Argument of a Complex Number

Q.

What is the amplitude of a complex number?

Q.

How do you find the principal argument of $z=3i$ $?$

Q.

The amplitude of $\frac{1}{i}$ is equal to

1. $-\frac{\pi }{2}$

2. 0

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

4. $-\pi$

Q.

The argument of $\frac{1-i\sqrt{3}}{1+i\sqrt{3}}$ is

1. ${60}^{\circ }$

2. ${120}^{\circ }$

3. ${210}^{\circ }$

4. ${240}^{\circ }$

Q. If $\frac{3+i\sin \theta }{4-i\cos \theta },\theta \in [0,2\pi ]$, is a real number, then an argument of $\sin \theta +i\cos \theta$ is :

1. $\pi -{\tan }^{-1}\left(\frac{4}{3}\right)$
2. $-{\tan }^{-1}\left(\frac{3}{4}\right)$
3. $\pi -{\tan }^{-1}\left(\frac{3}{4}\right)$
4. ${\tan }^{-1}\left(\frac{4}{3}\right)$

Q.

If $z=\frac{1+2i}{1-(1-i{)}^2}$, then arg (z) equals

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

2. 0

3. $\pi$

4. none of these

Q.

If $\sin 6\theta +\sin 4\theta +\sin 2\theta =0$, then the general value of $\theta$ is:

1. $\frac{n\mathrm{\pi }}{4},\mathrm{n}\pi \pm \frac{\pi }{3}$

2. $\frac{n\mathrm{\pi }}{4},\mathrm{n}\pi \pm \frac{\pi }{6}$

3. $\frac{n\mathrm{\pi }}{4},2\mathrm{n}\pi \pm \frac{\pi }{3}$

4. $\frac{n\mathrm{\pi }}{4},2\mathrm{n}\pi \pm \frac{\pi }{6}$

Q.

The amplitude of $\frac{1+i\sqrt{3}}{\sqrt{3}+i}$ is

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

2. $-\frac{\pi }{3}$

3. $-\frac{\pi }{6}$

4. $\frac{\pi }{6}$

Q. Let $z_1$ and $z_2$ be two complex numbers such that $\arg (z_1-z_2)=\frac{\pi }{4}$ and $z_1,z_2$ satisfy the equation $|z-3|=Re\left(z\right)$. Then the imaginary part of $z_1+z_2$ is equal to

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.

If $arg\left(z\right)$ less than $0$, then $arg(-z)-argz$ equal

1. $\pi$

2. $-\pi$

3. $-\pi /2$

4. $\pi /2$

Q. Let $f:[-10,10]\to R,$ where $f\left(x\right)=\sin x+\left[\frac{x^2}{a}\right]$ be an odd function. Then the set of values of parameter $a$ is (where $[.]$ denotes greatest integer function)

1. $(-10,10)-\left\{0\right\}$
2. $(0,10)$
3. $[100,\infty )$
4. $(100,\infty )$

Q.

Find the modulus and argument of the following complex numbers and hence express each of them in the poloar form :

$\left(i\right)1+i\left(ii\right)\sqrt{3}+i\left(iii\right)1-i\left(iv\right)\frac{1-i}{1+i}\left(v\right)\frac{1}{1+i}\left(vi\right)\frac{1+2i}{1-3i}\left(vii\right)sin{120}^0-icos{120}^0\left(viii\right)\frac{-16}{1+i\sqrt{3}}$

Q.

What is the domain of the function $f\left(x\right)=\frac{1}{\sqrt{\left|\tan x\right|-\tan x}}$is

1. $\left(2n\mathrm{\pi }+\left(\frac{\mathrm{\pi }}{4}\right),\mathrm{n\pi }+\left(\frac{\mathrm{\pi }}{4}\right)\right),n\in I$

2. $\left(n\mathrm{\pi }+\left(\frac{\mathrm{\pi }}{2}\right),\mathrm{n\pi }+\left(\mathrm{\pi }\right)\right),n\in I$

3. $\left(n\mathrm{\pi },\pi +\left(\frac{\mathrm{\pi }}{2}\right)\right),n\in I$

4. None of these

Q.

If $x+y=3-\cos 4\theta$ and $x-y=4\sin 2\theta$ then prove that $x^2+y^2-8x-8y-2xy+16=0$

Q. If $z_1,z_2,z_3$ are any three roots of the equation $z^6=(z+1{)}^6,$ then $\arg \left(\frac{z_1-z_3}{z_2-z_3}\right)$ can be equal to

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

Q.

