If f(x) = $\left|\mathrm{x} - 2\right|$ and g(x) = f(f(x)), then $\frac{\mathrm{dg}\left(\mathrm{x}\right)}{\mathrm{dx}}$ for x > 10 is

• 1 or - 1

• 1

• - 1

• None of these

If y = ${\sin }^{-1}\left(\frac{\sqrt{1 + \mathrm{x}} + \sqrt{1 - \mathrm{x}}}{2}\right)$, then the value of $\frac{\mathrm{dy}}{\mathrm{dx}}$ at x = 0 is

• 1/2

• - 1/2

• 0

• None of these

The distance (in metre) travelled by a vehicle in time t (in seconds) is given by the equation s = 3t³ + 2t² + t + 1. The difference in the acceleration between t = 2 and t = 4 is

• 36 m/s²

• 38 m/s²

• 45 m/s²

• 46 m/s²

If y² = P(x) be a cubic polynomial, then $2\frac{\mathrm{d}}{\mathrm{dx}}\left({\mathrm{y}}^3\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}\right)$ is equal to

• P'''(x) + P'(x)

• P''(x)P'''(x)

• P(x)P'''(x)

• constant

The set of points where the functton f(x) = $\mathrm{x}\left|\mathrm{x}\right|$ is differentiable is

• $\left(- \infty , \infty \right)$

• $\left(- \infty , 0\right) \cup \left(0, \infty \right)$

• $\left(0, \infty \right)$

• $\left[0, \infty \right]$

If $$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right) = \left\{\frac{1 - \sin \left(\mathrm{x}\right)}{{\left(\mathrm{\pi } - 2\mathrm{x}\right)}^2} . \frac{\log \left(\sin \left(\mathrm{x}\right)\right)}{\log \left(1 + {\mathrm{\pi }}^2 - 4\mathrm{\pi x} + 4{\mathrm{x}}^2\right)}, \mathrm{x} \ne \frac{\mathrm{\pi }}{2} \\ \mathrm{k} , \mathrm{x} = \frac{\mathrm{\pi }}{2}\right\} \end{gathered}$$ is continuous at x = $\frac{\mathrm{\pi }}{2}$, then k is equal to

• $- \frac{1}{16}$

• $- \frac{1}{32}$

• $- \frac{1}{64}$

• $- \frac{1}{28}$

The greatest value of f(x) = (x + 1)^1/3 - (x - 1)^1/3 on [0, 1] is

• 1

• 2

• 3

• $\frac{1}{3}$

248.

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

• 1

• - 1

• 0

• None of these

If f(x) = $$\begin{gathered} \left\{{\mathrm{x}}^{\mathrm{p}}\cos \left(\frac{1}{\mathrm{x}}\right), \mathrm{x} \ne 0 \\ 0 , \mathrm{x} = 0\right\} \end{gathered}$$ is differentiable at x = 0, then

• p < 0

• 0 < p < 1

• p = 1

• p > 1

Let f(x) = $$\begin{gathered} \left\{5^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\mathrm{x}$}\right.}, \mathrm{x} < 0 \\ \mathrm{\lambda }\left[\mathrm{x}\right], \mathrm{x} \ge 0\right\} \mathrm{and} \mathrm{\lambda } \in \mathrm{R} \end{gathered}$$, then at x = 0

• f is discontinuous

• f is continuous only, if $\mathrm{\lambda } = 0$

• f is continuous only, whatever $\mathrm{\lambda }$ maybe

• None of the above