The solution of ${\tan }^{-1}\left(\mathrm{x}\right) + 2{\cot }^{-1}\left(\mathrm{x}\right) = \frac{2\mathrm{\pi }}{3}$ is

• $- \frac{1}{\sqrt{3}}$

• $\frac{1}{\sqrt{3}}$

• $- \sqrt{3}$

• $\sqrt{3}$

The value of $\sin \left[2{\cos }^{-1}\left(\frac{\sqrt{5}}{3}\right)\right]$ is

• $\frac{\sqrt{5}}{3}$

• $\frac{2\sqrt{5}}{3}$

• $\frac{4\sqrt{5}}{9}$

• $\frac{2\sqrt{5}}{9}$

If A = $\left[\begin{array}{ccc}1& - 1& 1\\ 2& 1& - 3\\ 1& 1& 1\end{array}\right], 10\mathrm{B} = \left[\begin{array}{ccc}4& 2& 2\\ - 5& 0& \mathrm{\alpha }\\ 1& - 2& 3\end{array}\right]$ and B is the inverse of A, then the value of $\mathrm{\alpha }$ is

• 2

• 0

• 5

• 4

If A = $\left[\begin{array}{ccc}1& - 2& 2\\ 0& 2& - 3\\ 3& - 2& 4\end{array}\right]$, then A . adj (A) is equal to

• $\left[\begin{array}{ccc}5& 1& 1\\ 1& 5& 1\\ 1& 1& 5\end{array}\right]$

• $\left[\begin{array}{ccc}5& 0& 0\\ 0& 5& 0\\ 0& 0& 5\end{array}\right]$

• $\left[\begin{array}{ccc}8& 0& 0\\ 0& 8& 0\\ 0& 0& 8\end{array}\right]$

• $\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]$

If f : R $\to$ R is defined by f(x) = lxl, then

• f^-1(x) = - x

• f^-1 = $\frac{1}{\left|\mathrm{x}\right|}$

• the function f^-1(x) does not exist

• ${\mathrm{f}}^{-1}\left(\mathrm{x}\right) = \frac{1}{\mathrm{x}}$

If A = $\left[\begin{array}{ccc}0& \mathrm{x}& 16\\ \mathrm{x}& 5& 7\\ 0& 9& \mathrm{x}\end{array}\right]$is singular, then the possible values of x are

• 0, 12, - 12

• 0, 1, - 1

• 0, 4, - 4

• 0, 5, - 5

The value of $\left|\begin{array}{ccc}\mathrm{x}& \mathrm{p}& \mathrm{q}\\ \mathrm{p}& \mathrm{x}& \mathrm{q}\\ \mathrm{p}& \mathrm{q}& \mathrm{x}\end{array}\right|$ is

• x(x - p)(x - q)

• (x - p)(x - q)(x + p + q)

• (p - q)(x - q)(x - p)

• pq(x - p)(x - q)

8.

The range in which y = - x² + 6x - 3 is increasing is

• x < 3

• x > 3

• 7 < x < 8

• 5 < x < 6

OA and OB are two roads enclosmng an angle of 120°. X and Y start from 'O' at the same time. X travels along OA with a speed of 4 km/h and Y travels along OB with a speed of 3 km/h.The rate at which the shortest distance between X and Y is increasing after 1 h is

• $\sqrt{37} \mathrm{km} /\mathrm{h}$

• 37 km/h

• 13 km/h

• $\sqrt{13} \mathrm{km}/\mathrm{h}$

If the function f(x) = $$\begin{gathered} \left\{\frac{1 - \cos \left(\mathrm{x}\right)}{{\mathrm{x}}^2}, \mathrm{x} \ne 0 \\ \mathrm{k}, \mathrm{x} = 0\right\} \end{gathered}$$ is continuous at x = 0, then the value of k is

• 1

• 0

• $\frac{1}{2}$

• - 1