Complex numbers question bank final - KEA | Home

Complex numbers

One mark questions

1. Express the following in x+iy form

i)

ii)

iii)

iv)

v) vi) (1+2i)(2-3i)

vii)

viii)find x and y given that x+iy=

ix)If 3 4

?

x)Find the multiplicative inverse of 2-3i

xi)Find the multiplicative inverse of

xii) If z= Prove that ? 1 2. Find the Modulus and Amplitude of the following

iii)

3. Find 4. Simplify the following

i) cos

cos isin cos isin

Two mark questions:

1. If x+iy=

then Prove that x y

2. Find the Modulus and Amplitude of the following

i)1+iii) 1 3

iii 3 iv) 33 3 v) vi)

vii)1+cos+isin

viii) 1+cos

ix)1+

x)

xi)

3. Prove that

cos isin

4.

icot

5.

=itan

6. If z

are complex numbers, then prove the

following each carries 2 marks

i |z ? z |=|z | ? |z | ii)Amp z ? z =Amp z Amp z

iii

||

= iv)Amp

= Amp z

||

Amp z

7. Simplify the following

i) sin

ii)

iii)

iv) 1 cos isin V) 1 cos

8. Find the value of 1 cos 9. Find the value of

1 cos isin

1 cos isin

10.Prove that

1

11.If x+ 2cos and If y+ 2cos , then prove the

following(each carries 2 marks)

i)

=2cos( ) ii)

=2isin( )

iii)

=2cos(m

)

iv) xy =2cos

12. If , is an imaginary cube root of unity then show tha the following(each carries 2 marks)

i) (1+ - (1+ =0 ii) (1+

iii) (1+

iv) (1

= -8

v) (1+ vi) (1-

(1+

1

)4 = 16

vii) (1+ (1+ ) (1+ ) (1+ ) = 1

viii) (1+ 2 ix) 1 x) 1

1 = 64 1

2 = 16

2 27 0

xi) 1

+ 1

= 32

xii)

0

xiii) (3- ) (3- (3-

3

163

xiv) (1+ - 3 ? (1+ - 3 =0

xv) (1+ - (1+ - (1-

(1-

16

Four mark questions:

1. If z=cos+isin, then Prove that

2. If z=cos+isin, then Prove that

3. Prove the following

i) 1 3i 8

ii) 1 3i

1 3i 16

itann itan2

iii 1 3i 1 3i 128

iv 3 i 3 i 0

v) 3 i 3 i 0 323

Vi) 1 3i

1 3i

2

3. If 2-2x+4=0 thenProve that + =27

4. If cos+cos 0 and sin+sin 0, then Prove that

i) cos2+cos2 2cos ii) sin2+sin2 2sin

5. Find the cube roots of the following and represent them in Argand's diagram. Also find their continued product.

i)1+i ii) 8i iii) 43 4 iv) 3

v) 3

6. Find the fourth roots of the following and represent them in Argand's diagram. Also find their continued product.

i) 3

ii) 1 3 i iii)3 ? i

7. Find all the values of

8. Find all the values of z= 3

, and also

find their continued product and represent them

in Argand'sdiagram

9. Solve the equation z3=3 i 10. Solve x7-x4+x3-1=0 11. solve x5 ? x3 + x2 - 1=0. And represent cube

roots of unity in Argand diagram

Six mark questions:

1. State and Prove Demoivre's theorem

2. If cos

0

then

Prove the following

i) Cos3

3

ii) Sin3 +sin3

(iii) cos2

(iv) Sin2

3 3cos 33

3/2 3/2

v)cos( )+ cos( )+ cos( )=0

vi) sin( )+ sin

)+ sin( )=0

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