Let $a\ne b$ be two non-zero real numbers. Then the number of elements in the set $X=\{z\in \mathrm{ℂ}:Re(a{z}^2+bz)=a and Re(b{z}^2+az)=b\}$ is equal to

(a) 2                           (c) 0                            (c) 3                           (d) 1

Let $A=\{\theta \in (0,2\mathrm{\pi }):\frac{1+2i\sin \theta }{1-i\sin \theta }is purely imaginary\}.$ Then the sum of the elements in A is

(a) $2\pi$                            (b) $4\pi$                             (c) $\pi$                              (d) $3\pi$

For two non-zero complex numbers $z_1$ and $z_2$, if $Re\left(z_1{z}_2\right)=0$ and $Re(z_1+z_2)=0,$ then which of the following are possible?

(a) $Im\left(z_1\right)>0 and Im\left(z_2\right)>0$

(b) $Im\left(z_1\right)<0 and Im\left(z_2\right)>0$

(c) $Im\left(z_1\right)>0 and Im\left(z_2\right)<0$

(d) $Im\left(z_1\right)<0 and Im\left(z_2\right)<0$

Choose the correct answer from the options given below:

(a) A and B                       (b) B and C                       (c) B and D                      (d) A and C

Let $S=\{z\in \mathrm{ℂ}-\{i,2i\}:\frac{z^2+8iz-15}{z^2-3iz-2}\in \mathrm{ℝ}\}.$ If $\alpha -\frac{13}{11}i\in S,\alpha \in \mathrm{ℝ}-\left\{0\right\}, then 242{\alpha }^2 is equal to \_\_\_\_\_\_\_\_\_ .$

A) 1650

B) 2018

C) 1680

D) 2000

If for $z=\alpha +i\beta ,|z+2|=z+4(1+i), then \alpha +\beta$ and $\alpha \beta$ are the roots of the equation

(a) $x^2+7x+12=0$                      (b) $x^2+x-12=0$

(c) $x^2+3x-4=0$                         (d) $x^2+2x-3=0$ 

Let the complex number $z=x+iy$ be such that $\frac{2z-3i}{2z+i}$ is purely imaginary. If $x+y^2=0,$ then $y^4+y^2-y$ is equal to

(a) $\frac{3}{2}$                           (b) $\frac{2}{3}$                            (c) $\frac{3}{4}$                         (d) $\frac{4}{3}$

Let $S=\{z=x+iy:\frac{2z-3i}{4z+2i} is a real number\}.$ Then which of the following is NOT correct ?

(a) $(x,y)=(0,-\frac{1}{2})$                                        (b) $y+x^2+y^2\ne -\frac{1}{4}$

(c) $y\in (-\infty ,-\frac{1}{2})\cup (-\frac{1}{2},\infty )$                   (d) $x=0$

For $a\in \mathrm{ℂ},$ let $A=\{z\in \mathrm{ℂ}:Re(a+\overline{z})>Im(\overline{a}+z\left)\right\}$ and $B=\{z\in \mathrm{ℂ}:Re(a+\overline{z})<Im(\overline{a}+z\left)\right\}$. Then among the two statements:

(S1) : If Re (a), Im (a) > 0, then the set A contains all the real numbers

(S2) : If Re (a), Im (a) < 0, then the set B contains all the real numbers,

(a) only (S2) is true                                  (b) both are true

(c) only (S1) is true                                  (d) both are false

Let $S=\{z\in \mathrm{ℂ}:\overline{z}=i(z^2+Re\left(\overline{z}\right)\left)\right\}.$ Then $\sum _{z\in S}|z{|}^2$ is equal to

(a) 4                               (b) $\frac{5}{2}$                              (c) $\frac{7}{2}$                               (d) 3

If the set $\left\{Re\right(\frac{z-\overline{z}+z\overline{z}}{2-3z+5\overline{z}}):z\in C,Re(z)=3\}$ is equal to the interval $(\alpha ,\beta ]$, then 24 $(\beta -\alpha )$ is equal to

(a) 27                            (b) 36                          (c) 42                           (d) 30

Let a, b be two real numbers such that ab < 0. If the complex number $\frac{1+ai}{b+i}$ is of unit modulus and $a+ib$

lies on the circle $|z-1|=\left|2z\right|,$ then a possible value of $\frac{1+\left[a\right]}{4b}$, where $\left[t\right]$ is greatest integer functions, is

(a) $\frac{1}{2}$                                (b) $-\frac{1}{2}$                             (c) 0                           (d) $-1$

Let $z_1=2+3i$ and $z_2=3+4i$. The set $S=\{Z\in \mathrm{ℂ}:|z-z_1{|}^2-|z-z_2{|}^2=|z_1-z_2{|}^2\}$ represents a

(a) straight line with the sum of its intercepts on the coordinate axes equals -18

