Home » 9709 » Paper 32 Feb Mar 2020 Pure Math III – 9709/32/F/M/20

1.     (a)     Sketch the graph of $y \ = \ |x \ – \ 2|.$ $$\tag*{[1]}$$
(b)     Solve the inequality $|x \ – \ 2| \ < \ 3x \ - \ 4.$ $$\tag*{[3]}$$ Check here for my solution

2.       Solve the equation $$\ln ⁡3 + \ln ⁡(2x+5) \ = \ 2 \ln⁡ (x+2).$$
$$\tag*{[4]}$$

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3.     (a)     By sketching a suitable pair of graphs, show that the equation ${\small \ \sec x \ = \ 2 \ – \ \frac{1}{2} x} \$ has exactly one root in the interval ${\small \ 0 \ \leq x \ < \frac{1}{2} \pi }.$ $$\tag*{[2]}$$         (b)     Verify by calculation that this root lies between 0.8 and 1. $$\tag*{[2]}$$         (c)     Use the iterative formula $\ x_{n+1} \ = \ \cos^{-1}⁡ \big( \frac{2}{4 \ - \ x_{n} }\big) \$ to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places. $$\tag*{[3]}$$ Check here for my solution

4.       Find $$\ \displaystyle \int_{\frac{1}{6} π}^{\frac{1}{3} π} x \sec^{2} ⁡x \ \textrm{d}x.$$
$$\tag*{[7]}$$

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5.     (a)     Show that $$\frac{\cos ⁡3x}{\sin ⁡x} + \frac{\sin ⁡3x}{\cos ⁡x} \ = \ 2 \cot ⁡2x$$
$$\tag*{[4]}$$

(b)     Hence solve the equation $$\frac{\cos ⁡3x}{\sin ⁡x} + \frac{\sin ⁡3x}{\cos ⁡x} \ = \ 4,$$

for ${\small \ 0 < x < \pi }.$ $$\tag*{[3]}$$ Check here for my solution

6. The variables $x$ and $y$ satisfy the differential equation
$$\frac{\textrm{d}y}{\textrm{d}x} \ = \ \frac{1+4y^2}{e^{x}}.$$
It is given that $\ {\small y \ = \ 0 \ }$ when $\ {\small x \ = \ 1 \ }$.

(a)     Solve the differential equation, obtaining an expression for $y$ in terms of $x$.
$$\tag*{[7]}$$
(b)     State what happens to the value of $y$ as $x$ tends to infinity.
$$\tag*{[1]}$$

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7. The equation of a curve is $$x^{3} + 3 x{y}^{2} \ – \ y^{3} \ = \ 5.$$

(a)     Show that $\ \frac{\textrm{d}y}{\textrm{d}x} \ = \ \frac{x^2 + y^2}{y^2 \ – \ 2xy}.$
$$\tag*{[4]}$$
(b)     Find the coordinates of the points on the curve where the tangent is parallel to the $y$-axis.
$$\tag*{[5]}$$

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8. In the diagram, $\ {\small OABCDEFG} \$ is a cuboid in which $\ {\small \overrightarrow{OA} = 2 } \$ units, $\ {\small \overrightarrow{OC} = 3 } \$ units and $\ {\small \overrightarrow{OD} = 2 } \$ units. Unit vectors $\ {\small \mathbf{i}}$, $\ {\small \mathbf{j}} \$ and $\ {\small \mathbf{k}} \$ are parallel to $\ {\small \overrightarrow{OA}, \overrightarrow{OC} }\$ and $\ {\small \overrightarrow{OD} } \$ respectively. The point $M$ on $AB \$ is such that $\ MB \ = \ 2 AM$. The midpoint of $FG$ is $N$.

(a)     Express the vectors $\ {\small \overrightarrow{OM}} \$ and $\ {\small \overrightarrow{MN}} \$ in terms of $\ {\small \mathbf{i}}$, $\ {\small \mathbf{j}} \$ and $\ {\small \mathbf{k}} \$.
$$\tag*{[3]}$$
(b)     Find a vector equation for the line through $M$ and $N$.
$$\tag*{[2]}$$
(c)     Find the position vector of $P$, the foot of the perpendicular from $D$ to the line through $M$ and $N$.
$$\tag*{[4]}$$

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9.       Let $f(x) \ = \ \frac{2 + 11x \ – \ 10x^2}{(1+2x) (1 \ – \ 2x) (2+x) } . \\$

(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.     (a)   The complex numbers $v$ and $w$ satisfy the equations
$$v+iw \ = \ 5 \ {\small \textrm{and}} \ (1+2i)v \ – \ w \ = \ 3i.$$

Solve the equations for $v$ and $w$, giving your answers in the form $x +iy$, where $x$ and $y$ are real.
$$\tag*{[6]}$$
(b)   (i)   On an Argand diagram, sketch the locus of points representing complex numbers $z$ satisfying $|z \ – \ 2 \ – \ 3i| \ = \ 1.$
$$\tag*{[2]}$$
(ii)   Calculate the least value of arg $z$ for points on this locus.
$$\tag*{[2]}$$

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