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

Find the inverse of each of the matrices, if it exists.

`[(1,3,-2),(-3,0,-5),(2,5,0)]`

Find the inverse of each of the matrices, if it exists.

`[(2,0,-1),(5,1,0),(0,1,3)]`

In following cases, determine the direction cosines of the normal to the plane and the distance from the origin.

z = 2

In following cases, determine the direction cosines of the normal to the plane and the distance from the origin.

x + y + z = 1

In following cases, determine the direction cosines of the normal to the plane and the distance from the origin.

2x + 3y – z = 5

In following cases, determine the direction cosines of the normal to the plane and the distance from the origin.

5y + 8 = 0

Find the equation of the plane with intercept 3 on the y-axis and parallel to ZOX plane.

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

Find the coordinates of the point where the line through (5, 1, 6) and (3, 4, 1) crosses the YZ-plane

Find the coordinates of the point where the line through (5, 1, 6) and (3, 4, 1) crosses the ZX − plane.

Find the coordinates of the point where the line through (3, ­−4, −5) and (2, − 3, 1) crosses the plane 2x + y + z = 7).

The planes: 2x − y + 4z = 5 and 5x − 2.5y + 10z = 6 are

(A) Perpendicular

(B) Parallel

(C) intersect y-axis

(C) passes through `(0,0,5/4)`

Find the value of λ, if four points with position vectors `3hati + 6hatj+9hatk`, `hati + 2hatj + 3hatk`,`2hati + 3hatj + hatk` and `4hati + 6hatj + lambdahatk` are coplanar.

Find the coordinates of the point where the line through the points (3, - 4, - 5) and (2, - 3, 1), crosses the plane determined by the points (1, 2, 3), (4, 2,- 3) and (0, 4, 3)

If A is a 3 × 3 invertible matrix, then what will be the value of k if det(A^–1) = (det A)^k

Let `veca = hati + hatj + hatk = hati` and `vecc = c_1veci + c_2hatj + c_3hatk` then

1) Let `c_1 = 1` and `c_2 = 2`, find `c_3` which makes `veca, vecb "and" vecc`coplanar

2) if `c_2 = -1` and `c_3 = 1`, show that no value of `c_1`can make `veca, vecb and vecc` coplanar

if `A = ((2,3,1),(1,2,2),(-3,1,-1))`, Find `A^(-1)` and hence solve the system of equations 2x + y – 3z = 13, 3x + 2y + z = 4, x + 2y – z = 8

Give a condition that three vectors \[\vec{a}\], \[\vec{b}\] and \[\vec{c}\]  form the three sides of a triangle. What are the other possibilities?

Prove that a necessary and sufficient condition for three vectors \[\vec{a}\], \[\vec{b}\], \[\vec{c}\]  to be coplanar is that there exist scalars l, m, n not all zero simultaneously such that \[l \vec{a} + m \vec{b} + n \vec{c} = \vec{0} .\]