21.

For $0 \le \mathrm{x} \le \mathrm{\pi }$, the area between the curve y = sin(x) and x - axis is

• 1 sq unit

• 0 sq unit

• 2 sq unit

• - 1 sq unit

The order and power of differential equation

$\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2} + 7\frac{\mathrm{dy}}{\mathrm{dx}} + \int \mathrm{ydx} = \sin \left(\mathrm{x}\right)$ is

• 1, 3

• 3, 1

• 1, 2

• 2, 1

The solution of differential equation ${\mathrm{xcos}}^2\left(\mathrm{y}\right)\mathrm{dx} = {\mathrm{ycos}}^2\left(\mathrm{x}\right)\mathrm{dx}$ is

• $\mathrm{xtan}\left(\mathrm{x}\right) - \mathrm{ytan}\left(\mathrm{y}\right) - \log \left(\sec \left(\mathrm{x}\right)/\sec \left(\mathrm{y}\right)\right) = \mathrm{c}$

• $\mathrm{ytan}\left(\mathrm{x}\right) - \mathrm{xtan}\left(\mathrm{x}\right) - \log \left(\sec \left(\mathrm{x}\right).\sec \left(\mathrm{y}\right)\right) = \mathrm{c}$

• $\mathrm{xtan}\left(\mathrm{x}\right) - \mathrm{ytan}\left(\mathrm{y}\right) + \log \left(\sec \left(\mathrm{x}\right).\sec \left(\mathrm{y}\right)\right) = \mathrm{c}$

• None of the above

The intersection angle of the curve xy = a² and x² - y² = a² is

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{6}$

• $\frac{\mathrm{\pi }}{2}$

• $\frac{5\mathrm{\pi }}{6}$

The point of intersection of line $\frac{\mathrm{x} - 6}{- 1} = \frac{\mathrm{y} + 1}{0} = \frac{\mathrm{z} + 3}{4}$ and plane x + y - z = 3 is

• (2, 1, 0)

• (7, - 1, - 7)

• (1, 2, - 6)

• (5, - 1, 1)

At a point the addition of two active force is 18 N. If the magnitude of resultant is 12 N and meet at right angle. Then, magnitude of forces are

• 5 N, 13 N

• 6N, 12 N

• 8 N, 10 N

• None of these

The angle between two active forces P + Q and P  - Q is 2$\mathrm{\alpha }$. If their resultant make angle $\mathrm{\theta }$ with bisector of angle. Then

• $\mathrm{Pcos}\left(\mathrm{\theta }\right) = \mathrm{Qcos}\left(\mathrm{\alpha }\right)$

• $\mathrm{Ptan}\left(\mathrm{\theta }\right) = \mathrm{Qtan}\left(\mathrm{\alpha }\right)$

• $\mathrm{Qcos}\left(\mathrm{\theta }\right) = \mathrm{Pcos}\left(\mathrm{\alpha }\right)$

• $\mathrm{Qtan}\left(\mathrm{\theta }\right) = \mathrm{Ptan}\left(\mathrm{\alpha }\right)$

The vector $\overrightarrow{\mathrm{a}}$ is equal to

• $\left(\overrightarrow{\mathrm{a}} . \hat{\mathrm{i}}\right)\hat{\mathrm{i}} + \left(\overrightarrow{\mathrm{a}} . \hat{\mathrm{j}}\right)\hat{\mathrm{j}} + \left(\overrightarrow{\mathrm{a}} . \hat{\mathrm{k}}\right)\hat{\mathrm{k}}$

• $\left(\overrightarrow{\mathrm{a}} . \hat{\mathrm{j}}\right)\hat{\mathrm{i}} + \left(\overrightarrow{\mathrm{a}} . \hat{\mathrm{k}}\right)\hat{\mathrm{j}} + \left(\overrightarrow{\mathrm{a}} . \hat{\mathrm{i}}\right)\hat{\mathrm{k}}$

• $\left(\overrightarrow{\mathrm{a}} . \hat{\mathrm{k}}\right)\hat{\mathrm{i}} + \left(\overrightarrow{\mathrm{a}} . \hat{\mathrm{i}}\right)\hat{\mathrm{j}} + \left(\overrightarrow{\mathrm{a}} . \hat{\mathrm{j}}\right)\hat{\mathrm{k}}$

• $\left(\overrightarrow{\mathrm{a}} . \overrightarrow{\mathrm{a}}\right)\left(\hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}\right)$

If $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{c}}$ are the positton vectors of the vertices f an equilateral triangle whose orthocentre is at the origin, then

• $\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}} = \overrightarrow{0}$

• ${\overrightarrow{\mathrm{a}}}^2 = {\overrightarrow{\mathrm{b}}}^2 + {\overrightarrow{\mathrm{c}}}^2$

• $\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}} = \overrightarrow{\mathrm{c}}$

• None of these

If A and B are two events such that P(A) = $\frac{3}{4}$ and P(B) = $\frac{5}{8}$, then

• $\mathrm{P}\left(\mathrm{A} \cup \mathrm{B}\right) \ge \raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$4$}\right.$

• $\mathrm{P}\left(\mathrm{A}\text{'} \cap \mathrm{B}\right) \le 1/4$

• $\frac{3}{8} \le \mathrm{P}\left(\mathrm{A}\hspace{0.17em}\cap \mathrm{B}\right) \le \frac{5}{8}$

• All of the above