${\sin }^{-1}\left(\frac{1 + {\mathrm{x}}^2}{2\mathrm{x}}\right)$ is

• continuous but not differentiable at x = 1

• differentiable at x = 1

• neither continuous nor differentiable at x = 1

• continuous everywhere

If $\sqrt{{\mathrm{x}}^2 + {\mathrm{y}}^2} = {\mathrm{atan}}^{-1}\left(\frac{\mathrm{y}}{\mathrm{x}}\right), \mathrm{a} > 0, \mathrm{then} \frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}$ at x = 0 is

• 0

• $\frac{2}{\mathrm{a}}{\mathrm{e}}^{- \frac{\mathrm{\pi }}{2}}$

• $- \frac{2}{\mathrm{a}}{\mathrm{e}}^{- \frac{\mathrm{\pi }}{2}}$

• None of these

363.

Find C of Lagrange's mean value theorem for the function f(x) = 3x² + 5x + 7 in the interval [1, 3].

• $\frac{7}{3}$

• 2

• $\frac{3}{2}$

• $\frac{4}{3}$

For f(x) = (x - 1)^2/3, the mean value theorem is applicable to f(x) in the interval

• [2, 4]

• [0, 2]

• [- 2, 2]

• any finite interval

The differential coefficient of f(log(x)) with respect to x, where f(x) = log(x) is

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

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

• ${\left(\mathrm{xlog}\left(\mathrm{x}\right)\right)}^{- 1}$

• None of these

If f(x) = $$\begin{gathered} \left[\left[\mathrm{x}\right] + \left[- \mathrm{x}\right], \mathrm{x} \ne 2 \\ \mathrm{\lambda }, \mathrm{x} = 2\right] \end{gathered}$$, then f is continuous at x = 2, provided $\mathrm{\lambda }$ is equal to

• 1

• 0

• - 1

• 2

The derivative of ${\sec }^{-1}\left(\frac{1}{2{\mathrm{x}}^2 - 1}\right)$ with respect to $\sqrt{1 - {\mathrm{x}}^2}$ at x = 1/2 will be

• 1/4

• ${\sec }^{-1}\left(\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.\right)$

• 4

• 0

If $\mathrm{x} = {\mathrm{acos}}^3\left(\mathrm{\theta }\right), \mathrm{y} = {\mathrm{asin}}^3\left(\mathrm{\theta }\right), \mathrm{then} \sqrt{1 + {\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^2}$ is equal to

• ${\tan }^2\left(\mathrm{\theta }\right)$

• $\sec \left(\mathrm{\theta }\right)$

• ${\sec }^2\left(\mathrm{\theta }\right)$

• $\left|\sec \left(\mathrm{\theta }\right)\right|$

$\frac{\mathrm{d}}{\mathrm{dx}}{\cot }^{-1}\left(\mathrm{x}\right)$ is equal to

• $\frac{1}{1 + {\mathrm{x}}^2}$

• $\frac{- 1}{1 + {\mathrm{x}}^2}$

• $\frac{1}{\sqrt{1 + {\mathrm{x}}^2}}$

• $\frac{- 1}{\sqrt{1 + {\mathrm{x}}^2}}$

The value of $\sqrt{12}$. upto three places of decimals using the method of Newton-Raphson, will be

• 3.463

• 3.462

• 3.467

• None of these