The region represented by {z = x + iy ∈ C : |z|

Question 5:- The region represented by {z = x + iy ∈ C : |z| – Re(z) ≤ 1} is also given by the inequality :

(A) y² ≤ x + 1/2

(B) y² ≤ 2(x + 1/2)

(C) y² ≥ x + 1

(D) y² ≥ 2(x + 1)

Solution:- Given that,

z = x + iy

Also given that

|z| – Re(z) ≤ 1

⇒ $\sqrt{x^2 + y^2} - x \leq 1$

⇒ $\sqrt{x^2 + y ^2} \leq x + 1$

Squaring both side

⇒ x² + y² ≤ x² + 1 + 2x

⇒ y² ≤ 2x + 1

Answer is (B) y² ≤ 2(x + 1/2)

Question 1:- If z is complex number such that |z| ≥ 2 . then the minimum value of |z + 1/2|.

(a) is equal to 5/2

(b) lies int interval (1, 2)

(c) is strictly greater than 5/2

(d) is strictly greater than 3/2 but less than 5/2

Solution:- See full solution

Question 2:- Let complex numbers α and $1/\overline{\alpha}$ lies on circles $(x - x_0)^2 + (y-y_0)^2 = r^2$ and $(x - x_0)^2 + (y-y_0)^2 = 4r^2$ , respectively.

If $z_0 = x_0 + iy_0$ satisfies the equation $2|z_0|^2 = r^2 + 2$ , then |α| is equal to

(2013 Adv.)

Solution:- See full solution

Question 3:- If w = α + iβ where β ≠ 0 and z ≠ 1 satisfies that condition that $\left(\dfrac{w - \overline{w}z}{1 - z}\right)$ is purely real, then the set of values of z is