1. If $\theta$ is the angle which the line of $\bar{z} =\bar{z}_{0}+A\left(z-z_{0}\right)$    makes with the positive direction of real axis where A is a constant, then
a) $A=e^{2i\theta}$
b) $A=e^{-2i\theta}$
c) $A=e^{i\theta}$
d) $A=e^{-i\theta}$

Explanation:

2.The equation $\bar{z}=\bar{a}+\frac{r^{2}}{z-a} ,r>0$
represents
a) an ellipse
b) a parabola
c) a circle
d) a straight line through point $\bar{a}$

Explanation:

3. If $\bar{a}\neq b$  , then the equation $z\bar{z}+az+b\bar{z}+c=0$     represents
a) a circle
b) an ellipse
c) a straight line
d) finite number of point in C

Explanation:

4. If $\bar{a}=b,c\epsilon R$    and $\mid b\mid^{2}>c$   , then $z\bar{z}+az+b\bar{z}+c=0$     represents
a) a circle
b) a parabola
c) a straight line
d) finite number of points in C

Explanation: $\bar{z}z+\bar{a}\bar{z}+\bar{b}z+\bar{c}=0$      (2)

5. if $\bar{a}=b,c\epsilon R$     and $\mid b\mid^{2} < c$   , then the equation $z\bar{z} +az+b\bar{z}+c=0$
a) has no solution
b) exactly two solutions
c) infinite number of solution
d) none of these

Explanation:

6. if $z_{1}+z_{2}+z_{3}=0$    then $\mid z_{2}-z_{3}\mid^{2}+\mid z_{3}-z_{1}\mid^{2}+\mid z_{1}-z_{2}\mid^{2}$       equals
a) $\frac{1}{3}\mid z_{1}\mid^{2}+2\mid z_{2}\mid^{2}+2\mid z_{3}\mid^{2}$
b) $\frac{2}{3}(\mid z_{1}\mid^{2}+\mid z_{2}\mid^{2}+\mid z_{3}\mid^{2})$
c) $2(\mid z_{1}\mid^{2}+\mid z_{2}\mid^{2}+\mid z_{3}\mid^{2})$
d) $3(\mid z_{1}\mid^{2}+\mid z_{2}\mid^{2}+\mid z_{3}\mid^{2})$

Explanation:

7. if $\mid z_{1}\mid=\mid z_{2}\mid=\mid z_{3}\mid=\mid z_{4}\mid=1$     , and $z_{1}+ z_{2}+ z_{3}+ z_{4}=0$     then least value of the expression $E=\mid z_{1}-z_{2}\mid^{2}+\mid z_{2}-z_{3}\mid^{2}+\mid z_{3}-z_{4}\mid^{2}+\mid z_{4}-z_{1}\mid^{2}$
is
a) 6
b) 8
c) 10
d) 12

Explanation:

8. If $z_{1}+z_{2}+z_{3}+z_{4}=0$     , then the expression $\mid z_{1}-z_{2}\mid^{2}+\mid z_{2}-z_{3}\mid^{2}+\mid z_{3}-z_{4}\mid^{2}+\mid z_{4}-z_{1}\mid^{2}-2\left(\mid z_{1}\mid^{2}+\mid z_{2}\mid^{2}+\mid z_{3}\mid^{2}+\mid z_{4}\mid^{2}\right)$
is equal to 0, if and only if,
a) $z_{1}=-z_{3}$   and $z_{4}=-z_{2}$
b) $z_{1}=-z_{4}$   and $z_{2}=-z_{3}$
c) $z_{1}=z_{2}$   and $z_{3}=z_{4}$
d) $z_{1}=z_{3}$   and $z_{2}=z_{4}$

Explanation: From problem 7,

9. If $\mid u\mid=\mid v\mid=1,uv \neq -1$    ,and $z=\frac{u-v}{1+uv}$    then
a) $\mid z\mid=1$
b) $Re\left( z\right)=0$
c) $Im\left( z\right)=0$
d) $Re\left( z\right)=Im\left( z\right)$

10. If $\mid u\mid< 1, \mid v\mid < 1$   and $z=\frac{u-v}{1-\bar{u}v}$    then least value of $\mid z\mid$ is
a) $\frac{\mid u\mid-\mid v\mid}{1+\mid u\mid\mid v\mid}$
b) $\frac{\mid u\mid+\mid v\mid}{1-\mid u\mid\mid v\mid}$
c) $\frac{\parallel u\mid-\mid v\parallel}{1-\mid u\mid\mid v\mid}$