Properties of Conjugate of a Complex Number

Q. If $z_1,z_2$ and $z_3,z_4$ are two pairs of conjugate complex numbers, then $arg\left(\frac{z_1}{z_4}\right)+arg\left(\frac{z_2}{z_3}\right)$ equals

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

Q.

Find the value of each of the following.

$\left|3-8\right|$

Q. The minimum value of the function $f\left(x\right)=|2-|1-x||-1,$ where $|x|$ denotes the absolute value of $x,$ is

1. $-1$
2. $3$
3. $0$
4. $1$

Q. If $\overline{z}$ be the conjugate of the complex number z, then which of the following relations is false
[MP PET 1987]

1. $|z|=|\overline{z}|$
2. $z.\overline{z}|\overline{z}{|}^2$
3. $\overline{z_1}+\overline{z_2}=\overline{z_1}+\overline{z_2}$
4. $argz=arg\overline{z}$

Q. If $z$ is a complex number, then the conjugate of $z+2\overline{z}$ is

1. $2\overline{z}+z$
2. $\overline{z}+2z$
3. $\overline{z}-2z$
4. $2\overline{z}-z$

Q. If $z=x+iy(x,y\in R,x\ne -\frac{1}{2})$, the number of values of $z$ satisfying $|z{|}^n=z^2|z{|}^{n-2}+z|z{|}^{n-2}+1$, where $(n\in N,n>1)$ is

1. $0$
2. $1$
3. $2$
4. $3$

Q. If $z$ is a complex number satisfying $z+\overline{z}=0$, then

1. $Im\left(z\right)=0$
2. $Re\left(z\right)=0$
3. $Im\left(z\right)=1$
4. $Re\left(z\right)=1$

Q. If $(x+iy{)}^5=p+iq$, then the value of $\frac{2(y+ix{)}^5}{q+ip}$ is

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

Q. The conjugate of $(4+5i{)}^2$ is

1. $-9-40i$
2. $-9+40i$
3. $9-40i$
4. $9+40i$

Q. If $\alpha$ and $\beta$ are different complex numbers with $|\beta |=1,$ then the value of $\left|\frac{\beta -\alpha }{1-\overline{\alpha }\beta }\right|$ is

1. $0$
2. $1$
3. $\frac{1}{2}$
4. $2$

Q. If $\alpha$ and $\beta$ are different complex numbers with $|\beta |=1,$ then the value of $\left|\frac{\beta -\alpha }{1-\overline{\alpha }\beta }\right|$ is

1. $0$
2. $1$
3. $\frac{1}{2}$
4. $2$

Q. If $x+iy=\sqrt{\frac{a+ib}{c+id}},$ then $x^4+y^4+2x^2{y}^2$ is equal to

1. $\frac{(a+b{)}^2}{(c+d{)}^2}$
2. $\frac{a^2+b^2}{c^2+d^2}$
3. $\frac{(a-b{)}^2}{(c-d{)}^2}$
4. $\frac{a^2-b^2}{c^2-d^2}$

Q. $\int \sqrt{x^2+3x}dx$ is equal to
(where $C$ is integration constant)

1. $\frac{x}{2}\sqrt{x^2+3x}-\frac{9}{8}{\sin }^{-1}\left(\frac{2x+3}{2}\right)+C$
2. $\frac{x}{2}\sqrt{x^2+3x}-\frac{9}{8}\log |x+\frac{3}{2}+\sqrt{x^2+3x}|+C$
3. $\frac{2x+3}{4}\sqrt{x^2+3x}-\frac{9}{8}\log |x+\frac{3}{2}+\sqrt{x^2+3x}|+C$
4. $\frac{2x+3}{4}\sqrt{x^2+3x}-\frac{9}{8}{\sin }^{-1}\left(\frac{2x+3}{2}\right)+C$

Q. If the modulus of a complex number z = a + ib 5 units, then the modulus of a - ib is ___ units.

1. $\frac{1}{5}$
2. $\frac{2}{5}$
3. 5
4. Not possible to find

Q.

If $|z_1|=|z_2|=|z_3|=1$ and $z_1+z_2+z_3=0$
then area of the triangle whose vertices are $z_1,z_2,z_3$ is

1. $\frac{3\sqrt{3}}{4}$

2. $\frac{\sqrt{3}}{4}$

3. 1

4. 2