The point of intersection of the straight lines $\mathrm{r} = \left(3\hat{\mathrm{i}} - 4\hat{\mathrm{j}} + 5\hat{\mathrm{k}}\right) + \mathrm{\lambda }\left(- \hat{\mathrm{i}} - 2\hat{\mathrm{j}} + 2\hat{\mathrm{k}}\right)$ and $\frac{3 - \mathrm{x}}{- 1} = \frac{\mathrm{y} + 4}{2} = \frac{\mathrm{z} - 5}{7}$ is

• (- 3, - 4, - 5)

• (- 3, 4, 5)

• (- 3, 4, - 5)

• (3, - 4, 5)

The vector equation of the straight line $\frac{\mathrm{x} - 2}{- 1} = \frac{\mathrm{y}}{- 3} = \frac{1 - \mathrm{z}}{2}$ is

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

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

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

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

The straight line $\mathrm{r} = (\hat{\mathrm{i}} + \hat{\mathrm{j}} + 2\hat{\mathrm{k}}) + \mathrm{t}\left(2\hat{\mathrm{i}} + 5\hat{\mathrm{j}} + 3\hat{\mathrm{k}}\right)$ is parallel to the plane $\mathrm{r} . (2\hat{\mathrm{i}} + \hat{\mathrm{j}} - 3\hat{\mathrm{k}}) = 5$. Then, the distance between the straight line and the plane is

• $\frac{9}{\sqrt{14}}$

• $\frac{8}{\sqrt{14}}$

• $\frac{7}{\sqrt{14}}$

• $\frac{6}{\sqrt{14}}$

The distance between (2, 1, 0) and 2x + y + Zz + 5 = 0 is

• 10

• 10/3

• 10/9

• 5

The equation of the plane that passes through the points (1, 0, 2), (-1, 1, 2), (5, 0, 3) is

• x + 2y - 4z + 7 = 0

• x + 2y - 3z + 7 = 0

• x - 2y + 4z + 7 = 0

• 2y - 4z - 7 + x = 0

If a, b, c are vectors such that a + b + c = 0 and $\left|\mathrm{a}\right| = 7, \left|\mathrm{b}\right| = 5, \left|\mathrm{c}\right| = 3$, then the angle between c and b is

• $\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$3$}\right.$

• $\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$6$}\right.$

• $\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$4$}\right.$

• $\mathrm{\pi }$

177.

The angle between the pair of lines $\frac{\mathrm{x} - 2}{2} = \frac{\mathrm{y} - 1}{5} = \frac{\mathrm{z} + 3}{- 3}$ and $\frac{\mathrm{x} + 2}{- 1} = \frac{\mathrm{y} - 4}{8} = \frac{\mathrm{z} - 5}{4}$ is

• ${\cos }^{-1}\left(\frac{21}{9\sqrt{38}}\right)$

• ${\cos }^{-1}\left(\frac{23}{9\sqrt{38}}\right)$

• ${\cos }^{-1}\left(\frac{24}{9\sqrt{38}}\right)$

• ${\cos }^{-1}\left(\frac{26}{9\sqrt{38}}\right)$

The equation of the plane passing through the intersection of the planes x + 2y + 3z + 4 = 0 and 4x + 3y + 2z + 1 = 0 and the origin is :

• 3x + 2y + z + 1 = 0

• 3x + 2y + z = 0

• 2x + 3y + z = 0

• x + y + z = 0

If $\left(\frac{1}{2}, \frac{1}{3}, \mathrm{n}\right)$ are the direction cosines of a line, then the value of n is

• $\frac{\sqrt{23}}{6}$

• $\frac{23}{36}$

• $\frac{2}{3}$

• $\frac{3}{2}$

The equation of the plane passing through (2, 3, 4) and parallel to the plane 5x - 6y + 7z = 3 is :

• 5x - 6y + 7z + 20 = 0

• 5x - 6y + 7z - 20 = 0

• - 5x + 6y - 7z + 3 = 0

• 5x + 6y + 7z + 3 = 0