The value of f at x = 0 so that function $\mathrm{f}\left(\mathrm{x}\right) = \frac{2^{\mathrm{x}} - 2^{- \mathrm{x}}}{\mathrm{x}}$0, $\mathrm{x} \ne 0$ is continuous at x = 0, is

• 0

• log(2)

• 4

• log(4)

812.

If y = a^x . b^{2x - 1}, then $\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}$ is

• ${\mathrm{y}}^2\log \left({\mathrm{ab}}^2\right)$

• $\mathrm{y} . {\left(\log \left({\mathrm{ab}}^2\right)\right)}^2$

• y²

• $\mathrm{y} . {\left(\log \left({\mathrm{a}}^2\mathrm{b}\right)\right)}^2$

If f(x) = ${\sin }^{-1}\left(\frac{2\mathrm{x}}{1 + {\mathrm{x}}^2}\right)$, then f(x) is differentiable on

• [- 1, 1]

• R - {- 1, 1}

• R - (- 1, 1)

• None of these

The function f (x) = ${\mathrm{e}}^{- \left|\mathrm{x}\right|}$ is

• continuous everywhere but not differentiable at x = 0

• continuous and differentiable everywhere

• not continuous at x = 0

• None of the above

If y² = ax² + bx + c, where a, b, c are constants, then ${\mathrm{y}}^3\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}$ is equal to

• a constant

• a function of x

• a function of y

• a function of x and y both

The set of points where the function $\mathrm{f}\left(\mathrm{x}\right) = \left|\mathrm{x} - 1\right|{\mathrm{e}}^{\mathrm{x}}$ is differentiable, is

• R

• R - {1}

• R - {- 1}

• R - {0}

If $\mathrm{x} = \mathrm{\varphi }\left(\mathrm{t}\right), \mathrm{y} = \mathrm{\psi }\left(\mathrm{t}\right), \mathrm{then} \frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}$ is equal to

• $\frac{\mathrm{\varphi }\text{'}\mathrm{\psi }\text{'}\text{'} - \mathrm{\psi }\text{'}\mathrm{\varphi }\text{'}\text{'}}{{\left(\mathrm{\varphi }\text{'}\right)}^2}$

• $\frac{\mathrm{\varphi }\text{'}\mathrm{\psi }\text{'}\text{'} - \mathrm{\psi }\text{'}\mathrm{\varphi }\text{'}\text{'}}{{\left(\mathrm{\varphi }\text{'}\right)}^3}$

• $\frac{\mathrm{\varphi }\text{'}\text{'}}{\mathrm{\psi }\text{'}\text{'}}$

• $\frac{\mathrm{\psi }\text{'}\text{'}}{\mathrm{\varphi }\text{'}\text{'}}$

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

• 1

• - 1

• 0

• 2

If x = $\frac{1 - {\mathrm{t}}^2}{1 + {\mathrm{t}}^2}$ and y = $\frac{2\mathrm{at}}{1 +\hspace{0.17em}{\mathrm{t}}^2}$, then $\frac{\mathrm{dy}}{\mathrm{dx}}$ is equal to

• $\frac{\mathrm{a}\left(1 - {\mathrm{t}}^2\right)}{2\mathrm{t}}$

• $\frac{\mathrm{a}\left({\mathrm{t}}^2 - 1\right)}{2\mathrm{t}}$

• $\frac{\mathrm{a}\left({\mathrm{t}}^2 + 1\right)}{2\mathrm{t}}$

• $\frac{\mathrm{a}\left({\mathrm{t}}^2 - 1\right)}{\mathrm{t}}$

The value of $∆\log \left(\mathrm{f}\left(\mathrm{x}\right)\right) + {∆}^2\left(3^{\mathrm{x}}\right)$ is

• $\log \left[1 + \frac{∆\mathrm{f}\left(\mathrm{x}\right)}{\mathrm{f}\left(\mathrm{x}\right)}\right] + 4 . 3^{\mathrm{x}}$

• $\log \left[1 + \frac{∆\mathrm{f}\left(\mathrm{x}\right)}{\mathrm{f}\left(\mathrm{x}\right)}\right] + 3^{\mathrm{x}}$

• $\log \left[\frac{∆\mathrm{f}\left(\mathrm{x}\right)}{1 + \mathrm{f}\left(\mathrm{x}\right)}\right] + 4 . 3^{\mathrm{x}}$

• $\log \left[\frac{∆\mathrm{f}\left(\mathrm{x}\right)}{1 + \mathrm{f}\left(\mathrm{x}\right)}\right] + 3^{\mathrm{x}}$