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, i² = -1, then i¹ + i² + i³ + ....+ up to 1000 terms is equal to ________

(a)

1

(b)

-1

(c)

i

(d)

0

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 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 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

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

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

The amplitude of \(\frac{1}{i}\) is equal to _______

(a)

0

(b)

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

(c)

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

(d)

\(\pi \)

The value of (1+i)⁴ + (1-i)⁴ is __________

(a)

8

(b)

4

(c)

-8

(d)

-4

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 z = a + ib lies in quadrant then \(\frac { \bar { z } }{ z } \) also lies in the III quadrant if _________

(a)

a > b > 0

(b)

a < b < 0

(c)

b < a < 0

(d)

b > a > 0

\(\frac { 1+e^{ -i\theta } }{ 1+{ e }^{ i\theta } } \) =__________

(a)

cosθ + i sinθ

(b)

cosθ - i sinθ

(c)

sinθ - i cosθ

(d)

sinθ + icosθ

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 x = cosθ + i sinθ, then the value of xⁿ+\(\frac { 1 }{ { x }^{ n } } \) is ___________

(a)

2 cosθ

(b)

2i sin nθ

(c)

2i sin nθ

(d)

2i cos nθ

If ω is the cube root of unity, then the value of (1-ω) (1-ω²) (1-ω⁴) (1-ω⁸) is _________

(a)

9

(b)

-9

(c)

16

(d)

32

The points represented by 3 - 3i, 4 - 2i, 3 - i and 2 - 2i form _____ in the argand plane.

(a)

collinear points

(b)

Vertices of a parallelogram

(c)

Vertices of a rectangle

(d)

Vertices of a square

(1+i)³ = ______

(a)

3 + 3i

(b)

1 + 3i

(c)

3 - 3i

(d)

2i - 2

\(\frac { (cos\theta +isin\theta )^{ 6 } }{ (cos\theta -isin\theta )^{ 5 } } \) = ________

(a)

cos 11θ - isin 11θ

(b)

cos 11θ + isin 11θ

(c)

cosθ + i sinθ

(d)

\(cos\frac { 6\theta }{ 5 } +isin\frac { 6\theta }{ 5 } \)

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)

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)

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

The modular of \(\frac { (-1+i)(1-i) }{ 1+i\sqrt { 3 } } \) is ______

(a)

\(\sqrt{2}\)

(b)

2

(c)

1

(d)

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

The value of \(\frac { (cos{ 45 }^{ 0 }+isin{ 45 }^{ 0 })^{ 2 }(cos{ 30 }^{ 0 }-isin{ 30 }^{ 0 }) }{ cos{ 30 }^{ 0 }+isin{ 30 }^{ 0 } } \) is __________

(a)

\(\frac { 1 }{ 2 } +i\frac { \sqrt { 3 } }{ 2 } \)

(b)

\(\frac { 1 }{ 2 } -i\frac { \sqrt { 3 } }{ 2 } \)

(c)

\(-\frac { \sqrt { 3 } }{ 2 } +\frac { 1 }{ 2 } \)

(d)

\(\frac { \sqrt { 3 } }{ 2 } +\frac { 1 }{ 2 } \)

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

If z₁, z₂, z₃ are the vertices of a parallelogram, then the fourth vertex z₄ opposite to z₂ is _____

(a)

z₁ + z₂ - z₂

(b)

z₁ + z₂ - z₃

(c)

z₁ + z₂ - z₃

(d)

z₁ - z₂ - z₃

If x_r = \(cos\left( \frac { \pi }{ 2^{ r } } \right) +isin\left( \frac { \pi }{ 2^{ r } } \right) \) then x₁, x_2, x₃ ... x_∞ is _________

(a)

-∞

(b)

-2

(c)

-1

(d)

0