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

• 10

• 5

• 0

• - 5

If f(y) = e^y, g(y) = y, y > 0 and F(t) = ${\int }_0^{\mathrm{t}}\mathrm{f}\left(\mathrm{t} - \mathrm{y}\right) . \mathrm{g}\left(\mathrm{y}\right)\mathrm{dy}$, then 

• F(t) = 1 - e^{- t}(1 + t)

• F(t) = e^t - (1 + t)

• F(t) = te^t

• F(t) = te^{- t}

The area bounded by the X - axis, the curve y = f(x) and the lines x = 1, x = b and is equal to $\sqrt{{\mathrm{b}}^2 + 1} - \sqrt{2}$ for all b > 1, then f(x) is

• $\sqrt{\mathrm{x} - 1}$

• $\sqrt{\mathrm{x} + 1}$

• $\sqrt{{\mathrm{x}}^2 + 1}$

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

Let f(x) be a function satisfying f'(x) = f(x) with f(0) = 1 and g(x) be a function that satisfies f(x) + g(x) = x². Then, the value of the integeral ${\int }_0^1\mathrm{f}\left(\mathrm{x}\right)\mathrm{g}\left(\mathrm{x}\right)\mathrm{dx}$ is

• $\mathrm{e} - \frac{{\mathrm{e}}^2}{2} - \frac{5}{2}$

• $\mathrm{e} + \frac{{\mathrm{e}}^2}{2} - \frac{3}{2}$

• $\mathrm{e} - \frac{{\mathrm{e}}^2}{2} - \frac{3}{2}$

• $\mathrm{e} + \frac{{\mathrm{e}}^2}{2} + \frac{5}{2}$

${\int }_{- 1}^0\frac{\mathrm{dx}}{{\mathrm{x}}^2 + 2\mathrm{x} + 2}$ is equal to

• 0

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{2}$

• $\frac{- \mathrm{\pi }}{4}$

Family y = Ax + A³ ofcurve is represented by the differential equation ofdegree

• 3

• 2

• 1

• None of these

The degree and order of the differential equation of the family of all parabolas whose axis is X-axis, are respectively

• 2, 1

• 1, 2

• 3, 2

• 2, 3

Solution of the differential equation ydx - xdy = x²ydx

• ${\mathrm{ye}}^{{\mathrm{x}}^2} = {\mathrm{cx}}^2$

• ${\mathrm{ye}}^{- {\mathrm{x}}^2} = {\mathrm{cx}}^2$

• ${\mathrm{y}}^2{\mathrm{e}}^{{\mathrm{x}}^2} = {\mathrm{cx}}^2$

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

39.

The solution of the differential equation $\left(1 + {\mathrm{y}}^2\right) + \left(\mathrm{x} - {\mathrm{e}}^{{\tan }^{-1}\left(\mathrm{y}\right)}\right)\frac{d\mathrm{y}}{d\mathrm{x}} = 0$ is

• $\left(\mathrm{x} - 2\right) = {\mathrm{e}}^{2{\tan }^{-1}\left(\mathrm{y}\right)} + \mathrm{c}$

• $2{\mathrm{xe}}^{{\tan }^{-1}\left(\mathrm{y}\right)} = {\mathrm{e}}^{2{\tan }^{-1}\left(\mathrm{y}\right)} + \mathrm{c}$

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

• ${\mathrm{xe}}^{2{\tan }^{-1}\left(\mathrm{y}\right)} = {\mathrm{e}}^{{\tan }^{-1}\left(\mathrm{y}\right)} + \mathrm{c}$

Solution of the differential equation (x + y - 1)dx + (2x+ 2y - 3)dy = 0

• y + x + log (x + y - 2) = c

• y + 2x + log (x + y - 2) = c

• 2y + x + log (x + y - 2) = c

• 2y + 2x + log (x + y - 2) = c