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

Find the vector and cartesian equation of the plane passing through the point (2, 5, - 3), (-2, -3, 5) and (5, 3, -3). Also, find the point of intersection of this plane with the line passing through points (3, 1, 5) and (-1, -3, -1).

Vector equation of a line which passes through a point (3, 4, 5) and parallels to the vector `2hati + 2hatj - 3hatk`.

Find the vector and Cartesian equations of the plane passing through the points (2, 2 –1), (3, 4, 2) and (7, 0, 6). Also find the vector equation of a plane passing through (4, 3, 1) and parallel to the plane obtained above.

Find the vector equation of the plane that contains the lines `vecr = (hat"i" + hat"j") + λ (hat"i" + 2hat"j" - hat"k")` and the point (–1, 3, –4). Also, find the length of the perpendicular drawn from the point (2, 1, 4) to the plane thus obtained.

Find:

`int"x".tan^-1 "x"  "dx"`

If x = sin t, y = sin pt, prove that`(1-"x"^2)("d"^2"y")/"dx"^2 - "x" "dy"/"dx" + "p"^2"y" = 0`

Differentiate `tan^-1[(sqrt(1+"x"^2)-sqrt(1-"x"^2))/(sqrt(1+"x"^2) + sqrt(1-"x"^2))]`with respect to cos⁻¹x².

Show that a matrix which is both symmetric and skew symmetric is a zero matrix.

Express the matrix A as the sum of a symmetric and a skew-symmetric matrix, where A = `[(2, 4, -6),(7, 3, 5),(1, -2, 4)]`

Let A = `[(2, 3),(-1, 2)]`. Then show that A² – 4A + 7I = O. Using this result calculate A⁵ also.

If A and B are symmetric matrices of the same order, then (AB′ –BA′) is a ______.

If A and B are two skew-symmetric matrices of same order, then AB is symmetric matrix if ______.

If A = `[(0, 1),(1, 1)]` and B = `[(0, -1),(1, 0)]`, show that (A + B)(A – B) ≠ A² – B² 

Show that A′A and AA′ are both symmetric matrices for any matrix A.

If A = `[(cosalpha, sinalpha),(-sinalpha, cosalpha)]`, and A^–1 = A′, find value of α

If the matrix `[(0, "a", 3),(2, "b", -1),("c", 1, 0)]`, is a skew symmetric matrix, find the values of a, b and c.

If A, B are square matrices of same order and B is a skew-symmetric matrix, show that A′BA is skew-symmetric.

Express the matrix `[(2, 3, 1),(1, -1, 2),(4, 1, 2)]` as the sum of a symmetric and a skew-symmetric matrix.

The matrix `[(1, 0, 0),(0, 2, 0),(0, 0, 4)]` is a ______.

The matrix `[(0, -5, 8),(5, 0, 12),(-8, -12, 0)]` is a ______.