1. Let L be the straight line \[\bar{a}z+a\bar{z}+c=0,a \neq0\] \[c\epsilon R \] , and OA be the straight line joining O and
A(a), then

a) \[L\parallel OA\]

b) \[L\perp OA\]

c) L makes an angle of \[\pi/4\] with OA

d) none of these

Explanation:

2. Locus of point z so that z, i, and iz are collinear, is

a) a straight line

b) a circle

c) an ellipse

d) a rectangular hyperbola

Explanation:

3. The number of roots of the equation \[\mid z^{2}\mid -5 \mid z\mid +1 =0\] is

a) 0

b) 2

c) 4

d) infinite

Explanation:

4.The set of points in C satisfying the inequality
\[\mid arg \left(z\right)-\pi/2\mid <\pi/2\] is given by

a) \[\left\{z: Re \left(z\right)> 0\right\}\]

b) \[\left\{z: In \left(z\right)< 0\right\}\]

c) \[\left\{z: Im \left(z\right)> 0\right\}\]

d) \[\left\{z: Re \left(z\right)=Im \left(z\right)\right\}\]

Explanation:

5.If n \[\geq\] 3 and \[1, \alpha_{1},\alpha_{2},...\alpha_{n-1}\] are nth roots of unity,
then value of \[\sum_{1\leq i < j \leq n-1}\alpha_{i}\alpha_{j}\] is

a) 0

b) 1

c) -1

d) \[\left(-1\right)^{n}\]

Explanation:

6. The Quadrilateral formed by the point with affix 0, z, z + iz and iz is

a) parallelogram

b) rhombus

c) rectangle

d) square

Explanation:

7. If |z – 2| = 2|z – 1|, then \[3|z|^{2} -4 Re \left(z\right)\] equals

a) 0

b) -1

c) 1

d) none of these

Explanation:

8. Sum of the common roots of \[z^{2006}+z^{100}+1=0\] and
\[z^{3}+2z^{2}+2z+1=0\] is

a) 0

b) -1

c) 1

d) 2

Explanation:

9. If \[z_{1}\] and \[z_{2}\] lie on the same side the line \[\bar{a}z+a\bar{z}+b =0\] ,
where \[a \epsilon C, a\neq 0,b\epsilon R\] , then the ratio \[\frac{\bar{a}z_{1}+a\bar{z}_{1}+b }{\bar{a}z_{2}+a\bar{z}_{2}+b }\] is

a) purely imaginary

b) a positive real number

c) a negative real number

d) none of these

Explanation:

10.If \[z_{1}\] and \[z_{2}\] are the roots of the equation \[z^{2}+az+b=0\] , then prove that the origin, \[z_{1}\] and \[z_{2}\] form an
equilateral triangle if and only if

a) \[a^{2}=3b\]

b) \[b^{2}=3a\]

c) \[a^{2}+3b=0\]

d) \[b^{2}+3a=0\]

Explanation: