Further Complex Numbers (DP IB Maths: AA HL)

Topic Questions

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1.9 Further Complex Numbers

Medium

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Very Hard

3 marks

Consider $w = \frac{z_{1}}{z_{2}}$ , where $z_{1} = 2 + 2 \sqrt{3} i$ and $z_{2} = 2 + 2 i$ .

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

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4 marks

Write the complex numbers $z_{1}$ and $z_{2}$ in the form $r e^{i θ}, r \geq 0, - π < θ < π$ .

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3 marks

Express $w$ in the form $r e^{i θ}, r \geq 0, - π < θ < π .$

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6 marks

Solve the equation $z^{3} = 27 i$ , giving your answers in the form $a + b i$ .

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4 marks

Let $z_{1} = 6 cis (\frac{π}{6})$ and $z_{2} = 3 \sqrt{2} e^{i (\frac{π}{4})} .$

Giving your answers in the form $r cis θ,$ find

(i)

z_{1} z_{2}

(ii)

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

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2 marks

Write $z_{1}$ and $z_{2}$ in the form $a + b i .$

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2 marks

Find $z_{1} + z_{2},$ giving your answer in the form $a + b i .$

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2 marks

It is given that $z_{1}^{*}$ and $z_{2}^{*}$ are the complex conjugates of $z_{1}$ and $z_{2}$ respectively.

Find $z_{1}^{*} + z_{2}^{*},$ giving your answer in the form $a + b i .$

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2 marks

Let $z_{1} = 2 cis (\frac{π}{3})$ and $z_{2} = 2 + 2 i$ .

Express

(i)

z_{1}

in the form

a + b i

(ii)

z_{2}

in the form

r cis θ

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2 marks

Find $w_{1} = z_{1} + z_{2},$ giving your answer in the form $a + b i$ .

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3 marks

Find $w_{2} = z_{1} z_{2},$ giving your answer in the form $r cis θ .$

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2 marks

Sketch $w_{1}$ and $w_{2}$ on a single Argand diagram.

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3 marks

It is given that that $z_{1} = 2 e^{i (\frac{π}{3})}$ and $z_{2} = 3 cis (\frac{nπ}{12}), n \in ℤ^{+} .$

Find the value of $z_{1} z_{2}$ for $n = 3 .$

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3 marks

Find the least value of $n$ such that $z_{1} z_{2} \in ℝ^{+} .$

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5 marks

Consider the complex number $w = \frac{z_{1}}{z_{2}}$ where $z_{1} = 3 - \sqrt{3} i$ and $z_{2} = 2 c i s (\frac{2 π}{3}) .$

Express $w$ in the form $r c i s θ .$

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3 marks

Sketch $z_{1}, z_{2}$ and $w$ on the Argand diagram below.

q6b_1-9_ib-maths-aa-hl

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2 marks

Find the smallest positive integer value of $n$ such that $w^{n}$ is a real number.

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4 marks

Consider the complex number $z = - 1 + \sqrt{3} i$ .

Express $z$ in the form $r cis θ$ , where $r > 0$ and $- π < θ \leq π$ .

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4 marks

Find the three roots of the equation $z^{3} = - 1 + \sqrt{3} i$ , expressing your answers in the form $r cis θ$ , where $r > 0$ and $- π < θ \leq π$ .

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4 marks

Consider the equation $z^{4} - 1 = 15$ , where $z \in ℂ$ .

Find the four distinct roots of the equation, giving your answers in the form $a + b i$ , where $a, b \in ℝ .$

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2 marks

Represent the roots found in part (a) on the Argand diagram below.

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2 marks

Find the area of the polygon whose vertices are represented by the four roots on the Argand diagram.

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4 marks

Consider the complex numbers $w = 3 (\cos \frac{π}{3} - isin \frac{π}{3})$ and $z = 3 - \sqrt{3} i$ .

Write $w$ and $z$ in the form $r cis θ$ , where $r > 0$ and $- π < θ \leq π$ .

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2 marks

Find the modulus and argument of $z w$ .

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2 marks

Write down the value of $z w$ .

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10a

2 marks

Let $z = 12 + 16 i$ , where $a, b \in ℝ$ .

Verify that $4 + 2 i$ and $- 4 - 2 i$ are the second roots of $z$ .

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10b

4 marks

Hence, or otherwise, find two distinct roots of the equation $w^{2} + 4 w + (1 - 4 i) = 0$ , where $w \in ℂ$ . Give your answer in the form $a + b i$ , where $a, b \in ℝ$ .

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11a

1 mark

The complex numbers $ω_{1} = 3$ and $ω_{2} = 2 - 2 i$ are roots of the cubic equation $ω^{3} + p ω^{2} + q ω + r = 0,$ where $p, q, r \in ℝ .$

Write down the third root, $w_{3}$ , of the equation.

