Mathematics : Three Dimensional Geometry

The angle between the line r = (i + 2j + 3k) + $\mathrm{\lambda }$(2i + 3j + 4k) and the plane r - (i + j - 2k) = 0 is

• 0°

• 60°

• 30°

• 90°

The lines r = i + j - k + (3i - j) and r = 4j - k + µ (2i + 3k) intersect at the point

• (0, 0, 0)

• (0, 0, 1)

• (0, - 4, - 1)

• (4, 0, - 1)

An equation of the plane through the points (1, 0, 0) and (0, 2, 0) and at a distance $\frac{6}{7}$ units from the origin is

• 6x + 3y + z - 6 = 0

• 6x + 3y + 2z - 6= 0

• 6x + 3y + z + 6 = 0

• 6x + 3y + 2z + 6 = 0

The projection of a line segment on the axes are 9, 12 and 8. Then, the length of the line segment is

• 15

• 16

• 17

• 18

The straight line passing through the point (1, 0, - 2) and perpendicular to the plane x - 2y + 5z - 7 = 0 is

• $\frac{\mathrm{x} - 1}{1} = \frac{\mathrm{y}}{0} = \frac{\mathrm{z} - 5}{- 2}$

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

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

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

The equation of the plane passing through (1, 2, 3) and parallel to 3x - 2y + 4z = 5 is

• 3x - 2y + 4z = 11

• 3x - 2y + 4z = 0

• 3x - 2y + 4z = 10

• 3(x - 1) - 2(y - 2) + 4(z - 3) = 5

If the straight lines $\frac{\mathrm{x} - 2}{1} = \frac{\mathrm{y} - 3}{1} = \frac{\mathrm{z} - 4}{0}$ and $\frac{\mathrm{x} - 1}{\mathrm{k}} = \frac{\mathrm{y} - 4}{2} = \frac{\mathrm{z} - 5}{1}$ are copalnar then, the value of k is

• - 3

• 0

• 1

• - 2

The line $\frac{\mathrm{x} - {\mathrm{x}}_1}{0} = \frac{\mathrm{y} - {\mathrm{y}}_1}{1} = \frac{\mathrm{z} - {\mathrm{z}}_1}{2}$ is

• perpendicular to the x-axis

• perpendicular to the yz-plane

• parallel to the y-axis

• parallel to the xz-plane

The point which divides the line joining the points (1, 3, 4) and (4, 3, 1) internally in the ratio 2 : 1, is

• (2, - 3, 3)

• (2, 3, 3)

• $\left(\frac{5}{2}, 3, \frac{{\displaystyle 5}}{{\displaystyle 2}}\right)$

• (3, 3, 2)

The angle between the lines $\frac{\mathrm{x} - 7}{1} = \frac{\mathrm{y} + 3}{- 5} = \frac{\mathrm{z}}{3}$ and $\frac{2 - \mathrm{x}}{- 7} = \frac{\mathrm{y}}{2} = \frac{\mathrm{z} + 5}{1}$ is equal to

• $\frac{\mathrm{\pi }}{4}$

• $\frac{\mathrm{\pi }}{3}$

• $\frac{\mathrm{\pi }}{2}$

• $\frac{\mathrm{\pi }}{6}$