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1. Find the set of values of $ \ x \ $ for which $ \ 2({3}^{1 \ − \ 2x}) \lt 5x $. Give your answer in a simplified exact form.

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

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(b) State the set of values of $ \ x \ $ for which the expansion is valid.

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3. Express the equation $ \ \tan(\theta + {60}^{\circ}) = 2 + \tan({60}^{\circ} − \theta) \ $ as a quadratic equation in $ \ \tan \theta $, and hence solve the equation for $ \ {0}^{\circ} \le \theta \le {180}^{\circ} $.

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4. The curve with equation $ \ y = {\mathrm{e}}^{2x}(\sin x + 3 \cos x) \ $ has a stationary point in the interval $ \ 0^{\circ} \le x \le \pi$.

(a) Find the $ \ x $-coordinate of this point, giving your answer correct to 2 decimal places.

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(b) Determine whether the stationary point is a maximum or a minimum.

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5. (a) Find the quotient and remainder when $ \ 2x^3 \ − \ x^2 + 6x + 3 \ $ is divided by $ \ x^2 + 3 $.

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(b) Using your answer to part (a), find the exact value of $ \ \displaystyle \int_{1}^{3} \frac{2x^3 \ − \ x^2 + 6x + 3}{x^2 + 3} \ \mathrm{d}x $.

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6.

The diagram shows a circle with centre $ \ O \ $ and radius $ \ r $. The tangents to the circle at the points $ \ A \ $ and $ \ B \ $ meet at $ \ T \ $, and angle $ \ AOB \ $ is $ \ 2x \ $ radians. The shaded region is bounded by the tangents $ \ AT \ $ and $ \ BT $, and by the minor arc $ \ AB $. The area of the shaded region is equal to the area of the circle.

(a) Show that $ \ x \ $ satisfies the equation $ \ \tan x = \pi + x $.

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(b) This equation has one root in the interval $ \ 0 \lt x \lt \frac{1}{2} \pi $. Verify by calculation that this root lies between 1 and 1.4.

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(c) Use the iterative formula

$$ x_{n+1} = {\tan}^{-1} (\pi + x_n) $$

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

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7. Let $ \displaystyle \mathrm{f}(x) = \frac{\cos x}{1 + \sin x} $.

(a) Show that $ \ {\mathrm{f}}^{‘}(x) \lt 0 \ $ for all $x$ in the interval $ \ -\frac{1}{2} \pi \lt x \lt \frac{3}{2} \pi $.

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(b) Find $ \displaystyle \int_{\frac{1}{6} \pi}^{\frac{1}{2} \pi} \mathrm{f}(x) \mathrm{d}x $. Give your answer in a simplified exact form.

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8. A certain curve is such that its gradient at a point $(x, y)$ is proportional to $ \ \frac{y}{x\sqrt{x}}$. The curve passes through the points with coordinates $(1, 1)$ and $(4, \mathrm{e})$.

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

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(b) Describe what happens to $y$ as $x$ tends to infinity.

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9. With respect to the origin $O$, the vertices of a triangle $ABC$ have position vectors

$ \hspace{2em} \ {\small \overrightarrow{OA}} \ = \ 2 \mathbf{i} + 5\mathbf{k} $, $ \ {\small \overrightarrow{OB}} \ = \ 3 \mathbf{i} + 2 \mathbf{j} + 3\mathbf{k}$ and $ \ {\small \overrightarrow{OC}} \ = \ \mathbf{i} + \mathbf{j} + \mathbf{k} $.

(a) Using a scalar product, show that angle $ABC$ is a right angle.

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(b) Show that triangle $ABC$ is isosceles.

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(c) Find the exact length of the perpendicular from $O$ to the line through $B$ and $C$.

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10. (a) The complex number $u$ is defined by $ u = \large{\frac{3\mathrm{i}}{a + 2\mathrm{i}}}$, where $a$ is real.

(i) Express $u$ in the Cartesian form $x + \mathrm{i}y$, where $x$ and $y$ are in terms of $a$.

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(ii) Find the exact value of $a$ for which $\mathrm{arg} \ {u}^{*} = \frac{1}{3}\pi$.

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(b) (i) On a sketch of an Argand diagram, shade the region whose points represent complex numbers $z$ satisfying the inequalities

$$ |z \ – \ 2 \mathrm{i}| \le |z \ – \ 1 \ – \ \mathrm{i}| \enspace \mathrm{and} \enspace |z \ – \ 2 \ – \ \mathrm{i}| \le 2 . $$

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$\hspace{2em}$ (ii) Calculate the least value of $\mathrm{arg} \ z $ for points in this region.

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