Let f : R $\to$ R be a differentiable function and f(1) = 4. Then the value of $\underset{\mathrm{x}\to 1}{\lim }{\int }_4^{\mathrm{f}\left(\mathrm{x}\right)}\frac{2\mathrm{t}}{\mathrm{x} - 1}\mathrm{dt}$, if f'(1) = 2 is :

• 16

• 8

• 4

• 2

The solution of $\frac{\mathrm{dy}}{\mathrm{dx}}$ + y tan(x) = sec(x) is :

• $\mathrm{ysec}\left(\mathrm{x}\right) = \tan \left(\mathrm{x}\right) + \mathrm{c}$

• $\mathrm{ytan}\left(\mathrm{x}\right) = \sec \left(\mathrm{x}\right) + \mathrm{c}$

• $\tan \left(\mathrm{x}\right) = \mathrm{ytan}\left(\mathrm{x}\right) + \mathrm{c}$

• $\mathrm{xsec}\left(\mathrm{x}\right) = \tan \left(\mathrm{y}\right) + \mathrm{c}$

The solution of $\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\mathrm{ax} + \mathrm{h}}{\mathrm{by} + \mathrm{k}}$ represents a parabola, when :

• a = 0, b = 0

• a = 1, b = 2

• a = 0, b $\ne$ 0

• a = 2, b = 1

An integrating factor of the differential equation $\mathrm{x}\frac{\mathrm{dy}}{\mathrm{dx}} + \mathrm{ylog}\left(\mathrm{x}\right) = {\mathrm{xe}}^{\mathrm{x}}{\mathrm{x}}^{\frac{1}{2}\log \left(\mathrm{x}\right)}$, (x > 0) is :

• x^log(x)

• ${\left(\sqrt{\mathrm{x}}\right)}^{\log \left(\mathrm{x}\right)}$

• ${\left(\sqrt{\mathrm{e}}\right)}^{{\left(\log \left(\mathrm{x}\right)\right)}^2}$

• ${\mathrm{e}}^{{\mathrm{x}}^2}$

The solution of ${\mathrm{e}}^{\raisebox{1ex}{$\mathrm{dy}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{dx}$}\right.} = \left(\mathrm{x} + 1\right)$, y(0) = 3 is :

• y = xlog(x) - x + 2

• y = (x + 1)$\log \left|\mathrm{x} + 1\right| - \mathrm{x} + 3$

• $\mathrm{y} = \left(\mathrm{x} + 1\right)\log \left|\mathrm{x} + 1\right| + \mathrm{x} + 3$

• y = $\mathrm{xlog}\left(\mathrm{x}\right) + \mathrm{x} + 3$

Solution of the differential equation $\frac{\mathrm{dy}}{\mathrm{dx}}\tan \left(\mathrm{y}\right) = \sin \left(\mathrm{x} + \mathrm{y}\right) + \sin \left(\mathrm{x} - \mathrm{y}\right)$ is :

• $\sec \left(\mathrm{y}\right) + 2\cos \left(\mathrm{x}\right) = \mathrm{c}$

• $\sec \left(\mathrm{y}\right) - 2\cos \left(\mathrm{x}\right) = \mathrm{c}$

• $\cos \left(\mathrm{y}\right) - 2\sin \left(\mathrm{x}\right) = \mathrm{c}$

• $\tan \left(\mathrm{y}\right) - 2\sec \left(\mathrm{y}\right) = \mathrm{c}$

Solution of the differential equation $\frac{\mathrm{dy}}{\mathrm{dx}} + \frac{\mathrm{y}}{\mathrm{x}} = \sin \left(\mathrm{x}\right)$ is :

• $\mathrm{x}\left(\mathrm{y} + \cos \left(\mathrm{x}\right)\right) = \sin \left(\mathrm{x}\right) + \mathrm{c}$

• $\mathrm{x}\left(\mathrm{y} - \cos \left(\mathrm{x}\right)\right) = \sin \left(\mathrm{x}\right) + \mathrm{c}$

• $\mathrm{x}\left(\mathrm{ycos}\left(\mathrm{x}\right)\right) = \sin \left(\mathrm{x}\right) + \mathrm{c}$

• $\mathrm{x}\left(\mathrm{y} - \cos \left(\mathrm{x}\right)\right) = \cos \left(\mathrm{x}\right) + \mathrm{c}$

The solution of the differential equation $\mathrm{x}\frac{\mathrm{dy}}{\mathrm{dx}} + 2\mathrm{y} = {\mathrm{x}}^2$ is :

• $\mathrm{y} = \frac{{\mathrm{x}}^2 + \mathrm{c}}{4{\mathrm{x}}^2}$

• $\mathrm{y} = \frac{{\mathrm{x}}^2}{4} + \mathrm{c}$

• $\mathrm{y} = \frac{{\mathrm{x}}^2 + \mathrm{c}}{{\mathrm{x}}^2}$

• $\mathrm{y} = \frac{{\mathrm{x}}^4 + \mathrm{c}}{4{\mathrm{x}}^2}$

509.

y = - A cos(5x) + B sin(5x) satisfies the differential equation :

• $\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} + 10\frac{\mathrm{dy}}{\mathrm{dx}} + 25\mathrm{y} = 0$

• $\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} - 10\frac{\mathrm{dy}}{\mathrm{dx}} + 25\mathrm{y} = 0$

• $\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} - 25\mathrm{y} = 0$

• $\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} + 25\mathrm{y} = 0$

The order and degree of the differential equation $\sqrt{\sin \left(\mathrm{x}\right)}\left(\mathrm{dx} + \mathrm{dy}\right) = \sqrt{\cos \left(\mathrm{x}\right)}\left(\mathrm{dx} - \mathrm{dy}\right)$ is :

• (1, 2)

• (2, 2)

• (1, 1)

• (2, 1)