Equality of 2 Complex Numbers

Q.

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

1. 1

2. 2

3. 4

4. 3

Q.

Find the complex number z satisfying the equations $\left|\frac{z-12}{z-8i}\right|$=$\frac{5}{3}$, $\left|\frac{z-4}{z-8}\right|$=1

1. 6

2. 

3. 

4. None of these

Q.

If z is a complex number such that $z^2$ = ($\overline{z}{)}^2$, then

1. Either z is purely real or purely imaginary

2. z is purely real

3. z is purely imaginary

4. None of these

Q.

If z is a complex number, then z.$\overline{z}$ = 0 if and only if

1. z=0

2. Re(z)=0

3. Im(z)=0

4. None of these

Q. If $z^3+(3+2i)z+(-1+ia)=0$ has one real root, then the value of ${}^{\prime }{a}^{\prime }$ lies in the interval
$(a\in R)$

1. $(-2,1)$
2. $(-1,0)$
3. $(0,1)$
4. $(-2,3)$

Q. If $\sqrt{5-12i}+\sqrt{-5-12i}=z$, then principal value of $\arg z$ can be

1. $-\frac{\pi }{4}$
2. $\frac{\pi }{4}$
   
3. $\frac{3\pi }{4}$
4. $\frac{3\pi }{4}$

Q. The solutions of the equation $z^2+|z|=0$ are

1. $z=0$
2. $z=i$
3. $z=-i$
4. $z=1+i$

Q. Match the colums:

$\begin{array}{cc}Column1& Column2\\ \text{a. The set of points}z\text{satisfying}|z-i|z||=|z+i|z||& \text{p. an ellipse with eccentricity}\frac{4}{5}\\ \text{is contained in or equal to}& \\ \text{b. The set of points}z\text{satisfying}|z+4|+|z-4|=10& \text{q. the set of points}z\text{satisfying}\\ \text{is contained in or equal to}& \text{Img}z=0\\ \text{c. If}|w|=2,\text{then the set of points}z=w-\frac{1}{w}& \text{r. the set of points}z\text{satisfying}\\ \text{is contained in or equal to}& |\text{Img}z|\le 10\\ \text{d. If}|w|=1,\text{then the set of points}z=w+\frac{1}{w}& \text{s. the set of points}z\text{satisfying}\\ \text{is contained in or equal to}& |\text{Re}z|<2\\ & \text{t. the set of points}z\text{satisfying}\\ & |z|\le 3\end{array}$

1. $a-q,r;b-p;c-p,s,t;d-q,r,t$
2. $a-p,q,r;b-p;c-s,t;d-q,r,t$
3. $a-q,r;b-p,t;c-p,t;d-q,r$
4. $a-p,r;b-p,q;c-s,t;d-p,r$

Q. Let $z\in C$ with $\text{Im}\left(z\right)=10$ and it satisfies
$\frac{2z-n}{2z+n}=2i–1$ for some natural number $n$. Then

1. $n=20$ and $\text{Re}\left(z\right)=10$
2. $n=20$ and $\text{Re}\left(z\right)=-10$
3. $n=40$ and $\text{Re}\left(z\right)=10$
4. $n=40$ and $\text{Re}\left(z\right)=-10$

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