$\frac{\mathrm{d}}{\mathrm{dx}}\left[{\tan }^{-1}\left(\frac{\sqrt{1 + {\mathrm{x}}^2} + \sqrt{1 - {\mathrm{x}}^2}}{\sqrt{1 + {\mathrm{x}}^2} - \sqrt{1 - {\mathrm{x}}^2}}\right)\right]$ is equal to

• $\frac{- \mathrm{x}}{\sqrt{1 - {\mathrm{x}}^4}}$

• $\frac{\mathrm{x}}{\sqrt{1 - {\mathrm{x}}^4}}$

• $\frac{- 1}{2\sqrt{1 - {\mathrm{x}}^4}}$

• $\frac{1}{2\sqrt{1 - {\mathrm{x}}^4}}$

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

• $\mathrm{y}\left[\frac{1}{2\mathrm{x}} + \frac{4}{2\mathrm{x} + 3} - \frac{1}{2\left(\mathrm{x} + 1\right)}\right]$

• $\mathrm{y}\left[\frac{1}{3\mathrm{x}} + \frac{4}{2\mathrm{x} + 3} - \frac{1}{2\left(\mathrm{x} + 1\right)}\right]$

• $\mathrm{y}\left[\frac{1}{3\mathrm{x}} + \frac{4}{2\mathrm{x} + 3} - \frac{1}{\mathrm{x} + 1}\right]$

• None of the above

If y = x^y, then $\mathrm{x}\left(1 - \mathrm{ylog}\left(\mathrm{x}\right)\right)\frac{\mathrm{dy}}{\mathrm{dx}}$ is equal to

• x²

• y²

• xy²

• xy

Rolle's theorem is not applicable for the function f(x) = $\left|\mathrm{x}\right|$, where x $\in$ [- 1, 1] because

• the function f(x) is not continuous in the interval [- 1, 1]

• the function f(x) is not differentiable in the interval (- 1, 1)

• $\mathrm{f}\left(- 1\right) \ne \mathrm{f}\left(1\right)$

• $\mathrm{f}\left(- 1\right) = \mathrm{f}\left(1\right) \ne 0$

If the function f(x) = x³ - 6ax² + 5x satisfies the conditions of Lagrange's Mean Value theorem for the interval [1, 2] and the tangent to the curve y = f(x) at x = 7/4 is parallel to the chord that join the points ofintersection of the curve with the ordinates x = 1 and x = 2 . Then, the value of a is

• $\frac{35}{16}$

• $\frac{35}{48}$

• $\frac{7}{16}$

• $\frac{5}{16}$

If 2a + 3b + 6c = 0, then atleast one root of the equation ax² + bx + c = 0, lies in the interval

• (0, 1)

• (1, 2)

• (2, 3)

• None of these

If equation y + $\frac{{\mathrm{y}}^3}{3} + \frac{{\mathrm{y}}^5}{5} + ...\infty = 2\left[\mathrm{x} + \frac{{\displaystyle {\mathrm{x}}^3}}{{\displaystyle 3}} + \frac{{\displaystyle {\mathrm{x}}^5}}{{\displaystyle 5}} + ...\infty \right]$, then value of y is

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

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

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

• None of these

968.

The value of 'c' of Rolle's theorem for f(x) = e^x sin(x) in [0, x] is given by

• $\frac{\mathrm{\pi }}{4}$

• $\frac{3\mathrm{\pi }}{4}$

• $\frac{5\mathrm{\pi }}{6}$

• $\frac{\mathrm{\pi }}{2}$

${\int }_{- \mathrm{a}}^{\mathrm{a}}\mathrm{x}\sqrt{\left({\mathrm{a}}^2 - {\mathrm{x}}^2\right)}\mathrm{dx}$ is equal to

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{8}$

• 0

If y = ${\tan }^{-1}\left(\sqrt{\frac{1 - \cos \left(\mathrm{x}\right)}{1 + \cos \left(\mathrm{x}\right)}}\right)$, then $\frac{\mathrm{dy}}{\mathrm{dx}}$ will be

• $\sin \left(\mathrm{x}\right)\cos \left(\mathrm{x}\right)$

• $\frac{\mathrm{\pi }}{2}$

• $\frac{1}{2}$

• $\frac{1}{1 + {\cos }^2\left(\mathrm{x}\right)}$