41.

The integrating factor of the differential equation $\mathrm{x}\frac{\mathrm{dy}}{\mathrm{dx}} - \mathrm{y} = 2{\mathrm{x}}^2$ is

• $\frac{1}{\mathrm{x}}$

• x

• e^-x

• e^-y

By Simposon's rule taking n = 4, the value of the integral ${\int }_0^1\frac{\mathrm{dx}}{1 + {\mathrm{x}}^2}$ is equal to

• 0.785

• 0.788

• 0.781

• None of these

Thevalue of integral ${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\left(\sqrt{\tan \left(\mathrm{x}\right)} + \sqrt{\cot \left(\mathrm{x}\right)}\right)\mathrm{dx}$ is

• $\sqrt{2}\mathrm{\pi }$

• $\mathrm{\pi }$

• $\frac{\mathrm{\pi }}{2}$

• 0

The value of ${\int }_0^1\mathrm{xdx}$ by Trapezoidal rule taking x = 4 is

• 0.34375

• 0.5

• 0.38387

• 0.353367

The area of the figure bounded by the curves $\mathrm{y} = \left|\mathrm{x} - 1\right| \mathrm{and} \mathrm{y} = 3 - \left|\mathrm{x}\right|$ is

• 1 sq units

• 2 sq units

• 3 sq units

• 4 sq units

If $\frac{\mathrm{d}}{\mathrm{dx}}\mathrm{f}\left(\mathrm{x}\right) = 4{\mathrm{x}}^3 - \frac{3}{{\mathrm{x}}^4}$ such that f(2) = 0. Then, f(x) is

• ${\mathrm{x}}^3 + \frac{1}{{\mathrm{x}}^4} - \frac{129}{8}$

• ${\mathrm{x}}^4 + \frac{1}{{\mathrm{x}}^3} + \frac{129}{8}$

• ${\mathrm{x}}^3 + \frac{1}{{\mathrm{x}}^4} + \frac{129}{8}$

• ${\mathrm{x}}^4 + \frac{1}{{\mathrm{x}}^3} - \frac{129}{8}$

The order and degree of the differential equation ${\left(\frac{{\mathrm{d}}^3\mathrm{y}}{{\mathrm{dx}}^3}\right)}^2 - 3\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} + 2{\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^4 = {\mathrm{y}}^4$ are

• 1, 4

• 3, 4

• 2, 4

• 3, 2

If a = 2i - 3j + 6k and b = - 2i + 2i - k, then $\frac{\mathrm{Projection} \mathrm{of} \mathrm{a} \mathrm{on} \mathrm{b}}{\mathrm{Projection} \mathrm{of} \mathrm{b} \mathrm{on} \mathrm{a}}$ is equal to

• 1

• 7/3

• 3/7

• - 1/6

The solution of the differential equation (1 + y²) dx = (tan^-1((y) - x)dy is

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

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

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

• None of the above

The foot of perpendicular from the point (3, 4, 5) to the plane x + y + z = 9 is

• (2, 3, 4)

• (3, 5, - 2)

• (3, 5, 2)

• (3, 2, 4)