21.

If direction cosines of a vector of magnitude 3 are $\frac{2}{3}, - \frac{9}{3}, \frac{{\displaystyle 2}}{{\displaystyle 3}}$, then vector is

• $\hat{\mathrm{i}} - 2\hat{\mathrm{j}} + 2\hat{\mathrm{k}}$

• $2\hat{\mathrm{i}} + \hat{\mathrm{j}} + 2\hat{\mathrm{k}}$

• $\hat{\mathrm{i}} + 2\hat{\mathrm{j}} + 2\hat{\mathrm{k}}$

• None of these

Equation of line passing through the point (2, 3, 1) and parallel to the line of intersection of the planes x - 2y - z + 5 = 0 and x + y + 3z = 6 is

• $\frac{\mathrm{x} - 2}{5} = \frac{\mathrm{y} - 3}{4} = \frac{\mathrm{z} - 1}{3}$

• $\frac{\mathrm{x} - 2}{5} = \frac{\mathrm{y} - 3}{- 4} = \frac{\mathrm{z} - 1}{3}$

• $\frac{\mathrm{x} - 2}{4} = \frac{\mathrm{y} - 3}{3} = \frac{\mathrm{z} - 1}{2}$

• $\frac{\mathrm{x} - 2}{- 5} = \frac{\mathrm{y} - 3}{- 4} = \frac{\mathrm{z} - 1}{3}$

Foot of perpendicular drawn from the origin to the plane 2x - 3y + 4z = 29 is

• (7, - 1, 3)

• (5, - 1, 4)

• (5, - 2, 3)

• (2, - 3, 4)

$\int \frac{1}{{\mathrm{x}}^2{\left({\mathrm{x}}^4 + 1\right)}^{{\displaystyle \frac{3}{4}}}}\mathrm{dx}$ is equal to

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

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

• $\frac{- {\left(1 + {\mathrm{x}}^4\right)}^{{\displaystyle \frac{3}{4}}}}{\mathrm{x}} + \mathrm{C}$

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

A man takes a step forward probability 0.4 and one step backward with probability 0.6, then the probability that at the end of eleven steps he is one step away from the starting point, is

• ${}^{11}\mathrm{C}_5 \times {\left(0.12\right)}^5$

• ${}^{11}\mathrm{C}_5 \times {\left(0.48\right)}^5$

• ${}^{11}\mathrm{C}_6 \times {\left(0.72\right)}^6$

• ${}^{11}\mathrm{C}_6 \times {\left(0.24\right)}^5$

${\int }_0^{\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}\log \left(\frac{\sin \left(\mathrm{x}\right) + \cos \left(\mathrm{x}\right)}{\cos \left(\mathrm{x}\right)}\right)\mathrm{dx}$ is equal to

• ${\int }_0^{\raisebox{1ex}{${\displaystyle \mathrm{\pi }}$}\!\left/ \!\raisebox{-1ex}{${\displaystyle 4}$}\right.}\log \left(\frac{{\displaystyle \sin \left(\mathrm{x}\right) + \cos \left(\mathrm{x}\right)}}{{\displaystyle \cos \left(\mathrm{x}\right)}}\right)\mathrm{dx}$

• $\frac{{\displaystyle \mathrm{\pi }}}{{\displaystyle 4}}\log \left(2\right)$

• $\log \left(2\right)$

• $\frac{{\displaystyle \mathrm{\pi }}}{{\displaystyle 2}}\log \left(2\right)$

Area bounded by y = x³, y = 8 and x = 0 is

• 12 sq units

• 2 sq units

• 6 sq units

• 14 sq units

Let a = $\hat{\mathrm{i}} + 2\hat{\mathrm{j}} + \hat{\mathrm{k}}$, b = $\hat{\mathrm{i}} - \hat{\mathrm{j}} + \hat{\mathrm{k}}$ and c = $\hat{\mathrm{i}} + \hat{\mathrm{j}} - \hat{\mathrm{k}}$. A vector in the plane a and b whose projection on c is $\frac{1}{\sqrt{3}}$, is

• $\hat{\mathrm{i}} + 2\hat{\mathrm{j}} - 2\hat{\mathrm{k}}$

• $3\hat{\mathrm{i}} + \hat{\mathrm{j}} - 3\hat{\mathrm{k}}$

• $4\hat{\mathrm{i}} - \hat{\mathrm{j}} + 4\hat{\mathrm{k}}$

• $4\hat{\mathrm{i}} + \hat{\mathrm{j}} - 4\hat{\mathrm{k}}$

The probability distribution of X is

Find the value of k.

• 0.4

• 0.2

• 0.1

• 0.3

$\int \frac{{\sin }^2\left(\mathrm{x}\right)}{1 + \cos \left(\mathrm{x}\right)}\mathrm{dx}$ is equal to

• $\sin \left(\mathrm{x}\right) + \mathrm{C}$

• $\mathrm{x} + \sin \left(\mathrm{x}\right) + \mathrm{C}$

• $\cos \left(\mathrm{x}\right) + \mathrm{C}$

• $\mathrm{x} - \sin \left(\mathrm{x}\right) + \mathrm{C}$