The value of x satisfying the equation ${\tan }^{-1}\left(\mathrm{x}\right) + {\tan }^{-1}\left(\frac{2}{3}\right) + {\tan }^{-1}\left(\frac{7}{4}\right)$ is equal to

• $\frac{1}{2}$

• $- \frac{1}{2}$

• $\frac{3}{2}$

• $- \frac{1}{3}$

If ${\tan }^{-1}\left(\mathrm{x}\right) + {\tan }^{-1}\left(\mathrm{y}\right) = \frac{2\mathrm{\pi }}{3}$, then ${\cot }^{-1}\left(\mathrm{x}\right) + {\cot }^{-1}\left(\mathrm{y}\right)$ is equal to

• $\frac{\mathrm{\pi }}{2}$

• $\frac{1}{2}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\sqrt{3}}{2}$

$If $ {\tan }^{-1}\left(\mathrm{x}\right) + {\tan }^{-1}\left(\mathrm{y}\right) + {\tan }^{-1}\left(\mathrm{z}\right) = \mathrm{\pi }$\mathrm{}$ then x + y + z is :

• xyz

• 0

• 1

• 2xyz

If $\mathrm{\theta } = {\sin }^{-1}\left[\sin \left(- 600°\right)\right]$, then one of the possible values of $\mathrm{\theta }$ is :

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{2}$

• $\frac{2\mathrm{\pi }}{3}$

• $- \frac{2\mathrm{\pi }}{3}$

65.

Domain of the function f(x) = sin^-1(log₂(x))in the set of real numbers is :

• $\left\{\mathrm{x} :\hspace{0.17em}1 \le \mathrm{x} \le 2\right\}$

• $\left\{\mathrm{x} :\hspace{0.17em}1 \le \mathrm{x} \le 3\right\}$

• $\left\{\mathrm{x} :\hspace{0.17em}- 1 \le \mathrm{x} \le 2\right\}$

• $\left\{\mathrm{x} :\hspace{0.17em}\frac{1}{2} \le \mathrm{x} \le 2\right\}$

Which one of the following is true?

• $\sin \left({\cos }^{-1}\left(\mathrm{x}\right)\right) = \cos \left({\sin }^{-1}\left(\mathrm{x}\right)\right)$

• $\sec \left({\tan }^{-1}\left(\mathrm{x}\right)\right) = \tan \left({\sec }^{-1}\left(\mathrm{x}\right)\right)$

• $\cos \left({\tan }^{-1}\left(\mathrm{x}\right)\right) = \tan \left({\cos }^{-1}\left(\mathrm{x}\right)\right)$

• $\tan \left({\sin }^{-1}\left(\mathrm{x}\right)\right) = \sin \left({\tan }^{-1}\left(\mathrm{x}\right)\right)$

If ${\tan }^{-1}\left(\mathrm{a}\right) + {\tan }^{-1}\left(\mathrm{b}\right) = {\sin }^{-1}\left(1\right) - {\tan }^{-1}\left(\mathrm{c}\right)$, then

• a + b + c = abc

• ab + bc + ca = abc

• $\frac{1}{\mathrm{a}} + \frac{1}{\mathrm{b}} + \frac{1}{\mathrm{c}} - \frac{1}{\mathrm{abc}} = 0$

• ab + bc + ca = a + b + c

${\tan }^{-1}\left(\frac{\mathrm{m}}{\mathrm{n}}\right) - {\tan }^{-1}\left(\frac{\mathrm{m} - \mathrm{n}}{\mathrm{m} + \mathrm{n}}\right)$ is equal to :

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{8}$

For the principal value branch of the graph of the function $\mathrm{y} = {\sin }^{-1}\left(\mathrm{x}\right), - 1 \le \mathrm{x} \le 1$, which among the following is a true statement ?

• graph is symmetric about the x-axis

• graph is symmetric about the y-axis

• graph is not continuous

• the line x = 1 is a tangent

The range of the function $\sin \left({\sin }^{-1}\left(\mathrm{x}\right) + {\cos }^{-1}\left(\mathrm{x}\right)\right), \left|\mathrm{x}\right| \le 1$ :

• [- 1, 1]

• [1, - 1]

• {0}

• {1}