1. All the solutions of $Z^{2018}+\frac{1}{Z^{2018}}+Z^{2016}+\frac{1}{Z^{2016}}=0$
lie on the curve
a) $\mid Z\mid=1$
b) $\mid Z\mid=1/2018$
c) $\mid Z\mid=1/2016$
d) $\mid Z\mid=1/4034$

Explanation:

2. Let $z_{k}=a_{k}+ib_{k}\neq0,$     for $k=1,2...,2018$     . Then value of the expression
$E=\frac{\mid\left(\sum_{k=1}^{2018}\left(a_{k}-b_{k}\right)\right)\left(\sum_{k=1}^{2018}\left(a_{k}+b_{k}\right)\right)\mid}{\left(\sum_{k=1}^{2018}\mid z_{k} \mid\right)^{2}}$
a) is at least 2018
b) is at least 1009
c) is always less than 2
d) is always greater than 2

Explanation:

3. Let $\omega$ be a complex number such that $\mid\omega \mid =\sqrt{2}.$   Let $\alpha,\beta$  be solution of the equation
$z^{2}+i\left(\omega+i\bar{\omega}\right)z-2i=0$
then
a) $\mid\alpha\mid+\mid\beta\mid=2$
b) $\mid\alpha-\beta\mid=2$
c) $\mid\alpha\mid+\mid\beta\mid=2\sqrt{2}$
d) $\mid\alpha-\beta\mid=2\sqrt{2}$

Explanation:

4. If a, b are distinct complex numbers lying on the unit circle |z| = 1, then value of
$\mid\frac{a+b}{1+ab}\mid^{2}+\mid\frac{a-b}{1-ab}\mid^{2}$
a) cannot exceed 2
b) is at least 1
c) is at most 2
d) is at least 2

Explanation:

5. Let $a_{1}=2 , a_{n+1}=\frac{a_{n}-i}{a_{n}+i}$    for $n\geq 1$  , then $\sum_{k=1}^{3n}a_{k}$   is equal to
a) $\frac{n}{5}(13-19i)$
b) $\frac{1}{5}(13-19i)$
c) $n(2+3i)$
d) $n(3-2i)$

Explanation:

6. Let a, b, c and d be four distinct complex numbers such that |a| = |b| = |c| = 1. If $z=\frac{\left(a-c\right)\left(b-d\right)}{\left(a-d\right)\left(b-c\right)}$     is real, then
a) d is a real number
b) d = 0
c) d is a purely imaginary
d) |d| = 1

Explanation:

7. Let p and q be two complex numbers such that q $\neq$ 0. If the roots of the equation $x^{2}+px+q^{2}=0$    have the same absolute value, then p/q is
a) an integer
b) a rational number
c) a real number
d) a complex number

Explanation: Let x1, x2 be the roots of x2 + px + q2 = 0 and

8. If $\mid z^{3}+\frac{1}{z^{3}}\mid \leq2$   , then $\mid z+\frac{1}{z}\mid$   cannot exceed
a) 2
b) 1
c) $\sqrt{2}$
d) $\sqrt{2}-1$

Explanation:

9. If $\omega=\cos\frac{\pi}{n}+i\sin\frac{\pi}{n}$     then value of $1+\omega+\omega^{2}+....+\omega^{n-1}$     is
a) 1 + i
b) $1+i\tan\left(\pi/n\right)$
c) $1+i\cot\left(\pi/2n\right)$
d) 1 – i

d) $\sqrt{3}$