Home » 9709 » Paper 31 May June 2019 Pure Math III – 9709/31/M/J/19

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1.     Use the trapezium rule with 3 intervals to estimate the value of
$$\int_{0}^{3} | {2}^{x} \ – \ 4 | \ \textrm{d}x.$$
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2.     Showing all necessary working, solve the equation $\ \ln (2x \ − \ 3) = 2 \ln x − \ln (x \ − \ 1)$. Give your answer correct to 2 decimal place.
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3.     Find the gradient of the curve $\ x^3 + 3xy^2 − y^3 = 1 \$ at the point with coordinates $(1, 3)$.
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4.     By first expressing the equation $\cot \theta − \cot(\theta + 45^{\circ}) = 3 \$ as a quadratic equation in $\tan \theta$, solve the equation for $\ 0^{\circ} \lt \theta \lt 180^{\circ}$.
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5.     (i)     Differentiate $\displaystyle \frac{1}{\sin^{2} \theta}$ with respect to $\theta$.
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(ii)     The variables $x$ and $\theta$ satisfy the differential equation
$$x \tan \theta \frac{\textrm{d}x}{\textrm{d}\theta} + \textrm{cosec}^{2} \theta = 0,$$
for $0 \lt \theta \lt \frac{1}{2} \pi$ and $x \gt 0$. It is given that $x = 4$ when $\theta = \frac{1}{6} \pi$. Solve the differential equation, obtaining an expression for $x$ in terms of $\theta$.
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6.     (i)     By first expanding $\sin (2x + x)$, show that $\sin 3x \equiv 3 \sin x − 4 \sin^{3} x$.
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(ii)     Hence, showing all necessary working, find the exact value of $\displaystyle \int_{0}^{\frac{1}{3} \pi} \sin^{3}x \textrm{d}x$.
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7.     The diagram shows the curves $y = 4 \cos \frac{1}{2}x$ and $y = \frac{1}{4 \ – \ x}$, for $0 \leq x \lt 4$. When $x = a$, the tangents to the curves are perpendicular.

(i)     Show that $\ a = 4 \ – \ \sqrt{(2 \sin \frac{1}{2}a)}$.
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(ii)     Verify by calculation that a lies between 2 and 3.
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(iii)   Use an iterative formula based on the equation in part (i) to determine $a$ correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
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8.     Let $\displaystyle f(x) = \frac{16 \ – \ 17x}{(2+x)(3-x)^{2}}$

(i)     Express $f(x)$ in partial fractions.
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(ii)     Hence obtain the expansion of $f(x)$ in ascending powers of $x$, up to and including the term in $x^2$.
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9.     The diagram shows a set of rectangular axes $Ox$, $Oy$ and $Oz$, and four points $A$, $B$, $C$ and $D$ with position vectors $\ {\small \overrightarrow{OA}} \ = \ 3 \mathbf{i}$, $\ {\small \overrightarrow{OB}} \ = \ 3 \mathbf{i} + 4 \mathbf{j}$, $\ {\small \overrightarrow{OC}} \ = \ \mathbf{i} + 3\mathbf{j}$ and $\ {\small \overrightarrow{OD}} \ = \ 2\mathbf{i} + 3\mathbf{j} + 5\mathbf{k}$.

(i)     Find the equation of the plane $BCD$, giving your answer in the form $ax + by + cz = d$.
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(ii)     Calculate the acute angle between the planes $BCD$ and $OABC$.
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10.     The complex number $(\sqrt{3}) + \textrm{i}$ is denoted by $u$.
(i)     Express $u$ in the form $r{\textrm{e}}^{ \textrm{i} \theta}$, where $r \gt 0$ and $−\pi \lt \theta \leq \pi$, giving the exact values of $r$ and $\theta$. Hence or otherwise state the exact values of the modulus and argument of $u^4$.
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(ii)     Verify that $u$ is a root of the equation $z^3 − 8z + 8\sqrt{3} = 0$ and state the other complex root of this equation.
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(iii)     On a sketch of an Argand diagram, shade the region whose points represent complex numbers $z$ satisfying the inequalities $|z − u| \leq 2$ and $\textrm{Im} \ z \geq 2$, where $\textrm{Im} \ z$ denotes the imaginary part of $z$.
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