If ${\int }_0^{\mathrm{a}}\frac{{\mathrm{x}}^2 - 1}{1 - \mathrm{x}}\mathrm{dx} = - \frac{1}{2}$, then the value of a is equal to

• - 1

• 1

• 2

• - 2

The value of the integral ${\int }_0^1\mathrm{x}{\left(1 - \mathrm{x}\right)}^5\mathrm{dx}$ is equal to

• $\frac{1}{6}$

• $\frac{1}{7}$

• $\frac{6}{7}$

• $\frac{1}{42}$

If [x] denotes the greatest integer less than or equal to x, then the value of ${\int }_0^2\left(\left|\mathrm{x} - 2\right| + \left[\mathrm{x}\right]\right)\mathrm{dx}$ is equal to

• 2

• 3

• 1

• 4

${\int }_0^1{\mathrm{xe}}^{- 5\mathrm{x}}\mathrm{dx}$ is equal to

• $\frac{1}{25} - \frac{6{\mathrm{e}}^{- 5}}{25}$

• $\frac{1}{25} + \frac{6{\mathrm{e}}^{- 5}}{25}$

• $- \frac{1}{25} - \frac{6{\mathrm{e}}^{- 5}}{25}$

• $\frac{1}{25} + \frac{1}{25}{\mathrm{e}}^{- 5}$

The area bounded by the curve y = sin(x) between x = 0 and x = 2$\mathrm{\pi }$ is (in square units)

• 1

• 2

• 0

• 4

The differential equation representing the family of curves y² = 2c (x + $\sqrt{\mathrm{c}}$) where c is a positive parameter, is of

• order 1, degree 2

• order 1, degree 3

• order 2, degree 3

• order 2, degree 2

27.

An integrating factor of the differential equation $\left(1 + {\mathrm{x}}^2\right)\frac{\mathrm{dy}}{\mathrm{dx}} + \mathrm{xy} = \mathrm{x}$ is

• $\frac{\mathrm{x}}{1 + {\mathrm{x}}^2}$

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

• $\sqrt{1 + {\mathrm{x}}^2}$

• x

The solution of the differential equation $\mathrm{x}\frac{\mathrm{dy}}{\mathrm{dx}} + \mathrm{y} = \frac{1}{{\mathrm{x}}^2}$ at (1, 2) is

• x²y + 1 = 3x

• x²y + 1 = 0

• xy + 1 = 3x

• x²(y + 1) = 3x

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

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

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

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

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

If a is perpendicular to b, then the vector $\mathrm{a} \times \left\{\mathrm{a} \times \left\{\mathrm{a} \times \left(\mathrm{a} \times \mathrm{b}\right)\right\}\right\}$ is equal to

• ${\left|\mathrm{a}\right|}^2\mathrm{b}$

• $\left|\mathrm{a}\right|\mathrm{b}$

• ${\left|\mathrm{a}\right|}^3\mathrm{b}$

• ${\left|\mathrm{a}\right|}^4\mathrm{b}$