Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 11 - Complex Numbers>END-OF-CHAPTER REVIEW EXERCISE 11>Q 7

MEDIUM

AS and A Level

IMPORTANT

Earn 100

Find the complex number $w$ satisfying the equation $w^{*} - 2 - 2 i = 3 i w$ . Give your answer in the form $x + i y$ where $x$ and $y$ are real.

Important Questions on Complex Numbers

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 11 - Complex Numbers>END-OF-CHAPTER REVIEW EXERCISE 11>Q 7

On a single Argand diagram, sketch the loci

| z - 3 - 3 i | = 2

and

\arg (z - 3 - 3 i) = \frac{π}{3} .

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 11 - Complex Numbers>END-OF-CHAPTER REVIEW EXERCISE 11>Q 7

On a single Argand diagram, sketch the loci

| z - 3 - 3 i | = 2

and

\arg (z - 3 - 3 i) = \frac{π}{3} .

Hence determine the value of

z

that satisfies both loci, giving your answer in the form

\Rightarrow a - 9 a - 6 - 2 = 0

, where

x

and

y

are real.

MEDIUM

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 11 - Complex Numbers>END-OF-CHAPTER REVIEW EXERCISE 11>Q 8

Given

(x + i y)^{2} = 7 - (6 \sqrt{2}) i

Given that

x

and

y

are real numbers, find the values of

x

and the values of

y

.

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 11 - Complex Numbers>END-OF-CHAPTER REVIEW EXERCISE 11>Q 8

Show that

z - 3

is a factor of

2 z^{3} - 4 z^{2} - 5 z - 3 .

HARD

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 11 - Complex Numbers>END-OF-CHAPTER REVIEW EXERCISE 11>Q 8

Solve

2 z^{3} - 4 z^{2} - 5 z - 3 = 0 .

MEDIUM

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 11 - Complex Numbers>END-OF-CHAPTER REVIEW EXERCISE 11>Q 9

Given that

z_{1} = 5 - 3 i

and

z_{1} z_{2} = 21 + i,

find

z_{2},

giving your answer in the form

x + i y,

where

x

and

y

are real.

MEDIUM

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 11 - Complex Numbers>END-OF-CHAPTER REVIEW EXERCISE 11>Q 9

Solve

(3 z + 1)^{3} = - 27

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 11 - Complex Numbers>END-OF-CHAPTER REVIEW EXERCISE 11>Q 10

It is given that

w = 1

is a root of the equation

f (w) = 2 w^{4} + 5 w^{3} - 2 w^{2} + w - 6 = 0 .

Show that

w + 3

is a factor of

2 w^{4} + 5 w^{3} - 2 w^{2} + w - 6