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1. Solve the equation $ \ \ln(1+2^{x}) = 2$, giving your answer correct to 3 decimal places.

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2. Solve the inequality $ \ |x − 4| \lt 2|3x + 1|$.

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3. (i) By sketching suitable graphs, show that the equation $ \ \displaystyle \textrm{e}^{−\frac{1}{2}x} = 4 \ – \ x^{2} \ $ has one positive root and one negative root.

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(ii) Verify by calculation that the negative root lies between −1 and −1.5.

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

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4. (i) Express $ \ 8 \cos \theta − 15 \sin \theta \ $ in the form $ \ R \cos (\theta + \alpha)$, where $ \ R \gt 0 \ $ and $ \ 0 \lt \alpha \lt 90^{\circ}$, stating the exact value of $R$ and giving the value of $\alpha$ correct to 2 decimal places.

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(ii) Hence solve the equation

$$ 8 \cos 2x − 15 \sin 2x = 4, $$

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

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5. The curve with equation $ \ \displaystyle y = \textrm{e}^{−ax} \tan x $, where $a$ is a positive constant, has only one point in the interval $ \ 0 \lt x \lt \frac{1}{2} \pi \ $ at which the tangent is parallel to the $x$-axis. Find the value of $a$ and state the exact value of the $x$-coordinate of this point.

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6. The line $l$ has equation $ \ \mathbf{r} = \mathbf{i} + 2 \mathbf{j} \ − \ 3 \mathbf{k} + \lambda ( 2 \mathbf{i} \ − \ \mathbf{j} + \mathbf{k} ) $. The plane $p$ has equation $ \ 3x + y \ − \ 5z = 20.$

(i) Show that the line $l$ lies in the plane $p$.

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(ii) A second plane is parallel to $l$, perpendicular to $p$ and contains the point with position vector $ \ 3\mathbf{i} \ − \ \mathbf{j} + 2 \mathbf{k}$. Find the equation of this plane, giving your answer in the form $ \ ax + by + cz = d $.

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7. A water tank has vertical sides and a horizontal rectangular base, as shown in the diagram. The area of the base is $2 \ \textrm{m}^{2}$. At time $t = 0 $ the tank is empty and water begins to flow into it at a rate of $1 \ \textrm{m}^{3}$ per hour. At the same time water begins to flow out from the base at a rate of $0.2 \sqrt{h} \ \textrm{m}^3$

per hour, where $h \ \textrm{m}$ is the depth of water in the tank at time $t$ hours.

(i) Form a differential equation satisfied by $h$ and $t$, and show that the time $T$ hours taken for the depth of water to reach 4 m is given by

$$ T = \int_{0}^{4} \frac{10}{ 5 \ – \ \sqrt{h} } \textrm{d}h $$

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(ii) Using the substitution $u = 5 \ – \ \sqrt{h}$, find the value of $T$.

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8. The polynomial $ \ z^4 + 3z^2 + 6z + 10 \ $ is denoted by $ \ \textrm{p}(z)$. The complex number $ −1 + i \ $ is denoted by $u$.

(i) Showing all your working, verify that $u$ is a root of the equation $ \ \textrm{p}(z) = 0.$

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(ii) Find the other three roots of the equation $ \ \textrm{p}(z) = 0$.

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9. Let $ \displaystyle f(x) = \frac{ x(6 \ – \ x) }{ (2+x)(4+x)^{2}}. $

(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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10. The diagram shows the curve $ \ y = {(\ln x)}^{2}$. The $x$-coordinate of the point $P$ is equal to $\textrm{e}$, and the normal to the curve at $P$ meets the $x$-axis at $Q$.

(i) Find the $x$-coordinate of $Q$.

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(ii) Show that $ \ \displaystyle \int \ln x \ \textrm{d}x = x \ln x \ – \ x + c$, where $c$ is a constant.

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(iii) Using integration by parts, or otherwise, find the exact value of the area of the shaded region between the curve, the $x$-axis and the normal $PQ$.

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