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 is a non zero complex number, such that 2iz² = \(\bar { z } \) then |z| is

(a)

\(\cfrac { 1 }{ 2 } \)

(b)

1

(c)

2

(d)

3

The solution of the equation |z| - z = 1 + 2i is

(a)

\(\frac { 3 }{ 2 } -2i\)

(b)

\(-\frac { 3 }{ 2 } +2i\)

(c)

\(2-\frac { 3 }{ 2 } i\)

(d)

\(2+\frac { 3 }{ 2 } i\)

If (1+i)(1+2i)(1+3i)...(1+ni) = x + iy, then \(2\cdot 5\cdot 10...\left( 1+{ n }^{ 2 } \right) \) is

(a)

1

(b)

i

(c)

x²+y²

(d)

1+n²

The value of \(\left( \cfrac { 1+\sqrt { 3 } i}{ 1-\sqrt { 3}i } \right) ^{ 10 }\) is

(a)

\(cis\cfrac { 2\pi }{ 3 } \)

(b)

\(cis\cfrac { 4\pi }{ 3 } \)

(c)

\(-cis\cfrac { 2\pi }{ 3 }\)

(d)

\(-cis\cfrac { 4\pi }{ 3 }\)

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

If z = cos\(\frac { \pi }{ 4 } \) + i sin\(\frac { \pi }{ 6 } \), then ______

(a)

|z| = 1, arg(z) =\(\frac { \pi }{ 4 } \)

(b)

|z| = 1, arg(z) = \(\frac { \pi }{ 6 } \)

(c)

|z| = \(\frac { \sqrt { 3 } }{ 2 } \), arg(z) = \(\frac { 5\pi }{ 24 } \)

(d)

|z| = \(\frac { \sqrt { 3 } }{ 2 } \), arg (z) = tan^-1\(\left( \frac { 1 }{ \sqrt { 2 } } \right) \)

If x + iy = \(\frac { 3+5i }{ 7-6i } \), they y = ___________

(a)

\(\frac { 9 }{ 85 } \)

(b)

-\(\frac { 9 }{ 85 } \)

(c)

\(\frac { 53 }{ 85 } \)

(d)

none of these

If zⁿ = \(cos\frac { n\pi }{ 3 } +isin\frac { n\pi }{ 3 } \), then z₁, z₂ ..... z₆ is _________

(a)

1

(b)

-1

(c)

i

(d)

-i

The conjugate of \(\frac { 1+2i }{ 1-(1-i)^{ 2 } } \) is _______

(a)

\(\frac { 1+2i }{ 1-(1-i)^{ 2 } } \)

(b)

\(\frac { 5 }{ 1-(1-i)^{ 2 } } \)

(c)

\(\frac { 1-2i }{ 1+(1+i)^{ 2 } } \)

(d)

\(\frac { 1+2i }{ 1+(1-i)^{ 2 } } \)