Complex Numbers (DP IB Maths: AI HL)

Topic Questions

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1.5 Complex Numbers

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

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3 marks

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

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3 marks

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

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7 marks

Solve the following equations for $x$

(i)

x^{2} + 4 x + 5 = 0

(ii)

x^{2} = - 625

(iii)

x^{4} = 24 - 2 x^{2} .

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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 $w = a + b i$ .

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4 marks

Find the modulus and argument for $w$

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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 $z = a + b i$ .

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4 marks

Find the modulus and argument for $z$ .

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2 marks

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

Find

(i)

z + w

(ii)

w - z .

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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 $a + b i$ .

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4 marks

Find

(i)

z^{*} w

(ii)

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

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7 marks

Find all possible real values for $a$ and $b$ such that

(i)

a + b i = 8 i

(ii)

(2 + 3 i) (a + b i) = 13

(iii)

(a + i) (2 + b i) = - 6 + 22 i .

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7 marks

Consider the complex numbers $w = i z$ and $w + 2 z = 7 + 6 i$ .

Find

(i)

Re (w)

(ii)

Im (w)

(iii)

Re (z)

(iv)

Im (z)

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7 marks

It is given that $z_{1} = 3 + 4 i$ and $z_{2} = - 2 + 2 i$ .

Find

(i)

i z_{1} + z_{2}

(ii)

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

(iii)

i (z_{1} z_{2})

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8 marks

Find the complex numbers $z$ and $w$ such that

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

$w + i z^{*} = 5 + 16 i$

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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

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10b

3 marks

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

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11a

2 marks

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

Find $z w$ .

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11b

3 marks

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

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

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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.

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12a

4 marks

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

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

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12b

3 marks

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

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13a

1 mark

Consider the equation $x^{2} + b x + c = 0 .$

Write down an inequality, in terms of $b$ and $c,$ that shows the equation has no real solutions.

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13b

4 marks

$5 — 3 i$ is one solution to the equation $x^{2} + b x + c = 0 .$

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

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13c

1 mark

Let $z = c + b i .$

Find $z^{5}$ using technology.

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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 $b$ .

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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

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7 marks

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

Find

(i)

z_{1} + z_{2}

(ii)

z_{1} - z_{2}

(iii)

z_{1} z_{2}

(iv)

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

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7 marks

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

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

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6 marks

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

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.

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2 marks

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

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7 marks

Find all possible real values for $a$ and $b$ such that

(i)

(a + b i) (2 - 3 i) = 8 + i

(ii)

a (2 + b i) = b (- 6 + i)

(iii)

(2 a + 3 i) (3 + b i) = 12 + 21 i

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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}

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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}|

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8 marks

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

Find

(i)

|z|

(ii)

\arg w

(iii)

Re (z + w)

(iv)

Im (z - w)

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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 $b$ .

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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

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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.

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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 $A \hat{O} B .$

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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 $b$ .

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10b

3 marks

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

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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 $a, b \in ℝ$ .

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11b

2 marks

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

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12a

2 marks

Consider the complex number $z = \frac{\sqrt{2}}{2} + \frac{\sqrt{6}}{2} i$ .

Use technology to find the values of $z^{2}$ and $z^{3}$ . Give your answers in the form $a + b i$ , where $a, b \in ℝ$ .

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12b

3 marks

Draw $z, z^{2}$ and $z^{3}$ on an Argand diagram.

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12c

2 marks

Find the smallest integer $k > 3$ such that $z^{k}$ is a real number.

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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}}

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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.

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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})

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4 marks

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

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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 $a, b \in ℝ$ .

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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.

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6 marks

Consider a general complex number $z = x + i y$ , where $x, y \in ℝ$ , $z \in ℂ$ and $z \neq 0$ .

Show that

(i)

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

(ii)

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

(iii)

z z^{*} = {|z|}^{2}

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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 $Re (z)$ .

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4 marks

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

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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.

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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 $x$ .

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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 $a, b, p \in ℝ .$ .

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4 marks

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

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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 $b$ .

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2 marks

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

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2 marks

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

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7 marks

Consider the complex numbers $a - w = 2 z - i$ and $w - 2 z = b i - 1$ .

Find the values of $a$ and $b$ such that $Re (w) = Im (z)$ and $Re (w) = Re (z) + 1$ .

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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 $p$ .

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10b

3 marks

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

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10c

2 marks

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

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11a

3 marks

Consider the complex number $z = \frac{3}{2} + \frac{\sqrt{3}}{2} i$ .

Use technology to find the values of $z^{2}$ and $z^{3}$ . Give your answers in the form $a + b i$ , where $a, b \in ℝ$ .

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11b

3 marks

Draw $z, z^{2}$ and $z^{3}$ on an Argand diagram.

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11c

2 marks

Find the smallest integer $k > 3$ such that $z^{k}$ is purely imaginary.

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Related Revision Notes

Intro to Complex Numbers
Modulus & Argument
Introduction to Argand Diagrams
Geometry of Complex Numbers
Forms of Complex Numbers
Applications of Complex Numbers
Introduction to Matrices
Operations with Matrices
Determinants & Inverses
Solving Systems of Linear Equations with Matrices