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1. Find the quotient and remainder when $ 6x^4 + x^3 − x^2 + 5x − 6 $ is divided by $ 2x^2 − x + 1$.

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2.

The variables $x$ and $y$ satisfy the equation $y^2 = A{e}^{kx}$, where $A$ and $k$ are constants. The graph of $ \ln y$ against $x$ is a straight line passing through the points $(1.5, 1.2)$ and $(5.24, 2.7)$ as shown in the diagram.

Find the values of $A$ and $k$ correct to 2 decimal places.

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3. Find the exact value of $ \displaystyle \int_{1}^{4} x^{\frac{3}{2}} \ln x \ \mathrm{d}x $.

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4. A curve has equation $ y = \cos x \sin 2x$.

Find the $x$-coordinate of the stationary point in the interval $0 \lt x \lt \frac{1}{2} \pi$, giving your answer correct to 3 significant figures.

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5. (a) Express $\sqrt{2} \cos x − \sqrt{5} \sin x$ in the form $R \cos(x + \alpha)$, where $R \gt 0$ and $ \ {0}^{\circ} \lt \alpha \lt {90}^{\circ}$. Give the exact value of $R$ and the value of $\alpha$ correct to 3 decimal places.

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(b) Hence solve the equation $\sqrt{2} \cos 2\theta − \sqrt{5} \sin 2\theta = 1$, for $ \ {0}^{\circ} \lt \theta \lt {180}^{\circ}$.

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6.

The diagram shows the curve $ y = {\large \frac{x}{1 + 3x^4} }$, for $ x \gt 0$, and its maximum point $M$.

(a) Find the $x$-coordinate of $M$, giving your answer correct to 3 decimal places.

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(b) Using the substitution $ u = \sqrt{3} x^2$, find by integration the exact area of the shaded region bounded by the curve, the $x$-axis and the line $x = 1$.

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

$$ \hspace{2em} \frac{\mathrm{d}y}{\mathrm{d}x} = \frac{y-1}{(x+1)(x+3)} .$$

It is given that $y = 2$ when $x = 0$.

Solve the differential equation, obtaining an expression for $y$ in terms of $x$.

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8. (a) Solve the equation $ (1 + 2\mathrm{i})w + \mathrm{i}{w}^{*} = 3 + 5\mathrm{i}$. Give your answer in the form $x + \mathrm{i}y$, where $x$ and $y$ are real.

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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

$\hspace{2em} |z \ – \ 2 \ – \ 2\mathrm{i}| \le 1 \enspace $ and $ \enspace \mathrm{arg}(z \ – \ 4\mathrm{i}) \ge -\frac{1}{4}\pi $.

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$\hspace{2em}$ (ii) Find the least value of $\mathrm{Im} \ z$ for points in this region, giving your answer in an exact form.

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9.

The diagram shows the curves $y = \cos x $ and $y = \frac{k}{1 \ + \ x} $, where $k$ is a constant, for $ 0\le x \le \frac{1}{2}\pi $. The curves touch at the point where $x = p$.

(a) Show that $p$ satisfies the equation $ \tan p = \frac{1}{1 \ + \ p}$.

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(b) Use the iterative formula $p_{n+1} = {\tan}^{−1} \Big( \frac{1}{1 \ + \ p_{n}}\Big)$ to determine the value of $p$ correct to 3 decimal places. Give the result of each iteration to 5 decimal places.

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(c) Hence find the value of $k$ correct to 2 decimal places.

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10. With respect to the origin $O$, the points $A$ and $B$ have position vectors given by $ \overrightarrow{OA} \ = \ 6 \mathbf{i} + 2\mathbf{j} $ and $ \overrightarrow{OB} \ = \ 2 \mathbf{i} + 2\mathbf{j} + 3\mathbf{k} $. The midpoint of $OA$ is $M$. The point $N$ lying on $AB$, between $A$ and $B$, is such that $AN = 2NB$.

(a) Find a vector equation for the line through $M$ and $N$.

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The line through $M$ and $N$ intersects the line through $O$ and $B$ at the point $P$.

(b) Find the position vector of $P$.

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(c) Calculate angle $OPM$, giving your answer in degrees.

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