The angle between a line with direction ratio 2 : 2 : 1 and a line joining (3, 1, 4) to (7, 2, 12) is

• ${\cos }^{-1}\left(\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.\right)$

• ${\cos }^{-1}\left(\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right)$

• ${\tan }^{-1}\left(\raisebox{1ex}{$- 2$}\!\left/ \!\raisebox{-1ex}{$3$}\right.\right)$

• None of the above

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

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{6}$

253.

Direction cosines of the line $\frac{\mathrm{x} + 2}{2} = \frac{2\mathrm{y} - 5}{3}, \mathrm{z} = - 1$ are

• $\frac{4}{5}, \frac{3}{5}, 0$

• $\frac{3}{5}, \frac{4}{5}, \frac{1}{5}$

• $- \frac{3}{5}, \frac{4}{5}, 0$

• $\frac{4}{5}, - \frac{2}{5}, \frac{1}{5}$

The acute angle between the line r = $\left(\hat{\mathrm{i}} + 2\hat{\mathrm{j}} + \hat{\mathrm{k}}\right) + \mathrm{\lambda }\left(\hat{\mathrm{i}} + \hat{\mathrm{j}} + \hat{\mathrm{k}}\right)$ and the plane $\left(2\hat{\mathrm{i}} - \hat{\mathrm{j}} + \hat{\mathrm{k}}\right)$ = 5

• ${\cos }^{-1}\left(\frac{\sqrt{2}}{3}\right)$

• ${\sin }^{-1}\left(\frac{\sqrt{2}}{3}\right)$

• ${\tan }^{-1}\left(\frac{\sqrt{2}}{3}\right)$

• ${\sin }^{-1}\left(\frac{\sqrt{2}}{\sqrt{3}}\right)$

If A and B are foot of perpendicular drawn from point Q(a, b, c) to the planes yz and zx, then equation of plane through the points A, B and O is

• $\frac{\mathrm{x}}{\mathrm{a}} + \frac{\mathrm{y}}{\mathrm{b}} - \frac{\mathrm{z}}{\mathrm{c}} = 0$

• $\frac{\mathrm{x}}{\mathrm{a}} - \frac{\mathrm{y}}{\mathrm{b}} + \frac{\mathrm{z}}{\mathrm{c}} = 0$

• $\frac{\mathrm{x}}{\mathrm{a}} - \frac{\mathrm{y}}{\mathrm{b}} - \frac{\mathrm{z}}{\mathrm{c}} = 0$

• $\frac{\mathrm{x}}{\mathrm{a}} + \frac{\mathrm{y}}{\mathrm{b}} + \frac{\mathrm{z}}{\mathrm{c}} = 0$

If line joining points A and B having position vectors 6a - 4b + 4c and - 4c respectively and the line joining the points C and 0 having position vectors - a - 2b - 3c and a + 2b - 5c intersect, then point of intersection is

• B

• C

• D

• A

Direction ratios of the line which is perpendicular to the lines with direction ratios - 1, 2, 2 and 0, 2, 1 are

• 1, 1, 2

• 2, - 1, 2

• - 2, 1, 2

• 2, 1, - 2

If the angle between the planes $\mathrm{r} . \left(\mathrm{m}\hat{\mathrm{i}} - \hat{\mathrm{j}} + 2\hat{\mathrm{k}}\right)$ + 3 = 0 and $\mathrm{r} . \left(2\hat{\mathrm{i}} - \mathrm{m}\hat{\mathrm{j}} - \hat{\mathrm{k}}\right) - 5$ = 0 is $\frac{\mathrm{\pi }}{3}$, then m =

• 2

• $\pm 3$

• 3

• - 2

If the origin and the points P(2, 3, 4 ), Q(1, 2, 3) and R(x, y, z) are coplanar, then

• x - 2y - z = 0

• x + 2y + z = 0

• x - 2y + z = 0

• 2x - 2y + z = 0

If lines represented by equation px² - qy² = 0 are distinct, then

• pq > 0

• pq < 0

• pq = 0

• p + q = 0