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1. Expand ${(1 \ + \ 3x)}^{\frac{2}{3}}$ in ascending powers of $x$, up to and including the term in $x^3$, simplifying the coefficients.

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2. Solve the equation $4^x = 3 \ + \ {4}^{−x}$. Give your answer correct to 3 decimal places.

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3. The parametric equations of a curve are

$$ x = t \ + \ \ln(t \ + \ 2), \enspace y = (t \ – \ 1){\mathrm{e}}^{-2t},$$

where $t \gt -2$.

(a) Express $ {\large \frac{\mathrm{d}y}{ \mathrm{d}x} } $ in terms of $t$, simplifying your answer.

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(b) Find the exact $y$-coordinate of the stationary point of the curve.

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4. Let $ \mathrm{f}(x) = {\large \frac{15 \ – \ 6x}{(1 \ + \ 2x)(4 \ – \ x)}}$.

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

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(b) Hence find $\displaystyle \int_{1}^{2} \mathrm{f}(x) \ \mathrm{d}x$, giving your answer in the form $ \ln \Big({\large\frac{a}{b}}\Big) $ where $a$ and $b$ are integers.

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5. (a) By first expanding $ \tan (2\theta \ + \ 2\theta)$, show that the equation $ \tan 4\theta = {\large\frac{1}{2}} \tan \theta$ may be expressed as $ {\tan}^{4} \theta \ + \ 2 {\tan}^{2} \theta \ – \ 7 = 0$.

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(b) Hence solve the equation $ \tan 4\theta = {\large\frac{1}{2}} \tan \theta$, for ${0}^{\circ} \lt \theta \lt {180}^{\circ} $.

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6. (a) By sketching a suitable pair of graphs, show that the equation $ \cot {\large\frac{1}{2}x} = 1 \ + \ {\mathrm{e}}^{-x}$ has exactly one root in the interval $0 \lt x\le \pi$.

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(b) Verify by calculation that this root lies between 1 and 1.5.

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(c) Use the iterative formula

$$ x_{n+1} = 2{\tan}^{-1} \Big( {\large\frac{1}{1 \ + \ {\mathrm{e}}^{-x_{n}}} } \Big) $$

to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

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7.

For the curve shown in the diagram, the normal to the curve at the point $P$ with coordinates $(x, y)$ meets the $x$-axis at $N$. The point $M$ is the foot of the perpendicular from $P$ to the $x$-axis.

The curve is such that for all values of $x$ in the interval $0 \le x \lt \frac{1}{2} \pi$, the area of triangle $PMN$ is equal to $\tan x$.

(a) (i) Show that $ \frac{MN}{y} = \frac{\mathrm{d}y}{\mathrm{d}x}$.

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(ii) Hence show that $x$ and $y$ satisfy the differential equation $ \frac{1}{2} {y}^{2} \frac{\mathrm{d}y}{\mathrm{d}x} = \tan x$.

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(b) Given that $y = 1$ when $x = 0$, solve this differential equation to find the equation of the curve, expressing $y$ in terms of $x$.

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8.

The diagram shows the curve $y = {\large\frac{\ln x}{x^4} }$ and its maximum point $M$.

(a) Find the exact coordinates of $M$.

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(b) By using integration by parts, show that for all $a \gt 1$, $\displaystyle \int_{1}^{a} \frac{\ln x}{x^4} \ \mathrm{d}x \lt \frac{1}{9}$.

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9. The quadrilateral $ABCD$ is a trapezium in which $AB$ and $DC$ are parallel. With respect to the origin $O$, the position vectors of $A$, $B$ and $C$ are given by $ \ {\small \overrightarrow{OA}} \ = \ -\mathbf{i} + 2 \mathbf{j} + 3\mathbf{k} $, $ \ {\small \overrightarrow{OB}} \ = \ \mathbf{i} + 3 \mathbf{j} + \mathbf{k}$ and $ \ {\small \overrightarrow{OC}} \ = \ 2\mathbf{i} + 2\mathbf{j} \ – \ 3 \mathbf{k} $.

(a) Given that $ \ {\small \overrightarrow{DC}} = 3{\small \overrightarrow{AB}} \ $ find the position vector of $D$.

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

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(c) Find the distance between the parallel sides and hence find the area of the trapezium.

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10. (a) Verify that $ -1 \ + \ \sqrt{2}\mathrm{i} $ is a root of the equation $ z^4 \ + \ 3z^2 \ + \ 2z \ + \ 12 = 0$.

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(b) Find the other roots of this equation.

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