Mathematics : Three Dimensional Geometry

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

• ${\cos }^{-1}\left(\frac{- 3}{\sqrt{182}}\right)$

• ${\cos }^{-1}\left(\frac{5}{\sqrt{182}}\right)$

• ${\cos }^{-1}\left(\frac{3}{\sqrt{182}}\right)$

• ${\cos }^{-1}\left(\frac{- 5}{\sqrt{182}}\right)$

If a line makes angles $\mathrm{\alpha }, \mathrm{\beta }, \mathrm{\gamma } \mathrm{and} \mathrm{\delta }$ with the four diagonals of a cube, then${\cos }^2\left(\mathrm{\alpha }\right) + {\cos }^2\left(\mathrm{\beta }\right) + {\cos }^2\left(\mathrm{\gamma }\right) +\hspace{0.17em}{\cos }^2\left(\mathrm{\delta }\right)$ is

• 1

• 2

• 2/3

• 4/3

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

• $45°$

• $30°$

• $60°$

• $90°$

The ratio in which the join of (1, - 2,3) and (4, 2, - 1) is divided by the XOY plane is

• 1 : 3

• 3 : 1

• - 1 : 3

• None of these

The foot of perpendicular from the point (3, 4, 5) to the plane x + y + z = 9 is

• (2, 3, 4)

• (3, 5, - 2)

• (3, 5, 2)

• (3, 2, 4)

The area of the parallelogram having the diagonals 3i + j - 2k and i - 3j + 4k is

• 5 sq units

• $10\sqrt{3} \mathrm{sq} \mathrm{units}$

• $5\sqrt{3} \mathrm{sq} \mathrm{units}$

• 10 sq units

The ratio in which yz-plane divide the line joining the points A(3, 1, - 5) and B(1, 4, - 6) is

• - 3 : 1

• 3 : 1

• - 1 : 3

• 1 : 3

If ${}^{\mathrm{n}}\mathrm{P}_5 = 20{}^{\mathrm{n}}\mathrm{P}_3$, then the value of n is

• 7

• 5

• 8

• 9

The equation of a straight line parallel to the x-axis is given by

• $\frac{\mathrm{x} - \mathrm{a}}{1} = \frac{\mathrm{y} - \mathrm{b}}{1} = \frac{\mathrm{z} - \mathrm{c}}{1}$

• $\frac{\mathrm{x} - \mathrm{a}}{0} = \frac{\mathrm{y} - \mathrm{b}}{0} = \frac{\mathrm{z} - \mathrm{c}}{1}$

• $\frac{\mathrm{x} - \mathrm{a}}{0} = \frac{\mathrm{y} - \mathrm{b}}{1} = \frac{\mathrm{z} - \mathrm{c}}{1}$

• $\frac{\mathrm{x} - \mathrm{a}}{1} = \frac{\mathrm{y} - \mathrm{b}}{0} = \frac{\mathrm{z} - \mathrm{c}}{0}$

The shortest distance between the lines

$\frac{\mathrm{x} - 3}{1} = \frac{\mathrm{y} - 5}{- 2} = \frac{\mathrm{z} - 7}{1}$ and $\frac{\mathrm{x} + 1}{1} = \frac{\mathrm{y} + 1}{- 6} = \frac{\mathrm{z} + 1}{1}$ is

• $\frac{1}{2}\sqrt{29} \mathrm{units}$

• $2\sqrt{29} \mathrm{units}$

• $\sqrt{29} \mathrm{units}$

• $\frac{1}{4}\sqrt{29} \mathrm{units}$