COMPLEX NUMBERS WORKSHEET

Choose the correct or the most suitable answer from the given four alternatives :

Problem 1 :

iⁿ + i^{n + 1} + i^{n+ 2} + i^{n + 3} is

(1)  0   (2)  1  (3)  -1  (4)  i

Solution

Problem 2 :

(1)  1 + i    (2)  i   (3)  1   (4)   0

Solution

Problem 3 :

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

(1)  1/2 |z|²   (2)  |z|²    (3)  3/2  |z|²   (4)  2|z|²

Solution

Problem 4 :

The conjugate of a complex numbers is 1/(I - 2). Then, the complex number is

Solution

Problem 5 :

(1)   0   (2)  1  (3)  2  (4)  3

Solution

Problem 1 :

If z is a non zero complex number, such that 2iz² = z̄ then |z| is

(1)   1/2    (2)   1   (3)  2  (4)  3

Solution

Problem 2 :

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

(1)√3 - 2  (2)  √3 + 2  (3)  √5 - 2  (4) √5 + 2

Solution

Problem 3 :

If |z – 3/z| = 2, then the least value of |z| is

(1)1  (2)  2  (3)  3  (4)  5

Solution

Problem 4 :

If |z| = 1, then the value of (1 + z)/(1 + z̄) is

(1)  z   (2)  z̄   (3)  1/z  (4)  1

Solution

Problem 5 :

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

Solution

Problem 1 :

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

(1)   1   (2)   2   (3)   3   (4)   4

Solution

Problem 2 :

If z is a complex number such that z ϵ ℂ \ ℝ and z + 1/z ϵ ℝ, then |z| is

(1)   0   (2)   1   (3)   2   (4)   3

Solution

Problem 3 :

z₁, z₂ and z₃ are complex numbers such that z₁ + z₂ + z₃ = 0 and |z₁| = |z₂| = |z₃| = 1 then z₁² + z₂² + z₃² is

(1)  3   (2)   2   (3)   1   (4)   0

Solution

Problem 4 :

If (z - 1)/(z + 1) is purely imaginary, then |z| is 

(1)   1/2    (2)   1   (3)   2   (4)   3

Solution

Problem 5 :

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

(1)   real axis   (2)   imaginary axis   (3)   ellipse   (4)   circle

Solution

Problem 1 :

The principal argument of 3/(-1 + i) is

Solution

Problem 2 :

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

(1)  -110^º    (2)   -70^º   (3)   70^º   (4)   110^º

Solution

Problem 3 :

If (1 + i) (1 + 2i) (1 + 3i) … (1 + ni) = x + iy, then 2 ⋅ 5 ⋅ 10 … (1 + n²) is

(1)1   (2)   i   (3)   x² + y²    (4)   1 + n²

Solution

Problem 4 :

If ω ≠ 1 is a cubic root of unity (1 + ω)⁷ = A + Bω, then (A, B) equals

(1)   (1, 0)   (2)   (-1, 1)  (3)   (0,1)   (4)   (1, 1)

Solution

Problem 5 :

Solution

Problem 1 :

If α and β are the roots of x² + x + 1 = 0, then α²⁰²⁰ + β²⁰²⁰ is

(1)   -2   (2)  -1    (3)    1   (4)   2

Solution

Problem 2 :

(1)   -2   (2)   -1    (3)   1    (4)  2

Solution

Problem 3 :

(1)   1    (2)  -1   (3)  √3i    (4)   -√3i

Solution

Problem 4 :

Solution

Problem 5 :

Solution