Home » 9709 » Paper 33 May June 2020 Pure Math III – 9709/31/M/J/20

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1.     Solve the inequality $| 2x \ − \ 1| \gt 3 |x \ + \ 2|$.
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2.     Find the exact value of $\ \displaystyle \int_{0}^{1} (2 \ – \ x) {\mathrm{e}}^{-2x} \ \mathrm{d}x$.
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3.     (a)     Show that the equation
$$\ln (1 \ + \ \mathrm{e}^{-x}) \ + \ 2x = 0$$
can be expressed as a quadratic equation in $\mathrm{e}^x$.
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(b)     Hence solve the equation $\ln (1 \ + \ \mathrm{e}^{-x}) \ + \ 2x = 0$, giving your answer correct to 3 decimal places.
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4.     The equation of a curve is $\ y = x \ {\tan}^{-1} \big( \frac{1}{2} x\big)$ .

(a)     Find $\frac{\mathrm{d}y}{\mathrm{d}x}$.
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(b)     The tangent to the curve at the point where $x = 2$ meets the $y$-axis at the point with coordinates $(0, p)$. Find $p$.
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5.     By first expressing the equation
$$\tan \theta \ \tan (\theta \ + \ {45}^{\circ}) = 2 \cot 2\theta$$
as a quadratic equation in $\tan \theta$, solve the equation for ${0}^{\circ} \lt \theta \lt {90}^{\circ}$.
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6.    (a)     By sketching a suitable pair of graphs, show that the equation $x^5 = 2 + x$ has exactly one real root.
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(b)     Show that if a sequence of values given by the iterative formula
$$x_{n+1} = \Big(\frac{4{x}_{n}^{5} \ + \ 2}{5{x}_{n}^{4} \ – \ 1}\Big)$$
converges, then it converges to the root of the equation in part (a).
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(c)     Use the iterative formula with initial value $x_{1} = 1.5$ to calculate the root correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
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7.     Let $\displaystyle \mathrm{f}(x) = \frac{2}{(2x \ – \ 1)(2x \ + \ 1)}$.

(a)     Express $\mathrm{f}(x)$ in partial fractions.
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${(\mathrm{f}(x))}^{2} = \frac{1}{ {(2x – 1)}^{2}} – \frac{1}{(2x – 1)} + \frac{1}{(2x + 1)} + \frac{1}{ {(2x + 1)}^{2} }$
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(c)     Hence show that $\displaystyle \int_{1}^{2} {(\mathrm{f}(x))}^{2} \mathrm{d}x = \frac{2}{5} + \frac{1}{2} \ln \Big(\frac{5}{9}\Big)$.
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8.     Relative to the origin $O$, the points $A$, $B$ and $D$ have position vectors given by

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

A fourth point $C$ is such that $ABCD$ is a parallelogram.

(a)     Find the position vector of $C$ and verify that the parallelogram is not a rhombus.
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(b)     Find angle $BAD$, giving your answer in degrees.
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(c)     Find the area of the parallelogram correct to 3 significant figures.
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9.     (a)     The complex numbers $u$ and $w$ are such that

$$\hspace{2em} u \ – \ w = 2\mathrm{i} \enspace \mathrm{and} \enspace uw = 6.$$

Find $u$ and $w$, giving your answers in the form $x + \mathrm{i}y$, where $x$ and $y$ are real and exact.
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(b)     On a sketch of an Argand diagram, shade the region whose points represent complex numbers $z$ satisfying the inequalities

$$|z – 2 – 2\mathrm{i}| \le 2 , \ 0 \le \mathrm{arg} \ z \le \frac{1}{4} \pi \ \mathrm{and} \ \mathrm{Re} \ z \le 3.$$

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10. A tank containing water is in the form of a hemisphere. The axis is vertical, the lowest point is $A$ and the radius is $r$, as shown in the diagram. The depth of water at time $t$ is $h$. At time $t = 0$ the tank is full and the depth of the water is $r$. At this instant a tap at $A$ is opened and water begins to flow out at a rate proportional to $\sqrt{h}$. The tank becomes empty at time $t = 14$.

The volume of water in the tank is $V$ when the depth is $h$. It is given that $V = \frac{1}{ 3} \pi (3r{h}^{2} − h^3)$.

(a)     Show that $h$ and $t$ satisfy a differential equation of the form
$$\frac{ \mathrm{d}h }{ \mathrm{d}t } = – \frac{B}{ 2r{h}^{\frac{1}{2} } \ – \ {h}^{\frac{3}{2} } }$$
where $B$ is a positive constant.
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(b)     Solve the differential equation and obtain an expression for $t$ in terms of $h$ and $r$.
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