31.

The value of $\tan \left[2{\tan }^{-1}\left(\frac{1}{5}\right) - \frac{\mathrm{\pi }}{4}\right]$ is

• $\frac{17}{7}$

• $\frac{- 17}{7}$

• $\frac{7}{17}$

• $\frac{- 7}{17}$

The solution of the equation $\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{1}{\mathrm{x} + \mathrm{y} + 1}$ is

• x + y = Ce^y - 2

• x + y = Clog(y) - 4

• log(x+ y + 2) = Cy

• log(x + y + 2) = C + y

Image point of (1, 3, 4) in the plane 2x - y + z + 3 = 0 will be

• (3, 5, 2)

• (3, 5, - 2)

• (- 3, 5, 2)

• None of these

The number of vectors of unit length perpendicular to vectors a = $\hat{\mathrm{i}} + \hat{\mathrm{j}}$ and b = $\hat{\mathrm{j}} + \hat{\mathrm{k}}$

• infinite

• one

• two

• three

By trapezoidal rule, the approximate value of the integral ${\int }_0^6\frac{\mathrm{dx}}{1 + {\mathrm{x}}^2}$ is

• 1.3128

• 1.4108

• 1.4218

• None of these

The solution of differential equation (ylog(x) - 1)ydx = xdy is

• $\mathrm{y}\left(\log \left({\mathrm{e}}^{\mathrm{x}}\right) + \mathrm{Cx}\right) = 1$

• $\left(\log \left(\frac{\mathrm{x}}{\mathrm{e}}\right) + \mathrm{Cx}\right)\mathrm{x} = \mathrm{y}$

• $\left(\log \left({\mathrm{Cx}}^2\right) + {\mathrm{ex}}^2\right)\mathrm{y} = \mathrm{x}$

• None of these

If m things are distributed among a men and b women. Then, the chance that the number of things received by men is odd, is

• $\frac{{\left(\mathrm{b} - \hspace{0.17em}\mathrm{a}\right)}^{\mathrm{m}} - {\left(\mathrm{b} + \mathrm{a}\right)}^{\mathrm{m}}}{2{\left(\mathrm{b} + \mathrm{a}\right)}^{\mathrm{m}}}$

• $\frac{{\left(\mathrm{b} + \hspace{0.17em}\mathrm{a}\right)}^{\mathrm{m}} - {\left(\mathrm{b} - \mathrm{a}\right)}^{\mathrm{m}}}{2{\left(\mathrm{b} + \mathrm{a}\right)}^{\mathrm{m}}}$

• $\frac{{\left(\mathrm{b} + \hspace{0.17em}\mathrm{a}\right)}^{\mathrm{m}} - {\left(\mathrm{b} - \mathrm{a}\right)}^{\mathrm{m}}}{{\left(\mathrm{b} + \mathrm{a}\right)}^{\mathrm{m}}}$

• None of these

The value of the integral $\mathrm{I} = \int \left(\sqrt{\tan \left(\mathrm{x}\right)} + \sqrt{\cot \left(\mathrm{x}\right)}\right)\mathrm{dx}, \mathrm{where} \mathrm{x} \in \left(0, \frac{\mathrm{\pi }}{2}\right)$, is

• $\sqrt{2}{\sin }^{-1}\left(\cos \left(\mathrm{x}\right) - \sin \left(\mathrm{x}\right)\right) + \mathrm{C}$

• $\sqrt{2}{\sin }^{-1}\left(\sin \left(\mathrm{x}\right) - \cos \left(\mathrm{x}\right)\right) + \mathrm{C}$

• $\sqrt{2}{\sin }^{-1}\left(\cos \left(\mathrm{x}\right) + \sin \left(\mathrm{x}\right)\right) + \mathrm{C}$

• $- \sqrt{2}{\sin }^{-1}\left(\sin \left(\mathrm{x}\right) + \cos \left(\mathrm{x}\right)\right) + \mathrm{C}$

The solution of the differential equation $\sqrt{\mathrm{a} + \mathrm{x}}\frac{\mathrm{dy}}{\mathrm{dx}} + \mathrm{xy} = 0$ is

• $\mathrm{y} = {\mathrm{Ce}}^{\frac{2}{3}\left(2\mathrm{a} - \mathrm{x}\right)\sqrt{\mathrm{x} + \mathrm{a}}}$

• $\mathrm{y} = {\mathrm{Ce}}^{\frac{2}{3}\left(\mathrm{a} - \mathrm{x}\right)\sqrt{\mathrm{x} + \mathrm{a}}}$

• $\mathrm{y} = {\mathrm{Ce}}^{\frac{2}{3}\left(2\mathrm{a} + \mathrm{x}\right)\sqrt{\mathrm{x} + \mathrm{a}}}$

• $\mathrm{y} = {\mathrm{Ce}}^{- \frac{2}{3}\left(2\mathrm{a} - \mathrm{x}\right)\sqrt{\mathrm{x} + \mathrm{a}}}$

If $\mathrm{a} = \frac{\hat{\mathrm{i}} - 2\hat{\mathrm{j}}}{\sqrt{5}} \mathrm{and} \mathrm{b} = \frac{2\hat{\mathrm{i}} + \hat{\mathrm{j}} + 3\hat{\mathrm{k}}}{\sqrt{14}}$ are vectors in space, then the value of $\left(2\mathrm{a} +\hspace{0.17em}\mathrm{b}\right) . \left[\left(\mathrm{a} \times \mathrm{b}\right) \times \left(\mathrm{a} - 2\mathrm{b}\right)\right]$ is

• 0

• 1

• 5

• 4