31.

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

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

• $\frac{3}{2}$

• $\frac{4}{3}$

The length of the perpendicular from the point (1 2, 3) on the line $\frac{\mathrm{x} - 6}{3} = \frac{\mathrm{y} - 7}{2} = \frac{\mathrm{z} - 7}{- 2}$ is

• 3 units

• 4 units

• 5 units

• 7 units

The equation of the plane passing through the intersection ofthe planes 2x - 3y + z - 4 = 0 and x - y + z + 1 = 0 and perpendicular to the plane x + 2y - 3z + 6 = 0 is

• x - 5y + 3z - 23 = 0

• x - 5y - 3z - 23 = 0

• x + 5y - 3z + 23 = 0

• x - 5y + 3z + 23 = 0

The angle between the lines $\frac{\mathrm{x} - 2}{3} = \frac{\mathrm{y} + 1}{- 2} = \mathrm{z} = 2$ and $\frac{{\displaystyle \mathrm{x} - 1}}{{\displaystyle 1}} = \frac{{\displaystyle \mathrm{y} + 3}}{{\displaystyle 3}} = \frac{{\displaystyle \mathrm{z} + 5}}{2}$ is

• ${\cos }^{-1}\left(\frac{- 3}{\sqrt{182}}\right)$

• ${\cos }^{-1}\left(\frac{5}{\sqrt{182}}\right)$

• ${\cos }^{-1}\left(\frac{3}{\sqrt{182}}\right)$

• ${\cos }^{-1}\left(\frac{- 5}{\sqrt{182}}\right)$

The scalar A · {(B + C) x (A + B + C)} equals

• [ABC][BCA]

• [ABC]

• 0

• None of these

The points with position vectors 60i + 3j, 40i - 8 j and ai - 52j are collinear, if

• a = - 40

• a = 40

• a = - 20

• a = 20

$\int \frac{\mathrm{dx}}{\mathrm{x}\left({\mathrm{x}}^{\mathrm{n}} + 1\right)}$ is equal to

• $\frac{1}{\mathrm{n}}{\log }_{\mathrm{e}}\left(\frac{{\mathrm{x}}^{\mathrm{n}}}{{\mathrm{x}}^{\mathrm{n}} + 1}\right) + \mathrm{C}$

• $- \frac{1}{\mathrm{n}}{\log }_{\mathrm{e}}\left(\frac{{\mathrm{x}}^{\mathrm{n}} + 1}{{\mathrm{x}}^{\mathrm{n}}}\right) + \mathrm{C}$

• $\frac{1}{\mathrm{n}}{\log }_{\mathrm{e}}\left(\frac{{\mathrm{x}}^{\mathrm{n}}}{{\mathrm{x}}^{\mathrm{n}} + 1}\right) + \mathrm{C}$

• None of the above

Evaluate $\int \frac{\mathrm{dx}}{\mathrm{x}\left({\mathrm{x}}^5 + 2\right)}$

• $\frac{1}{10}\log \left|\frac{{\mathrm{x}}^5 + 1}{{\mathrm{x}}^5 + 2}\right| + \mathrm{C}$

• $\frac{1}{5}\log \left|\frac{{\mathrm{x}}^5}{{\mathrm{x}}^5 + 2}\right| + \mathrm{C}$

• $\frac{1}{10}\log \left|\frac{{\mathrm{x}}^5}{{\mathrm{x}}^5 + 2}\right| + \mathrm{C}$

• $\frac{1}{10}\log \left|\frac{{\mathrm{x}}^5 + 1}{{\mathrm{x}}^5 + 2}\right| + \mathrm{C}$