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

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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 \ + \ 1 \ – \ \mathrm{i}| \le 1 \enspace \mathrm{and} \enspace \mathrm{arg} (z \ – \ 1) \le {\large \frac{3}{4}\pi} . $

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3. The variables $x$ and $y$ satisfy the equation $ x = A({3}^{-y})$, where $A$ is a constant.

(a) Explain why the graph of $y$ against $\ln x$ is a straight line and state the exact value of the gradient of the line.

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It is given that the line intersects the $y$-axis at the point where $y = 1.3$.

(b) Calculate the value of $A$, giving your answer correct to 2 decimal places.

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4. Using integration by parts, find the exact value of $\displaystyle \int_{0}^{2} {\tan}^{-1} \Big(\frac{1}{2}x \Big) \ \mathrm{d}x$.

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5. The complex number $ u $ is given by $ u = 10 \ – \ 4\sqrt{6} \mathrm{i}$.

Find the two square roots of $u$, giving your answers in the form $a + \mathrm{i}b$, where $a$ and $b$ are real and exact.

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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 curve is such that the gradient at a general point with coordinates $(x, y)$ is proportional to $ {\large \frac{y}{\sqrt{ x + 1 } } }$.

The curve passes through the points with coordinates $(0, 1)$ and $(3, \mathrm{e})$.

By setting up and solving a differential equation, find the equation of the curve, expressing $y$ in terms of $x$.

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8. The equation of a curve is $ y = {\mathrm{e}}^{-5x} \ {\tan}^{2}x$ for $ – \frac{1}{2} \pi \lt x \lt \frac{1}{2} \pi $.

Find the $x$-coordinates of the stationary points of the curve. Give your answers correct to 3 decimal places where appropriate.

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9. Let $ \mathrm{f}(x) = {\large \frac{14 \ – \ 3x \ + \ 2x^2}{(2 \ + \ x)(3 \ + \ x^2)}}$.

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

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(b) Hence obtain the expansion of $ \mathrm{f}(x) $ in ascending powers of $x$, up to and including the term in $x^2$.

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10.

(a) Given that the sum of the areas of the shaded sectors is 90% of the area of the trapezium, show that $x$ satisfies the equation $x = 0.9(2 \ − \ \cos x) \sin x$.

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(b) Verify by calculation that $x$ lies between 0.5 and 0.7.

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(c) Show that if a sequence of values in the interval $0 \lt x \lt \frac{1}{2} \pi$ given by the iterative formula

$$ x_{n+1} = {\cos}^{-1} \Big( 2 \ – \ \frac{x_{n}}{0.9 \sin x_{n}} \Big)$$

converges, then it converges to the root of the equation in part (a).

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(d) Use this iterative formula to determine $x$ correct to 2 decimal places. Give the result of each iteration to 4 decimal places.

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11. With respect to the origin $O$, the points $A$ and $B$ have position vectors given by $ \ {\small \overrightarrow{OA}} \ = \ 2 \mathbf{i} \ – \ \mathbf{j} \ $ and $ \ {\small \overrightarrow{OB}} \ = \ \mathbf{j} \ – \ 2 \mathbf{k} $.

(a) Show that $OA = OB$ and use a scalar product to calculate angle $AOB$ in degrees.

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The midpoint of $AB$ is $M$. The point $P$ on the line through $O$ and $M$ is such that $PA : OA = \sqrt{7} : 1$.

(b) Find the possible position vectors of $P$.

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