Purely Imaginary

Q. If $\frac{2z_1}{3z_2}$ is a purely imaginary number, then $\left|\frac{z_1-z_2}{z_1+z_2}\right|$ =
[MP PET 1993]

1. 3/2

2. 1

3. 2/3
4. 4/9

Q. $LetA=\{\theta \in (-\frac{\pi }{2},\pi ):\frac{3+2i\sin \theta }{1-2i\sin \theta }\text{is purely imaginary}\}$. The sum of all the elements in $A$ is :

1. $\pi$
2. $\frac{2\pi }{3}$
3. $\frac{3\pi }{4}$
4. $\frac{5\pi }{6}$

Q. A value of $\theta$ for which $\frac{2+3isin\theta }{1-2isin\theta }$ is purely imaginary, is:

1. $si{n}^{-1}\left(\frac{\sqrt{3}}{4}\right)$
2. $si{n}^{-1}\left(\frac{1}{\sqrt{3}}\right)$
3. $\frac{\pi }{3}$
4. $\frac{\pi }{6}$

Q. What is the fundamental period of the function $y={\sin }^{-1}\left(\sin x\right)$.

1. $2\pi$
2. $4\pi$
3. $3\pi$
4. $\pi$

Q.

If $g\left\{f\right(x\left)\right\}=|sinx|$ and $f\left\{g\right(x\left)\right\}=(sin\sqrt{x}{)}^2$, then

1. $f\left(x\right)=si{n}^{2}x,g\left(x\right)=\sqrt{x}$

2. $f\left(x\right)=sinx,g\left(x\right)=|x|$

3. $f\left(x\right)=x^2,g\left(x\right)=sin\sqrt{x}$

4. f and g cannot be determined

Q. If $\frac{z-i}{z+i}(z\ne -i)$ is a purely imaginary number, then $z.\overline{z}$ is equal to

1. \N
2. 1
3. 2
4. None of these

Q. A value of $\theta$ for which $z=\frac{2+3isin\theta }{1-2isin\theta }$ is purely imaginary, is

1. $\frac{\pi }{3}$
2. $\frac{\pi }{6}$
3. $si{n}^{-1}\left(\frac{\sqrt{3}}{4}\right)$
4. $si{n}^{-1}\left(\frac{1}{\sqrt{3}}\right)$

Q. If $\frac{z-i}{z+i}(z\ne -i)$ is a purely imaginary number, then $z.\overline{z}$ is equal to

1. \N
2. 1
3. 2
4. None of these