71.

The equation of one of the curves whose slope at any point is equal to y + 2x is

• y = 2(e^x + x - 1)

• y = 2(e^x - x - 1)

• y = 2(e^x - x + 1)

• y = 2(e^x + x + 1)

Solution of the differential equation xdy - ydx = 0 represents a

• parabola

• circle

• hyperbola

• straight line

The value of integral ${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{\sin }^5\left(\mathrm{x}\right)d\mathrm{x}$ is

• $\frac{4}{15}$

• $\frac{8}{5}$

• $\frac{8}{15}$

• $\frac{4}{5}$

If $\frac{d}{d\mathrm{x}}\left\{\mathrm{f}\right(\mathrm{x}\left)\right\} = \mathrm{g}\left(\mathrm{x}\right), \mathrm{then} {\int }_{\mathrm{a}}^{\mathrm{b}}\mathrm{f}\left(\mathrm{x}\right)\mathrm{g}\left(\mathrm{x}\right)\mathrm{dx}$ is equal to

• $\frac{1}{2}\left[{\mathrm{f}}^2\left(\mathrm{b}\right) - {\mathrm{f}}^2\left(\mathrm{a}\right)\right]$

• $\frac{1}{2}\left[{\mathrm{g}}^2\left(\mathrm{b}\right) - {\mathrm{g}}^2\left(\mathrm{a}\right)\right]$

• f(b) - f(a)

• $\frac{1}{2}\left[\mathrm{f}\left({\mathrm{b}}^2\right) - \mathrm{f}\left({\mathrm{a}}^2\right)\right]$

If I₁ = ${\int }_0^{3\mathrm{\pi }}\mathrm{f}\left({\cos }^2\left(\mathrm{x}\right)\right)d\mathrm{x}$

and I₂$= {\int }_0^{\mathrm{\pi }}\mathrm{f}\left({\cos }^2\left(\mathrm{x}\right)\right)d\mathrm{x}$, then

• I₁ = I₂

• 3I₁ = I₂

• I₁ = 3I₂

• I₁ = 5I₂

The value of I = ${\int }_{- \raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\left|\sin \left(\mathrm{x}\right)\right|d\mathrm{x}$ is

• 0

• 2

• - 2

• - 2 < 1 < 2

If I = ${\int }_0^1\frac{\mathrm{dx}}{1 +\hspace{0.17em}{\mathrm{x}}^{{\displaystyle \raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}$, then

• ${\log }_{\mathrm{e}}\left(2\right) < 1 < \raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.$

• ${\log }_{\mathrm{e}}\left(2\right) > 1$

• $\mathrm{I} = \raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.$

• $\mathrm{I} = {\log }_{\mathrm{e}}\left(2\right)$

The area enclosed by y = 3x - 5y = 0, x = 3 and x = 5 is

• 12 sq units

• 13 sq units

• $13\frac{1}{2}$ sq units

• 14 sq units

The area bounded by the parabolas y = 4x², $\mathrm{y} = \frac{{\mathrm{x}}^2}{9}$ and the line y = 2 is

• $\frac{5\sqrt{2}}{3}$ sq units

• $\frac{10\sqrt{2}}{3}$ sq units

• $\frac{15\sqrt{2}}{3}$sq units

• $\frac{20\sqrt{2}}{3}$ sq units

In a triangle PQR, $\angle \mathrm{PQR}, \angle \mathrm{R} = . \mathrm{If} \tan \left(\frac{\mathrm{P}}{2}\right) \mathrm{and} \tan \left(\frac{\mathrm{Q}}{2}\right)$ are roots of ax² + bx + c = 0, where a$\ne$ 0, then which one is true?

• c = a + b

• a = b + c

• b = a + c

• b = c