PROBLEMS ON PROPERTIES OF COMPLEX NUMBERS

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 :

Given, |z₁| = 1, |z₂| = 2, |z₃| = 3 

|9z₁z₂ + 4z₁z₃ + z₂z₃| = 12

So, option (2) is correct.

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 :

Given, z is a complex number.

So, option (2) is correct.

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 :

Given, z₁, z₂ and z₃ are complex numbers.

z₁ + z₂ + z₃ = 0 and |z₁| = |z₂| = |z₃| = 1

To find : z₁² + z₂² + z₃²

z₁² + z₂² + z₃² = z₁²+ z₂² + z₃² - 2(z₁z₂ + z₂z₃ + z₃z₁)

z₁² + z₂² + z₃² = (z₁ + z₂ + z₃)² - 2(z₁z₂ + z₂z₃ + z₃z₁)

z₁² + z₂² + z₃² = - 2(z₁z₂ + z₂z₃ + z₃z₁)

z₁² + z₂² + z₃² = -2z₁z₂z₃(1/z₁ + 1/z₂ + 1/z₃) --- (1)

|z₁| = 1

|z₁|² = 1

z₁z̄₁ = 1

1/z₁ = z̄₁ similarly 1/z₂ = z̄₂, 1/z₃ = z̄₃

1/z₁ = z̄₁, 1/z₂ = z̄₂ and 1/z₃ = z̄₃ substitute in (1) becomes,

So, option (4) is correct.

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 :

|z + 2| = |z - 2|

|x + iy + 2| = |x + iy - 2|

|x + 2 + iy|² = |x - 2 + iy|²

(x + 2)² + y² = (x - 2)² + y²

x² + y² + 4x = x² + 4 - 4x

x² + y² + 4x - x² - 4 + 4x = 0

8x = 0

x = 0

So, option (2) is correct.