If $\underset{\mathrm{x}\to 0}{\lim }\left(\frac{\cos \left(4\mathrm{x}\right) + \mathrm{acos}\left(2\mathrm{x}\right) + \mathrm{b}}{{\mathrm{x}}^4}\right)$ is finite, then the values of a, b are respectively :

• 5, - 4

• - 5, - 4

• - 4, 3

• 4, 5

$\mathrm{A} = \left[\begin{array}{ccc}1& 2& 2\\ 2& 1& 2\\ 2& 2& 1\end{array}\right] \mathrm{then} {\mathrm{A}}^3 - 4{\mathrm{A}}^2 - 6\mathrm{A} \mathrm{is} \mathrm{equal} \mathrm{to} :$

• 0

• A

• - A

• I

If A is an invertible matrix of order n, then the determinant of adj A is equal to : 

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

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

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

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

54.

$\left|\begin{array}{ccc}\log \left(\mathrm{e}\right)& \log \left({\mathrm{e}}^2\right)& \log \left({\mathrm{e}}^3\right)\\ \log \left({\mathrm{e}}^2\right)& \log \left({\mathrm{e}}^3\right)& \log \left({\mathrm{e}}^4\right)\\ \log \left({\mathrm{e}}^3\right)& \log \left({\mathrm{e}}^4\right)& \log \left({\mathrm{e}}^5\right)\end{array}\right| \mathrm{is} \mathrm{equal} \mathrm{to} :$

• 0

• 1

• 4log(e)

• 5log(e)

$\mathrm{If} {\mathrm{x}}^{\mathrm{y}} = {\mathrm{y}}^{\mathrm{x}}, \mathrm{then} \mathrm{x}\left(\mathrm{x} - \mathrm{ylog}\left(\mathrm{x}\right)\right)\frac{\mathrm{dy}}{\mathrm{dx}} \mathrm{is} \mathrm{equal} \mathrm{to} :$

• $\mathrm{y}\left(\mathrm{y} - \mathrm{xlog}\left(\mathrm{y}\right)\right)$

• $\mathrm{y}\left(\mathrm{y} + \mathrm{xlog}\left(\mathrm{y}\right)\right)$

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

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

$\mathrm{f}\left(\mathrm{x}\right) = {\mathrm{e}}^{\mathrm{x}}\sin \left(\mathrm{x}\right), \mathrm{then} {\mathrm{f}}^{\left(6\right)}\left(\mathrm{x}\right) = ?$

• ${\mathrm{e}}^{6\mathrm{x}}\sin \left(6\mathrm{x}\right)$

• $- 8{\mathrm{e}}^{\mathrm{x}}\cos \left(\mathrm{x}\right)$

• $8{\mathrm{e}}^{\mathrm{x}}\sin \left(\mathrm{x}\right)$

• $8{\mathrm{e}}^{\mathrm{x}}\cos \left(\mathrm{x}\right)$

$$\begin{gathered} \mathrm{If} \mathrm{f}\left(\mathrm{x}\right) = \left\{\frac{1 - \sqrt{2}\sin \left(\mathrm{x}\right)}{\mathrm{\pi } - 4\mathrm{x}} \mathrm{if} \mathrm{x} \ne \frac{\mathrm{\pi }}{4} \\ \mathrm{a} \mathrm{if} \mathrm{x} = \frac{\mathrm{\pi }}{4}\right\} \\ \mathrm{is} \mathrm{continuous} \mathrm{at} \frac{\mathrm{\pi }}{4}, \mathrm{then} \mathrm{a} \mathrm{is} \mathrm{equal} \mathrm{to} : \end{gathered}$$

• 4

• 2

• 1

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

If $\mathrm{\theta }$ is the angle between the curves xy = 2 and x² + 4y = 0 and x² + 4y = 0, then tan$\mathrm{\theta }$ is equal to :

• 1

• - 1

• 2

• 3

In the interval(- 3, 3) the function f(x) = $\frac{\mathrm{x}}{3} + \frac{3}{\mathrm{x}}, \mathrm{x} \ne 0$ is

• increasing

• decreasing

• neither increasing nor decreasing

• partly increasing and partly decreasing

The perimeter of a sector is a constant. If its area is to be maximum, the sectorial angle is :

• $\frac{{\mathrm{\pi }}^{\mathrm{c}}}{6}$

• $\frac{{\mathrm{\pi }}^{\mathrm{c}}}{4}$

• $4^{\mathrm{c}}$

• $2^{\mathrm{c}}$