$\text{If}z=cos\frac{\pi }{4}+isin\frac{\pi }{6},\text{then}$

1. $|z|=1,arg\left(z\right)=\frac{\pi }{4}$

2. $|z|=1,arg\left(z\right)=\frac{\pi }{6}$

3. $|z|=\frac{\sqrt{3}}{2},arg\left(z\right)=\frac{5\pi }{24}$

4. $|z|=\frac{\sqrt{3}}{2},arg\left(z\right)=ta{n}^{-1}\frac{1}{\sqrt{2}}$

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

Q.

If $y={\tan }^{-1}\left(\frac{\sqrt{1+x^2}-\sqrt{1-x^2}}{\sqrt{1+x^2}+\sqrt{1-x^2}}\right)$, then $\frac{dy}{dx}$is equal to

1. $\frac{x^2}{\sqrt{1-x^4}}$

2. $\frac{x^2}{\sqrt{1+x^4}}$

3. $\frac{x}{\sqrt{1+x^4}}$

4. $\frac{x}{\sqrt{1-x^4}}$

Q.

If$n=1,2,3....,$then $\cos \left(\alpha \right)\cos \left(2\alpha \right)\cos \left(3\alpha \right)·····\cos \left(2^{n-1}\alpha \right)$ is equal to.

1. $\frac{\sin \left(2n\alpha \right)}{2n\sin \left(\alpha \right)}$

2. $\frac{\sin \left(2^n\alpha \right)}{2^n\sin \left(2^{n-1}\alpha \right)}$

3. $\frac{\sin \left(4^{n-1}\alpha \right)}{4^{n-1}\sin \left(4^{n-1}\alpha \right)}$

4. $\frac{\sin \left(2^n\alpha \right)}{2^n\sin \left(\alpha \right)}$

Q.

If $\theta$ is the amplitude of $\frac{a+ib}{a-ib},\text{then tan}\theta$

1. $\frac{2a}{a^2+b^2}$

2. $\frac{2ab}{a^2-b^2}$

3. $\frac{a^2-b^2}{a^2+b^2}$

4. none of these

Q.

If $\sin \left(\theta \right)=\frac{2t}{1+t^2}$ and $\left(\theta \right)$ lies in the second quadrant , then $\cos \left(\theta \right)$ is equals to

1. $\left(\frac{1-t^2}{1+t^2}\right)$

2. $\left(\frac{t^2-1}{1+t^2}\right)$

3. $\frac{-\left|1-t^2\right|}{\left(1+t^2\right)}$

4. $\frac{\left(1+t^2\right)}{\left|1-t^2\right|}$

Q.

$\left(2+\sqrt{3}\right)\sin \left(\theta \right)+2\cos \left(\theta \right)$ lies between

1. $-\sqrt{3}\mathrm{and}\sqrt{3}$

2. $-\left(2+\sqrt{3}\right)\mathrm{and}\left(2+\sqrt{3}\right)$

3. $-\left(2+\sqrt{5}\right)\mathrm{and}\left(2+\sqrt{5}\right)$

4. none of these

Q.

$\text{If}z=\frac{1+7i}{(2-i{)}^2},\text{then}$

1. $\text{amp (z)}=\frac{3\pi }{4}$

2. $|z|=2$

3. $|z|=\frac{1}{2}$

4. $\text{amp (z)}=\frac{\pi }{4}$

Q. If $z_1,z_2$ are two complex numbers such that $|z_1|=|z_2|=2$ and $\arg z_1+\arg z_2=0$, then $z_1{z}_2=$

Q. The function $f\left(x\right)=\frac{4-x^2}{4x-x^3}$
(a) discontinuous at only one point
(b) discontinuous exactly at two points
(c) discontinuous exactly at three points
(d) none of these

Q.

Find the modulus and argument of mentioned below complex number and hence express in polar form

$\frac{-16}{1+i\sqrt{3}}$

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.

Identify the like terms in the following-

$$\begin{gathered} 4ab,7a,8b,-3a^2{b}^2,-4ba,-50b,-75,15b^2{a}^2,-7a^2,45,150a,70ba,10a^{2}b \\ -15a{b}^2,3b{a}^2,-21a^2,7b^{2}a \end{gathered}$$

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. $-{120}^o$