81.

The value of the integral ${\int }_{- \raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}\sin \left(- 4\mathrm{x}\right)d\mathrm{x}$ is

• $- \frac{8}{3}$

• $\frac{3}{2}$

• $\frac{8}{3}$

• None of these

The solution of the differential equation $\mathrm{xdy} - \mathrm{ydx} = \sqrt{{\mathrm{x}}^2 + {\mathrm{y}}^2}\mathrm{dx}$ is

• $\mathrm{x} + \sqrt{{\mathrm{x}}^2 + {\mathrm{y}}^2} = {\mathrm{cx}}^2$

• $\mathrm{y} - \sqrt{{\mathrm{x}}^2 + {\mathrm{y}}^2} = {\mathrm{cx}}^2$

• $\mathrm{x} - \sqrt{{\mathrm{x}}^2 + {\mathrm{y}}^2} = \mathrm{cx}$

• $\mathrm{y} + \sqrt{{\mathrm{x}}^2 + {\mathrm{y}}^2} = {\mathrm{cx}}^2$

The differential equation of all non-vertical lines in a plane is

• $\frac{d^2\mathrm{y}}{d{\mathrm{x}}^2} = 0$

• $\frac{d^2\mathrm{x}}{d{\mathrm{y}}^2} = 0$

• $\frac{d\mathrm{y}}{d\mathrm{x}} = 0$

• $\frac{d\mathrm{x}}{d\mathrm{y}} = 0$

$\int {\left(\frac{\mathrm{x} + 2}{\mathrm{x} + 4}\right)}^2{\mathrm{e}}^{\mathrm{x}}\mathrm{dx}$ is equal to

• ${\mathrm{e}}^{\mathrm{x}}\left(\frac{\mathrm{x}}{\mathrm{x} + 4}\right) +\hspace{0.17em}\mathrm{C}$

• ${\mathrm{e}}^{\mathrm{x}}\left(\frac{\mathrm{x} + 2}{\mathrm{x} + 4}\right) +\hspace{0.17em}\mathrm{C}$

• ${\mathrm{e}}^{\mathrm{x}}\left(\frac{\mathrm{x} - 2}{\mathrm{x} + 4}\right) +\hspace{0.17em}\mathrm{C}$

• ${\mathrm{e}}^{\mathrm{x}}\left(\frac{2{\mathrm{xe}}^{\mathrm{x}}}{\mathrm{x} + 4}\right) +\hspace{0.17em}\mathrm{C}$

The value of ${\int }_3^5\frac{{\mathrm{x}}^2}{{\mathrm{x}}^2 - 4}d\mathrm{x}$

• $2 - {\log }_{\mathrm{e}}\left(\frac{15}{7}\right)$

• $2 + {\log }_{\mathrm{e}}\left(\frac{15}{7}\right)$

• 2 + 4${\log }_{\mathrm{e}}\left(3\right) - 4{\log }_{\mathrm{e}}\left(7\right) + 4{\log }_{\mathrm{e}}\left(5\right)$

• $2 - {\tan }^{-1}\left(\frac{5}{7}\right)$

${\int }_0^{\infty }\frac{d\mathrm{x}}{{\left(\mathrm{x} +\hspace{0.17em}\sqrt{{\mathrm{x}}^2 +\hspace{0.17em}1}\right)}^3}$ is equal to

• $\frac{3}{8}$

• $\frac{1}{8}$

• $\frac{- 3}{8}$

• None of these

A common tangent to 9x² - 16y² = 144 and x² + y² = 9 is

• $\mathrm{y} = \frac{3}{\sqrt{7}}\mathrm{x} + \frac{15}{\sqrt{7}}$

• $\mathrm{y} = 3\sqrt{\frac{2}{7}}\mathrm{x} + \frac{15}{\sqrt{7}}$

• $\mathrm{y} = 2\sqrt{\frac{3}{7}}\mathrm{x} + 15\sqrt{7}$

• None of these

The differential equation of all parabolas whose axes are parallel to y-axis is

• $\frac{d^3\mathrm{y}}{d{\mathrm{x}}^3} = 0$

• $\frac{d^2\mathrm{x}}{d{\mathrm{y}}^2} = 0$

• $\frac{d^3\mathrm{y}}{d{\mathrm{x}}^3}+ \frac{d^2\mathrm{x}}{d{\mathrm{y}}^2} = 0$

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

The locus of a point P which moves such that 2PA = 3PB, where A(0, 0) and B(4,- 3) are points, is

• 5x² - 5y² - 72x + 54y + 225 = 0

• 5x² + 5y² - 72x + 54y + 225 = 0

• 5x² + 5y² + 72x - 54y + 225 = 0

• 5x² + 5y² - 72x - 54y - 225 = 0

The points P is equidistant from A(1, 3), B (- 3, 5) and C(5, - 1), then PA is equal to

• 5

• $5\sqrt{5}$

• 25

• $5\sqrt{10}$