The value of $\sin \left({\sin }^{-1}\left(\frac{1}{3}\right) + {\sec }^{-1}\left(3\right)\right) + \cos \left({\tan }^{-1}\left(\frac{1}{2}\right) + {\tan }^{-1}\left(2\right)\right)$ is :

• 1

• 2

• 3

• 4

If f(x) = x$\left[\sqrt{\mathrm{x}} - \sqrt{\mathrm{x} +\hspace{0.17em}1}\right]$, then :

• f(x) is continuous but not differentiable at x = 0

• f(x) is not differentiable at x = 0

• f(x) is differentiable at x = 0

• None of the above

Let A, B and C be n x n matrices. Which one of the following is a correct statement?

• If AB = AC, then B = C

• If A³ + 2A² + 3A + 51 = 0, then A is invertible

• If A² = 0,then A = 0

• None of these

The value of ${\cot }^{-1}\left(9\right) + {\csc }^{-1}\left(\frac{\sqrt{41}}{4}\right)$ is given by :

• 0

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{2}$

• ${\tan }^{-1}\left(2\right)$

If the line ax + by + c = 0 is a normal to the curve y =1, then :

• a > 0, b > 0

• a > 0, b < 0

• a < 0, b < 0

• Data is insufficient

If f(x) = cos(log(x)), then f(x)f(y) $- \frac{1}{2}\left[\mathrm{f}\left(\frac{\mathrm{x}}{\mathrm{y}}\right) + \mathrm{f}\left(\mathrm{xy}\right)\right]$ has the value :

• - 1

• $\frac{1}{2}$

• - 2

• zero

If y = 3^{x - 1} + 3^{-x - 1} (x real), then the least value ofy is

• 2

• 6

• 2/3

• None of these

8.

${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\frac{\sqrt{\cot \left(\mathrm{x}\right)}}{\sqrt{\cot \left(\mathrm{x}\right)} + \sqrt{\tan \left(\mathrm{x}\right)}}\mathrm{dx}$ is equal to :

• 1

• - 1

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{4}$

Area bounded by the curve y = log_e(x), x = 0, y $\le$ 0 and x - axis is :

• 1 sq unit

• $\frac{1}{2}$ sq unit

• 2 sq unit

• None of these

If ${\left|\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}\right|}^2 +\hspace{0.17em}{\left|\overrightarrow{\mathrm{a}} . \overrightarrow{\mathrm{b}}\right|}^2 = 144 \mathrm{and} \left|\overrightarrow{\mathrm{a}}\right| = 4, \mathrm{then} \left|\overrightarrow{\mathrm{b}}\right|$ is equal to

• 12

• 3

• 8

• 4