The value of maxima of ${\left(\frac{1}{\mathrm{x}}\right)}^{\mathrm{x}}$ is

• ${\left(\frac{1}{\mathrm{e}}\right)}^{\mathrm{e}}$

• e^e

• e

• e^1/e

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

• $\frac{1}{\sqrt{5}}$

• $\sqrt{\frac{3}{10}}$

• $\sqrt{\frac{5}{2}}$

• $1 - \sqrt{\frac{5}{2}}$

If r² = x² + y² + z² and ${\tan }^{-1}\left(\frac{\mathrm{yz}}{\mathrm{xr}}\right) + {\tan }^{-1}\left(\frac{\mathrm{xz}}{\mathrm{yr}}\right) = \frac{\mathrm{\pi }}{2} - {\tan }^{-1}\left(\mathrm{\varphi }\right)$, then

• $\mathrm{\varphi } = \frac{\mathrm{zr}}{\mathrm{xy}}$

• $\mathrm{\varphi } = \frac{\mathrm{xy}}{\mathrm{zr}}$

• $\mathrm{\varphi } = \frac{\mathrm{x} + \mathrm{y}}{\mathrm{zr}}$

• $\mathrm{\varphi } = \frac{\mathrm{yz}}{\mathrm{xr}} + \frac{\mathrm{xz}}{\mathrm{yr}}$

4.

The value of determinant $\left|\begin{array}{ccc}\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\mathrm{a}$}\right.& \mathrm{bc}& {\mathrm{a}}^3\\ \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\mathrm{b}$}\right.& \mathrm{ca}& {\mathrm{b}}^3\\ \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\mathrm{c}$}\right.& \mathrm{ab}& {\mathrm{c}}^3\end{array}\right|$

• a²b²c²(a - b)(b - c)(c - a)

• 0

• (a - b)(b - c)(c - a)

• None of the above

If $\left[\begin{array}{cc}3& - 1\\ 0& 6\end{array}\right]\left[\begin{array}{c}3\mathrm{x}\\ 1\end{array}\right] + \left[\begin{array}{c}- 2\mathrm{x}\\ 3\end{array}\right] = \left[\begin{array}{c}8\\ 9\end{array}\right]$, then the value of x is

• $- \frac{3}{8}$

• 7

• $- \frac{2}{9}$

• None of these

Consider A and B two square matrices of same order. Select the correct alternative.

• $\left|\mathrm{AB}\right| \mathrm{must} \mathrm{be} \mathrm{greater} \mathrm{than} \left|\mathrm{A}\right|$

• $\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right] \mathrm{is} \mathrm{not} \mathrm{unit} \mathrm{matrix}$

• $\left|\mathrm{A} + \mathrm{B}\right| \mathrm{must} \mathrm{be} \mathrm{greater} \mathrm{than} \left|\mathrm{A}\right|$

• If AB = 0, either A or B must be zero matrix

Function f : N $\to$ N, f(x) = 2x + 3 is

• many-one onto function

• many-one into function

• one-one onto function

• one-one into function

If domain of the function f(x) = x² - 6x + 7 is ($- \infty , \infty$), then its range is

• [ - 2, 3]

• $\left(- \infty , 2\right)$

• $\left(- \infty , \infty \right)$

• $[ - 2, \infty )$

A point particle moves along a straight line such that x = $\sqrt{\mathrm{t}}$, where t is time. Then, ratio of acceleration to cube of the velocity is

• - 1

• - 0.5

• - 3

• - 2

If function f(x) = $$\begin{gathered} \left\{\mathrm{xsin}\left(\frac{1}{\mathrm{x}}\right); \mathrm{x} \ne 0 \\ \mathrm{a}; \mathrm{x} = 0\right\} \end{gathered}$$ is continuous at x = 0, then the value of a is

• 0

• 1

• - 1

• None of these