81.

Suppose f is such that f(- x) = - f(x), for every x and ${\int }_0^1\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx} = 5$, then ${\int }_{- 1}^0\mathrm{f}\left(\mathrm{t}\right)\mathrm{dt}$ is equal to

• 10

• 5

• 0

• - 5

The general solution of the differential equation ydx + (1 + x²) tan^{- 1}(x)dy = 0, is

• ${\mathrm{ytan}}^{-1}\left(\mathrm{x}\right) = \mathrm{c}$

• ${\mathrm{xtan}}^{-1}\left(\mathrm{y}\right) = \mathrm{c}$

• $\mathrm{y} + {\tan }^{-1}\left(\mathrm{x}\right) = \mathrm{c}$

• $\mathrm{x} + {\tan }^{-1}\left(\mathrm{y}\right) = \mathrm{c}$

Solution of differential equation $\frac{\mathrm{dy}}{\mathrm{dx}} = 2\mathrm{xy}$ is

• y = ${\mathrm{ce}}^{{\mathrm{x}}^2}$

• y² = 2x² + c

• y = ${\mathrm{ce}}^{- {\mathrm{x}}^2}$

• y = x² + c

The second order differential equation is

• y' + x = y²

• y'y'' + y = sin(x)

• y''' + y'' + y = 0

• y' = y

The solution of the differential equation $\frac{\mathrm{dy}}{\mathrm{dx}} - \frac{\mathrm{y}}{\mathrm{x}} = 1$ is

• x²log_e(x) + y = c

• xlog_e(x) + cx = y

• x²log_e(x) - y = c

• xlog_e(x) + y = cx

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

• - x

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

• $\sqrt{\left(1 - {\mathrm{x}}^2\right)}$

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

Withthehelp of Trapezoidal rule fornumerical integration and the following table

the value of ${\int }_0^1\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}$ is

• 0.35342

• 0.34375

• 0.34457

• 0.33334

The maximum value of z = 4x + 2y subject to the constraints 2x + 3y $\le$ 18, x + y $\ge$ 10, x, y $\ge$ 0

• 36

• 40

• 20

• None of these

The maximum value of $\mathrm{\mu }$ = 3x + 4y, subject to the conditions x + y $\le$ 40, x + 2y $\le$ 60, x, y $\ge$ 0, is

• 130

• 140

• 40

• 120