1. A particle P starts from the point $z_{0}=1+2i$   , where $i=\sqrt{-1}$   . It moves first horizontally away from the origin by 5 units and then vertically away from the origin by 3 units to reach a point $z_{1}$ . From $z_{1}$ the particle moves $\sqrt{2}$ units in the direction of $\hat{i}+\hat{j}$  and then it moves through an angle $\pi/2$ in the anticlockwise direction on a circle with centre at origin, to reach point $z_{2}$. The point $z_{2}$ is given by
a) 6+7i
b) -7+6i
c) 7+6i
d) -6+7i

Explanation:

2. If P( $\omega$ ) lies on the chord joining A, B where A(a), B(b) lie on the unit circle |z| = 1, then $\bar{\omega}$   is equal to
a) $\frac{a+b-\omega}{a+b}$
b) $\frac{a+b-\omega}{ab}$
c) $\frac{1}{2}\left(a+b\right)-\frac{\omega}{ab}$
d) $\frac{1}{ab}\left(a+b+\omega\right)$

Explanation: Suppose P( $\omega$ ) divides the join of A, B in the ratio k : 1, then

3. Number of solution of
$\mid z-7-i\mid=3\sqrt{2}$            (1)
and $\mid z-1-7i\mid\leq3\sqrt{2}$          (2)
is
a) 0
b) 1
c) 2
d) 3

Explanation: $\mid z-7-i\mid=3\sqrt{2}$    represents a circle C1 with centre at A(7 + i) and radius r1 = $3\sqrt{2}$

4. Let $z=\begin{bmatrix}7+5i & 3+3i \\8+5i & 5+2i \end{bmatrix}$
then det $\left(Z\bar{Z}\right)$ , (where $\bar{Z}$  is the matrix obtained by taking Z) is equal to
a) 256
b) 121
c) 36
d) 25

Explanation:

5. Let a, b, c be three points lying on the circle $\mid z\mid=1$  and suppose $\alpha \epsilon\left(0,\pi/2\right)$   be such that $a+b\cos\alpha+c\sin\alpha=0$     , then
a) $b=\pm ic$
b) $2a^{2}+b^{2}+c^{2}=0$
c) $a^{2}+b^{2}+c^{2}=0$
d) b=c

Explanation:

6. Let $x_{1},x_{2},x_{3},x_{4}\epsilon C$     be such that
$x_{1}\left(x_{1}-x_{2}\right)\left(x_{1}-x_{3}\right)\left(x_{1}-x_{4}\right)=4$
$x_{2}\left(x_{2}-x_{1}\right)\left(x_{2}-x_{3}\right)\left(x_{2}-x_{4}\right)=4$
$x_{3}\left(x_{3}-x_{1}\right)\left(x_{3}-x_{2}\right)\left(x_{3}-x_{4}\right)=4$
$x_{4}\left(x_{4}-x_{1}\right)\left(x_{4}-x_{2}\right)\left(x_{4}-x_{3}\right)=4$
a) $x_{1}+x_{2}+x_{3}+x_{4}=0$
b) $x_{1}x_{2}x_{3}x_{4}=-1$
c) $x_1^2+x_2^2+x_3^2+x_4^2=0$
d) All of the Above

Explanation: Note that none of xi is zero and no two of xi’s are equal.

7. Let $z_{1},z_{2}$ be two non-zero complex numbers such that $\left(\mid z_{1}\mid-\mid z_{2}\mid\right)^{2}+ \mid z_{1}z_{2}\mid=\bar{z}_{1}z_{2}+z_{1}\bar{z}_{2}$
then $\mid\frac{z_{2}}{z_{1}}\mid$  lies in the interval
a) $\left[\frac{1}{2}\left(3-\sqrt{5}\right),\frac{1}{2}\left(3+\sqrt{5}\right)\right]$
b) $\left[\frac{1}{2}\left(\sqrt{5}-1\right),\frac{1}{2}\left(\sqrt{5}+1\right)\right]$
c) $\left[1,\frac{1}{2}\left(3+\sqrt{5}\right)\right]$
d) $\left[\frac{1}{2}\left(3-\sqrt{5}\right),1\right]$

Explanation:

8. Suppose A(a), B(b) lie on the unit circle |z| = 1. If $\omega$ is a complex number such that$\bar{\omega}=\frac{b+a-\omega}{ab}$     , then $P\left(\omega\right)$ lies on
a) chord AB
b) perpendicular bisector of chord AB
c) unit circle |z| = 1
d) the circle |z| = 1/2

Explanation:

9. If $\mid z_{1}\mid\neq1$    , $\mid\frac{z_{1}-z_{2}}{1-\bar{z}_{1}z_{2}}\mid=1$     , then
a) $\mid z_{2}\mid=1$
b) $\mid z_{2}\mid=0$
c) $\mid z_{2}\mid>1$
d) $0<\mid z_{2}\mid<1$

10. Let $z_{1}, z_{2}, z_{3}\epsilon C$
$f\left(z_{1},z_{2},z_{3}\right)=sgn \begin{bmatrix}1 &1&1 \\z_{1} &z_{2}&z_{3}\\\bar{z}_{1}& \bar{z}_{2} & \bar{z}_{3} \end{bmatrix}$
where sgn x = $\frac{x}{\mid x\mid}$   if $x\neq 0$  and sgn(0)=0, then for $a\neq 0$
a) $f\left(z_{1},z_{2},z_{3}\right)=f\left(az_{1},az_{2},az_{3}\right)$
b) $f\left(z_{1},z_{2},z_{3}\right)=f\left(az_{2},az_{1},az_{3}\right)$
c) $f\left(z_{1},z_{2},z_{3}\right)=f\left(az_{3},az_{2},az_{1}\right)$