Conjugate of a Complex Number

Q.

The complex numbers sinx+icos2x and cosx-isin2x are conjugate to each other for

1. 

2. 

3. x=0

4. No value of x

Q. The conjugate of complex number $\frac{2-3i}{4-i}$ is .

1. $\frac{2+3i}{4-i}$
2. $\frac{11+10i}{17}$
3. $\frac{11-10i}{17}$
4. $\frac{2+3i}{4i}$

Q.

The conjugate of the complex number $\frac{2+5i}{4-3i}$ is

1. 

2. 

3. 

4. 

Q. If $i{z}^3+z^2-z+i=0$ then show that |z|=1.

Q.

The area (in sq. units) of the region $\left\{(x,y):x\ge 0,x+y\le 3,x^2\le 4y\text{and}y\le 1+x\right\}$ is:

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

2. $\frac{7}{3}$

3. $\frac{5}{2}$

4. $\frac{59}{12}$

Q.

The conjugate of complex number $\frac{2-3i}{4-i}$ is_______.

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

2. $\frac{11+10i}{17}$

3. $\frac{11-10i}{17}$

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

Q. Find the equation of the curve formed by the set of all those points the sum of whose distances from the points A(4, 0, 0) and B(-4, 0, 0) is 10 units.

Q.

If $z_1$ and $z_2$ are two roots of the equation $z^2+az+b=0$, $z$ being a complex number. Further assume that the origin, $z_1$ and $z_2$ form an equilateral triangle. Then

1. $a^2=b$

2. $a^2=2b$

3. $a^2=3b$

4. $a^2=4b$

Q. Find the multiplicative inverse of the complex number $4-3i$

Q.

For the complex number z, which of the following is true?

1. $z+\overline{z}$ is real and $z\overline{z}$ is imaginary

2. $z+\overline{z}$ is imaginary and $z\overline{z}$ is real

3. Both $z+\overline{z}$ and $z\overline{z}$ are real.

4. Both $z+\overline{z}$ and $z\overline{z}$ are imaginary numbers

Q.

The values of x and y for which the numbers 3+i$x^2$y and $x^2$ +y+4i are conjugate complex can be

1. (1, -2) or (-2, 1)

2. (1, 2) or (-1, -2)

3. None of these

4. (-2, -1) or (2, -1)

Q. If x – iy = prove that .

Q.

If $i{z}^3+z^2-z+i=0,$ then show that $\left|z\right|=1.$

Or

Find the real values of x and y, if $\frac{x-1}{3+i}+\frac{y-1}{3-i}=i.$

Q. If $z=\frac{1+2i\cos \theta }{4+3i\sin \theta }$, where $\theta \in (0,\frac{\pi }{2})$, then which of the following is/are INCORRECT?

1. there exist one value of $\theta$ for which $Re\left(z\right)=0$
2. there exist $2$ values of $\theta$ for which $Re\left(z\right)=0$
3. there exist one value of $\theta$ for which $Im\left(z\right)=0$
4. there exist $2$ values of $\theta$ for which $Im\left(z\right)=0$

Q.

If one root of the equation $x^4+4x^3+6x^2+4x+5$ is $\sqrt{-1}$ . Find the sum of the squares of other three roots.

Q. Find the real numbers x and y if ( x – iy ) (3 + 5 i ) is the conjugate of –6 – 24 i .

Q.

Find the value of a, if the 2 is one root of the equation $a^2$$x^2$ - 4ax + 4 = 0

__

Q. Let $z$ be a complex number such that both real and imaginary parts of $\frac{z}{20}$ and $\frac{20}{\overline{z}}$ lies between $[0,1]$. Then the area of the region in the complex plane that consists of all points $z$ is
Assume $\pi =3.14$

Q.

If \omega = $\alpha$ + i$\beta$ where $\alpha$, $\beta$ are real, $\beta$ $\ne$ 0 and z $\ne$ 1 satisfies the condition that $\frac{\omega -\overline{\omega }z}{1-z}$ is purely real then the set of values of z is

1. 

2. 

3. 

4. 

Q.

The conjugate of the complex number $\frac{(2+3i)}{4i}$ is _____.

1. (3 + 2i)/4

2. (3 - 2i)/4

3. (2i - 3)/4

4. (2i - 3)/2

Q. The set of all $\alpha \in R$, for which $w=\frac{1+(1-8\alpha )z}{1-z}$ is a purely imaginary number, for all $z\in C$ satisfying $|z|=1$ and $Re\left(z\right)\ne 1$, is

1. an empty set
2. $\left\{0\right\}$
3. $\{0,\frac{1}{4},-\frac{1}{4}\}$
4. equal to $R$

Q. If $\frac{\sqrt{5+12i}+\sqrt{5-12i}}{\sqrt{5+12i}-\sqrt{5-12i}}=a+ib,b<0$, then the value of $a-2b$ is

Q.

The conjugate of the complex number $\frac{(2+3i)}{4i}$ is _____.

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

Q. Let $z_1=2-i,z_2=-2+i$.
Find
(i) $Re\left(\frac{z_1{z}_2}{\overline{z_1}}\right)$
(ii) $Im\left(\frac{1}{z_1\overline{z_1}}\right)$

Q. If $z_1=2-i,z_2=1+i$, find $\left|\frac{z_1+z_2+1}{z_1-z_2+1}\right|$

Q. If $a>0$ and $z=\frac{(1+i{)}^2}{a-i}$, has magnitude $\sqrt{\frac{2}{5}}$ , then $\overline{z}$ is equal to :

1. $-\frac{1}{5}-\frac{3}{5}i$
2. $-\frac{1}{5}+\frac{3}{5}i$
3. $-\frac{3}{5}-\frac{1}{5}i$
4. $\frac{1}{5}-\frac{3}{5}i$

Q.

The values of x and y for which the numbers 3+i$x^2$y and $x^2$ +y+4i are conjugate complex are

1. (-2, -1) or (2, -1)

2. (1, -2) or (-2, 1)

3. (1, 2) or (-1, -2)

4. None of these

Q. If $a$ is a complex number such that $|a|=1$ and $a{z}^2+z+1=0$ has one purely imaginary root, then $\cos \left(\arg \right(a\left)\right)$ is

1. $\frac{\sqrt{2}-1}{2}$
2. $\frac{\sqrt{3}+1}{2}$
3. $\frac{\sqrt{5}-1}{2}$
4. $\frac{\sqrt{7}-1}{2}$

Q.

If z is a complex number $\overline{z^{-1}}\left(\overline{z}\right)$ = , then

1. 1

2. -1

3. 0

4. None of these