Commerce (English Medium) Class 12 - CBSE Question Bank Solutions for Mathematics

If the vertices A, B and C of ∆ABC have position vectors (1, 2, 3), (−1, 0, 0) and (0, 1, 2), respectively, what is the magnitude of ∠ABC? 

If A, B and C have position vectors (0, 1, 1), (3, 1, 5) and (0, 3, 3) respectively, show that ∆ ABC is right-angled at C. 

If a line makes angles of 90°, 60° and 30° with the positive direction of x, y, and z-axis respectively, find its direction cosines

If a line has direction ratios 2, −1, −2, determine its direction cosines.

Find the direction cosines of the line passing through two points (−2, 4, −5) and (1, 2, 3) .

Using direction ratios show that the points A (2, 3, −4), B (1, −2, 3) and C (3, 8, −11) are collinear.

Find the direction cosines of the sides of the triangle whose vertices are (3, 5, −4), (−1, 1, 2) and (−5, −5, −2).

Find the angle between the vectors with direction ratios proportional to 1, −2, 1 and 4, 3, 2.

Find the angle between the vectors whose direction cosines are proportional to 2, 3, −6 and 3, −4, 5.

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