Pure Mathematics Practice: Complex Numbers — Answers

Questions: Complex Numbers Drills.

Question 1

(a)

$$z+w=4+2i,$$

and

$$zw=(3-2i)(1+4i)=11+10i.$$

(b) Multiplying numerator and denominator by $\bar{w}=1-4i$,

$$\frac{z}{w}=\frac{(3-2i)(1-4i)}{1^2+4^2}=-\frac{5}{17}-\frac{14}{17}i.$$

Question 2

(a)

$$|z|=\sqrt{(-1{)}^2+(\sqrt{3}{)}^2}=2.$$

(b) The point lies in quadrant II with reference angle $\pi /3$, so

$$\arg z=\frac{2\pi }{3}.$$

(c)

$$\bar{z}=-1-\sqrt{3}i.$$

Thus $\bar{z}$ is the reflection of $z$ in the real axis. The two points have equal modulus and opposite imaginary parts.

Question 3

(a) Since the polynomial has real coefficients, the conjugate root $2-i$ must also occur.

(b)

$$\begin{array}{rl}(x-(2+i))(x-(2-i))& =((x-2)-i)((x-2)+i)\\ & =(x-2{)}^2+1\\ & =x^2-4x+5.\end{array}$$

Question 4

(a)

$$iz=i(x+yi)=-y+xi.$$

(b) The Argand point $(x,y)$ moves to $(-y,x)$. This is a rotation through $90^{\circ }$ anticlockwise about the origin, with distance from the origin unchanged.