Equality of 2 Complex Numbers

Q. If $(x+iy)(2-3i)=4+i$, then values of $x$ and $y$ are

1. $x=\frac{5}{11},y=\frac{7}{11}$
2. $x=\frac{5}{13},y=\frac{7}{13}$
3. $x=\frac{5}{13},y=\frac{14}{13}$
4. $x=\frac{5}{11},y=\frac{7}{13}$

Q. If $u+iv=(x+iy{)}^3$ then $\frac{u}{x}+\frac{v}{y}=$

1. $4(x^2+y^2)$
2. $4(x^2-y^2)$
3. $2(x^2+y^2)$
4. $2(x^2-y^2)$

Q. If $(\frac{1}{1+2i}+\frac{3}{1-i})\left(\frac{3-2i}{1+3i}\right)$ is reducible to $a+ib$, then values of $a$ and $b$ are

1. $a=\frac{7}{10},b=\frac{-13}{5}$
2. $a=\frac{7}{10},b=\frac{-11}{5}$
3. $a=\frac{7}{11},b=\frac{-13}{10}$
4. $a=\frac{7}{11},b=\frac{-13}{5}$

Q. The complex number $z$ satisfying the equation $|z|=z+1+2i$ is

1. $\frac{3}{2}-2i$
2. $\frac{3}{2}+2i$
3. $\frac{2}{3}-2i$
4. $\frac{3}{2}-\frac{i}{2}$

Q. If $z$ is a complex number and $\overline{z}$ is it's conjugate, then the number of solution(s) of $z^3+\overline{z}=0$ is

Q. The area bounded by $|\arg \left(z\right)|\le \frac{\pi }{4}$ and $|z-1|<|z-3|$ is

1. $1$ sq. unit
2. $2$ sq. units
3. $4$ sq. units
4. $8$ sq. units

Q. Let $a,b,x$ and $y$ be real numbers such that $a-b=1$ and $y\ne 0$. If the complex number $z=x+iy$ satisfies $Im\left(\frac{az+b}{z+1}\right)=y$, then which of the following is(are) possible value(s) of $x$?

1. $-1+\sqrt{1-y^2}$
2. $-1-\sqrt{1-y^2}$
3. $1+\sqrt{1+y^2}$
4. $1-\sqrt{1+y^2}$

Q. If $z$ is a complex number, then the number of solution(s) for the equation $z^2=\overline{z}$ is

1. $1$
2. $2$
3. $4$
4. $5$

Q.

The number of solutions of the equation $z^2$ + $\overline{z}$ = 0 is

1. 1

2. 2

3. 3

4. 4

Q. If $|z^2-1|=|z{|}^2+1$, then $z$ lies on

1. a circle
2. a parabola
3. an ellipse
4. a straight line