Properties of Conjugate of a Complex Number

Q.

Express the complex numbers in the form of a + ib:

$i^9+i^{19}$

Q.

Solve the equation, $z^2=\overline{z},$ where z is a complex number.

Q.

Express the complex numbers in the form of a + ib:

${(\frac{1}{3}+3i)}^3$

Q.

Among the complex number z satisfying the condition $|z+1-i|\le 1$, the complex number having the least positive argument is:

1. -1+i

2. -i

3. None

4. 1 - i

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.

ctrlshift and alt are known as

Q. Find the angles between the lines $\sqrt{3}x+y=1$ and $x+\sqrt{3}y=1$

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

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

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

Q.

If $|z-i|=1$ and arg $\left(z\right)=\theta$ where $\theta \in (0,\frac{\pi }{2})$, then
$cot\theta -\frac{2}{z}$

1. 2i

2. -i

3. i

4. 1+i

Q.

If $|z_1+z_2|=|z_1|+|z_2|$ where $z_1$ and $z_2$ are different non-zero complex numbers, then :

1. 

2. 

3. 

4. 

Q.

If $x-iy=\sqrt{\frac{a-id}{c-id}},$ prove that $(x^2+y^2{)}^2=\frac{a^2+b^2}{c^2+d^2}$.

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 $\overline{z}$ 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 $k>0,|z|=k$ and $w=\frac{z-k}{z+k}$, then $Re\left(w\right)$ equals

1. 0

2. k

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

4. $-\frac{1}{k}$

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

Q.

In a $\Delta$ ABC, $2a{­}^2+4b^2$ + $c^2=4ab+2ac,$then cos B is equal to

1. 0

2. 1/8

3. 3/8

4. 7/8

Q.

If $w=\alpha +i\beta ,where\beta \ne 0andz\ne 1,$ satisfies the condition that $\left(\frac{w-\overline{w}z}{1-z}\right)$ is purely real, then the set of values of z is

1. $|z|=1,z\ne 2$

2. $z=\overline{z}$

3. None of these

4. $|z|=1,andz\ne 1$

Q.

If |$z_1|=|z_2|=|z_3|=|z_4|=1$ and $z_1+z_2+z_3+z_4=0$
then least value of the expression
$E=|z_1-z_2{|}^2+|z_2-z_3{|}^2+|z_3-z_4{|}^2+|z_4-z_1{|}^2$ is

1. 6

2. 8

3. 10

4. 12

Q. The roots $Z_1,Z_2,Z_3$ of the equation $x^3+3p{x}^2+3qx+r=0$ (p, q, r are complex numbers) correspond to
points A, B and C, then triangle ABC is equilateral, if

1. 
2. 
3. 
4. 

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.

Solve the equation $z+\sqrt{2}|z+1|+i=0$ for complex value of z.

Q.

The abscissae of A and B are the roots of the equation $x^2+2ax-b^2=0$ and their coordinates are the roots of the

equation y^2 + 2py - q^2 = 0. The equation of the circle with AB as diameter

1. 

2. 

3. 

4. 

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. 1

2. 2

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

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

Q. Suppose $a,b,c$ are three distinct real numbers. Let
$P\left(x\right)=\frac{(x-b)(x-c)}{(a-b)(a-c)}+\frac{(x-c)(x-a)}{(b-c)(b-a)}+\frac{(x-a)(x-b)}{(c-a)(c-b)}$.
When simplified, $P\left(x\right)$ becomes

1. $1$
2. $x$
3. $\frac{x^2+(a+b+c)(ab+bc+ca)}{(a-b)(b-c)(c-a)}$
4. $0$

Q. Given (f(x)=In (secx+tanx)then f(x) is

1. odd
2. even
3. neither even nor odd
4. none of these

Q.

If $k>0,|z|=k$ and $w=\frac{z-k}{z+k}$, then $Re\left(w\right)$ equals

1. 0

2. k

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

4. $-\frac{1}{k}$

Q.

What is a modulus example?

Q.

If $|z-i|=1$ and arg $\left(z\right)=\theta$ where $\theta \in (0,\frac{\pi }{2})$, then
$cot\theta -\frac{2}{z}$

1. 2i

2. i

3. -i

4. 1+i

Q.

Let $z_1=2-i,z_2=-2+i$. Find

$\left(i\right)Re\left(\frac{z_1{z}_2}{\overline{z_1}}\right)$

$\left(ii\right)Im\left(\frac{1}{z_1\overline{z_1}}\right)$