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 z_{1}, z_{2} and z_{3} 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, z

_{1}and z

_{2}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

Explanation:

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 |z_{1}| = |z_{2}| = |z_{3}| > 0, then orthocentre
of \[\triangle ABC \] is

a) \[z_{1}+z_{2}+z_{3}\]

b) \[z_{1}z_{2}z_{3}\]

c) 0

d) 1

Explanation: