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

The area of the triangle formed by the complex numbers z, iz and z+iz in the Argand’s diagram is

(a)

\(\cfrac { 1 }{ 2 } \left| z \right| ^{ 2 }\)

(b)

|z|²

(c)

\(\cfrac { 3 }{ 2 } \left| z \right| ^{ 2 }\)

(d)

2|z|²

The conjugate of a complex number is \(\cfrac { 1 }{ i-2 } \). Then the complex number is

(a)

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

(b)

\(\cfrac { -1 }{ i+2 } \)

(c)

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

(d)

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

If \(z=\cfrac { \left( \sqrt { 3 } +i \right) ^{ 3 }\left( 3i+4 \right) ^{ 2 } }{ \left( 8+6i \right) ^{ 2 } } \) , then |z| is equal to 

(a)

0

(b)

1

(c)

2

(d)

3

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 \(\left| z-\frac { 3 }{ z } \right| =2\), then the least value |z| is

(a)

1

(b)

2

(c)

3

(d)

5

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 |z₁| = 1, |z₂| = 2, |z₃| = 3 and |9z₁z₂ + 4z₁z₃ + z₂z₃| = 12, then the value of |z₁+z₂+z₃| is 

(a)

1

(b)

2

(c)

3

(d)

4

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

z₁, z₂ and z₃ are complex number such that z₁ + z₂ + z₃ = 0 and |z₁| = |z₂| = |z₃| = 1 then z₁² + z₂² + z₃³ is

(a)

3

(b)

2

(c)

1

(d)

0

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

(a)

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

(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)

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

The principal argument of (sin 40°+i cos 40°)⁵ is

(a)

−110°

(b)

−70°

(c)

70°

(d)

110°

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 \(\omega \neq 1\) is a cubic root of unity and \(\left( 1+\omega \right) ^{ 7 }=A+B\omega \), then (A, B) equals

(a)

(1, 0)

(b)

(−1, 1)

(c)

(0, 1)

(d)

(1, 1)

The principal argument of the complex number \(\frac { \left( 1+i\sqrt { 3 } \right) ^{ 2 } }{ 4i\left( 1-i\sqrt { 3 } \right) } \) is

(a)

\(\frac { 2\pi }{ 3 } \)

(b)

\(\frac { \pi }{ 6 } \)

(c)

\(\frac { 5\pi }{ 6 } \)

(d)

\(\frac { \pi }{ 2 } \)

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

The product of all four values of \(\left( cos\cfrac { \pi }{ 3 } +isin\cfrac { \pi }{ 3 } \right) ^{ \frac { 3 }{ 4 } }\) 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)

\(-\sqrt { 3i } \)

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 \(\omega =cis\cfrac { 2\pi }{ 3 } \), 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