Home » 9709 » Paper 33 May June 2021 Pure Math III – 9709/33/M/J/21

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1.     Expand ${(1 \ + \ 3x)}^{\frac{2}{3}}$ in ascending powers of $x$, up to and including the term in $x^3$, simplifying the coefficients.
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2.     Solve the equation $4^x = 3 \ + \ {4}^{−x}$. Give your answer correct to 3 decimal places.
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3.     The parametric equations of a curve are
$$x = t \ + \ \ln(t \ + \ 2), \enspace y = (t \ – \ 1){\mathrm{e}}^{-2t},$$
where $t \gt -2$.

(a)     Express ${\large \frac{\mathrm{d}y}{ \mathrm{d}x} }$ in terms of $t$, simplifying your answer.
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(b)     Find the exact $y$-coordinate of the stationary point of the curve.
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4.     Let $\mathrm{f}(x) = {\large \frac{15 \ – \ 6x}{(1 \ + \ 2x)(4 \ – \ x)}}$.

(a)     Express $\mathrm{f}(x)$ in partial fractions.
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(b)     Hence find $\displaystyle \int_{1}^{2} \mathrm{f}(x) \ \mathrm{d}x$, giving your answer in the form $\ln \Big({\large\frac{a}{b}}\Big)$ where $a$ and $b$ are integers.
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5.     (a)     By first expanding $\tan (2\theta \ + \ 2\theta)$, show that the equation $\tan 4\theta = {\large\frac{1}{2}} \tan \theta$ may be expressed as ${\tan}^{4} \theta \ + \ 2 {\tan}^{2} \theta \ – \ 7 = 0$.
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(b)     Hence solve the equation $\tan 4\theta = {\large\frac{1}{2}} \tan \theta$, for ${0}^{\circ} \lt \theta \lt {180}^{\circ}$.
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6.     (a)     By sketching a suitable pair of graphs, show that the equation $\cot {\large\frac{1}{2}x} = 1 \ + \ {\mathrm{e}}^{-x}$ has exactly one root in the interval $0 \lt x\le \pi$.
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(b)     Verify by calculation that this root lies between 1 and 1.5.
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(c)     Use the iterative formula
$$x_{n+1} = 2{\tan}^{-1} \Big( {\large\frac{1}{1 \ + \ {\mathrm{e}}^{-x_{n}}} } \Big)$$
to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
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7.

For the curve shown in the diagram, the normal to the curve at the point $P$ with coordinates $(x, y)$ meets the $x$-axis at $N$. The point $M$ is the foot of the perpendicular from $P$ to the $x$-axis.

The curve is such that for all values of $x$ in the interval $0 \le x \lt \frac{1}{2} \pi$, the area of triangle $PMN$ is equal to $\tan x$.

(a)     (i)     Show that $\frac{MN}{y} = \frac{\mathrm{d}y}{\mathrm{d}x}$.
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(ii) Hence show that $x$ and $y$ satisfy the differential equation $\frac{1}{2} {y}^{2} \frac{\mathrm{d}y}{\mathrm{d}x} = \tan x$.
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(b)     Given that $y = 1$ when $x = 0$, solve this differential equation to find the equation of the curve, expressing $y$ in terms of $x$.
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8.

The diagram shows the curve $y = {\large\frac{\ln x}{x^4} }$ and its maximum point $M$.

(a)     Find the exact coordinates of $M$.
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(b)     By using integration by parts, show that for all $a \gt 1$, $\displaystyle \int_{1}^{a} \frac{\ln x}{x^4} \ \mathrm{d}x \lt \frac{1}{9}$.
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9.     The quadrilateral $ABCD$ is a trapezium in which $AB$ and $DC$ are parallel. With respect to the origin $O$, the position vectors of $A$, $B$ and $C$ are given by $\ {\small \overrightarrow{OA}} \ = \ -\mathbf{i} + 2 \mathbf{j} + 3\mathbf{k}$, $\ {\small \overrightarrow{OB}} \ = \ \mathbf{i} + 3 \mathbf{j} + \mathbf{k}$ and $\ {\small \overrightarrow{OC}} \ = \ 2\mathbf{i} + 2\mathbf{j} \ – \ 3 \mathbf{k}$.

(a)     Given that $\ {\small \overrightarrow{DC}} = 3{\small \overrightarrow{AB}} \$ find the position vector of $D$.
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(b)     State a vector equation for the line through $A$ and $B$.
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(c)     Find the distance between the parallel sides and hence find the area of the trapezium.
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10.     (a)     Verify that $-1 \ + \ \sqrt{2}\mathrm{i}$ is a root of the equation $z^4 \ + \ 3z^2 \ + \ 2z \ + \ 12 = 0$.
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(b)     Find the other roots of this equation.
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