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

1a

1 mark

A complex number $z$ may be written in modulus-argument form as

$z = r (\cos θ + i \sin θ)$

where $r = | z |$ is the modulus of $z$ , and $θ = \arg z$ is the argument of $z$ .

According to Euler’s relation, the equation $e^{i θ} = \cos θ + i \sin θ$ is true for any real number $θ$ . Therefore it is also possible to write a complex number in exponential form as

$z = r e^{i θ}$

where again $r = | z |$ is the modulus of $z$ , and $θ = \arg z$ is the argument of $z$ .

The modulus-argument form of a complex number is $6 (\cos 2 + i \sin 2)$ . Write that complex number in exponential form.

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1b

4 marks

$3 + 3 i$ is another complex number.

(i)

Calculate the modulus and argument of

3 + 3 i

, giving your answers as exact values.

(ii)

Use your answers to part (i) to write

3 + 3 i

in both modulus-argument form and exponential form.

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1

4 marks

Express the following complex numbers in exponential form:

(i)

3 (2 \cos 2 - 2 i \sin (- 2))

(ii)

- 2 + 2 \sqrt{3} i

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1

4 marks

$z_{1} = 6 e^{i}$

$z_{2} = 3 e^{- 2 i}$

(i)

Work out

z_{1} z_{2}

and

\frac{z_{1}}{z_{2}}

, giving your answers in exponential form.

(ii)

Express your answers to part (i) as complex numbers in modulus-argument form.

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1

4 marks

Express the following complex numbers in exponential form:

(i)

- 5 (\cos 2 - i \sin 2)

(ii)

(\sqrt{2} - \sqrt{6}) - (\sqrt{2} + \sqrt{6} i)

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CIE A Level Maths: Pure 3

Topic Questions

8.3 Further Complex Numbers