51.

Area bounded by the curve y = log (x - 2), x-axis and x = 4 is equal to

• 2log(2) + 1

• log(2) - 1

• log(2) + 1

• 2log(2) - 1

Let f(x) = x² - 2. If ${\int }_3^6\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx} = 3\mathrm{f}\left(\mathrm{c}\right)$ for some c $\in$ (3, 6),  then the value of c is equal to

• $\sqrt{12}$

• $\sqrt{21}$

• $\sqrt{19}$

• $\sqrt{17}$

${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\frac{\sin \left(2\mathrm{x}\right)}{1 + 2{\cos }^2\left(\mathrm{x}\right)}\mathrm{dx}$ is equal to

• $\frac{1}{2}\log \left(2\right)$

• $\log \left(2\right)$

• $\frac{1}{2}\log \left(3\right)$

• $\log \left(3\right)$

${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\frac{\mathrm{dx}}{1 +\hspace{0.17em}{\tan }^3\left(\mathrm{x}\right)}$ is equal to

• 1

• $\mathrm{\pi }$

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{4}$

The solution of the differential equation $\log \left(\mathrm{x}\right)\frac{\mathrm{dy}}{\mathrm{dx}} + \frac{\mathrm{y}}{\mathrm{x}} = \sin \left(2\mathrm{x}\right)$ is

• $\mathrm{ylog}\left|\mathrm{x}\right| = \mathrm{C} - \frac{1}{2}\cos \left(\mathrm{x}\right)$

• $\mathrm{ylog}\left|\mathrm{x}\right| = \mathrm{C} + \frac{1}{2}\cos \left(2\mathrm{x}\right)$

• $\mathrm{ylog}\left|\mathrm{x}\right| = \mathrm{C} - \frac{1}{2}\cos \left(2\mathrm{x}\right)$

• $\mathrm{xylog}\left|\mathrm{x}\right| = \mathrm{C} - \frac{1}{2}\cos \left(2\mathrm{x}\right)$

If xy = A sin(x) + B cos(x) is the solution of  the differential equation $\mathrm{x}\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} - 5\mathrm{a}\frac{\mathrm{dy}}{\mathrm{dx}} + \mathrm{xy} = 0$, then the value of a is

• $\frac{2}{5}$

• $\frac{5}{2}$

• $\frac{- 2}{5}$

• $\frac{- 5}{2}$

The solution of the differential equation $\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{3{\mathrm{e}}^{2\mathrm{x}} + 3{\mathrm{e}}^{4\mathrm{x}}}{{\mathrm{e}}^{\mathrm{x}} + {\mathrm{e}}^{- \mathrm{x}}}$ is

• $\mathrm{y} = {\mathrm{e}}^{3\mathrm{x}} + \mathrm{C}$

• $\mathrm{y} = 2{\mathrm{e}}^{2\mathrm{x}} + \mathrm{C}$

• $\mathrm{y} = {\mathrm{e}}^{\mathrm{x}} + \mathrm{C}$

• $\mathrm{y} = {\mathrm{e}}^{4\mathrm{x}} + \mathrm{C}$

The order and degree of the differential equation of the family of circles of fixed  radius r with centres on the y-axis, are respectively

• 2, 2

• 2, 3

• 1, 1

• 1, 2