Complex Numbers (DP IB Maths: AA HL)

Topic Questions

4 hours33 questions

Medium

Hard

Very Hard

2 marks

Consider the complex numbers $z_{1} = 2 + 2 i$ and $z_{2} = 2 + 2 \sqrt{3} i$ .

Sketch $z_{1}$ and $z_{2}$ on the Argand diagram below, be sure to include an appropriate scale.

q1a_1-8_complex-numbers_medium_ib-maths-aa-hl

How did you do?

3 marks

Find the modulus of $z_{1}$ and $z_{2}$ .

How did you do?

3 marks

Find the argument of $z_{1}$ and $z_{2}$ .

How did you do?

Did this page help you?

2 marks

Let $w_{1} = z_{1} z_{2}$ , where $z_{1} = 5 + i$ and $z_{2} = 1 + 2 i$ .

Express $w$ in the form $w = a + b i$ .

How did you do?

4 marks

Find the modulus and argument for $w$

How did you do?

Did this page help you?

3 marks

Let $z = \frac{w_{1}}{w_{2}}$ , where $w_{1} = 4 - i$ and $w_{2} = 1 - 2 i$ .

Express $z$ in the form $z = a + b i$ .

How did you do?

4 marks

Find the modulus and argument for $z$ .

How did you do?

Did this page help you?

2 marks

Consider the complex numbers $z = 3 - 4 i$ and $w = 7 - 2 i$ .

Find

(i)

z + w

(ii)

w - z .

How did you do?

2 marks

Let $z^{*}$ and $w^{*}$ represent the complex conjugates of $z$ and $w$ , respectively.

Write down $z^{*}$ and $w^{*}$ , giving your answers in the form $a + b i$ .

How did you do?

4 marks

Find

(i)

z^{*} w

(ii)

\frac{w^{*}}{z} .

How did you do?

Did this page help you?

10a

5 marks

Let $z = 3 + 8 i$ and $w = 4 - 4 i$ .

Find $θ$ , the angle shown on the diagram below.

q10a_1-8_complex-numbers_medium_ib-maths-aa-hl

How did you do?

10b

3 marks

Find the area of the triangle formed in the diagram above.

How did you do?

Did this page help you?

11a

2 marks

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

Find $z w$ .

How did you do?

11b

3 marks

Sketch $z, w and z w$ on the Argand diagram below.

q11b_1-8_complex-numbers_medium_ib-maths-aa-hl

How did you do?

11c

4 marks

Let $θ$ be the angle between $z$ and $z w$ and $ϕ$ be the angle between $w$ and $z w$ .

Find the angles $θ$ and $ϕ$ , giving your answers in degrees.

How did you do?

Did this page help you?

12a

4 marks

Let $w = \frac{z + 1}{z^{*} + 1}$ , where $z = a + b i, a, b \in ℝ$ .

Write $w$ in the form $x + y i, x, y \in ℝ .$

How did you do?

12b

3 marks

Determine the conditions under which $w$ is purely imaginary.

How did you do?

Did this page help you?

3 marks

Consider the quadratic equation $z^{2} - 8 z + 25 = 0, z \in ℂ$ .

The roots of the equation are $z_{1} = a + b i$ and $z_{2} = a - b i$ where $a, b \in ℤ .$

Find the value of $a$ and $b$ .

How did you do?

4 marks

Sketch $z_{1}, z_{2}, z_{1} + z_{2}$ and $z_{1} - z_{2}$ on the Argand diagram below, be sure to include an appropriate scale.

q1b-1-8-ib-aa-hl-complex-numbers-hard-maths_dig

How did you do?

Did this page help you?

6 marks

Consider the complex numbers $z_{1} = 1 - 2 i$ and $z_{2} = - 3 + 5 i$ .

Work out the following:

(i)

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

(ii)

Im (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.

How did you do?

2 marks

Write down the complex conjugate of $z_{2}$ and describe the geometrical relationship between $z_{2}$ and ${z_{2}}^{*}$ .

How did you do?

Did this page help you?

3 marks

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

(i)

Re (z) = \frac{z + z *}{2}

(ii)

Im (z) = \frac{z - z^{*}}{2 i}

How did you do?

6 marks

For the complex numbers $z_{1} = a_{1} + b_{1} i$ and $z_{2} = a_{2} + b_{2} i$ , where $a_{1}, a_{2}, b_{1}, b_{2} \in ℝ$ , show that

|z_{1} z_{2}| = |z_{1}| |z_{2}|

How did you do?

Did this page help you?

4 marks

Consider the complex numbers $z_{1} = a - 6 i, z_{2} = 1 + b i$ and $z_{1} z_{2} = - 17 - 9 i$ where $a, b \in ℝ$ .

Find the possible values of $a$ and $b$ .

How did you do?

2 marks

Using the answers gained in part (a), write down values for $c$ and $d$ that will satisfy the equation

- (3 + i) (c + d i) = 17 - 9 i

How did you do?

Did this page help you?

