The maximum value of $\mathrm{f}\left(\mathrm{x}\right) = \frac{\log \left(\mathrm{x}\right)}{\mathrm{x}}\left(\mathrm{x} \ne 0, \mathrm{x} \ne 1\right)$ is

• e

• $\frac{1}{\mathrm{e}}$

• e²

• $\frac{1}{{\mathrm{e}}^2}$

2.

If g(x) is the inverse function of f(x) and$\mathrm{f}\text{'}\left(\mathrm{x}\right) = \frac{1}{1 +\hspace{0.17em}{\mathrm{x}}^4}$, then g'(x) is

• 1 + [g(x)]⁴

• 1 - [g(x)]⁴

• 1 + [f(x)]⁴

• $\frac{1}{1 + {\left[\mathrm{g}\left(\mathrm{x}\right)\right]}^4}$

The inverse of the matrix $\left[\begin{array}{ccc}1& 0& 0\\ 3& 3& 0\\ 5& 2& - 1\end{array}\right]$ is

• $- \frac{1}{3}\left[\begin{array}{ccc}- 3& 0& 0\\ 3& 1& 0\\ 9& 2& - 3\end{array}\right]$

• $- \frac{1}{3}\left[\begin{array}{ccc}- 3& 0& 0\\ 3& - 1& 0\\ - 9& - 2& 3\end{array}\right]$

• $- \frac{1}{3}\left[\begin{array}{ccc}3& 0& 0\\ 3& - 1& 0\\ - 9& - 2& 3\end{array}\right]$

• $- \frac{1}{3}\left[\begin{array}{ccc}- 3& 0& 0\\ - 3& - 1& 0\\ - 9& - 2& 3\end{array}\right]$

If the function f(x) = ${\left[\tan \left(\frac{\mathrm{\pi }}{4} + \mathrm{x}\right)\right]}^{\frac{1}{\mathrm{x}}}$ for $\mathrm{x} \ne 0$ is  = K for x = 0 continuous at x = 0, then K = ?

• e

• e^{- 1}

• e²

• e^{- 2}

For a invertible matrix A if A(adjA) = $\left[\begin{array}{cc}10& 0\\ 0& 10\end{array}\right]$ then $\left|\mathrm{A}\right|$ =

• 100

• - 100

• 10

• - 10

If x = f(t) and y = g(t) are differentiable functions of t, then $\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}$ is

• $\frac{\mathrm{f}\text{'}\left(\mathrm{t}\right) . \mathrm{g}\text{'}\text{'}\left(\mathrm{t}\right) - \mathrm{g}\text{'}\left(\mathrm{t}\right) . \mathrm{f}\text{'}\text{'}\left(\mathrm{t}\right)}{{\left[\mathrm{f}\text{'}\left(\mathrm{t}\right)\right]}^3}$

• $\frac{\mathrm{f}\text{'}\left(\mathrm{t}\right) . \mathrm{g}\text{'}\text{'}\left(\mathrm{t}\right) - \mathrm{g}\text{'}\left(\mathrm{t}\right) . \mathrm{f}\text{'}\text{'}\left(\mathrm{t}\right)}{{\left[\mathrm{f}\text{'}\left(\mathrm{t}\right)\right]}^2}$

• $\frac{\mathrm{g}\text{'}\left(\mathrm{t}\right) . \mathrm{f}\text{'}\text{'}\left(\mathrm{t}\right) - \mathrm{f}\text{'}\left(\mathrm{t}\right) . \mathrm{g}\text{'}\text{'}\left(\mathrm{t}\right)}{{\left[\mathrm{f}\text{'}\left(\mathrm{t}\right)\right]}^3}$

• $\frac{\mathrm{g}\text{'}\left(\mathrm{t}\right) . \mathrm{f}\text{'}\text{'}\left(\mathrm{t}\right) + \mathrm{f}\text{'}\left(\mathrm{t}\right) . \mathrm{g}\text{'}\text{'}\left(\mathrm{t}\right)}{{\left[\mathrm{f}\text{'}\left(\mathrm{t}\right)\right]}^3}$

If $\mathrm{\alpha } \mathrm{and} \mathrm{\beta }$ are roots of the equation x² + 5$\left|\mathrm{x}\right|$ - 6 = 0, then the value of $\left|{\tan }^{-1}\left(\mathrm{\alpha }\right) - {\tan }^{-1}\left(\mathrm{\beta }\right)\right|$ is

• $\frac{\mathrm{\pi }}{2}$

• 0

• $\mathrm{\pi }$

• $\frac{\mathrm{\pi }}{4}$

If the volume of spherical ball is increasing at the rate of 4$\mathrm{\pi }$ cm³/s, then the rate of change of its surface area when the volume is 288 $\mathrm{\pi }$ cm³, is

• $\frac{4}{3}\mathrm{\pi } {\mathrm{cm}}^2/\mathrm{s}$

• $\frac{2}{3}\mathrm{\pi } {\mathrm{cm}}^2/\mathrm{s}$

• $4\mathrm{\pi } {\mathrm{cm}}^2/\mathrm{s}$

• $2\mathrm{\pi } {\mathrm{cm}}^2/\mathrm{s}$

If f(x) = $$\begin{gathered} \left\{= \log {\left({\sec }^2\left(\mathrm{x}\right)\right)}^{{\cot }^2\left(\mathrm{x}\right)} \mathrm{for} \mathrm{x} \ne 0 \\ = \mathrm{K} \mathrm{for} \mathrm{x} = 0\right\} \end{gathered}$$ is continuous at x = 0, then K is

• e^{- 1}

• 1

• e

• 0

If the inverse of the matrix $\left[\begin{array}{ccc}\mathrm{\alpha }& 14& - 1\\ 2& 3& 1\\ 6& 2& 3\end{array}\right]$ does not exist, then the value of $\mathrm{\alpha }$ is

• 1

• - 1

• 0

• - 2