31.

$$\begin{gathered} \mathrm{If} \mathrm{f}\left(\mathrm{x}\right) = \left\{{\mathrm{x}}^2, \mathrm{x} \le 0 \\ 2 \sin \left(\mathrm{x}\right), \mathrm{x} > 0\right\}, \mathrm{then} \mathrm{x} = 0 \mathrm{is} \end{gathered}$$

• point of minima

• point of maxima

• point of discontinuity

• None of the above

The system of equations 2x + y - 5 = 0, x - 2y + 1 = 0, 2x - 14y - a= 0, is consistent. Then, a is equal to

• 1

• 2

• 5

• None of these

Evaluate $\int \frac{{\mathrm{x}}^2 + 4}{{\mathrm{x}}^4 + 16}\mathrm{dx}$.

• $\frac{1}{2\sqrt{2}}{\tan }^{-1}\left(\frac{{\mathrm{x}}^2 - 4}{2\mathrm{x}\sqrt{2}}\right) + \mathrm{C}$

• $\frac{1}{2\sqrt{2}}{\tan }^{-1}\left(\frac{{\mathrm{x}}^2 - 4}{2\sqrt{2}}\right) + \mathrm{C}$

• $\frac{1}{2\sqrt{2}}{\tan }^{-1}\left(\frac{{\mathrm{x}}^2 - 4}{\mathrm{x}\sqrt{2}}\right) + \mathrm{C}$

• None of these

Evaluate ${\int }_{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}^{\raisebox{1ex}{$3\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}\frac{1}{1 + \cos \left(\mathrm{x}\right)}\mathrm{dx}$

• 2

• - 2

• 1/2

• - 1/2

The projection of the vector i - 2j + k on the vector 4i - 4j + 7k is

• $\frac{5\sqrt{6}}{10}$

• $\frac{19}{9}$

• $\frac{9}{19}$

• $\frac{\sqrt{6}}{19}$

If a, b, c are three non-zero vectors such that a + b+ c = 0 and m = a . b + b . c + c · a, then

• m < 0

• m > 0

• m = 0

• m = 3

If X is a poisson variate such that P (X = 1) = P (X = 2), then P (X = 4) is equal to

• $\frac{1}{2{\mathrm{e}}^2}$

• $\frac{1}{3{\mathrm{e}}^2}$

• $\frac{2}{3{\mathrm{e}}^2}$

• $\frac{1}{{\mathrm{e}}^2}$

The area enclosed by y = 3x - 5, y = 0, x = 3 and x = 5 is

• 12 sq units

• 13 sq units

• $13\frac{1}{2}$ sq units

• 14 sq units

The order and degree of the differential equation ${\left(1 + 4\frac{\mathrm{dy}}{\mathrm{dx}}\right)}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.} = 4\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}$ are respectively

• $1, \frac{2}{3}$

• 3, 2

• 2, 3

• $2, \frac{2}{3}$

The solution of the differential equation $\frac{\mathrm{dy}}{\mathrm{dx}} = {\left(4\mathrm{x} + \mathrm{y} + 1\right)}^2$, is

• (4x + y + 1) = tan (2x + C)

• (4x + y + 1)² = 2 tan (2x + C)

• (4x + y + 1)³ = 3 tan (2x + C)

• (4x + y + 1) = 2 tan (2x + C)