2 marks

Consider the complex numbers $z = 3 + 5 i and w = - 2 + 3 i$ .

Represent the complex numbers $z$ and $w$ on an Argand diagram.

How did you do?

5 marks

The points $z + w$ and $z - w$ are represented by the points $A$ and $B$ on the Argand diagram respectively.

Find the angle $A \hat{O} B .$

How did you do?

Did this page help you?

10a

4 marks

Consider the complex numbers $z = - 4 - 3 i, w = a i$ and $\frac{z}{w} = b + 2 a i$ , where $a, b \in ℝ$ .

Find the possible values of $a$ and $b$ .

How did you do?

10b

3 marks

Find the modulus of $\frac{w}{z}$ .

How did you do?

Did this page help you?

11a

4 marks

Let $ω_{1} = 3 - i and ω_{2} = 1 + 2 i$ .

Given that $\frac{1}{ω_{1}} + \frac{1}{ω_{2}} = \frac{1}{z}$ , express $z$ in the form $a + b i$ , where $a, b \in ℝ$ .

How did you do?

11b

2 marks

Find $ω_{1} ω_{2} z^{*}$ , giving your answer in the form $a + b i$ , where $a, b \in ℝ .$

How did you do?

Did this page help you?

4 marks

Consider the complex numbers $z_{1} = \sqrt{3} + 2 i$ and $z_{2} = i - 3 \sqrt{3}$ .

Find

(i)

u = z_{1} z_{2}

(ii)

v = \frac{z_{1}}{z_{2}}

How did you do?

3 marks

The complex numbers $u$ and $v$ are represented by the points $A$ and $B$ respectively on an Argand diagram with origin $O$ .

Determine whether the angle made by $OA$ with the positive horizontal axis is greater than or less than the angle made by $OB$ with the positive horizontal axis. Give a reason for your answer.

How did you do?

Did this page help you?

4 marks

Consider the complex number $z = - a + \frac{3}{4} i$ .

Write down, in terms of $a$ ,

(i)

Re (z^{2})

(ii)

Im (z^{3})

How did you do?

4 marks

In the case where $a = 2$ , find the modulus and argument of $z^{3}$ .

How did you do?

Did this page help you?

3 marks

Consider the complex numbers $z_{1} = i - \frac{1}{2}$ and $z_{2} = \frac{1}{2} - \frac{3}{i}$ .

Express $z_{2}$ in the form $a + b i$ , where $a, b \in ℝ$ .

How did you do?

6 marks

Find

(i)

{z_{1}}^{*} z_{2}

(ii)

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

(iii)

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

, giving your answer as an exact value.

How did you do?

Did this page help you?

4 marks

Consider the equation $z w - w + i z + 1 = 0$ , where $w, z \in ℂ$ , $w = x + i y$ .

Find an expression in terms of $x$ and $y$ for $Re (z)$ .

How did you do?

4 marks

Find in terms of $x$ given that $z$ is purely real.

How did you do?

Did this page help you?

5 marks

Consider the complex numbers $z_{1} = \frac{3 - i}{1 - 2 i}$ and $z_{2} = - 3 i + 1$ .

Find the modulus of $\frac{z_{1}}{{z_{2}}^{*}}$ giving your answer as an exact value.

How did you do?

2 marks

The argument of $\frac{z_{1}}{{z_{2}}^{*}}$ is given as $θ = \tan^{- 1} x$ , where $0 < θ < 2 π$ . Find the value of $x$ .

How did you do?

Did this page help you?

3 marks

Consider the complex numbers $z = \frac{v}{w}, v = 1 - p i and w = 3 i - 2$

Express $z$ in the form $a + b i$ , where $a, b, p \in ℝ .$ .

How did you do?

4 marks

In the case where $z$ is purely imaginary, represent $v, w$ and $z$ on an Argand diagram.

How did you do?

Did this page help you?

4 marks

Consider the complex numbers $z = \frac{a - 3 i}{2 + i}, w = a + b i$ and $\frac{z}{w} = 1 + 2 i$ where $a, b \in ℝ .$

Find the values of $a$ and $b$ .

How did you do?

2 marks

Find the modulus of $\frac{w}{z}$ , giving your answer as an exact value.

How did you do?

2 marks

Find the argument of $\frac{w}{z}$ , giving your answer in the range $- π \leq \arg \frac{w}{z} \leq π$ .

How did you do?

Did this page help you?

10a

3 marks

Consider the complex numbers $z_{1} = 5 + p i$ , $z_{2} = a + b i$ and $\frac{z_{1}}{z_{2}} = - 1 + i$ , where $z \in ℂ$
and $a, b \in ℝ$ .

Find the values of $a$ and $b$ in terms of $p$ .

How did you do?

10b

3 marks

Given that $|z_{2}| = \sqrt{73}$ , find the possible values of $p$ .

How did you do?

10c

2 marks

Given additionally that $\arg (z_{2}) = 2.78$ radians correct to 2 decimal places, determine the exact value of $Im (z_{2})$ .

How did you do?

Did this page help you?