If $\mathrm{x} = {\mathrm{e}}^{\mathrm{t}}\sin \left(\mathrm{t}\right), \mathrm{y} = {\mathrm{e}}^{\mathrm{t}}\cos \left(\mathrm{t}\right)$, then $\frac{d^2\mathrm{y}}{d{\mathrm{x}}^2}$ at $\mathrm{x} = \mathrm{\pi }$ is

• $2{\mathrm{e}}^{\mathrm{\pi }}$

• $\frac{1}{2}{\mathrm{e}}^{\mathrm{\pi }}$

• $\frac{1}{2{\mathrm{e}}^{\mathrm{\pi }}}$

• $\frac{2}{{\mathrm{e}}^{\mathrm{\pi }}}$

The value of $\frac{d\mathrm{y}}{d\mathrm{x}} \mathrm{at} \mathrm{x} = \frac{\mathrm{\pi }}{2}$, where y is given by $\mathrm{y} = {\mathrm{x}}^{\sin \left(\mathrm{x}\right)} + \sqrt{\mathrm{x}}$, is

• $1 + \frac{1}{\sqrt{2\mathrm{\pi }}}$

• 1

• $\frac{1}{\sqrt{2\mathrm{\pi }}}$

• $1 - \frac{1}{\sqrt{2\mathrm{\pi }}}$

The value of ${\int }_0^{\mathrm{\pi }}\left|\cos \left(\mathrm{x}\right)\right|d\mathrm{x}$ is

• $2\mathrm{\pi }$

• 2

• $\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$\mathrm{\pi }$}\right.$

• $\mathrm{\pi }$

The value of ${\int }_{- 3}^3\left({\mathrm{ax}}^5 + {\mathrm{bx}}^3 + \mathrm{cx} + \mathrm{k}\right)d\mathrm{x}$ where a, b, c, k are constant, depends only on

• a and k

• a and b

• a, b and c

• k

The value of the integral ${\int }_{- \mathrm{a}}^{\mathrm{a}}\frac{{\mathrm{xe}}^{{\mathrm{x}}^2}}{1 + {\mathrm{x}}^2}d\mathrm{x}$ is

• ${\mathrm{e}}^{{\mathrm{a}}^2}$

• 0

• ${\mathrm{e}}^{- {\mathrm{a}}^2}$

• a

The order and degree of the following differential equation ${\left[1 + {\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^2\right]}^{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$2$}\right.} = \frac{d^3\mathrm{y}}{d{\mathrm{x}}^3}$ are respectively

• 3, 2

• 3, 10

• 2, 3

• 3, 5

77.

The differential equation of the family of circles passing through the fixed points (a, 0) and (- a, 0) is

• y₁(y² - x²) + 2xy + a² = 0

• y₁y² + xy + a²x² = 0

• y₁(y² - x² + a²) + 2xy = 0

• y₁(y² + x²) - 2xy + a² =  0

The differential equation of the family of curves y = e^2x(acos(x) + bsin(x)), where a and b are arbitrary constants, is given by

• y₂ - 4y₁ + 5y = 0

• 2y₂ - y₁ + 5y = 0

• y₂ + 4y₁ - 5y = 0

• y₂ - 2y₁ + 5y = 0

The area enclosed between the curve y = 1 + x², the y-axis and the straight line y = 5 is given by

• $\frac{14}{3}$ unit

• $\frac{7}{3}$ unit

• 5 sq unit

• $\frac{16}{3}$

A person draws out two balls successively from a bag containing 6 red and 4 white balls. The probability that at least one of them will be red, is

• $\frac{78}{90}$

• $\frac{30}{90}$

• $\frac{48}{90}$

• $\frac{12}{90}$