Complex Numbers | CIE A Level Maths: Pure 3 Topic Questions 2020

1a

2 marks

$z_{1} = 3 + i$ and $z_{2} = - 5 + 2 i$

(i)

Work out

z_{1} + z_{2}

and

z_{1} - z_{2} .

(ii)

Use your answers to part (i) to write down

Re (z_{1} + z_{2})

and

Im (z_{1} - z_{2}) .

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1b

2 marks

By first writing $z_{1} \times z_{2} = (3 + i) (- 5 + 2 i)$ , and then expanding brackets and using the fact that $i^{2} = - 1$ , show that

$z_{1} \times z_{2} = - 17 + i$

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1

6 marks

Given that $z_{1} = 1 - 2 i$ and $z_{2} = - 3 + 5 i$ , work out the following:

(i)

R e (z_{2} - z_{1})

(ii)

I m (z_{1} z_{2})

(iii)

{(\frac{z_{1}}{z_{2}})}^{*}

For part (iii) give your answer in the form $a + b i$ , where $a$ and $b$ are real numbers.

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1

3 marks

Given that $z_{1} = 2 - 3 i$ and $z_{2} = - 7 + i$ , work out the following:

(i)

z_{1} \times z_{2}

(ii)

R e (z_{1} + z_{2})

(iii)

I m (z_{2} - z_{1})

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1

3 marks

For a general complex number $z = x + i y$ , where $x, y \in ℝ$ and $z \neq 0$ , show that

(i)

R e (\frac{1}{z} + \frac{1}{z^{*}}) = \frac{2 x}{x^{2} + y^{2}}

(ii)

I m (\frac{1}{z} + \frac{1}{z^{*}}) = 0

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CIE A Level Maths: Pure 3