$\mathrm{What} \mathrm{s} \mathrm{the} \mathrm{value} \mathrm{of} {\sin }^{-1}\left(\frac{4}{5}\right) + {\sec }^{-1}\left(\frac{5}{4}\right) - \frac{\mathrm{\pi }}{2} = ?$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{2}$

• $\mathrm{\pi }$

• 0

$\mathrm{If} {\sin }^{-1}\left(\frac{2\mathrm{p}}{1 + {\mathrm{p}}^2}\right) - {\cos }^{-1}\left(\frac{1 - {\mathrm{q}}^2}{1 + {\mathrm{q}}^2}\right) = {\tan }^{-1}\left(\frac{2\mathrm{x}}{1 - {\mathrm{x}}^2}\right), \mathrm{then} \mathrm{what} \mathrm{is} \mathrm{x} = ?$

• $\frac{\mathrm{p} + \mathrm{q}}{1 + \mathrm{pq}}$

• $\frac{\mathrm{p} - \mathrm{q}}{1 + \mathrm{pq}}$

• $\frac{\mathrm{pq}}{1 + \mathrm{pq}}$

• $\frac{\mathrm{p} + \mathrm{q}}{1 - \mathrm{pq}}$

3.

What is the principal value of ${\sin }^{-1}\left(\sin \left(\frac{2\mathrm{\pi }}{3}\right)\right)$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{2\mathrm{\pi }}{3}$

$\mathrm{What} \mathrm{is} {\tan }^{-1}\left(\frac{1}{4}\right) + {\tan }^{-1}\left(\frac{3}{5}\right) = ?$

• 0

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{2}$

Consider the following values of x :

1.  2,  2. $-$ 4, 

3. $\frac{1}{6}, 4. - \frac{1}{4}$

Which of the above values of x is/are the solution(s) of the equation

${\tan }^{-1}\left(2\mathrm{x}\right) + {\tan }^{-1}\left(3\mathrm{x}\right) = \frac{\mathrm{\pi }}{4}$

• 3 only

• 2 and 3 only

• 1 and 4 only

• 4 only

The angle between the two lines lx + my + n = 0 and l'x + m'y + n' - 0 is given by ${\tan }^{-1}\left(\mathrm{\theta }\right)$. What is $\mathrm{\theta }$ equal to ?

• $\left|\frac{\mathrm{lm}\text{'} - \mathrm{l}\text{'}\mathrm{m}}{\mathrm{ll}\text{'} - \mathrm{mm}\text{'}}\right|$

• $\left|\frac{\mathrm{lm}\text{'} + \mathrm{l}\text{'}\mathrm{m}}{\mathrm{ll}\text{'} + \mathrm{mm}\text{'}}\right|$

• $\left|\frac{\mathrm{lm}\text{'} - \mathrm{l}\text{'}\mathrm{m}}{\mathrm{ll}\text{'} + \mathrm{mm}\text{'}}\right|$

• $\left|\frac{\mathrm{lm}\text{'} + \mathrm{l}\text{'}\mathrm{m}}{\mathrm{ll}\text{'} - \mathrm{mm}\text{'}}\right|$

The value of ${\sin }^{-1}\left(\frac{3}{5}\right) + {\tan }^{-1}\left(\frac{1}{7}\right) = ?$

• 0

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{2}$

The principal value of ${\sin }^{-1}\left(\mathrm{x}\right)$ lies in the interval :

• $\left( - \frac{\mathrm{\pi }}{2}, - \frac{\mathrm{\pi }}{2}\right)$

• $\left[ - \frac{\mathrm{\pi }}{2}, \frac{\mathrm{\pi }}{2}\right]$

• $\left[0, \frac{\mathrm{\pi }}{2}\right]$

• $\left[0, \mathrm{\pi }\right]$

Consider the following statements :

$$\begin{gathered} 1. \mathrm{There} \mathrm{exists} \mathrm{\theta } \in \left( - \frac{\mathrm{\pi }}{2}, \frac{\mathrm{\pi }}{2}\right) \mathrm{for} \mathrm{which} {\tan }^{-1}\left(\tan \left(\mathrm{\theta }\right)\right) \ne \mathrm{\theta } \\ 2. {\sin }^{-1}\left(\frac{1}{3}\right) - {\sin }^{-1}\left(\frac{1}{5}\right) = {\sin }^{-1}\left(\frac{2\sqrt{2}\left(\sqrt{3} - 1\right)}{15}\right) \\ \mathrm{Which} \mathrm{of} \mathrm{the} \mathrm{above} \mathrm{statements} \mathrm{is}/\mathrm{are} \mathrm{correct} ? \end{gathered}$$

• 1 only

• 2 only

• both 1 and 2

• neither 1 nor 2

$$\begin{gathered} \mathrm{Consider} \mathrm{the} \mathrm{following} \mathrm{statements} : \\ 1. {\tan }^{-1}\left(\mathrm{x}\right) + {\tan }^{-1}\left(\frac{1}{\mathrm{x}}\right) = \mathrm{\pi } \\ 2. \mathrm{There} \mathrm{exist} \mathrm{x}, \mathrm{y} \in \left[ - 1, 1\right], \mathrm{where}, \mathrm{x} \ne \mathrm{y} \mathrm{such} \mathrm{that} {\sin }^{-1}\left(\mathrm{x}\right) + {\cos }^{-1}\left(\mathrm{y}\right) = \frac{\mathrm{\pi }}{2} \end{gathered}$$

Which of the above statements is/are correct ?

• 1 only

• 2 only

• both 1 and 2 

• neither 1 nor 2