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

If |z - 2 + i | ≤ 2, then the greatest value of |z| is

(a)

\(\sqrt { 3 } -2\)

(b)

\(\sqrt { 3 } +2\)

(c)

\(\sqrt { 5 } -2\)

(d)

\(\sqrt { 5 } +2\)

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

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 z is a complex number such that \(z \in \mathbb{C} \backslash \mathbb{R}\) and \(z+\frac { 1 }{ z } \epsilon R\), then |z| is

(a)

0

(b)

1

(c)

2

(d)

3

If z = x + iy is a complex number such that |z+2| = |z−2|, then the locus of z is

(a)

real axis

(b)

imaginary axis

(c)

ellipse

(d)

circle

The principal argument of \(\cfrac { 3 }{ -1+i } \) is

(a)

\(\cfrac { -5\pi }{ 6 } \)

(b)

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

(c)

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

(d)

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

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

If \(\alpha \) and \(\beta \) are the roots of x²+x+1 = 0, then \({ \alpha }^{ 2020 }+{ \beta }^{ 2020 }\) is

(a)

-2

(b)

-1

(c)

1

(d)

2

If \(\omega \neq 1\) is a cubic root of unity and \(\left| \begin{matrix} 1 & 1 & 1 \\ 1 & { -\omega }^{ 2 }-1 & { \omega }^{ 2 } \\ 1 & { \omega }^{ 2 } & { \omega }^{ 7 } \end{matrix} \right| \) = 3k, then k is equal to 

(a)

1

(b)

-1

(c)

\(\sqrt { 3i } \)

(d)

\(-\sqrt { 3i } \)

If \(\omega =cis\cfrac { 2\pi }{ 3 } \), then the number of distinct roots of \(\left| \begin{matrix} z+1 & \omega & { \omega }^{ 2 } \\ \omega & z+{ \omega }^{ 2 } & 1 \\ { \omega }^{ 2 } & 1 & z+\omega \end{matrix} \right| \)=0

(a)

1

(b)

2

(c)

3

(d)

4

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

If a = cos θ + i sin θ, then \(\frac { 1+a }{ 1-a } \) = ___________

(a)

cot \(\frac { \theta }{ 2 } \)

(b)

cot θ

(c)

i cot \(\frac { \theta }{ 2 } \)

(d)

i tan\(\frac { \theta }{ 2 } \)

.If a = 3+i and z = 2-3i, then the points on the Argand diagram representing az, 3az and - az are _________

(a)

Vertices of a right angled triangle

(b)

Vertices of an equilateral triangle

(c)

Vertices of an isosceles

(d)

Collinear

The least positive integer n such that \(\left( \frac { 2i }{ 1+i } \right) ^{ n }\) is a positive integer is ____________

(a)

16

(b)

8

(c)

4

(d)

2

If z = \(\frac { 1 }{ (2+3i)^{ 2 } } \) then |z| = ____________

(a)

\(\frac { 1 }{ 13 } \)

(b)

\(\frac { 1 }{ 5} \)

(c)

\(\frac { 1 }{ 12 } \)

(d)

none of these

If z = 1-cos θ + i sin θ, then |z| = _____________

(a)

2 sin\(\frac { 1 }{ 3 } \)

(b)

2 cos\(\frac { \theta }{ 2 } \)

(c)

2|sin\(\frac { \theta }{ 2 } \)|

(d)

2|cos\(\frac { \theta }{ 2 } \)|

If z = \(\frac { 1 }{ 1-cos\theta -isin\theta } \), the Re(z) = ___________

(a)

0

(b)

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

(c)

cot\(\frac { \theta }{ 2 } \)

(d)

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

The complex number z which satisfies the condition \(\left| \frac { 1+z }{ 1-z } \right| \)  = 1 lies on _________

(a)

circle x²+ y² = 1

(b)

x-axis

(c)

y-axis

(d)

the lines x+y = 1

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

If a = cos α + i sin α, b = -cos β + i sin β then \(\left( ab-\frac { 1 }{ ab } \right) \) is _________

(a)

-2i sin(α - β)

(b)

2i sin(α - β)

(c)

2 cos(α - β)

(d)

-2 cos(α - β)

If x = cos θ + i sin θ, then xⁿ + \(\frac { 1 }{ { x }^{ n } } \) is ______

(a)

2 cos nθ

(b)

2 i sin nθ

(c)

2ⁿ cosθ

(d)

2ⁿ i sinθ

Re(z)

(1)

\(\frac { z+\bar { z } }{ 2 } \)

Im(z)

(2)

\(\frac { z-\bar { z } }{ 2 } \)

z is real

(3)

2\(\sqrt { 2 } \)

z is imaginary

(4)

|\(\bar { z } \)|

|z|

(5)

z =\(\bar { z } \)

|2+2i|

(6)

z = -\(\bar { z } \)