Home » 9709 » Paper 32 Feb Mar 2021 Pure Math III – 9709/32/F/M/21

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1.     Solve the equation $\ln (x^3 − 3) = 3 \ln x \ − \ \ln 3$. Give your answer correct to 3 significant figures.
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2.     The polynomial $ax^3 + 5x^2 − 4x + b$, where $a$ and $b$ are constants, is denoted by $\mathrm{p}(x)$. It is given that $(x + 2)$ is a factor of $\mathrm{p}(x)$ and that when $\mathrm{p}(x)$ is divided by $(x + 1)$ the remainder is 2.

Find the values of $a$ and $b$.
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3.     By first expressing the equation $\tan(x + {45}^{\circ}) = 2 \cot x + 1$ as a quadratic equation in $\tan x$, solve the equation for ${0}^{\circ} \lt x \lt {180}^{\circ}$.
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4.     The variables x and y satisfy the differential equation
$$(1 \ – \ \cos x) \frac{\mathrm{d}y}{\mathrm{d}x} = y \sin x.$$
It is given that $y = 4$ when $x = \pi$.

(a)     Solve the differential equation, obtaining an expression for $y$ in terms of $x$.
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(b)     Sketch the graph of $y$ against $x$ for $0 \lt x \lt 2\pi$.
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5.     (a)     Express $\sqrt{7} \sin x + 2 \cos x$ in the form $R \sin(x + \alpha)$, where $R \gt 0$ and ${0}^{\circ} \lt \alpha \lt {90}^{\circ}$. State the exact value of $R$ and give $\alpha$ correct to 2 decimal places.
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(b)    Hence solve the equation $\sqrt{7} \sin 2\theta + 2 \cos 2\theta = 1$, for ${0}^{\circ} \lt \theta \lt {180}^{\circ}$.
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6.    Let $\displaystyle \mathrm{f}(x) = \frac{ 5a }{ (2x \ – \ a)(3a \ – \ x) }$ where $a$ is a positive constant.

(a)     Express $\mathrm{f}(x)$ in partial fractions.
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(b)     Hence show that $\displaystyle \int_{a}^{2a} \mathrm{f}(x) \ \mathrm{d}x = \ln 6$.
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7.     Two lines have equations $\ \mathbf{r} = \begin{pmatrix} 1 \\[1pt] 3 \\[1pt] 2 \end{pmatrix} + s\begin{pmatrix} 2 \\[1pt] -1 \\[1pt] 3 \end{pmatrix} \$ and $\ \mathbf{r} = \begin{pmatrix} 2 \\[1pt] 1 \\[1pt] 4 \end{pmatrix} + t \begin{pmatrix} 1 \\[1pt] -1 \\[1pt] 4 \end{pmatrix}$.

(a)     Show that the lines are skew.
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(b)     Find the acute angle between the directions of the two lines.
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8.     The complex numbers $u$ and $v$ are defined by $u = −4 \ + \ 2\mathrm{i}$ and $v = 3 \ + \ \mathrm{i}$.

(a)     Find ${\large\frac{u}{v}}$ in the form $x + \mathrm{i}y$, where $x$ and $y$ are real.
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(b)     Hence express ${\large\frac{u}{v}}$ in the form $r { \mathrm{e} }^{ \mathrm{i}\theta}$, where $r$ and $\theta$ are exact.
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In an Argand diagram, with origin $O$, the points $A$, $B$ and $C$ represent the complex numbers $u$, $v$ and $2u \ + \ v$ respectively.

(c)     State fully the geometrical relationship between $OA$ and $BC$.
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(d)     Prove that angle $AOB = \frac{3}{4} \pi$.
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9.     Let $\displaystyle \mathrm{f}(x) = \frac{ {\mathrm{e}}^{2x} \ + \ 1 }{ {\mathrm{e}}^{2x} \ – \ 1 }$, for $x \gt 0$.

(a)     The equation $x = \mathrm{f}(x)$ has one root, denoted by $a$.

Verify by calculation that $a$ lies between 1 and 1.5.
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(b)     Use an iterative formula based on the equation in part (a) to determine $a$ correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
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(c)     Find ${ \mathrm{f} }^{‘}(x)$. Hence find the exact value of $x$ for which ${ \mathrm{f} }^{‘}(x) = −8$.
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10. The diagram shows the curve $y = \sin 2x \ {\cos}^{2} x$ for $0 \le x \le \frac{1}{ 2} \pi$, and its maximum point $M$.

(a)     Using the substitution $u = \sin x$, find the exact area of the region bounded by the curve and the $x$-axis.
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(b)     Find the exact $x$-coordinate of $M$.
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