IBMathsDPAnalysis & Approaches (AA)HLExam Questions1. Number & AlgebraComplex Numbers

Complex Numbers (DP IB Analysis & Approaches (AA): HL): Exam Questions

4 hours33 questions
Easy
Medium
Hard
Very Hard
1a
2 marks

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

Plot z_{1} and z_{2} on the Argand diagram below. Mark a suitable scale on each axis.

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

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1b
3 marks

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

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1c
3 marks

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

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2a
2 marks

Let w = z_{1} z_{2}, where z_{1} = 5 + \mathrm{i} and z_{2} = 1 + 2 \mathrm{i}.

Express w in the form a + b \mathrm{i}, where a, \; b \in \mathbb{R}.

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2b
3 marks

Find the modulus and the argument of w.

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3a
3 marks

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

Express z in the form a + b \mathrm{i}, where a, \; b \in \mathbb{R}.

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3b
3 marks

Find the modulus and the argument of z.

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4a
2 marks

Consider the complex numbers z = 3 - 4 \mathrm{i} and w = 7 - 2 \mathrm{i}.

Find

(i) z + w

(ii) w - z.

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4b
2 marks

Let z^{*} and w^{*} be the complex conjugates of z and w respectively.

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

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4c
4 marks

Find

(i) z^{*} w

(ii) \frac{w^{*}}{z}.

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5
7 marks

Let z_{1} = 3 + 4 \mathrm{i} and z_{2} = - 2 + 2 \mathrm{i}.

Find

(i) \mathrm{i} z_{1} + z_{2}

(ii) \frac{z_{1}}{\mathrm{i} z_{2}}

(iii) \mathrm{i} \left(z_{1} z_{2}\right).

How did you do?

6
7 marks

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

Find

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

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

(iii) z_{1} z_{2}

(iv) \frac{z_{1}}{z_{2}}.

How did you do?

7a
6 marks

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

Find

(i) \text{Re} \left(z_{2} - z_{1}\right)

(ii) \text{Im} \left(z_{1} z_{2}\right)

(iii) \left(\frac{z_{1}}{z_{2}}\right)^{*}, giving your answer in the form a + b \mathrm{i}, where a, \; b \in \mathbb{R}.

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7b
2 marks

Write down the complex conjugate of z_{2}, and describe the geometrical relationship between z_{2} and z_{2}^{*} on an Argand diagram.

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1
7 marks

Solve each equation for x \in \mathbb{C}.

(i) x^{2} + 4 x + 5 = 0

(ii) x^{2} = - 625

(iii) x^{4} = 24 - 2 x^{2}

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2
7 marks

Find all possible real values of a and b such that

(i) a + b \mathrm{i} = 8 \mathrm{i}

(ii) \left(2 + 3 \mathrm{i}\right) \left(a + b \mathrm{i}\right) = 13

(iii) \left(a + \mathrm{i}\right) \left(2 + b \mathrm{i}\right) = - 6 + 22 \mathrm{i}.

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3
7 marks

The complex numbers z and w satisfy w = \mathrm{i} z and w + 2 z = 7 + 6 \mathrm{i}.

Find

(i) \text{Re} \left(w\right)

(ii) \text{Im} \left(w\right)

(iii) \text{Re} \left(z\right)

(iv) \text{Im} \left(z\right).

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4
8 marks

Find the complex numbers z and w such that

2 z - \mathrm{i} w^{*} = 5 + 7 \mathrm{i}

w + \mathrm{i} z^{*} = 5 + 16 \mathrm{i}

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5a
4 marks

Let z = 3 + 8 \mathrm{i} and w = 4 - 4 \mathrm{i}. The diagram below shows z, w and z + w on an Argand diagram.

Find \theta, the angle shown on the diagram.

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

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5b
3 marks

Find the area of the triangle shown in the diagram.

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6a
2 marks

Let z = - 1 - 3 \mathrm{i} and w = 1 + \mathrm{i}.

