1. If $f\left(z_{1},z_{2},z_{3}\right)=sgn \begin{bmatrix}1 &1&1 \\z_{1} &z_{2}&z_{3}\\\bar{z}_{1}& \bar{z}_{2} & \bar{z}_{3} \end{bmatrix}$
, then for a $\epsilon$ C,
a) $f\left(z_{1},z_{2},z_{3}\right)=f\left(z_{1}+a,z_{2}+a,z_{3}+a\right)$
b) $f\left(z_{1},z_{2},z_{3}\right)=\mid a\mid f\left(z_{1},z_{2},z_{3}\right)$
c) $f\left(z_{1},z_{2},z_{3}\right)=\mid a\mid f\left(a_{1},a_{2},a_{3}\right)$
d) none of these

Explanation:

2.If $z_{1},z_{2},z_{3}$  are three complex numbers, then $\mid z_{2}+z_{3}\mid^{2}+\mid z_{3}+z_{1}\mid^{2}+\mid z_{1}+z_{2}\mid^{2}$       equal
a) $2\mid z_{1}+z_{2}+z_{3}\mid^{2}$
b) $\mid z_{1}+z_{2}+z_{3}\mid^{2}$
c) $2\mid z_{1}+z_{2}+z_{3}\mid^{2}-\mid z_{1}\mid^{2}-\mid z_{2}\mid^{2}-\mid z_{3}\mid^{2}$
d) none of these

Explanation:

3. If $\mid z\mid=1,\mid a\mid\neq 1$   and $x=z/\left( z-a \right)\left(1-z\bar{a}\right)$    , then
a) x < 0
b) x > 0
c) $x \geq 1$
d) 0 < x < 1

Explanation:

4. Let $z_{1},z_{2}\epsilon C$   and $x= \mid z_{1}z_{2}\mid-Re \left(z_{1}z_{2}\right)-\frac{1}{2}\mid \bar{z}_{1}-z_{2}\mid^{2}+\frac{1}{2}\left(\mid z_{2}\mid-\mid z_{1}\mid\right)^{2}$
a) x < 0
b) x= 0
c) $x \geq 1$
d) 0 < x < 1

Explanation:

5. If the ratio$\frac{1-z}{1+z}$   is purely imaginary, then
a) $|z| \leq 1/2$
b) |z| = 1
c) |z| > 1
d) 0 < |z| < 1

Explanation:

6. If $\mid a\mid \neq\mid b\mid$   , then the equation $az+b\bar{z}+c=0$     represents
a) a circle
b) an ellipse
c) a straight line
d) a point

Explanation:

7. If $\mid a\mid =\mid b\mid$   and $\bar{a}c \neq b\bar{c}$   , then the equation $az+b\bar{z}+c=0$    has
a) no solution
b) exactly one solution
c) finitely many solutions
d) infinitely many solutions

Explanation: From problem 6, we get

8. If $\mid a\mid=\mid b\mid\neq 0$   and $\bar{a}c \neq b\bar{c}$   , then $az+b\bar{z}+c=0$    represents
a) a circle
b) an ellipse
c) a straight line
d) a point

Explanation:

9.The equation $\bar{z} =\bar{z}_{0}+A\left(z-z_{0}\right)$    where A is a constant , represents
a) straight line
b) a circle
c) a point
d) none of these

Explanation:

10.If m is slope of the straight line in above Question 9, then
a) $A=\frac{1+im}{1-im}$
b) $A=\frac{1-im}{1+im}$
c) $A=\frac{i+m}{1-mi}$
d) $A=\frac{i-m}{1+mi}$