The general solution of the equation $\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{{\mathrm{y}}^2 - \mathrm{x}}{2\mathrm{y}\left(\mathrm{x} + 1\right)}$ is

• y² = (1 + x)log(1 + x) - c

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

• ${\mathrm{y}}^2 = \left(1 - \mathrm{x}\right)\log \left(\frac{\mathrm{c}}{\left(1 - \mathrm{x}\right)}\right) - 1$

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

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

• ${\log }_{\mathrm{e}}\left|\tan \left(\frac{\mathrm{y}}{2}\right)\right| = - 2\sin \left(\frac{\mathrm{x}}{2}\right) +\hspace{0.17em}\mathrm{C}$

• ${\log }_{\mathrm{e}}\left|\tan \left(\frac{\mathrm{y}}{4}\right)\right| = 2\sin \left(\frac{\mathrm{x}}{2}\right) +\hspace{0.17em}\mathrm{C}$

• ${\log }_{\mathrm{e}}\left(\tan \left(\frac{\mathrm{y}}{2}\right)\right) = - 2\sin \left(\frac{\mathrm{x}}{2}\right) +\hspace{0.17em}\mathrm{C}$

• ${\log }_{\mathrm{e}}\left(\tan \left(\frac{\mathrm{y}}{4}\right)\right) = - 2\sin \left(\frac{\mathrm{x}}{2}\right) +\hspace{0.17em}\mathrm{C}$

23.

If a, b, c are three vectors such that [a b c] = 5, then the value of [a x b, b x c, c x a] is

• 15

• 25

• 20

• 10

The point of inflection of the function y = ${\int }_0^{\mathrm{x}}\left({\mathrm{t}}^2 - 3\mathrm{t} + 2\right)\mathrm{dt}$ is

• $\left(\frac{3}{2}, \frac{3}{4}\right)$

• $\left(- \frac{3}{2}, - \frac{3}{4}\right)$

• $\left(- \frac{1}{2}, - \frac{3}{2}\right)$

• $\left(\frac{1}{2}, \frac{3}{2}\right)$

The value of integral $\int \frac{\mathrm{dx}}{\mathrm{x}\sqrt{{\mathrm{x}}^2 - {\mathrm{a}}^2}}$

• $\mathrm{c} - \frac{1}{\mathrm{a}}{\sin }^{-1}\left(\frac{\mathrm{a}}{\left|\mathrm{x}\right|}\right)$

• $\mathrm{c} - \frac{1}{\mathrm{a}}{\cos }^{-1}\left(\frac{\mathrm{a}}{\left|\mathrm{x}\right|}\right)$

• ${\sin }^{-1}\left(\frac{\mathrm{a}}{\left|\mathrm{x}\right|}\right) + \mathrm{c}$

• $\mathrm{c} + \frac{1}{\mathrm{a}}{\sin }^{-1}\left(\frac{\mathrm{a}}{\left|\mathrm{x}\right|}\right)$

The function y specified implicitly by the relation ${\int }_0^{\mathrm{y}}{\mathrm{e}}^{\mathrm{t}}\mathrm{dt} + {\int }_0^{\mathrm{x}}\cos \left(\mathrm{t}\right)\mathrm{dt}$ = 0 satisfies the differential equation

• ${\mathrm{e}}^{2\mathrm{y}}\left(\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} + {\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^2\right) = \sin \left(\mathrm{x}\right)$

• ${\mathrm{e}}^{\mathrm{y}}\left(\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} + {\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^2\right) = \sin \left(2\mathrm{x}\right)$

• ${\mathrm{e}}^{\mathrm{y}}\left(2\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} + {\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^2\right) = \sin \left(\mathrm{x}\right)$

• ${\mathrm{e}}^{\mathrm{y}}\left(\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} + {\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^2\right) = \sin \left(\mathrm{x}\right)$