Home » 9709 » Paper 32 Feb March 2019 Pure Math III – 9709/32/F/M/19

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1.     (i)     Show that the equation $\ \log_{10} (x \ − \ 4) = 2 \ − \ \log_{10} x \$ can be written as a quadratic equation in $x$.
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(ii)     Hence solve the equation $\ \log_{10} (x \ − \ 4) = 2 \ − \ \log_{10} x$, giving your answer correct to 3 significant figures.
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2.     The sequence of values given by the iterative formula
$$x_{n+1} = \frac{2x_{n}^{6}+12x_{n}}{3x_{n}^{5}+8},$$
with initial value $x_{1} = 2$, converges to $\alpha$.
(i)     Use the formula to calculate $\alpha$ correct to 4 decimal places. Give the result of each iteration to 6 decimal places.
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(ii)     State an equation satisfied by $\alpha$ and hence find the exact value of $\alpha$.
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3.     (i)     Given that $\ \sin(\theta + 45^{\circ}) + 2 \cos(\theta + 60^{\circ}) = 3 \cos \theta$, find the exact value of $\tan \theta$ in a form involving surds. You need not simplify your answer.
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(ii)     Hence solve the equation $\ \sin(\theta + 45^{\circ}) + 2 \cos(\theta + 60^{\circ}) = 3 \cos \theta \$ for $\ 0^{\circ} \lt \theta \lt 360^{\circ}$.
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4.     Show that $\displaystyle \int_{1}^{4} x^{ -\frac{3}{2} } \ln x \ \textrm{d}x = 2 \ – \ \ln 4.$
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5.     The variables $x$ and $y$ satisfy the relation $\sin y = \tan x$, where $-\frac{1}{2} \pi \lt y \lt \frac{1}{2} \pi.$ Show that
$$\frac{\textrm{d}y}{\textrm{d}x} = \frac{1}{ \cos x \sqrt{ (\cos 2x) }} .$$
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6.     The variables $x$ and $y$ satisfy the differential equation
$$\frac{\textrm{d}y}{\textrm{d}x} = k y^{3} \textrm{e}^{-x},$$
where $k$ is a constant. It is given that $\ y = 1 \$ when $\ x = 0 \$, and that $\ y = \sqrt{\textrm{e}} \$ when $\ x = 1$. Solve the differential equation, obtaining an expression for $y$ in terms of $x$.
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7.     (a)     Showing all working and without using a calculator, solve the equation
$$(1 + \textrm{i})z^{2} \ – \ (4 + 3 \textrm{i})z + 5 + \textrm{i} = 0.$$
Give your answers in the form $x + \textrm{i}y$, where $x$ and $y$ are real.
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(b)     The complex number $u$ is given by
$$u = -1 \ – \ \textrm{i}.$$
On a sketch of an Argand diagram show the point representing $u$. Shade the region whose points represent complex numbers satisfying the inequalities $\ |z| \lt |z \ – \ 2 \textrm{i}| \$ and $\ \frac{1}{4} \pi \lt \textrm{arg}(z \ – \ u) \lt \frac{1}{2} \pi .$
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8.     Let $\displaystyle f(x) = \frac{12+12x-4x^{2}}{(2+x)(3 \ – \ 2x)}.$

(i)     Express $f(x)$ in partial fractions.
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(b)     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.     Two planes have equations $\ 2x + 3y \ − \ z = 1 \$ and $\ x \ − \ 2y + z = 3.$

(i)     Find the acute angle between the planes.
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(ii)     Find a vector equation for the line of intersection of the planes.
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10.     The diagram shows the curve $\ y = \sin^{3} x \sqrt{ (\cos x)} \$ for $\ 0 \leq x \leq \frac{1}{2} \pi$, and its maximum point $M$.

(i)     Using the substitution $\ u = \cos x$, find by integration the exact area of the shaded region bounded by the curve and the $x$-axis.
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(ii)     Showing all your working, find the $x$-coordinate of $M$, giving your answer correct to 3 decimal places.
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