Polar Representation of a Complex Number

Q.

Prove $si{n}^{-1}\left(x\right)+co{s}^{-1}\left(x\right)=\left(\frac{\mathrm{\pi }}{2}\right)$

Q.

The differential coefficient of ${\tan }^{-1}\sqrt{\frac{1-x^2}{1+x^2}}$ with respect to ${\cos }^{-1}\left(x^2\right)$ is

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

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

3. $1$

4. $0$

Q.

The differential coefficient of ${\tan }^{-1}\left(\frac{\sqrt{1+x^2}-1}{x}\right)$ with respect to ${\tan }^{-1}x$ is

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

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

3. $1$

4. None of these

Q.

The real part of $(1-\cos \left(\mathrm{\theta }\right)+2i\sin {\left(\mathrm{\theta }\right)}^{-1}$ is

1. $\frac{1}{(3+5\cos \left(\theta \right))}$

2. $\frac{1}{(5-3\cos \left(\theta \right))}$

3. $\frac{1}{(3-5\cos \left(\theta \right))}$

4. $\frac{1}{(5+3\cos \left(\theta \right))}$

Q.

If sin$\alpha$+sin$\beta$+sin$\gamma$=0= cos$\alpha$+cos$\beta$+cos$\gamma$,

value of $si{n}^2$$\alpha$+$si{n}^2$$\beta$+$si{n}^2$$\gamma$

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

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

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

4. 1

Q.

Find the value of $e^{6^{deg}i}$ $\times$ $e^{9^{deg}i}$ $\times$ $e^{{13}^{deg}i}$ $\times$ $e^{{17}^{deg}i}$ (where deg = degree)

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

Q.

The trigonometric form of $z=(1-icot8{)}^3$ (where $i=\sqrt{-1}$) is

1. $cose{c}^{3}8.e^{i(24-\frac{3\pi }{2})}$

2. $cose{c}^{3}8.e^{-i(24-\frac{3\pi }{2})}$

3. $cose{c}^{3}8.e^{i(36-\frac{\pi }{2})}$

4. $cose{c}^{2}8.e^{-\left(24i\frac{+\pi }{2}\right)}$

Q.

If a=cos$\alpha$+isin$\alpha$, b=cos$\beta$+isin$\beta$,

c=cos$\gamma$+isin$\gamma$ and $\frac{b}{c}$+$\frac{c}{a}$+$\frac{a}{b}$=1, Then

cos($\beta$-$\gamma$)+cos($\gamma$-$\alpha$)+cos($\alpha$-$\beta$) is equal to

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

Q.

The value of $\frac{(e^{2\theta }-1)}{(e^{2\theta }+1)}$ is

1. $coth\theta$

2. $coth2\theta$

3. $\tan h\theta$

4. $\tan h2\theta$

Q. $\text{If}{z}_1=a+ib\text{and}{z}_2=c+id$ are complex numbers such that $|z_1|=|z_2|=1\text{and}Re\left(z_1\overline{z_2}\right)=0$, then the pair of complex numbers $w_1=a+ic\text{and}{w}_2=b+id$ satisfies

1. $|w_1|=1$
2. $|w_2|=1$
3. $Re\left(w_1\overline{w_2}\right)=0$
4. All of these

Q.

The maximum distance from the origin of coordinates to the point z satisfying the equation $|z+\frac{1}{z}|$=a is

1. $\frac{1}{2}$($\sqrt[2]{2(a^2+1)}$+a)

2. $\frac{1}{2}$($\sqrt[2]{2(a^2+2)}$+a)

3. $\frac{1}{2}$($\sqrt[2]{2(a^2+4)}$+a)

4. None of these

Q.

The number of non-zero integral solutions of the equation $|1-i{|}^x$=$2^x$ is

1. zero

2. 1

3. 2

4. None of these

Q.

$\sqrt{\left(\frac{1+\sin \theta }{1-\sin \theta }\right)}$ =

1. $\tan \theta -sec\theta$

2. $\tan \theta +sec\theta$

3. $sec\theta -\tan \theta$

4. none of these

Q.

If sin$\alpha$+sin$\beta$+sin$\gamma$=0= cos$\alpha$+cos$\beta$+cos$\gamma$,

value of $si{n}^2$$\alpha$+$si{n}^2$$\beta$+$si{n}^2$$\gamma$

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

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

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

4. 1