1. If $\mid z-25i\mid\leq 15$    , then |maximum arg (z) – minimum arg (z)| equals
a) $2\cos^{-1}\left(3/5\right)$
b) $2\cos^{-1}\left(4/5\right)$
c) $\pi/2+\cos^{-1}\left(3/5\right)$
d) $\sin^{-1}\left(3/5\right)-\cos^{-1}\left(3/5\right)$

Explanation: If $\mid z-25i\mid\leq 15$    ,then z lies either in the interior and or on the boundary of the circle with centre at C (25i) and radius equal to 15. From the Figure, it is clear that argument is least for point A and argument is greatest for point B.

2. If $\mid z_{1}\mid=\mid z_{2}\mid=\mid z_{3}\mid=1$    and $z_{1}+z_{2}+z_{3}=0$    , then area of the triangle whose vertices are $z_{1},z_{2},z_{3}$  is
a) $3\sqrt{3}/4$
b) $\sqrt{3}/4$
c) 1
d) 2

Explanation:

3. Shaded region is given by

a) $\mid z +2\mid\geq 6,0\leq arg \left(z\right)\leq\frac{\pi}{6}$
b) $\mid z+2\mid\geq 6,0\leq arg \left(z\right)\leq\frac{\pi}{3}$
c) $\mid z+2\mid\geq 6,0\leq arg \left(z\right)\leq\frac{\pi}{2}$
d) none of these.

Explanation: Note that AB = 6 and

4. If $-\pi < arg \left(z\right)< -\frac{\pi}{2}$   , then arg $\bar{z}$ - arg $(-\bar{z})$ is
a) $\pi$
b) $-\pi$
c) $\frac{\pi}{2}$
d) $-\frac{\pi}{2}$

Explanation: Let arg (z) = $\theta$ , then arg $\overline{z}$ = – $\theta$ and arg (– $\overline{z}$ ) = $-\pi-\theta$

5. If $k>1,\mid z_{1}\mid< k$    and $\mid\frac{k- z_{1}\bar{ z_{2}}}{ z_{1}-k z_{2}}\mid=1$    , then
a) $\mid z_{2}\mid< k$
b) $\mid z_{2}\mid= k$
c) $z_{2}=0$
d) $\mid z_{2}\mid= 1$

Explanation:

6. If $k>0,\mid z\mid=\mid w\mid=k$    , and $\alpha=\frac{z-\bar{w}}{k^{2}+z\bar{w}}$    then Re $\left(\alpha\right)$ equals
a) 0
b) k/2
c) k
d) 2k/3

Explanation:

7. Roots of the equation are $\left(z+1\right)^{5}=\left(z-1\right)^{5}$
are
a) $\pm i\tan\left(\frac{\pi}{5}\right),\pm i\tan\left(\frac{2\pi}{5}\right)$
b) $\pm i\cot\left(\frac{\pi}{5}\right),\pm i\cot\left(\frac{2\pi}{5}\right)$
c) $\pm i\cot\left(\frac{\pi}{5}\right),\pm i\tan\left(\frac{2\pi}{5}\right)$
d) none of these

Explanation: For z $\neq$ 1, (1) can be written as

8. Let $z_{1},z_{2},z_{3}$  be three distinct complex numbers lying on a circle with centre at the origin such that $z_{1}+z_{2}z_{3} , z_{2}+z_{3}z_{1}$    and $z_{3}+z_{1}z_{2}$   are real numbers, then $z_{1}z_{2}z_{3}$  equals
a) -1
b) 0
c) 1
d) i

Explanation: As z1, z2, z3 lie on a circle with centre at the origin,

9. If the points z, – iz and 1 are collinear then z lies on
a) a straight line
b) a circle
c) an ellipse
d) a pair of straight lines

10. If $\mid z-i\mid=1$   and arg $\left(z\right)=\theta$   where $\theta \epsilon \left(0 ,\pi/2\right)$    , then
$\cot\theta-\frac{2}{z}$    equals
Explanation: $\mid z-i\mid=1$   represents a circle with centre at i and radius 1. As 0 < $\theta$ < $\pi$/2, $\theta$ lies on the semi-circle in the first quadrant.