Polar Representation of a Complex Number

Q.

If $x={\cos }^{-1}\frac{1}{\sqrt{1+t^2}}\mathrm{and}y={\sin }^{-1}\frac{t}{\sqrt{1+t^2}}$, then $\frac{dy}{dx}$ is equal to:

1. $0$

2. $\tan t$

3. $1$

4. $\sin t\cos t$

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 solution of equations ${\cos }^2\left(\theta \right)+\sin \left(\theta \right)+1=0$ lies in the interval

1. $\left(-\frac{\mathrm{\pi }}{4},\frac{\mathrm{\pi }}{4}\right)$

2. $\left(\frac{\mathrm{\pi }}{4},\frac{3\mathrm{\pi }}{4}\right)$

3. $\left(\frac{3\mathrm{\pi }}{4},\frac{5\mathrm{\pi }}{4}\right)$

4. $\left(\frac{5\mathrm{\pi }}{4},\frac{7\mathrm{\pi }}{4}\right)$

Q. 1+ 1

Q. Convert the given complex number in polar form:

Q.

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

1. 

2. 

3. 

4. None of these

Q.

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

cos2$\alpha$+cos2$\beta$+cos2$\gamma$ equals

1. 0

2. 1

3. 

4. 

Q.

If -1+$\sqrt{-3}$=r$e^{i\theta }$, then $\theta$ is equal to

1. 

2. 

3. 

4. 

Q. 6. 3

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. 

3. 

4. 

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. Convert the following in the polar form: (i) , (ii)

Q.

If $|z|=1$ and $z\ne \pm 1$, then all the values of $\frac{z}{1-z^2}$ lies on

1. The Y - axis

2. $a$ line not passing through the origin

3. $|z|=\sqrt{2}$

4. The X - axis

Q. Find the number of non-zero integral solution of the equation ${|1-i|}^x=2^x$

Q. Convert the given complex number in polar form: i

Q. 3, Iii

Q. $\sin x=\frac{1}{4}$, x in quadrant II. Find the value of $\sin \frac{x}{2}$

Q.

A standard hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is drawn along with its auxiliary circle. A point P $(asec\theta ,btan\theta )$ is taken. A perpendicular is dropped from P to the x axis which meets at the axis at R. A tangent is drawn from R to auxiliary circle. Which angle is equal to θ

1. 

2. 

3. 

4. 

Q.

The polar form of $\frac{-\sqrt{3}}{2}-\frac{i}{2}$ (where i = $\sqrt{-1}$) is

1. $cos\frac{\pi }{6}+isin\frac{\pi }{6}$

2. $cos\frac{5\pi }{6}+isin\frac{5\pi }{6}$

3. $cos\left(\frac{-\pi }{6}\right)+isin\left(\frac{-\pi }{6}\right)$

4. $cos\left(\frac{-5\pi }{6}\right)+isin\left(\frac{-5\pi }{6}\right)$

Q. If $|z|=1$ and $z\ne \pm 1$, then all the values of $\frac{z}{1-z^2}$ lie on

1. The X - axis
2. The Y - axis
3. $a$ line not passing through the origin
4. $|z|=\sqrt{2}$

Q.

$\alpha$ + $\beta$ - $\gamma$ = $\pi$ Then $si{n}^2\beta$ - $si{n}^2\gamma$ =

1. () ()

2. sin sin

3. cos cos

4. Both A & B

Q.

If -1+$\sqrt{-3}$=r$e^{i\theta }$, then $\theta$ is equal to

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

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

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

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

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. S, 717i

Q. Let $z_1,z_2,z_3$​ be three distinct complex numbers lying on a circle whose centre is at the origin. If $z_i+z_j{z}_k,$ ​where $i,j,k\in \{1,2,3\}$ and $i\ne j\ne k$ are real numbers, then the value of $4(z_1\times z_2\times z_3)$ is