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11b

4 marks

Find the values of $p, q$ and $r$ .

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11c

4 marks

Express $w_{1},$ $w_{2}$ and $w_{3}$ in the form $r c i s θ$ .

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4 marks

Consider the equation $z^{2} + p z - 2 p - 1 = 0,$ where $z \in ℂ, p \in ℝ$ .

Find the value of $p$ for which one of the two distinct roots is $z_{1} = 2 + \sqrt{3} i$ .

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4 marks

Find the range of values of $p$ for which the equation has two distinct, real roots.

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4 marks

Consider the complex number $ω = - 1 + 4 i$ .

Show that $ω$ is a root of the cubic equation

z^{3} + 5 z^{2} + 23 z + 51 = 0

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4 marks

Find the other two roots of the cubic equation in part (a).

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5 marks

Consider $z = cis θ$ where $z \in ℂ, z \neq 1$ .

Show that $Re (\frac{1 + z}{1 - z}) = 0$ .

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5 marks

Consider the equation ${(z - 2)}^{2} = i, z \in ℂ$ .

(i)

Verify that

ω_{1} = 2 + e^{i \frac{π}{4}}

is a root of this equation.

(ii)

Find the second root of the equation, expressing your answer in the form

ω_{2} = a + e^{i θ}

where

a \in ℝ

and

θ > 0

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3 marks

The roots $ω_{1}$ and $ω_{2}$ are represented by the points A and B respectively on an Argand diagram.

Find AB.

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8 marks

Consider the equation $z^{4} + (1 - 4 i) z^{2} - 4 i = 0$ , where $z \in ℂ$ .

Find the four distinct roots of the equation, giving your answers in the form $a + b i$ where $a, b \in ℝ$ .

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3 marks

Consider the complex numbers $w_{1} = \frac{z_{1}}{z_{2}}, z_{1} = \frac{\sqrt{2} e^{- \frac{π}{3} i}}{3}$ and $z_{2} = 2 - 2 \sqrt{3} i$ .

Express

(i)

z_{1}

in the form

a + b i

(ii)

z_{2}

in the form

r cis θ

, where

r > 0

and

- π < θ < π

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2 marks

Find the exact value of $w_{1}$ .

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2 marks

Find $w_{2} = z_{1} z_{2}$ , giving your answer in the form $r cis θ$ , where $r > 0$ and $- π < θ < π$ .

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1 mark

Without drawing an Argand diagram, describe the geometrical relationship between $z_{1}$ and $z_{2}$ .

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5 marks

$z = \frac{\sqrt{3}}{2} i - \frac{1}{2}$

Find all the powers $z^{n}$ .

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3 marks

Find the area of the shape made by the powers $z^{n}$ when plotted on an Argand diagram.

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2 marks

Let $z = \cos θ + i \sin θ$ .

Write down the value of $z z^{*}$ .

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5 marks

Let $z_{1} = r_{1} (\cos θ_{1} + i \sin θ_{1}) a n d z_{2} = r_{2} (\cos θ_{2} + i \sin θ_{2})$ .

Prove the results

(i)

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

(ii)

\arg \frac{z_{1}}{z_{2}} = \arg z_{1} - \arg z_{2}

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1 mark

Using the results from part (b), describe fully the geometrical interpretation of dividing $z_{1}$ by $z_{2}$ .

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3 marks

Sketch $ω_{1}, ω_{2} and ω_{3}$ on the Argand diagram below.

q9b-1-9-ib-aa-hl-further-complex-numbers-hard-maths-dig

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4 marks

$ω_{1}, ω_{2}$ and $ω_{3}$ represent the vertices of a triangle.

Find the area of the triangle.

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10a

5 marks

The complex numbers $z_{1} = a, z_{2} = 3 - 2 i$ and $z_{3}$ are roots of the cubic equation $z^{3} + p z^{2} + q z - 26 = 0,$ where $a, p, q \in ℝ$ .

Find the values of $a, p$ and $q$ .

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10b

3 marks

Express $z_{1}, z_{2}$ and $z_{3}$ in the form $r e^{i θ}$ , where $r > 0$ and $0 < θ \leq 2 π$ .

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8 marks

Let $f (z) = z^{4} + a z^{3} + 6 z^{2} + b z + 65$ , where $a$ and $b$ are real constants.

Given that $z = 3 + 2 i$ is a root of the equation $f (z) = 0$ , show the roots $f (z) = 0$ on the Argand diagram below.

q11-1-9-ib-aa-hl-further-complex-numbers-hard-maths-dig

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6 marks

Consider the equation $p z^{3} + q z^{2} + 8 p^{3} z + 5 q = 0$ , where $z \in ℂ, p \in ℝ$ .

Given that one of the distinct roots is $z_{1} = \frac{5}{2}$ , find

(i)

the values of

p

and

q

(ii)

the roots

z_{1}

and

z_{2}

, giving your answers in the form

a + b i

, where

a, b \in ℝ

How did you do?

