Equality of 2 Complex Numbers

Q.

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

1. 2

2. 1

3. 3

4. 4

Q.

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

1. z is purely imaginary

2. Either z is purely real or purely imaginary

3. None of these

4. z is purely real

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. 

2. 

3. None of these

4. 6

Q.

If $\alpha$ is a complex constant such that $\alpha$$z^2$ + z + $\overline{\alpha }$ = 0 has a real root.Find the value of real root.

1. 1

2. -1

3. 0

4. All of these

Q. $z_1=a+ib$ and $z_2=c+id$ are two complex numbers then $z_1>z_2$ is meaningful if

1. $a>c$ and $b>d$
2. $a+c>b+d$
3. $a=c=0$ and $b>d$
4. $a>c$ and $b=d=0$

Q. Complex number $z$ satisfying the equation $z^2\times (1+i)=-7+17i,$ is/are

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

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 $z$ 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.

If $\sqrt{a+ib}$ = x+iy, then possible value of $\sqrt{a+ib}$ is

1. 

2. 

3. 

4. 

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 $(x+iy)(2-3i)=4+i$, 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. The complex number z is purely imaginary if

1. $z\tilde{z}$ is real
2. $z=\tilde{z}$
3. None of these
4. $z+\tilde{z}=0$

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. The number of complex number(s) $z$ satisfying $|z-3-i|=|z-9-i|$ and $|z-3+3i|=3$ is

1. one
2. two
3. four
4. infinitely many