Complex numbers

WFM01Topic 1.13 questions

Arithmetic of complex numbers, Argand diagrams, modulus-argument form and roots of polynomials.

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Original practice questions written in the style of the exam.

ar g (z^{2})

\arg z = -\dfrac{\pi}{3}

B1

z^2 = 4 - 8\sqrt{3}\,i - 12 = -8 - 8\sqrt{3}\,i

(1)

B1

\arg(z^2) = -\dfrac{2\pi}{3}

(1)

z

−

10

(a) Show that $z = 1 + 2i$ is a root of $f(z) = 0$ .

(b) Hence find the other two roots of $f(z) = 0$ .

Mark scheme

M1 Finds $(1+2i)^2 = -3 + 4i$ and $(1+2i)^3 = -11 - 2i$ (1)
A1 Substitutes to get $-11 - 2i + 12 - 16i + 9 + 18i - 10 = 0$ (1)
B1 $1 - 2i$ is a root since the coefficients are real (1)
M1 Quadratic factor $(z - 1 - 2i)(z - 1 + 2i) = z^2 - 2z + 5$ , then divides or compares coefficients (1)
A1 $f(z) = (z^2 - 2z + 5)(z - 2)$ , so the third root is (1)

2

−

2z+

5

z = 2