Home » 9709 » Paper 32 May June 2021 Pure Math III – 9709/32/M/J/21

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1.     Solve the inequality $|2x \ − \ 1| \lt 3|x \ + \ 1|$.
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2.     On a sketch of an Argand diagram, shade the region whose points represent complex numbers z
satisfying the inequalities $|z \ + \ 1 \ – \ \mathrm{i}| \le 1 \enspace \mathrm{and} \enspace \mathrm{arg} (z \ – \ 1) \le {\large \frac{3}{4}\pi} .$
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3.     The variables $x$ and $y$ satisfy the equation $x = A({3}^{-y})$, where $A$ is a constant.

(a)     Explain why the graph of $y$ against $\ln x$ is a straight line and state the exact value of the gradient of the line.
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It is given that the line intersects the $y$-axis at the point where $y = 1.3$.

(b)     Calculate the value of $A$, giving your answer correct to 2 decimal places.
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4.     Using integration by parts, find the exact value of $\displaystyle \int_{0}^{2} {\tan}^{-1} \Big(\frac{1}{2}x \Big) \ \mathrm{d}x$.
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5.     The complex number $u$ is given by $u = 10 \ – \ 4\sqrt{6} \mathrm{i}$.

Find the two square roots of $u$, giving your answers in the form $a + \mathrm{i}b$, where $a$ and $b$ are real and exact.
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6.     (a)     Prove that $\mathrm{cosec} \ 2\theta \ − \ \cot 2 \theta \equiv \tan \theta$.
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(b)     Hence show that $\displaystyle \int_{\frac{1}{4} \pi}^{\frac{1}{3} \pi} (\mathrm{cosec} \ 2\theta \ − \ \cot 2 \theta) \ \mathrm{d}\theta = \frac{1}{2} \ln 2$.

7.     A curve is such that the gradient at a general point with coordinates $(x, y)$ is proportional to ${\large \frac{y}{\sqrt{ x + 1 } } }$.

The curve passes through the points with coordinates $(0, 1)$ and $(3, \mathrm{e})$.

By setting up and solving a differential equation, find the equation of the curve, expressing $y$ in terms of $x$.
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8.     The equation of a curve is $y = {\mathrm{e}}^{-5x} \ {\tan}^{2}x$ for $– \frac{1}{2} \pi \lt x \lt \frac{1}{2} \pi$.

Find the $x$-coordinates of the stationary points of the curve. Give your answers correct to 3 decimal places where appropriate.
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9.     Let $\mathrm{f}(x) = {\large \frac{14 \ – \ 3x \ + \ 2x^2}{(2 \ + \ x)(3 \ + \ x^2)}}$.

(a)     Express $\mathrm{f}(x)$ in partial fractions.
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(b)     Hence obtain the expansion of $\mathrm{f}(x)$ in ascending powers of $x$, up to and including the term in $x^2$.
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10.

(a)     Given that the sum of the areas of the shaded sectors is 90% of the area of the trapezium, show that $x$ satisfies the equation $x = 0.9(2 \ − \ \cos x) \sin x$.
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(b)     Verify by calculation that $x$ lies between 0.5 and 0.7.
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(c)     Show that if a sequence of values in the interval $0 \lt x \lt \frac{1}{2} \pi$ given by the iterative formula
$$x_{n+1} = {\cos}^{-1} \Big( 2 \ – \ \frac{x_{n}}{0.9 \sin x_{n}} \Big)$$
converges, then it converges to the root of the equation in part (a).
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(d)     Use this iterative formula to determine $x$ correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
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11.     With respect to the origin $O$, the points $A$ and $B$ have position vectors given by $\ {\small \overrightarrow{OA}} \ = \ 2 \mathbf{i} \ – \ \mathbf{j} \$ and $\ {\small \overrightarrow{OB}} \ = \ \mathbf{j} \ – \ 2 \mathbf{k}$.

(a)     Show that $OA = OB$ and use a scalar product to calculate angle $AOB$ in degrees.
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The midpoint of $AB$ is $M$. The point $P$ on the line through $O$ and $M$ is such that $PA : OA = \sqrt{7} : 1$.

(b)     Find the possible position vectors of $P$.
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