21.

Let A be a non-singular square matrix. Then,$\left|\mathrm{adj} \mathrm{A}\right|$ is equal to

• $\left|\mathrm{A}\right|$

• ${\left|\mathrm{A}\right|}^{\mathrm{n} - 1}$

• ${\left|\mathrm{A}\right|}^{\mathrm{n} - 2}$

• None of these

If P is a 3 x 3 matrix such that P^T = 2P + I, where p^T is the transpose of P and I is 3 x 3 identity, then there exists a column matrix X = $\left[\begin{array}{c}\mathrm{x}\\ \mathrm{y}\\ \mathrm{z}\end{array}\right] \ne \left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]$ such that PX is equal to

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

• X

• 2X

• - X

If the tangent at (1, 1) on y = x(2 - x)² meets the curve again at P, then P is

• (4, 4)

• (- 1, 2)

• (9/4, 3/8)

• None of these

If $$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right) = \left\{{\mathrm{ax}}^2 - \mathrm{b}, \mathrm{a} \le \mathrm{x} < 1 \\ 2, \mathrm{x} = 1 \\ \mathrm{x} + 1, 1 < \mathrm{x} \le 2\right\} \end{gathered}$$ Then, the value of the pair (a, b) for which f(x) cannot be continuous at x = 1, is

• (2, 0)

• (1, - 1)

• (4, 2)

• (1, 1)

The range of the function

$\mathrm{f}\left(\mathrm{x}\right) = \tan \left(\sqrt{\frac{{\mathrm{\pi }}^2}{9} - {\mathrm{x}}^2}\right)$ is

• [0, $\sqrt{3}$]

• (0, $\sqrt{3}$)

• $[0, \sqrt{3})$

• $(0, \sqrt{3}]$

If ${\sin }^{-1}\left(\mathrm{x}\right) + {\sin }^{-1}\left(1 - \mathrm{x}\right) = {\cos }^{-1}\left(\mathrm{x}\right)$, then x equals

• 1, - 1

• 1, 0

• 0, 1/2

• None of these

The differential equation $\frac{d\mathrm{y}}{d\mathrm{x}} + \frac{{\mathrm{y}}^2}{{\mathrm{x}}^2} = \mathrm{y}/\mathrm{x}$ has the solution

• x = y(log(x) + C)

• y = x(log(y) + C)

• x = (y + C)log(x)

• y = (x + C)log(y)

The mean and the variance of a binomial distribution are 4 and 2, respectively. Then, the probability of 2 succeses is

• 28/256

• $\frac{219}{256}$

• 128/256

• 37/256

The value of ${\int }_{- 2}^3\left|1 - {\mathrm{x}}^2\right|d\mathrm{x}$, is

• $\frac{1}{3}$

• 14/3

• 7/3

• 28/3

$\int \frac{1}{\tan \left(\mathrm{x}\right) + \cot \left(\mathrm{x}\right) + \sec \left(\mathrm{x}\right) + \csc \left(\mathrm{x}\right)}\mathrm{dx}$ is equal to

• $\frac{1}{2}\left(\sin \left(\mathrm{x}\right) + \cos \left(\mathrm{x}\right) + \mathrm{x} \right)$ + C

• $\frac{1}{2}\left(\sin \left(\mathrm{x}\right) - \cos \left(\mathrm{x}\right) - \mathrm{x} \right)$ + C

• $\frac{1}{2}\left(\cos \left(\mathrm{x}\right) - \mathrm{x} + \sin \left(\mathrm{x}\right) \right)$ + C

• None of the above