WFM02/01
June 2026 | Time: 1 hour 30 minutes | Total Marks: 75
Given that $z^3 = -8$,
(a) find the roots of the equation $z^3 = -8$, giving your answers in the form $a + ib$. (3)
(b) Show that the points representing these roots form the vertices of an equilateral triangle and find its area. (3)
$$\frac{|2x - 1|}{x + 2} < 1$$
Solve the inequality. (7)
$$\mathrm{f}(r) = \frac{2}{(2r-1)(2r+1)(2r+3)}$$
(a) Express $\mathrm{f}(r)$ in partial fractions. (2)
(b) Hence, using the method of differences, show that
$$\sum_{r=1}^{n} \mathrm{f}(r) = \frac{2n(n+2)}{3(4n^{2}+8n+3)}$$
(4)
(c) State the value of $\displaystyle \lim_{n \to \infty} \sum_{r=1}^{n} \mathrm{f}(r)$. (1)
Given the differential equation
$$\frac{\mathrm{d}y}{\mathrm{d}x} + \frac{2}{x}y = x^{2}, \qquad x > 0$$
where $y = 2$ at $x = 1$,
(a) find an integrating factor. (2)
(b) solve the differential equation, giving $y$ in terms of $x$. (5)
(c) find the value of $y$ at $x = 2$, giving your answer to 3 significant figures. (3)
The function $\mathrm{f}$ is defined by $\mathrm{f}(x) = \ln(1 + \sin x)$.
(a) Find the values of $\mathrm{f}(0)$, $\mathrm{f}'(0)$, $\mathrm{f}''(0)$ and $\mathrm{f}'''(0)$. (4)
(b) Hence find the Maclaurin series for $\mathrm{f}(x)$, up to and including the term in $x^{3}$. (5)
$$\frac{\mathrm{d}^{2}y}{\mathrm{d}x^{2}} - 4\frac{\mathrm{d}y}{\mathrm{d}x} + 13y = 5e^{2x}$$
Given that $y = 2$ and $\dfrac{\mathrm{d}y}{\mathrm{d}x} = 3$ at $x = 0$,
(a) find the complementary function. (4)
(b) find a particular integral. (4)
(c) Hence find the particular solution. (5)
The transformation $T$ from the $z$-plane to the $w$-plane is given by
$$w = \frac{z}{z-1}, \qquad z \neq 1$$
A circle $C$ in the $z$-plane is given by $|z| = 2$.
(a) Show that the image of $C$ under $T$ is a circle. (5)
(b) Find the centre and radius of this image circle. (3)
(c) Sketch the image circle in the $w$-plane, stating the coordinates of its centre and its radius. (3)
A curve $C$ has polar equation
$$r = 3(1 + \cos\theta), \qquad 0 \leq \theta < 2\pi$$
(a) Find the polar coordinates of the points on $C$ where the tangent is parallel to the initial line, giving your answers in the form $(r, \theta)$. (5)
(b) Find the area enclosed by the curve $C$. (7)