71.

The area (in square unit) of the region enclosed by the curves y = x and y = x³ is

• $\frac{1}{12}$

• $\frac{1}{6}$

• $\frac{1}{3}$

• 1

The differential equation obtained by eliminating the arbitrary constants a and b from xy = ae^x + be^{- x} is

• $\mathrm{x}\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} +\hspace{0.17em}2\frac{\mathrm{dy}}{\mathrm{dx}} - \mathrm{xy} = 0$

• $\mathrm{x}\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} +\hspace{0.17em}2\mathrm{y}\frac{\mathrm{dy}}{\mathrm{dx}} - \mathrm{xy} = 0$

• $\mathrm{x}\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} +\hspace{0.17em}2\frac{\mathrm{dy}}{\mathrm{dx}} + \mathrm{xy} = 0$

• $\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} +\hspace{0.17em}\frac{\mathrm{dy}}{\mathrm{dx}} - \mathrm{xy} = 0$

$\mathrm{The} \mathrm{solution} \mathrm{of} \left(\mathrm{x} + \mathrm{y} +\hspace{0.17em}1\right)\frac{\mathrm{dy}}{\mathrm{dx}} = 1 \mathrm{is}$

• y = (x + 2) + ce^x

• y = - (x + 2) + ce^x

• x = - (y + 2) + ce^y

• x = (y + 2)² + ce^y

$\mathrm{The} \mathrm{solution} \mathrm{of} \frac{\mathrm{dy}}{\mathrm{dx}} = \frac{{\mathrm{y}}^2}{\mathrm{xy} - {\mathrm{x}}^2} \mathrm{is}$

• ${\mathrm{e}}^{\raisebox{1ex}{$\mathrm{y}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{x}$}\right.} = \mathrm{kx}$

• ${\mathrm{e}}^{\raisebox{1ex}{$\mathrm{y}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{x}$}\right.} = \mathrm{ky}$

• ${\mathrm{e}}^{\raisebox{1ex}{$\mathrm{x}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{y}$}\right.} = \mathrm{kx}$

• ${\mathrm{e}}^{\raisebox{1ex}{$ - \mathrm{y}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{x}$}\right.} = \mathrm{ky}$

$\mathrm{The} \mathrm{solution} \mathrm{of} \frac{\mathrm{dy}}{\mathrm{dx}} +\hspace{0.17em}1 = {\mathrm{e}}^{\mathrm{x} +\hspace{0.17em}\mathrm{y}} \mathrm{is}$

• ${\mathrm{e}}^{ - \left(\mathrm{x} + \mathrm{y}\right)} +\hspace{0.17em}\mathrm{x} + \mathrm{c} = 0$

• ${\mathrm{e}}^{ - \left(\mathrm{x} + \mathrm{y}\right)} -\hspace{0.17em}\mathrm{x} + \mathrm{c} = 0$

• ${\mathrm{e}}^{ \left(\mathrm{x} + \mathrm{y}\right)} +\hspace{0.17em}\mathrm{x} + \mathrm{c} = 0$

• ${\mathrm{e}}^{ \left(\mathrm{x} + \mathrm{y}\right)} -\hspace{0.17em}\mathrm{x} + \mathrm{c} = 0$

The number of ways of arranging 8 men and 4 women around a circular table such that no two women can sit together is

• 8!

• 4!

• 8!4!

• $7!{}_8\mathrm{P}_4$

If a polygon of n sides has 275 diagonals, then n is equal to

• 25

• 35

• 20

• 15

If cos(A - B) = 3/5 and tan(A)tan(B) = 2, then which one of the following is true ?

• $\sin \left(\mathrm{A} + \mathrm{B}\right) = \frac{1}{5}$

• $\sin \left(\mathrm{A} + \mathrm{B}\right) = - \frac{1}{5}$

• $\cos \left(\mathrm{A} - \mathrm{B}\right) = \frac{1}{5}$

• $\cos \left(\mathrm{A} + \mathrm{B}\right) = - \frac{1}{5}$

If a, b, c are in AP, b - a, c - b and a are in GP, then a: b : c is

• 1 : 2 : 3

• 1 : 3 : 5

• 2 : 3 : 4

• 1 : 2 : 4

The angle of elevation of an object from a point P on the level ground is $\mathrm{\alpha }$. Moving d metres on the ground towards the object, the angle of elevation is found to be $\mathrm{\beta }$. Then the height (in metres) of the object is

• $\mathrm{dtan}\left(\mathrm{\alpha }\right)$

• $\mathrm{dcot}\left(\mathrm{\beta }\right)$

• $\frac{\mathrm{d}}{\cot \left(\mathrm{\alpha }\right) + \cot \left(\mathrm{\beta }\right)}$

• $\frac{\mathrm{d}}{\cot \left(\mathrm{\alpha }\right) - \cot \left(\mathrm{\beta }\right)}$