A particle is dropped from a height 12 g metre and 4 s after anotherparticle is projected from the ground towards it with a velocity 4g ms. The time after which the second particle meets first is

• 4 s

• 2 s

• $\frac{1}{2} \mathrm{s}$

• 1 s

A uniform ladder rests in limiting equilibrium with its lower end on a rough horizontal plane with coefficient of friction µ and its upper end against a smooth vertical wall. If $\mathrm{\theta }$ is the inclination of the ladder with the wall, then $\mathrm{\theta }$ is equal to

• ${\tan }^{-1}\left(\mathrm{u}\right)$

• ${\cot }^{-1}\left(\mathrm{u}\right)$

• ${\cot }^{-1}\left(2\mathrm{u}\right)$

• ${\tan }^{-1}\left(2\mathrm{u}\right)$

A plane which passes through the point (3, 2, 0) and the line $\frac{\mathrm{x} - 3}{1} = \frac{\mathrm{y} - 6}{5} = \frac{3 - 4}{4}$ is

• x - y + z = 1

• x + y + z = 5

• x + 2y - z = 0

• 2x - y + z = 5

If the planes x + 2y + kz = 0 and 2x + y - 2z = 0, are at right angles, then the value of k is

• 2

• - 2

• $\frac{1}{2}$

• $- \frac{1}{2}$

The ratio in which the line joining (2, 4, 5), (3, 5, - 4) is divided by the yz-plane is

• 2 : 3

• 3 : 2

• - 2 : 3

• 4 : - 3

The equation of line of intersection of planes 4x + 4y - 5z = 12, 8x + 12y - 13z = 32can be written as :

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

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

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

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

The equation of the plane, which makes with co-ordinate axes, a triangle with its centroid $\left(\mathrm{\alpha }, \mathrm{\beta }, \mathrm{\gamma }\right)$ is :

• $\mathrm{\alpha x} + \mathrm{\beta y} + \mathrm{\gamma z} = 3$

• $\mathrm{\alpha x} + \mathrm{\beta y} + \mathrm{\gamma z} = 1$

• $\frac{\mathrm{x}}{\mathrm{\alpha }} + \frac{\mathrm{y}}{\mathrm{\beta }} + \frac{\mathrm{z}}{\mathrm{\gamma }} = 3$

• $\frac{\mathrm{x}}{\mathrm{\alpha }} + \frac{\mathrm{y}}{\mathrm{\beta }} + \frac{\mathrm{z}}{\mathrm{\gamma }} = 1$

A variable plane moves so that sum of the reciprocals of its intercepts on the co-ordinate axes is 1/2. Then the plane passes through :

• $\left(\frac{1}{2}, \frac{1}{2}, - \frac{1}{2}\right)$

• (- 1, 1, 1)

• (2, 2, 2)

• (0, 0, 0)

The direction cosines l, m, n of two lines are connected by the relations l + m + n = 0, lm = 0, then the angle between them is :

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

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

• $\raisebox{1ex}{$\mathrm{\pi }$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$

• 0

620.

The equation of the plane passing through three non-collinear points $\overrightarrow{\mathrm{a}}, \overrightarrow{\mathrm{b}}, \overrightarrow{\mathrm{c}}$ is :

• $\overrightarrow{\mathrm{r}} . \left(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}} + \overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}\right) = 0$

• $\overrightarrow{\mathrm{r}} . \left(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}} + \overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}\right) = \left[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}\right]$

• $\overrightarrow{\mathrm{r}} . \left(\overrightarrow{\mathrm{a}} \times \left(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}\right)\right) = \left[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}\right]$

• $\overrightarrow{\mathrm{r}} . \left(\overrightarrow{\mathrm{a}} + \overrightarrow{\mathrm{b}} + \overrightarrow{\mathrm{c}}\right) = \left[\overrightarrow{\mathrm{a}} \overrightarrow{\mathrm{b}} \overrightarrow{\mathrm{c}}\right]$