iⁿ+iⁿ⁺¹+iⁿ⁺²+iⁿ⁺³ is

(a)

0

(b)

1

(c)

-1

(d)

i

 The value of \(\sum_{n=1}^{13}\left(i^{n}+i^{n-1}\right)\) is

(a)

1+ i

(b)

i

(c)

1

(d)

0

If |z| = 1, then the value of \(\frac { 1+z }{ 1+\overline { z } }\) is

(a)

z

(b)

\(\bar { z } \)

(c)

\(\cfrac { 1 }{ z } \)

(d)

1

If \(\frac { z-1 }{ z+1 } \) is purely imaginary, then |z| is

(a)

\(\frac { 1 }{ 2 } \)

(b)

1

(c)

2

(d)

3

The value of (1+i) (1+i²) (1+i³) (1+i⁴) is ____________

(a)

2

(b)

0

(c)

1

(d)

i

If \(\sqrt { a+ib } \)  = x + iy, then possible value of \(\sqrt { a-ib }\) is ___________

(a)

x²+y²

(b)

\(\sqrt { { x }^{ 2 }+{ y }^{ 2 } } \)

(c)

x+iy

(d)

x-iy

i^-1 =
(i) \(\frac{1}{i}\)
(ii) i
(iii) -i
(4) \(\frac { 1 }{ { i }^{ 2 } } \)

When z = x + iy, then iz is
(1) x-iy
(2) i(x+iy)
(3) -y+ix
(4) Rotation of z by 90° in the counter clockwise direction

Simplify the following
i¹⁹⁴⁷+ i¹⁹⁵⁰

If z₁ and z₂ are two complex numbers, such that |z₁| = Iz₂|, then is it necessary that z₁ = z₂?

Find Re (z) and im (z) if z = 5i¹¹ + 7i³

Simplify the following i⁷

If \(\frac { 1+z }{ 1-z } =cos2\theta +isin2\theta \), show that z = i tan\(\theta\)

Show that \(\left| \frac { z-3 }{ z+3 } \right| \) = 2 represent a circle.

 If z = x + iy is a complex number such that Im \(\left( \frac { 2z+1 }{ iz+1 } \right) =0\) show that the locus of z is 2x²+ 2y²+ x - 2y = 0

Prove that the values of \(\sqrt [ 4 ]{ -1 } arr\ \pm \frac { 1 }{ \sqrt { 2 } } \left( 1\pm i \right) \). Let z = (-1)

If 1, ω, ω² are the cube roots of unity then show that (1+5ω²+ω⁴) (1+5ω+ω²) (5+ω+ω⁵) = 64

Find the radius and centre of the circle \(z\bar { z } \)-(2+3i)z-(2-3i)\(\bar { z } \)+9 = 0 where z is a complex number.