Express z w in the form a + b \mathrm{i}, where a, \; b \in \mathbb{R}.

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6b
3 marks

Plot z, w and z w on the Argand diagram below.

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

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6c
4 marks

Let \theta be the angle between z and z w, and let \phi be the angle between w and z w, both measured at the origin.

Find \theta and \phi, giving your answers in degrees.

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7a
3 marks

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

The roots of the equation are z_{1} = a + b \mathrm{i} and z_{2} = a - b \mathrm{i}, where a, \; b \in \mathbb{Z} and b > 0.

Find the values of a and b.

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7b
4 marks

Plot z_{1}, z_{2}, z_{1} + z_{2} and z_{1} - z_{2} on an Argand diagram.

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8
5 marks

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

Find the modulus and the argument of z_{1} \left(z_{2}\right)^{*}.

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9
8 marks

The complex numbers z and w satisfy w = 2 \mathrm{i} z and w - z = 5 - 5 \mathrm{i}.

Find

(i) \left|z\right|

(ii) \text{arg} \left(w\right)

(iii) \text{Re} \left(z + w\right)

(iv) \text{Im} \left(z - w\right).

How did you do?

10a
2 marks

Consider the complex numbers z = 3 + 5 \mathrm{i} and w = - 2 + 3 \mathrm{i}.

Plot z and w on an Argand diagram.

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10b
5 marks

The points \mathrm{A} and \mathrm{B} on the Argand diagram represent z + w and z - w respectively.

Find \mathrm{A} \hat{\mathrm{O}} \mathrm{B}.

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11a
4 marks

Consider the complex numbers z = - 4 - 3 \mathrm{i} and w = a \mathrm{i}, where a \in \mathbb{R}. It is given that \frac{z}{w} = b + 2 a \mathrm{i}, where b \in \mathbb{R}.

Find the possible values of a and b.

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11b
2 marks

Find the modulus of w over z.

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12a
4 marks

Let \omega_{1} = 3 - \mathrm{i} and \omega_{2} = 1 + 2 \mathrm{i}.

Given that \frac{1}{\omega_{1}} + \frac{1}{\omega_{2}} = \frac{1}{z}, express z in the form a + b \mathrm{i}, where a, \; b \in \mathbb{R}.

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12b
2 marks

Find \omega_{1} \omega_{2} z^{*}, giving your answer in the form a + b \mathrm{i}, where a, \; b \in \mathbb{R}.

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13a
4 marks

Consider the complex number z = - a + \frac{3}{4} \mathrm{i}, where a \in \mathbb{R}.

Find, in terms of a,

(i) \text{Re} \left(z^{2}\right)

(ii) \text{Im} \left(z^{3}\right).

How did you do?

13b
4 marks

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

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14a
2 marks

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

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

How did you do?

14b
6 marks

Find

(i) \left(z_{1}\right)^{*} z_{2}

(ii) \frac{z_{2}}{z_{1}}

(iii) \left|\frac{z_{2}}{z_{1}}\right|, giving your answer as an exact value.

How did you do?

15
6 marks

Consider a general complex number z = x + y \mathrm{i}, where x, \; y \in \mathbb{R} and z \neq 0.

Show that

(i) \text{Re} \left(\frac{1}{z} + \frac{1}{z^{*}}\right) = \frac{2 x}{x^{2} + y^{2}}

(ii) \text{Im} \left(\frac{1}{z} + \frac{1}{z^{*}}\right) = 0

(iii) z z^{*} = \left|z\right|^{2}.

How did you do?

16a
5 marks

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

Find the modulus of \frac{z_{1}}{\left(z_{2}\right)^{*}}, giving your answer as an exact value.

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16b
2 marks

The argument of \frac{z_{1}}{\left(z_{2}\right)^{*}} is \theta, where 0 < \theta < 2 \pi.