2 marks

On an Argand diagram $z_{1}, z_{2}$ and $z_{3}$ are represented by the points A, B and C respectively.

Find the area of the triangle ABC.

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4 marks

Consider the complex numbers $z$ and $w$ , where $z = \sqrt{3} - i$ , $Im (\frac{z^{2}}{w}) = 0, |\frac{z^{2}}{w}| = \frac{1}{2} |z|$ .

Use geometrical reasoning to find the two possibilities for $w$ , giving your answers in exponential form.

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8 marks

Consider the equation $z^{4} - 5 a z^{3} + 25 a z^{2} - 20 a b z + 24 a b = 0$ , where $a, b \in ℤ$ and $z \in ℂ$ .

Given that one root is $a + a i$ and another root is $b + b i$ , find the possible values of $a$ and $b$ .

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3 marks

Consider the complex number $z = 1 + \sqrt{3} i$ .

Use De Moivre’s theorem to find the value of $z^{3}$ .

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5 marks

Use the principle of mathematical induction to prove, for all $n \in ℕ$ , that

{(\cos θ + i \sin θ)}^{n} = \cos n θ + i \sin n θ

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2 marks

Show that the result in part (b) is true for all $n \in ℤ .$

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5 marks

Consider the equation ${(z + a)}^{5} + 1 = 0, z \in ℂ$ .

Given that the product of the roots is 31, find the roots of the equation, expressing your answers in the form $ω_{n} = b + e^{i θ}$ , where $b \in ℝ$ and $θ > 0$ .

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3 marks

Let S be the sum of the roots found in part (a).

Show that $Im (S) = 0$ and find the value of $Re (S)$ .

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1 mark

The roots $ω_{1}, ω_{2}, . . ., ω_{5}$ are represented on an Argand diagram.

Describe the geometrical shape made by the five roots.

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8 marks

Consider the equations $u^{*} + 2 v = 2 i$ and $i u + v^{*} = 3$ , where $u, v \in ℂ$ . Find $\frac{u}{v}$ giving your answer in the form $r e^{i θ}$ , where $r > 0$ and $0 < θ < 2 π$ .

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8 marks

By first expressing $1 + \sqrt{3} i$ and $- 1 + i$ in the form $r cis θ$ where $r > 0$ and $- π < θ \leq π$ , show that $\tan \frac{5 π}{12} = 2 + \sqrt{3}$ .

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5 marks

Consider the complex number $z = 1 + \cos 2 θ + i \sin 2 θ$ .

Find the modulus and argument of z.

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2 marks

Solve $z = 0$ for $- π < θ \leq π$ .

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4 marks

Let $z = e^{i θ}$ .

Show that

(i)

z + z^{2} + z^{3} + \dots + z^{n} = \cos θ + \cos 2 θ + \dots + \cos n θ + i (\sin θ + \sin 2 θ + \dots + \sin n θ)

(ii)

(2 - z) (2 - z^{*}) = 5 - 4 \cos θ

How did you do?

4 marks

Use the results found in part (a) to find the sum of the infinite series

\frac{\sin θ}{2} + \frac{\sin 2 θ}{2^{2}} + \frac{\sin 3 θ}{2^{3}} + \frac{\sin 4 θ}{2^{4}} + \dots

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10a

5 marks

The primary square root of a complex number $z$ is defined as $\sqrt{z} = x + i y$ , where $x, y \in ℝ$ and $x \geq 0$ . If $x = 0$ then the value for $y$ is chosen such that $y \geq 0$ . Note that the other square root of $z$ will then be given by $- \sqrt{z} = - x - i y$ .

Show that

x = \sqrt{\frac{Re (z) + \sqrt{{(Re (z))}^{2} + {(Im (z))}^{2}}}{2}}

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10b

2 marks

Given that $x > 0$ , derive a formula for $y$ in terms of $x$ and $Im (z)$ , and explain why $y$ in this case will always have the same sign (positive, negative, or zero) as $Im (z)$ .

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10c

2 marks

Hence show that in general

y = \pm \sqrt{\frac{- Re (z) + \sqrt{{(Re (z))}^{2} + {(Im (z))}^{2}}}{2}}

with the choice of the positive or negative value being dependent on the properties of $z$ .

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10d

3 marks

Explain what must be true of $z$ for each of the following to be true:

(i)

x = 0, y \neq 0

(ii)

x \neq 0, y = 0

(iii)

x = 0, y = 0

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Related Revision Notes

Geometry of Complex Numbers
Forms of Complex Numbers
Complex Roots of Polynomials
De Moivre's Theorem
Roots of Complex Numbers
Systems of Linear Equations
Algebraic Solutions
Equations of a Straight Line
Quadratic Functions
Factorising & Completing the Square