Argument of a Complex Number

Q.

If z is a complex number of unit modulus and argument $\theta$, find arg $\left(\frac{1+z}{1+\overline{z}}\right)$

Q.

If $\mathit{A}\mathbf{}\mathbf{+}\mathbf{}\mathit{B}\mathbf{}\mathbf{+}\mathbf{}\mathit{C}\mathbf{}\mathbf{=}\mathbf{}\mathbf{\pi }$ then prove that $\sin \left(A\right)+\sin \left(B\right)+\sin \left(C\right)=4\cos \left(\frac{A}{2}\right)\times \cos \left(\frac{B}{2}\right)\times \cos \left(\frac{C}{2}\right)$

Q.

If $z$ and $\omega$ are complex numbers such that $\left|z\omega \right|=1$, $\arg \left(z\right)-\arg \left(\omega \right)=\frac{3\mathrm{\pi }}{2}$ then find $\arg \left(\frac{1-2\overline{z}\omega }{1+3\overline{z}\omega }\right)$

1. $\frac{\mathrm{\pi }}{4}$

2. $-\frac{\mathrm{\pi }}{4}$

3. $\frac{3\mathrm{\pi }}{4}$

4. $-\frac{3\mathrm{\pi }}{4}$

Q.

Principal value of argument of-2+2i is:

1. 

2. 

3. 

4. 

Q.

The principal amplitude of ${\left(\sin \left(40°\right)+i\cos \left(40°\right)\right)}^5$

Q.

The domain of ${\sin }^{-1}\left[{\log }_2\frac{x}{12}\right]$ is ?

1. $\left[2,12\right]$

2. $\left[-1,1\right]$

3. $\left[\left(\frac{1}{3}\right),24\right]$

4. $\left[\left(\frac{2}{3}\right),24\right]$

5. $\left[6,24\right]$

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.

Find the modulus and argument of the complex number $\frac{1+2i}{1-3i}$

Q. If $\cos \alpha =-\frac{12}{13},\cot \beta =\frac{24}{7},{90}^{\circ }$ < $\alpha$ < ${180}^{\circ }$ and ${180}^{\circ }$ < $\beta$ < ${270}^{\circ },$ then the quadrant in which $\alpha +\beta$ lies

1. I
2. II
3. III
4. IV

Q. Find the modulus and the argument of the complex number $z=-\sqrt{3}+i$

Q. Argument and modulus of $\frac{1+i}{1-i}$ are respectively
[RPET 1984; MP PET 1987; Karnataka CET 2001]

1. $\frac{\pi }{2}$ and $\sqrt{2}$
2. 0 and $\sqrt{2}$
3. $\frac{\pi }{2}$ and 1
4. $\frac{-\pi }{2}$ and 1

Q.

If z is a complex number of unit modulus and argument $\theta$, arg $\left(\frac{1+z}{1+\overline{z}}\right)$ is equal to

1. - θ

2. (π/2)- θ

3. θ

4. π- θ

Q. The eccentricity of the ellipse 4x² + 9y² + 8x + 36y + 4 = 0 is

(a) $\frac{5}{6}$

(b) $\frac{3}{5}$

(c) $\frac{\sqrt{2}}{3}$

(d) $\frac{\sqrt{5}}{3}$

Q. Find the modulus and the argument of the complex number

Q. If complex numbers $z_1$ and $z_2$ both satisfy $z+\overline{z}=2|z-1|$ and $\arg (z_1-z_2)=\frac{\pi }{3},$ then find the value of $\text{Im}(z_1+z_2)$. (where $\text{Im}\left(z\right)$ denotes the imaginary part of $z$)

1. $\sin \frac{\pi }{3}$
2. $\text{cosec}\frac{\pi }{3}$
3. $\tan \frac{\pi }{3}$
4. $\cot \frac{\pi }{3}$

Q. For a complex number a + ib where b = 0 and a < 0, the argument will be

1. $\pi$
2. None
3. 0
4. -$\pi$

Q. Find the modulus and the argument of the complex number $z=-1-i\sqrt{3}$

Q. If $z$ and $\omega$ be two non-zero compex numbers such that $|z|=|\omega |$ and $arg\left(z\right)+arg\left(\omega \right)=\pi$, then $z$ equals

1. $-\omega$
2. $\overline{\omega }$
3. $-\overline{\omega }$
4. $\omega$

Q. If $z_1$ and $z_2$ are two non-zero complex numbers such that $|z_1-z_2|=||z_1|-|z_2||$ then $\arg \left(z_1\right)-\arg \left(z_2\right)=$

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

Q. The eccentricity of the ellipse 9x² + 25y² − 18x − 100y − 116 = 0, is
(a) 25/16
(b) 4/5
(c) 16/25
(d) 5/4

Q. Which one of the following is not a function?

(a) {(x, y) : x, y ∈ R, x² = y}
(b) {(x, y) : x, y ∈, R, y² = x}
(c) {(x, y) : x, y ∈ R, x² = y³}
(d) {(x, y) : x, y ∈, R, y = x³}

Q. if z and $\omega$ be two non-zero compex numbers such that $|z|=|\omega |$ and arg(z) + arg$\left(\omega \right)=\pi$, then z equals

1. $-\omega$
2. $-\overline{\omega }$
3. $\overline{\omega }$
4. $\omega$

Q.

Let α & β be the roots of \({x^2}\) – 6x – 2= 0 which α >β . if $a_n$= ${\alpha }^n$ –${\beta }^n$ for n$\ge$1, then the

value of $\frac{a_{10}-2a_8}{2a_9}$ is

1. 4

2. 1

3. 2

4. 3

Q. If z is a complex number of unit modulus and argument $\theta$, then arg$\left(\frac{1+z}{1+\overline{z}}\right)$ is equal to

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

Q. Find the modulus and argument of the complex number $\frac{1+2i}{1-3i}$

Q. The eccentricity of the conic 9x² + 25y² = 225 is
(a) 2/5
(b) 4/5
(c) 1/3
(d) 1/5
(e) 3/5

Q. The eccentricity of the conic 9x² + 25y² = 225 is
(a) 2/5
(b) 4/5
(c) 1/3
(d) 1/5
(e) 3/5

Q. If $z$ is a complex number of unit modulus and argument $\theta$, then arg$\left(\frac{1+z}{1+\overline{z}}\right)$ is equal to

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

Q. The argument of the complex number -1 + i $\sqrt{3}$ is
[MP PET 1994]

1. $-{60}^o$
2. ${60}^o$
3. ${120}^o$
4. $-{120}^o$

Q. Let A = (x, y, z) and B = (a, b). Find the total number of relations from A into B.