(b) hyperbola with eccentricity 2

(c) hyperbola with the length of the transverse axis 7

(d) straight line with the sum of its intercepts on the coordinate axes equals 14

Let $z$ be a complex number such that $\left|\frac{z-2i}{z+i}\right|=2,z\ne -i.$ Then $z$ lies on the circle of radius 2 and centre

(a) (2, 0)                          (b) (0, 2)                         (c) (0, 0)                         (d) (0, -2)

The complex number $z=\frac{i-1}{\cos {\displaystyle \frac{\mathrm{\pi }}{3}}+i\sin {\displaystyle \frac{\mathrm{\pi }}{3}}}$ is equal to

(a) $\sqrt{2}i(\cos \frac{5\mathrm{\pi }}{12}-i\sin \frac{5\mathrm{\pi }}{12})$                      (b) $\cos \frac{\mathrm{\pi }}{12}-i\sin \frac{\mathrm{\pi }}{12}$

(c) $\sqrt{2}(\cos \frac{\mathrm{\pi }}{12}+i\sin \frac{\mathrm{\pi }}{12})$                         (d) $\sqrt{2}(\cos \frac{5\mathrm{\pi }}{12}+i\sin \frac{5\mathrm{\pi }}{12})$

Let $\alpha =8-14i,A=\{z\in \mathrm{ℂ}:\frac{\alpha z-\overline{\alpha }\overline{z}}{z^2-{\left(\overline{z}\right)}^2-112i}=1\}$ and 

$B=\{z\in \mathrm{ℂ}:|z+3i|=4\}$. Then $\sum _{z\in A\cap B}(Rez-Imz)$ is equal to _________ .

Let $w_1$ be the point obtained by the rotation of $z_1=5+4i$ about the origin through a right angle in the anticlockwise direction, and $w_2$ be the

point obtained by the rotation of $z_2=3+5i$ about the origin through a right angle in the clockwise direction. Then the principal argument of $w_1-w_2$ is equal to

(a) $-\mathrm{\pi }+{\tan }^{-1}\frac{33}{5}$                           (b) $\mathrm{\pi }-{\tan }^{-1}\frac{33}{5}$

(c) $-\mathrm{\pi }+{\tan }^{-1}\frac{8}{9}$                             (d) $\mathrm{\pi }-{\tan }^{-1}\frac{8}{9}$

Let C be the circle in the complex plane with centre $z_0=\frac{1}{2}(1+3i)$ and radius $r=1.$ Let $z_1=1+i$ and the complex number $z_2$ be outside

the circle C such that $|z_1-z_0||z_2-z_0|=1.$ If $z_0,z_1 and z_2$ are collinear, then the smaller value of ${\left|z_2\right|}^2$ is equal to

(a) $\frac{5}{2}$                        (b) $\frac{7}{2}$                          (c) $\frac{3}{2}$                            (d) $\frac{13}{2}$

If the center and radius of the circle $\left|\frac{z-2}{z-3}\right|=2$ are respectively $(\alpha ,\beta )$ and $\gamma ,$ then $3(\alpha +\beta +\gamma )$ is equal to

(a) 12                              (b) 11                               (c) 10                              (d) 9                        

For all $z\in C$ on the curve $C_1:\left|z\right|=4,$ let the locus of the point $z+\frac{1}{z}$ be the curve $C_2$. Then:

(a) the curve $C_1$ lies inside $C_2$

(b) the curves $C_1$ and $C_2$ intersect at 4 points

(c) the curve $C_2$ lies inside $C_1$

(d) the curves $C_1$ and $C_2$ intersect at 2 points

For $\alpha ,\beta ,z\in \mathrm{ℂ}$ and $\lambda >1,$ if $\sqrt{\lambda -1}$ is the radius of the circle $|z-\alpha {|}^2+|z-\beta {|}^2=2\lambda ,$ then $|\alpha -\beta |$ is equal to ___________ .

(A) 2

(B) 5

(C) 6

(D) 1

Let $w=z\overline{z}+k_{1}z+k_{2}iz+\lambda (1+i),k_1,k_2\in \mathrm{ℝ}.$ Let $Re\left(w\right)=0$ be the circle C of radius 1 in the first quadrant 

touching the line $y=1$ and the $y$-axis. If the curve $Im\left(w\right)=0$ intersects C at A and B, then 30${\left(AB\right)}^2$ is equal to ____________ .

(A) 5

(B) 10

(C) 24

(D) 11

Let $z=1+i$ and $z_1=\frac{1+i\overline{z}}{\overline{z}(1-z)+{\displaystyle \frac{1}{z}}}.$ Then $\frac{12}{\mathrm{\pi }}arg\left(z_1\right)$ is equal to ____________ .

(A) 9

(B) 10

(C) 14

(D) 20