Let * be a binary operation defined on R by a * b = $\frac{\mathrm{a} + \mathrm{b}}{4}, \forall \mathrm{a}, \mathrm{b} \in \mathrm{R}$, then the operation * is

• commutative and associative

• commutative but not associative

• associative but not commutative

• neither associative nor commutative

2.

The value of ${\sin }^{-1}\left(\cos \left(\frac{53\mathrm{\pi }}{5}\right)\right)$ is

• $\frac{3\mathrm{\pi }}{5}$

• $\frac{- 3\mathrm{\pi }}{5}$

• $\frac{\mathrm{\pi }}{10}$

• $\frac{- \mathrm{\pi }}{10}$

If $3{\tan }^{-1}\left(\mathrm{x}\right) + {\cot }^{-1}\left(\mathrm{x}\right) = \mathrm{\pi }$, then x is equal to

• 0

• 1

• - 1

• 1/2

The simplified form of ${\tan }^{-1}\left(\frac{\mathrm{x}}{\mathrm{y}}\right) - {\tan }^{-1}\left(\frac{\mathrm{x} - \mathrm{y}}{\mathrm{x} + \mathrm{y}}\right)$ is equal to

• 0

• $\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.$

• $\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$

• $\mathrm{\pi }$

If x, y,z are all different and not equal to zero and $\left|\begin{array}{ccc}1+ \mathrm{x}& 1& 1\\ 1& 1 + \mathrm{y}& 1\\ 1& 1& 1 + \mathrm{z}\end{array}\right|$ = 0, then the value of x^-1 + y^-1 + z^-1 is equal to

• xyz

• x^-1y^-1z^-1

• - x - y - z

• - 1

If A is any square matrix of order 3 x 3, then $\left|3\mathrm{A}\right|$ is equal to

• $3\left|\mathrm{A}\right|$

• $\frac{1}{3}\left|\mathrm{A}\right|$

• $27\left|\mathrm{A}\right|$

• $9\left|\mathrm{A}\right|$

If y = ${\mathrm{e}}^{{\sin }^{-1}\left({\mathrm{t}}^2 - 1\right)} \mathrm{and} \mathrm{x} = {\mathrm{e}}^{{\sec }^{-1}\left(\frac{1}{{\mathrm{t}}^2 - 1}\right)}, \mathrm{then} \frac{\mathrm{dy}}{\mathrm{dx}}$ is equal to

• $\frac{\mathrm{x}}{\mathrm{y}}$

• $\frac{- \mathrm{y}}{\mathrm{x}}$

• $\frac{\mathrm{y}}{\mathrm{x}}$

• $\frac{- \mathrm{x}}{\mathrm{y}}$

If A = $\frac{1}{\mathrm{\pi }}\left[\begin{array}{cc}{\sin }^{-1}\left(\mathrm{\pi x}\right)& {\tan }^{-1}\left(\frac{\mathrm{x}}{\mathrm{\pi }}\right)\\ {\sin }^{-1}\left(\frac{{\displaystyle \mathrm{x}}}{{\displaystyle \mathrm{\pi }}}\right)& {\cot }^{-1}\left(\mathrm{\pi x}\right)\end{array}\right]$, B = $\frac{1}{\mathrm{\pi }}\left[\begin{array}{cc}- {\cos }^{-1}\left(\mathrm{\pi x}\right)& {\tan }^{-1}\left(\frac{\mathrm{x}}{\mathrm{\pi }}\right)\\ {\sin }^{-1}\left(\frac{{\displaystyle \mathrm{x}}}{{\displaystyle \mathrm{\pi }}}\right)& - {\tan }^{-1}\left(\mathrm{\pi x}\right)\end{array}\right]$, then A - B is equal to

• I

• 0

• 2I

• $\frac{1}{2}\mathrm{I}$

If x^y = e^{x - y}, then $\frac{\mathrm{dy}}{\mathrm{dx}}$ is equal to

• $\frac{\log \left(\mathrm{x}\right)}{\log \left(\mathrm{x} - \mathrm{y}\right)}$

• $\frac{{\mathrm{e}}^{\mathrm{x}}}{{\mathrm{x}}^{\mathrm{x} - \mathrm{y}}}$

• $\frac{\log \left(\mathrm{x}\right)}{{\left(1 + \log \left(\mathrm{x}\right)\right)}^2}$

• $\frac{1}{\mathrm{y}} - \frac{1}{\mathrm{x} - \mathrm{y}}$

If A is a matrix of order m x n and B is  a matrix, such that AB' and B'A are both defined, then the order of the matrix B is

• m x m

• n x n

• n x m

• m x n