Complex Numbers.

Question 1

What is i ^ 4n+1 ?

Answer 1

i

Question 2

What is i^4n+2

Answer 2

-1

Question 3

What is i^4n+3

Answer 3

-i

Question 4

What is i^4n+4

Answer 4

1

Question 5

In complex numbers when is an even integer and when is n an odd integer for i^n

Answer 5

i=(-1)^n/2 if n is an even integer and

i=(-1)^ n-1/2 is n is an odd integer

Question 6

What is z1+z2 equal to ? ; z1/z2 are complex numbers.

Answer 6

z1+z2=(x1+x2)+i(y1+y2)

Question 7

What is z1.z2 equal to?

; z1/z2 are complex numbers.

Answer 7

z1.z2=(x1x2-y1y2)+i(x1y2+x2y1)

Question 8

What is z1/z2 equal to ?

; z1/z2 are complex numbers.

Answer 8

z1/z2= = [(x1x2+y1y2)+i(x2y2-x1y2)]/x₂²+y₂²

Question 9

Conjugate of a complex number z=x+iy

Answer 9

z(conjugate) = x -iy
Re(z) = z-z(conjugate)/2
Im(z)=z-z(conjugate)/2i

Question 10

What is modulus of complex number z=x + iy

Answer 10

|z|= √x²+y²

Question 11

What is multiplication inverse 1/z of a complex number z?

Answer 11

1/z= x÷x²+y² + i(-y)÷x²+y²

Question 12

How is nth root of unity found out ?

Answer 12

z= cos 2kπ÷n + i sin 2kπ÷n 
k= 0,1,2.....(n-1)

Question 13

On what curve does the nth root of unity lie ?

Answer 13

The nth root of unity lies on a unit circle and divides the circumference into n equal parts.

Question 14

What is the eq of cube root of unity ?

Answer 14

1+ω^r+ ω^ 2r = 0 if r ain’t a multiple of 3 and 3 if r is a multiple of 3.
ω^3r=1
ω^3r+1= ω
ω^3r+2=ω²

Question 15

For a complex number z, what is the centroid and incentre of triangle ABC?

Answer 15

Centroid :
z= (z1+z2+z3)/3
Incentre:
z= az1+bz2+cz3/a+b+c

Question 16

General eq of a straight line in a complex plane

Answer 16

a(bar)z+az(bar)+b=0

Question 17

Complex slope of complex line a(bar)z+az(bar)+b=0?

Answer 17

W= -a/ a(bar)

Question 18

Real slope of complex line a(bar)z+az(bar)+b=0 ?

Answer 18

m= -i x (a+ a(bar) )/(a - a(bar) )

Question 19

Eq of a straight line passing through two complex points ?

Answer 19

z= tz1 + (1-t)z2

Question 20

Length of perpendicular from a point to a line a(bar)z+az(bar)+b=0?

Answer 20

l =

|a(bar)z1+az1(bar)+b|/2|a|

Question 21

Eq of a straight line parallel to a complex line a(bar)z+az(bar)+b=0?

Answer 21

a(bar)z+az(bar)+λ=0

Question 22

Eq of a line perpendicular to a given complex line a(bar)z+az(bar)+b=0?

Answer 22

az(bar)-a(bar)z+ iλ=0

Question 23

Eq of a perpendicular bisector of a line in complex plane ?

Answer 23

z(z1(bar)-z2(bar))+z(bar)(z1-z2)= |z1|²−|z2|².

;z is a point on alone segment and z1z2 are two points making a line segment.

Question 24

In complex numbers , what is x²+xy+y² equal to ?

Answer 24

x²+xy+y²=(x-yω) (x+yω)

Question 25

In complex numbers , what is x²-xy+y² equal to ?

Answer 25

x²-xy+y²=(x-yω) (x+yω^)

Question 26

In complex numbers, what is x²+y² equal to ?

Answer 26

x²+y²= (x+iy) (x-iy)

Question 27

In complex numbers what is x³+y³ equal to ?

Answer 27

x³+y³=

x+y) (x+yω) (x+yω²

Question 28

In complex numbers what is x³- y³ equal to ?

Answer 28

x³-y³= (x-y) (x-yω) (x-yω²)