Find the exact value of \tan \theta.

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17a
3 marks

Consider the complex numbers v = 1 - p \mathrm{i} and w = 3 \mathrm{i} - 2, where p \in \mathbb{R}, and let z = \frac{v}{w}.

Express z in the form a + b \mathrm{i}, where a, \; b \in \mathbb{R}, giving a and b in terms of p.

How did you do?

17b
4 marks

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

How did you do?

1
7 marks

Find all possible real values of a and b such that

(i) \left(a + b \mathrm{i}\right) \left(2 - 3 \mathrm{i}\right) = 8 + \mathrm{i}

(ii) a \left(2 + b \mathrm{i}\right) = b \left(- 6 + \mathrm{i}\right)

(iii) \left(2 a + 3 \mathrm{i}\right) \left(3 + b \mathrm{i}\right) = 12 + 21 \mathrm{i}.

How did you do?

2a
3 marks

For a general complex number z = x + y \mathrm{i}, where x, \; y \in \mathbb{R}, show that

(i) \text{Re} \left(z\right) = \frac{z + z^{*}}{2}

(ii) \text{Im} \left(z\right) = \frac{z - z^{*}}{2 \mathrm{i}}.

How did you do?

2b
5 marks

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

\left|z_{1} z_{2}\right| = \left|z_{1}\right| \left|z_{2}\right|.

How did you do?

3a
5 marks

Consider the complex numbers z_{1} = a - 6 \mathrm{i} and z_{2} = 1 + b \mathrm{i}, where a, \; b \in \mathbb{R}.

Given that z_{1} z_{2} = - 17 - 9 \mathrm{i}, find the possible values of a and b.

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3b
2 marks

Using your answers to part (a), write down values of c and d that satisfy the equation

- \left(3 + \mathrm{i}\right) \left(c + d \mathrm{i}\right) = - 17 - 9 \mathrm{i}

How did you do?

4a
4 marks

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

Write w in the form x + y \mathrm{i}, where x, \; y \in \mathbb{R}.

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4b
3 marks

Determine the conditions on a and b for which w is purely imaginary.

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5a
4 marks

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

Find

(i) u = z_{1} z_{2}

(ii) v = \frac{z_{1}}{z_{2}}.

How did you do?

5b
2 marks

Determine whether \text{arg} \left(u\right) is greater than or less than \text{arg} \left(v\right), where - \pi < \text{arg} \left(u\right) \leq \pi and - \pi < \text{arg} \left(v\right) \leq \pi. Give a reason for your answer.

How did you do?

6a
4 marks

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

Find the values of a and b.

How did you do?

6b
2 marks

Find the modulus of w over z, giving your answer as an exact value.

How did you do?

6c
2 marks

Find the argument of \frac{w}{z}, giving your answer in the range - \pi < \text{arg} \left(\frac{w}{z}\right) \leq \pi.

How did you do?

7a
3 marks

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

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

How did you do?

7b
3 marks

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

How did you do?

7c
2 marks

Given also that \text{arg} \left(z_{2}\right) = 2.78, correct to two decimal places, determine the exact value of \text{Im} \left(z_{2}\right).

How did you do?

1a
4 marks

Consider the equation z w - w + \mathrm{i} z + 1 = 0, where w, \; z \in \mathbb{C} and w = x + y \mathrm{i}, with x, \; y \in \mathbb{R}.

Find an expression in terms of x and y for \text{Re} \left(z\right).

How did you do?

1b
4 marks

Given that z is purely real, find z in terms of x.

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2
7 marks

The complex numbers z and w satisfy a - w = 2 z - \mathrm{i} and w - 2 z = b \mathrm{i} - 1, where a, \; b \in \mathbb{R}.

Find the values of a and b such that \text{Re} \left(w\right) = \text{Im} \left(z\right) and \text{Re} \left(w\right) = \text{Re} \left(z\right) + 1.

How did you do?