31.

$\int \frac{\mathrm{dx}}{\sin \left(\mathrm{x}\right) - \cos \left(\mathrm{x}\right) + \sqrt{2}}$ equals to

• $- \frac{1}{\sqrt{2}}\tan \left(\frac{\mathrm{x}}{2} + \frac{\mathrm{\pi }}{8}\right)$ + C

• $\frac{1}{2}\tan \left(\frac{\mathrm{x}}{2} + \frac{\mathrm{\pi }}{8}\right)$ + C

• $\frac{1}{\sqrt{2}}\cot \left(\frac{\mathrm{x}}{2} + \frac{\mathrm{\pi }}{8}\right)$ + C

• $- \frac{1}{\sqrt{2}}\cot \left(\frac{\mathrm{x}}{2} + \frac{\mathrm{\pi }}{8}\right)$ + C

The value of I = ${\int }_0^1\mathrm{x}\left|\mathrm{x} - \frac{1}{2}\right|\mathrm{dx}$

• $\frac{1}{3}$

• $\frac{1}{4}$

• $\frac{1}{8}$

• None of these

The value of integral ${\int }_0^1\sqrt{\frac{1 - \mathrm{x}}{1 +\hspace{0.17em}\mathrm{x}}}d\mathrm{x}$ is

• $\frac{\mathrm{\pi }}{2} + 1$

• $\frac{\mathrm{\pi }}{2} - 1$

• - 1

• 1

If a plane meets the coordinate axes at A, B and C in such a way that the centroid of $∆$ABC is at the point (1, 2, 3), then equation of the plane is

• $\frac{\mathrm{x}}{1} + \frac{\mathrm{y}}{2} + \frac{\mathrm{z}}{3} = 1$

• $\frac{\mathrm{x}}{3} + \frac{\mathrm{y}}{6} + \frac{\mathrm{z}}{9} = 1$

• $\frac{\mathrm{x}}{1} + \frac{\mathrm{y}}{2} + \frac{\mathrm{z}}{3} = \frac{1}{3}$

• None of these

Area lying in the first quadrant and bounded by the circle x² + y² = 4, the line x = √3y and x - axis is

• π sq units

• π/2 sq units

• π/3 sq units

• None of these

The general solution of the differential equation (1 + y²) dx + (1 + x)dy = 0 is

• x - y = C(1 - xy)

• x - y = C(1 + xy)

• x + y = C(1 - xy)

• x + y = C(1 + xy)

The order and degree of the differential equation $\mathrm{\rho } = \frac{{\left[1 + {\left({\displaystyle \frac{\mathrm{dy}}{\mathrm{dx}}}\right)}^2\right]}^{{\displaystyle \raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}{\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}}$ are, respectively

• 2, 2

• 2, 3

• 2, 1

• None of these

The solution of the differential equation

$\frac{\mathrm{dy}}{\mathrm{dx}} + \frac{2\mathrm{yx}}{1 +\hspace{0.17em}{\mathrm{x}}^2} = \frac{1}{{\displaystyle {\left(1 +\hspace{0.17em}{\mathrm{x}}^2\right)}^2}}$ is

• $\mathrm{y}\left(1 + {\mathrm{x}}^2\right) = \mathrm{C} + {\tan }^{-1}\left(\mathrm{x}\right)$

• $\frac{\mathrm{y}}{1 +\hspace{0.17em}{\mathrm{x}}^2} = \mathrm{C} + {\tan }^{-1}\left(\mathrm{x}\right)$

• $\mathrm{y} \log \left(1 + {\mathrm{x}}^2\right) = \mathrm{C} + {\tan }^{-1}\left(\mathrm{x}\right)$

• y(1 + x²) = C + ${\sin }^{-1}\left(\mathrm{x}\right)$

The probability that atleast one of the events A and B occurs is 0.6. If A and B occurs simultaneously with probability 0.2,then $\mathrm{P}\left(\overline{\mathrm{A}}\right) + \mathrm{P}\left(\overline{\mathrm{B}}\right)$ is

• 0.4

• 0.8

• 1.2

• 1.4

If 3p and 4p are resultant of force 5p,then angle between 3p and 5p is

• ${\sin }^{-1}\left(\frac{3}{5}\right)$

• ${\sin }^{-1}\left(\frac{4}{5}\right)$

• $90°$

• None of these