1. If $z_{1},z_{2},z_{3}$  are complex numbers such that $\mid z_{1}\mid=\mid z_{2}\mid=\mid z_{3}\mid=\mid\frac{1}{z_{1}}+\frac{1}{z_{2}}+\frac{1}{z_{3}}\mid = 1$        , then $\mid z_{1}+z_{2}+z_{3}\mid$    is
a) equal to 1
b) less than 1
c) greater than 3
d) equal to 3

Explanation:

2.If $\omega$ is an imaginary cube root of unity, then value of the expression $2\left(1+\omega\right)\left(1+\omega^{2}\right)+3\left(2+\omega\right)\left(2+\omega^{2}\right)+...+(n+1)\left(n+\omega\right)\left(n+\omega^{2}\right)$
is
a) $\frac{1}{4}n^{2}\left(n+1\right)^{2}+n$
b) $\frac{1}{4}n^{2}\left(n+1\right)^{2}-n$
c) $\frac{1}{4}n\left(n+1\right)^{2}-n$
d) $\frac{1}{4}n\left(n+1\right)^{2}-2n$

Explanation: rth term of the given expression is

3. The greatest and the least value of $\mid z_{1}+z_{2}\mid$   if $z_{1}=24+7i$    and $\mid z_{2}\mid=6$   are respectively
a) 31, 19
b) 25, 19
c) 31, 25
d) 31, 29

Explanation:

4. For complex numbers z1, z2 and z3 satisfying $\frac{ z_{1}-z_{3}}{ z_{2}-z_{3}}=\frac{1-i\sqrt{3}}{2}$     are the vertices of a triangle which is
a) isosceles
b) right-angled and isosceles
c) equilateral
d) obtuse-angled and isosceles

Explanation:

5. If a > 0, and the equation $\mid z-a^{2}\mid+\mid z-2a\mid =3$      represents an ellipse, then a lies in
a) (1, 3)
b) $\left(\sqrt{2}, \sqrt{3}\right)$
c) (0, 3)
d) $\left(1, \sqrt{3}\right)$

Explanation: The equation $\mid z-a^{2}\mid+\mid z-2a\mid =3$      will represent

6. If $\theta$ is real and $z_{1},z_{2}$  are connected by $z_1^2+z_2^2+2z_{1},z_{2} cos \theta=0$     , then triangle with vertices $0,z_{1}$ and $z_{2}$  is
a) equilateral
b) right angled
c) isosceles
d) none of these

Explanation:

Thus, triangle with vertices 0, z1 and z2 are vertices of an isosceles triangle.

7. If $\mid z-\frac{4}{z}\mid=2$   , then the greatest value of $\mid z\mid$ is
a) $1+\sqrt{2}$
b) $2+\sqrt{2}$
c) $\sqrt{3}+1$
d) $\sqrt{5}+1$

Explanation:

8. Reflection of the line $\bar{a}z+a\bar{z}=0$    in the real axis is
a) $\bar{a}\bar{z}+ az =0$
b) $\frac{\bar{a}}{a}=\frac{\bar{z}}{z}$
c) $\left(a+\bar{a}\right)\left(z+\bar{z}\right)=0$
d) none of these

Explanation:

9. If $z_{1}$ and $z_{2}$ are two non-zero complex numbers such that $\mid z_{1}+z_{2}\mid= \mid z_{1}\mid+\mid z_{2}\mid$     then arg $\left(z_{1}\right)-$   arg $\left(z_{2}\right)$ is equal to
a) $-\pi$
b) $-\pi/2$
c) $\pi/2$
d) 0

10. If $A\left(z_{1}\right) , B\left(z_{2}\right)$   and $C\left(z_{3}\right)$  are the vertices of a triangle such that |z1| = |z2| = |z3| > 0, then orthocentre of $\triangle ABC$   is
a) $z_{1}+z_{2}+z_{3}$
b) $z_{1}z_{2}z_{3}$