$, $C$ and $D$ with position vectors $ \ {\small \overrightarrow{OA}} \ = \ 3 \mathbf{i}$, $ \ {\small \overrightarrow{OB}} \ = \ 3 \mathbf{i} + 4 \mathbf{j}$, $ \ {\small \overrightarrow{OC}} \ = \ \mathbf{i} + 3\mathbf{j}$ and $ \ {\small \overrightarrow{OD}} \ = \ 2\mathbf{i} + 3\mathbf{j} + 5\mathbf{k}$.

(i) Find the equation of the plane $BCD$, giving your answer in the form $ax + by + cz = d$.

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(ii) Calculate the acute angle between the planes $BCD$ and $OABC$.

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10. The complex number $(\sqrt{3}) + \textrm{i}$ is denoted by $u$.

(i) Express $u$ in the form $r{\textrm{e}}^{ \textrm{i} \theta}$, where $r \gt 0$ and $−\pi \lt \theta \leq \pi$, giving the exact values of $r$ and $\theta$. Hence or otherwise state the exact values of the modulus and argument of $u^4$.

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(ii) Verify that $u$ is a root of the equation $z^3 − 8z + 8\sqrt{3} = 0$ and state the other complex root of this equation.

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(iii) On a sketch of an Argand diagram, shade the region whose points represent complex numbers $z$ satisfying the inequalities $|z − u| \leq 2$ and $\textrm{Im} \ z \geq 2$, where $\textrm{Im} \ z$ denotes the imaginary part of $z$.

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