Home » 9709 » Paper 31 May June 2020 Pure Math III – 9709/31/M/J/20

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1.     Find the set of values of $\ x \$ for which $\ 2({3}^{1 \ − \ 2x}) \lt 5x$. Give your answer in a simplified exact form.
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2.     (a)     Expand $\ {(2 \ − \ 3x)}^{−2} \$ in ascending powers of $\ x$, up to and including the term in $\ x^2$, simplifying the coefficients.
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(b)     State the set of values of $\ x \$ for which the expansion is valid.
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3.     Express the equation $\ \tan(\theta + {60}^{\circ}) = 2 + \tan({60}^{\circ} − \theta) \$ as a quadratic equation in $\ \tan \theta$, and hence solve the equation for $\ {0}^{\circ} \le \theta \le {180}^{\circ}$.
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4.     The curve with equation $\ y = {\mathrm{e}}^{2x}(\sin x + 3 \cos x) \$ has a stationary point in the interval $\ 0^{\circ} \le x \le \pi$.

(a)     Find the $\ x$-coordinate of this point, giving your answer correct to 2 decimal places.
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(b)     Determine whether the stationary point is a maximum or a minimum.
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5.     (a)     Find the quotient and remainder when $\ 2x^3 \ − \ x^2 + 6x + 3 \$ is divided by $\ x^2 + 3$.
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(b)     Using your answer to part (a), find the exact value of $\ \displaystyle \int_{1}^{3} \frac{2x^3 \ − \ x^2 + 6x + 3}{x^2 + 3} \ \mathrm{d}x$.
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6.

The diagram shows a circle with centre $\ O \$ and radius $\ r$. The tangents to the circle at the points $\ A \$ and $\ B \$ meet at $\ T \$, and angle $\ AOB \$ is $\ 2x \$ radians. The shaded region is bounded by the tangents $\ AT \$ and $\ BT$, and by the minor arc $\ AB$. The area of the shaded region is equal to the area of the circle.

(a)     Show that $\ x \$ satisfies the equation $\ \tan x = \pi + x$.
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(b)     This equation has one root in the interval $\ 0 \lt x \lt \frac{1}{2} \pi$. Verify by calculation that this root lies between 1 and 1.4.
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(c)     Use the iterative formula
$$x_{n+1} = {\tan}^{-1} (\pi + x_n)$$
to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
$$\tag*{[3]}$$

7.     Let $\displaystyle \mathrm{f}(x) = \frac{\cos x}{1 + \sin x}$.

(a)     Show that $\ {\mathrm{f}}^{‘}(x) \lt 0 \$ for all $x$ in the interval $\ -\frac{1}{2} \pi \lt x \lt \frac{3}{2} \pi$.
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(b)     Find $\displaystyle \int_{\frac{1}{6} \pi}^{\frac{1}{2} \pi} \mathrm{f}(x) \mathrm{d}x$. Give your answer in a simplified exact form.
$$\tag*{[4]}$$

8.     A certain curve is such that its gradient at a point $(x, y)$ is proportional to $\ \frac{y}{x\sqrt{x}}$. The curve passes through the points with coordinates $(1, 1)$ and $(4, \mathrm{e})$.

(a)     By setting up and solving a differential equation, find the equation of the curve, expressing $y$ in terms of $x$.
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(b)     Describe what happens to $y$ as $x$ tends to infinity.
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9.     With respect to the origin $O$, the vertices of a triangle $ABC$ have position vectors

$\hspace{2em} \ {\small \overrightarrow{OA}} \ = \ 2 \mathbf{i} + 5\mathbf{k}$, $\ {\small \overrightarrow{OB}} \ = \ 3 \mathbf{i} + 2 \mathbf{j} + 3\mathbf{k}$ and $\ {\small \overrightarrow{OC}} \ = \ \mathbf{i} + \mathbf{j} + \mathbf{k}$.

(a)     Using a scalar product, show that angle $ABC$ is a right angle.
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(b)     Show that triangle $ABC$ is isosceles.
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(c)     Find the exact length of the perpendicular from $O$ to the line through $B$ and $C$.
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10.     (a)     The complex number $u$ is defined by $u = \large{\frac{3\mathrm{i}}{a + 2\mathrm{i}}}$, where $a$ is real.

(i)     Express $u$ in the Cartesian form $x + \mathrm{i}y$, where $x$ and $y$ are in terms of $a$.
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(ii)     Find the exact value of $a$ for which $\mathrm{arg} \ {u}^{*} = \frac{1}{3}\pi$.
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(b)     (i)     On a sketch of an Argand diagram, shade the region whose points represent complex numbers $z$ satisfying the inequalities
$$|z \ – \ 2 \mathrm{i}| \le |z \ – \ 1 \ – \ \mathrm{i}| \enspace \mathrm{and} \enspace |z \ – \ 2 \ – \ \mathrm{i}| \le 2 .$$
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$\hspace{2em}$ (ii)     Calculate the least value of $\mathrm{arg} \ z$ for points in this region.
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