Science (English Medium) कक्षा १२ - CBSE Question Bank Solutions for Mathematics

Find `1/2` (A + A')  and  `1/2` (A -A') When `A = [(0, a, b),(-a,0,c),(-b,-c,0)]`

Express the following matrices as the sum of a symmetric and a skew symmetric matrix:

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

Express the following matrices as the sum of a symmetric and a skew symmetric matrix:

`[(6, -2,2),(-2,3,-1),(2,-1,3)]`

Express the following matrices as the sum of a symmetric and a skew symmetric matrix:

`[(3,3,-1),(-2,-2,1),(-4,-5,2)]`

Express the following matrices as the sum of a symmetric and a skew symmetric matrix:

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

If A and B are symmetric matrices, prove that AB − BA is a skew symmetric matrix.

Show that the matrix B'AB is symmetric or skew symmetric according as A is symmetric or skew symmetric.

Find the values of x, y, z if the matrix `A = [(0,2y,z),(x,y,-z),(x , -y,z)]` satisfy the equation A'A = I.

If the matrix A is both symmetric and skew symmetric, then ______.

Find the Cartesian equation of the following planes:

`vecr.(hati + hatj-hatk) = 2`

Find the Cartesian equation of the following planes:

`vecr.(2hati + 3hatj-4hatk) = 1`

Find the Cartesian equation of the following planes:

`vecr.[(s-2t)hati + (3 - t)hatj + (2s + t)hatk] = 15`

In the following cases, find the coordinates of the foot of the perpendicular drawn from the origin.

2x + 3y + 4z – 12 = 0

In the following cases, find the coordinates of the foot of the perpendicular drawn from the origin.

3y + 4z – 6 = 0

In the following cases, find the coordinates of the foot of the perpendicular drawn from the origin.

x + y + z = 1

In the following cases, find the coordinates of the foot of the perpendicular drawn from the origin.

5y + 8 = 0

Find the vector and Cartesian equation of the planes that passes through the point (1, 0, −2) and the normal to the plane is `hati + hatj - hatk`

Find the vector and Cartesian equation of the planes that passes through the point (1, 4, 6) and the normal vector to the plane is `hati -2hatj +  hatk`

Show that all the diagonal elements of a skew symmetric matrix are zero.

Find the equation of the plane through the line of intersection of `vecr*(2hati-3hatj + 4hatk) = 1`and `vecr*(veci - hatj) + 4 =0`and perpendicular to the plane `vecr*(2hati - hatj + hatk) + 8 = 0`. Hence find whether the plane thus obtained contains the line x − 1 = 2y − 4 = 3z − 12.