31.

If a = 3, b = 4, c = 5 each one of a, b and c is perpendicular to the sum of the remaining, then $\left|\mathrm{a} + \mathrm{b} + \mathrm{c}\right|$ is equal to

• $\frac{5}{\sqrt{2}}$

• $\frac{2}{\sqrt{5}}$

• $5\sqrt{2}$

• $\sqrt{5}$

Area lying between the curves y² = 2x and y = x is

• $\frac{2}{3} \mathrm{sq} \mathrm{unit}$

• $\frac{1}{3} \mathrm{sq} \mathrm{unit}$

• $\frac{1}{4} \mathrm{sq} \mathrm{unit}$

• $\frac{3}{4} \mathrm{sq} \mathrm{unit}$

If P$\mathrm{P}\left(\mathrm{A} \cap \mathrm{B}\right) = \frac{7}{10} \mathrm{and} \mathrm{P}\left(\mathrm{B}\right) = \frac{17}{20}$, where P(A/B) stands for probability, then P(A/B) is equal to

• $\frac{7}{8}$

• $\frac{{\displaystyle 17}}{{\displaystyle 20}}$

• $\frac{14}{17}$

• $\frac{1}{8}$

The vaue of ${\int }_2^8\frac{\sqrt{10 - \mathrm{x}}}{\sqrt{\mathrm{x}} + \sqrt{10 - \mathrm{x}}}\mathrm{dx}$ is

• 10

• 0

• 8

• 3

The value of x, if x$\left(\hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}\right)$ is a unit vector, is

• $\pm \frac{1}{\sqrt{3}}$

• $\pm \sqrt{3}$

• $\pm 3$

• $\pm \frac{1}{3}$

The vector equation of the plane, which is at a distance of $\frac{3}{\sqrt{14}}$, from the origin and the normal from the origin is $2\hat{\mathrm{i}} - 3\hat{\mathrm{j}} + \hat{\mathrm{k}}$ is

• $\mathrm{r} . \left(2\hat{\mathrm{i}} - 3\hat{\mathrm{j}} + \hat{\mathrm{k}}\right) = 3$

• $\mathrm{r} . \left(\hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}\right) = 9$

• $\mathrm{r} . \left(\hat{\mathrm{i}} + 2\hat{\mathrm{j}}\right) = 3$

• $\mathrm{r} . \left(2\hat{\mathrm{i}} + \hat{\mathrm{k}}\right) = 3$

Find the coordinates of the foot of the perpendicular drawn from the origin to the plane 5y + 8 = 0.

• $\left(0, - \frac{18}{5}, 2\right)$

• $\left(0, \frac{8}{5}, 2\right)$

• $\left(\frac{8}{25}, 0, 0\right)$

• $\left(0, - \frac{8}{5}, 0\right)$

If $\cos \left(\mathrm{\alpha }\right), \cos \left(\mathrm{\beta }\right), \cos \left(\mathrm{\gamma }\right)$ are the direction cosines of a vector a, then $\cos \left(2\mathrm{\alpha }\right) + \cos \left(2\mathrm{\beta }\right) + \cos \left(2\mathrm{\gamma }\right)$ is equal to

• 2

• 3

• - 1

• 0

Integrating factor of x$\frac{\mathrm{dy}}{\mathrm{dx}} - \mathrm{y} = {\mathrm{x}}^4 - 3\mathrm{x}$ is

• x

• log(x)

• $\frac{1}{\mathrm{x}}$

• - x