Pure Mathematics Practice: Complex Numbers

Core 9758 practice

These four questions total 23 marks. They use Cartesian complex arithmetic and prescribed Argand geometry only; polar form, de Moivre methods and complex loci are not assessed.

Answers: Complex Numbers Answers.

Question 1: Cartesian Form

Let $z=3-2i$ and $w=1+4i$.

(a) Find $z+w$ and $zw$. [3]

(b) Express $z/w$ in the form $a+bi$, where $a,b\in \mathbb{R}$. [4]

[Total: 7]

Question 2: Modulus, Argument and Conjugate

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

(a) Find $|z|$. [2]

(b) State the principal value of $\arg z$ in $(-\pi ,\pi ]$. [2]

(c) Describe the position of $\bar{z}$ relative to $z$ on an Argand diagram. [2]

[Total: 6]

Question 3: Complex Roots of a Real Polynomial

A polynomial with real coefficients has a root $2+i$.

(a) State another root that must occur. [1]

(b) Form the real quadratic factor corresponding to these two roots. [4]

[Total: 5]

Question 4: Multiplication by $i$

Let $z=x+yi$, where $x,y\in \mathbb{R}$.

(a) Express $iz$ in Cartesian form. [2]

(b) Explain the geometric effect of multiplying by $i$ on an Argand diagram. [3]

[Total: 5]