Let A and B are two square matrices such that AB = A and BA = B, then A² equals to :

• B

• A

• I

• O

$\left|\begin{array}{ccc}\mathrm{x} - 2& 2\mathrm{x} - 3& 3\mathrm{x} - 4\\ \mathrm{x} - 4& 2\mathrm{x} - 9& 3\mathrm{x} - 16\\ \mathrm{x} - 8& 2\mathrm{x} - 27& 3\mathrm{x} - 64\end{array}\right|$ = 0, then x is equal to :

• - 2

• 3

• - 4

• 4

If X = $\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]$, then Xⁿ, for n $\in$ , is equal to :

• 2^{n - 1}X

• n²X

• nX

• 2^{n + 1}X

If $\left[\begin{array}{cc}2& 1\\ 3& 2\end{array}\right]\mathrm{A}\left[\begin{array}{cc}- 3& 2\\ 5& - 3\end{array}\right] = \left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$, then the matrix A is equal to :

• $\left[\begin{array}{cc}1& 1\\ 1& 0\end{array}\right]$

• $\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]$

• $\left[\begin{array}{cc}1& 0\\ 1& 1\end{array}\right]$

• $\left[\begin{array}{cc}0& 1\\ 1& 1\end{array}\right]$

If a, b, c, d, e and f are in GP, then the value of $\left|\begin{array}{ccc}{\mathrm{a}}^2& {\mathrm{d}}^2& \mathrm{x}\\ {\mathrm{b}}^2& {\mathrm{e}}^2& \mathrm{y}\\ {\mathrm{c}}^2& {\mathrm{f}}^2& \mathrm{z}\end{array}\right|$ depends on :

• x and y

• x and z

• y and z

• independent of x, y and z

Let X = $\left[\begin{array}{c}\mathrm{x}\\ \mathrm{y}\\ \mathrm{z}\end{array}\right]$, D = $\left[\begin{array}{c}3\\ 5\\ 11\end{array}\right]$ and A = $\left[\begin{array}{ccc}1& - 1& - 2\\ 2& 1& 1\\ 4& - 1& - 2\end{array}\right]$, if X = A^{- 1}D, then X is equal to :

• $\left[\begin{array}{c}1\\ 0\\ 2\end{array}\right]$

• $\left[\begin{array}{c}\frac{8}{3}\\ \frac{- 1}{3}\\ 0\end{array}\right]$

• $\left[\begin{array}{c}\frac{- 8}{3}\\ 1\\ 0\end{array}\right]$

• $\left[\begin{array}{c}\frac{8}{3}\\ \frac{1}{3}\\ - 1\end{array}\right]$

If ${\sin }^{-1}\left(\frac{5}{\mathrm{x}}\right) + {\sin }^{-1}\left(\frac{12}{\mathrm{x}}\right) = \frac{\mathrm{\pi }}{2}$, then x is equal to :

• $\frac{7}{13}$

• $\frac{4}{3}$

• 13

• $\frac{13}{7}$

18.

The length of the longest size rectangle of maximum area that can be inscribed in a semicircle of radius 1, so that 2 vertices lie on the diameter, is :

• $\sqrt{2}$

• 2

• $\sqrt{3}$

• $\frac{\sqrt{2}}{3}$

The sum of the series ${\tan }^{-1}\left(\frac{1}{1 + 1 + 1^2}\right) + {\tan }^{-1}\left(\frac{1}{2 + 2 + 2^2}\right) + {\tan }^{-1}\left(\frac{1}{3 + 3 + 3^2}\right) + ... \infty$ is equal to :

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{6}$

The avlue of $\sec \left[{\tan }^{-1}\left(\frac{\mathrm{b} + \mathrm{a}}{\mathrm{b} - \mathrm{a}}\right) - {\tan }^{-1}\left(\frac{\mathrm{a}}{\mathrm{b}}\right)\right]$ is :

• 2

• $\sqrt{2}$

• 4

• 1