PUC Science 2nd PUC Class 12 - Karnataka Board PUC Question Bank Solutions for Mathematics

Find the acute angle between the lines whose direction ratios are proportional to 2 : 3 : 6 and 1 : 2 : 2.

Show that the points (2, 3, 4), (−1, −2, 1), (5, 8, 7) are collinear.

Show that the line through points (4, 7, 8) and (2, 3, 4) is parallel to the line through the points (−1, −2, 1) and (1, 2, 5).

Show that the line through the points (1, −1, 2) and (3, 4, −2) is perpendicular to the line through the points (0, 3, 2) and (3, 5, 6).

Show that the line joining the origin to the point (2, 1, 1) is perpendicular to the line determined by the points (3, 5, −1) and (4, 3, −1).

Find the angle between the lines whose direction ratios are proportional to a, b, c and b − c, c − a, a− b.

If the coordinates of the points A, B, C, D are (1, 2, 3), (4, 5, 7), (−4, 3, −6) and (2, 9, 2), then find the angle between AB and CD.

Find the direction cosines of the lines, connected by the relations: l + m +n = 0 and 2lm + 2ln − mn= 0.

Find the angle between the lines whose direction cosines are given by the equations
(i) l + m + n = 0 and l² + m² − n² = 0

Find the angle between the lines whose direction cosines are given by the equations

2l − m + 2n = 0 and mn + nl + lm = 0

Find the angle between the lines whose direction cosines are given by the equations

 l + 2m + 3n = 0 and 3lm − 4ln + mn = 0

Find the angle between the lines whose direction cosines are given by the equations

2l + 2m − n = 0, mn + ln + lm = 0

Define direction cosines of a directed line.

What are the direction cosines of X-axis?

What are the direction cosines of Y-axis?

What are the direction cosines of Z-axis?

Write the distances of the point (7, −2, 3) from XY, YZ and XZ-planes.

Write the distance of the point (3, −5, 12) from X-axis?

Write the ratio in which YZ-plane divides the segment joining P (−2, 5, 9) and Q (3, −2, 4).

A line makes an angle of 60° with each of X-axis and Y-axis. Find the acute angle made by the line with Z-axis.