Test:- Complex Number - 1 - Mathematics MCQ

so, smallest positive integer is 2 for which iⁿ will be an integer as i = -1

ANSWER :- d

Solution :- z = x + iy = (-16 + 30i)^1/2

Squaring it we get 

x² - y² + 2xyi = -16 + 30 i

x² - y² = -16……………………………….(1)

2xy = 30   =>y = 15/x

Put the value of y in eq(1)

x²- 225/x² = -16

x⁴ + 16x² -225 = 0

x⁴ + 25x² - 9x² - 225 = 0

(x² + 25) (x² - 9) = 0

x = +-3 and y = +-5

x + iy = +-(3 + 5i)

Let z be the purely imaginary number other than 0 so for z = x + iy, y ≠ 0 but x = 0 so, arg will be tan^-1 y/ 0 =  tan^-1 ∞ = π/2

Let z be the purely imaginary number with (lm)z < 0, so arg(z) is, θ = - π/2

Given z is purely real is z=a+i(0) for some a∈R

⇒z=a

given Re(z)<0

⇒a<0

∴ z can be represented as  

∴Arg(z)=π

Let z = x+iy

zˉ = x−iy

arg(z) = xy​,arg(zˉ) = −xy​

arg(z) + arg(zˉ) = 0

= i.e. in the 2nd quadrant

i² + i⁴ + i⁶ + ... up to (2n + 1) terms = -1 
Note: For i² + i⁴ + i⁶ + ...up to n terms
If n is even then sum = 0
If n is odd then sum = -1
hence i² + i⁴ + i⁶ + ... (2n + 1) term is = -1

Real part of (1 - cosθ + 2: cos θ)^-1

So, real part is 

 sin x + i cos 2x and cos x - i sin 2x, and their conjugates
sin x - i cos 2x and cos x + i sin 2x will be equal if
cos x: = sin x: ...(i)
=>tan.x:=1
cos 2x sin 2x: ...(ii)
=> tan 2.x = 1
tan x = tan 45

which cannot be satisfied simultaneously
=> no value of x exist.

x² + y² = 2

Complex number z, if and only if

= (w)⁵⁸ + (w²)⁵⁸ = w + w² = 1

| z - 3 | = 6 in the argand plane is
| z - (-3)| = 6 which is the circle where r = 6. and centre (-3,0)

 ,
extended portion of line segment.