Three Dimensional Geometry Class 12 Mcq Test (121101)

Which equation represents the Cartesian equation of a line passing through points \( P(1,2,3) \) and \( Q(4,5,6) \)?

a) \( \frac{x-1}{3} = \frac{y-2}{3} = \frac{z-3}{3} \)

b) \( \frac{x-1}{4} = \frac{y-2}{5} = \frac{z-3}{6} \)

c) \( \frac{x-1}{3} = \frac{y-2}{3} = \frac{z-3}{4} \)

d) \( \frac{x-1}{4} = \frac{y-2}{5} = \frac{z-3}{6} \)

Two lines with direction ratios \( 2, 1, -1 \) and \( 3, -1, 2 \) are:

a) Parallel.

b) Perpendicular.

c) Skew.

d) Intersecting.

The direction ratios of a line are determined by:

a) Ratios of its equation coefficients

b) Components of its direction vector

c) Ratios of its coordinates

d) Components of its equation

Skew lines are lines that:

a) Are parallel but do not intersect.

b) Lie in the same plane but do not intersect.

c) Are not coplanar and do not intersect.

d) Intersect at a right angle.

What do the direction cosines of a line indicate?

a) The angles between the line and the coordinate axes.

b) The ratios of the direction components of the line.

c) The perpendicular distances from the line to the coordinate axes.

d) The inclinations of the line with the coordinate axes.

Which equation represents the vector equation of a line passing through points \( P(1,1,1) \) and \( Q(2,2,2) \)?

a) \( r = \langle 1,1,1 \rangle + \lambda \langle 1,1,1 \rangle \)

b) \( r = \langle 1,1,1 \rangle + \lambda \langle 2,2,2 \rangle \)

c) \( r = \langle 1,1,1 \rangle + \lambda \langle 1,2,1 \rangle \)

d) \( r = \langle 1,1,1 \rangle + \lambda \langle 1,1,2 \rangle \)

The direction cosines of a line are given by:

a) Ratios of its coordinates

b) Components of its equation

c) Ratios of its coefficients

d) Components of its direction vector

Skew lines are:

a) Parallel lines

b) Coplanar lines

c) Intersecting lines

d) Non-intersecting and non-parallel lines

Which equation represents the vector equation of a line parallel to the vector \( \langle 1, -2, 3 \rangle \) passing through the point \( (4,5,6) \)?

a) \( r = \langle 4,5,6 \rangle + \lambda \langle 1,-2,3 \rangle \)

b) \( r = \langle 4,5,6 \rangle + \lambda \langle -1,2,-3 \rangle \)

c) \( r = \langle 1,-2,3 \rangle + \lambda \langle 4,5,6 \rangle \)

d) \( r = \langle 4,5,6 \rangle + \lambda \langle 3,2,-1 \rangle \)