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1. (i) Show that the equation $ \ \log_{10} (x \ − \ 4) = 2 \ − \ \log_{10} x \ $ can be written as a quadratic equation in $x$.

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(ii) Hence solve the equation $ \ \log_{10} (x \ − \ 4) = 2 \ − \ \log_{10} x $, giving your answer correct to 3 significant figures.

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2. The sequence of values given by the iterative formula

$$ x_{n+1} = \frac{2x_{n}^{6}+12x_{n}}{3x_{n}^{5}+8},$$

with initial value $x_{1} = 2$, converges to $\alpha$.

(i) Use the formula to calculate $\alpha$ correct to 4 decimal places. Give the result of each iteration to 6 decimal places.

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(ii) State an equation satisfied by $\alpha$ and hence find the exact value of $\alpha$.

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3. (i) Given that $ \ \sin(\theta + 45^{\circ}) + 2 \cos(\theta + 60^{\circ}) = 3 \cos \theta $, find the exact value of $\tan \theta$ in a form involving surds. You need not simplify your answer.

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(ii) Hence solve the equation $ \ \sin(\theta + 45^{\circ}) + 2 \cos(\theta + 60^{\circ}) = 3 \cos \theta \ $ for $ \ 0^{\circ} \lt \theta \lt 360^{\circ}$.

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4. Show that $\displaystyle \int_{1}^{4} x^{ -\frac{3}{2} } \ln x \ \textrm{d}x = 2 \ – \ \ln 4.$

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5. The variables $x$ and $y$ satisfy the relation $ \sin y = \tan x$, where $ -\frac{1}{2} \pi \lt y \lt \frac{1}{2} \pi.$ Show that

$$ \frac{\textrm{d}y}{\textrm{d}x} = \frac{1}{ \cos x \sqrt{ (\cos 2x) }} . $$

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

$$ \frac{\textrm{d}y}{\textrm{d}x} = k y^{3} \textrm{e}^{-x}, $$

where $k$ is a constant. It is given that $ \ y = 1 \ $ when $ \ x = 0 \ $, and that $ \ y = \sqrt{\textrm{e}} \ $ when $ \ x = 1 $. Solve the differential equation, obtaining an expression for $y$ in terms of $x$.

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7. (a) Showing all working and without using a calculator, solve the equation

$$ (1 + \textrm{i})z^{2} \ – \ (4 + 3 \textrm{i})z + 5 + \textrm{i} = 0. $$

Give your answers in the form $x + \textrm{i}y$, where $x$ and $y$ are real.

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(b) The complex number $u$ is given by

$$ u = -1 \ – \ \textrm{i}. $$

On a sketch of an Argand diagram show the point representing $u$. Shade the region whose points represent complex numbers satisfying the inequalities $ \ |z| \lt |z \ – \ 2 \textrm{i}| \ $ and $ \ \frac{1}{4} \pi \lt \textrm{arg}(z \ – \ u) \lt \frac{1}{2} \pi .$

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8. Let $ \displaystyle f(x) = \frac{12+12x-4x^{2}}{(2+x)(3 \ – \ 2x)}. $

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

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(b) 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. Two planes have equations $ \ 2x + 3y \ − \ z = 1 \ $ and $ \ x \ − \ 2y + z = 3.$

(i) Find the acute angle between the planes.

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(ii) Find a vector equation for the line of intersection of the planes.

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10. The diagram shows the curve $ \ y = \sin^{3} x \sqrt{ (\cos x)} \ $ for $ \ 0 \leq x \leq \frac{1}{2} \pi $, and its maximum point $M$.

(i) Using the substitution $ \ u = \cos x $, find by integration the exact area of the shaded region bounded by the curve and the $x$-axis.

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(ii) Showing all your working, find the $x$-coordinate of $M$, giving your answer correct to 3 decimal places.

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