Home » 9709 » Paper 31 Oct Nov 2020 Pure Math III – 9709/31/O/N/20

1.       Solve the inequality
$$\ 2 \ – \ 5x \ > \ 2|x \ – \ 3|.$$ $$\tag*{[4]}$$

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2.       On a sketch of an Argand diagram, shade the region whose points represent complex numbers $z$ satisfying the inequalities
$$|z| \ \geq 2 \ {\small\textrm{ and} } \ |z \ – \ 1 + i| \ \leq \ 1.$$ $$\tag*{[4]}$$

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3.       The parametric equations of a curve are
\begin{multline}
\shoveleft \hspace{2em} x \ = \ 3 \ – \ \cos 2\theta, \\
\shoveleft \hspace{2em} y \ = \ 2\theta + \sin 2\theta, \\
\shoveleft {\small\textrm{ for } \ 0 < \theta < \frac{1}{2}\pi }\\[5pt] \shoveleft {\small\textrm{ Show that } \ \frac{\text{d}y}{\text{d}x} \ = \ \cot \theta. } \\ \end{multline} $$\tag*{[5]}$$ Check here for my solution

4.       Solve the equation
$$\hspace{1em}\small \log_{10} ⁡(2x+1) \ = \ 2 \log_{10} ⁡(x+1) \ – \ 1.$$
Give your answers correct to 3 decimal places. $$\tag*{[6]}$$

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5.     (a)     By sketching a suitable pair of graphs, show that the equation $\ \textrm{cosec} {\small \ x \ = \ 1 + e^{-\frac{1}{2} x}} \$ has exactly two roots in the interval $\ \small 0 < x < \pi.$ $$\tag*{[2]}$$     (b)     The sequence of values given by the iterative formula $$x_{n+1} \ = \ \pi \ - \ \sin^{-1} \Big( \frac{1}{e^{- \frac{1}{2} x_{n} } + 1 } \Big),$$ with initial value $\small \ x_{1} \ = \ 2$, converges to one of these roots.  Use the formula to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places. $$\tag*{[3]}$$ Check here for my solution

6.     (a)     Express $\sqrt{6} \ \cos⁡ \theta + 3 \sin⁡ \theta \$ in the form $R \cos⁡ (\theta \ – \ \alpha)$, where $R > 0$ and $0^{\circ} < \alpha < 90^{\circ}.$ State the exact value of $$R$$ and give $$\alpha$$ correct to 2 decimal places. $$\tag*{[3]}$$     (b)     Hence solve the equation $$\sqrt{6} \ \cos⁡ \frac{1}{3}x + 3 \sin⁡ \frac{1}{3}x \ = \ 2.5$$, for $\ 0^{\circ} < \alpha < 360^{\circ}.$ $$\tag*{[4]}$$ Check here for my solution

7.     (a)     Verify that $\ -1 + \sqrt{5} i \$ is a root of the equation $\ 2x^{3}+x^{2}+6{x} \ – \ 18 \ = \ 0.$ $$\tag*{[3]}$$
(b)     Find the other roots of this equation. $$\tag*{[4]}$$

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8.       The coordinates ($x$, $y$) of a general point of a curve satisfy the differential equation
$$x \frac{\textrm{d}y}{\textrm{d}x} \ = \ (1 \ – \ 2x^{2} )y, \\$$
for ${\small \ x > 0 }$. It is given that ${\small y \ = \ 1 }$ when ${\small x \ = \ 1. }$
Solve the differential equation, obtaining an expression for $y$ in terms of $x$. $$\tag*{[6]}$$

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9.       Let $f(x) \ = \ \frac{8+5x+12x^2}{(1-x) (2+3x)^{2} } . \\$

(a)     Express f(x) in partial fractions. $$\tag*{[5]}$$
(b)     Hence obtain the expansion of ${\small f(x) }$ in ascending powers of $x$, up to and including the term in $x^{2}$. $$\tag*{[5]}$$

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10.       The diagram shows the curve $\ y \ = \ (2 \ – \ x) e^{-\frac{1}{2} x},$ and its minimum point $M$.

(a)     Find the exact coordinates of $M$. $$\tag*{[5]}$$
(b)     Find the area of the shaded region bounded by the curve and the axes. Give your answer in terms of $\textrm{e}$. $$\tag*{[5]}$$

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11.       Two lines have equations $\ \mathbf{r} = \mathbf{i} + 2\mathbf{j} + \mathbf{k} + \lambda (a\mathbf{i}+2\mathbf{j} \ – \ \mathbf{k}) \$ and $\ \mathbf{r} = 2\mathbf{i} + \mathbf{j} \ – \ \mathbf{k} + \mu (2\mathbf{i} \ – \ \mathbf{j}+\mathbf{k}),$ where $a$ is a constant.

(a)     Given that the two lines intersect, find the value of $a$ and the position vector of the point of intersection. $$\tag*{[5]}$$
(b)     Given instead that the acute angle between the directions of the two lines is $\ \cos^{-1}⁡ \big( \frac{1}{6} \big),$ find the two possible values of $a$.
$$\tag*{[6]}$$

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