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

Show that height of the cylinder of greatest volume which can be inscribed in a right circular cone of height h and semi-vertical angle α is one-third that of the cone and the greatest volume of the cylinder is `(4)/(27) pi"h"^3 tan^2 α`.

Find: `int sec^2 x /sqrt(tan^2 x+4) dx.`

Find: `intsqrt(1 - sin 2x) dx, pi/4 < x < pi/2`

Find: `int sin^-1 (2x) dx.`

Solve the differential equation: x dy - y dx = `sqrt(x^2 + y^2)dx,` given that y = 0 when x = 1.

If x = a t⁴ y = 2a t² then `("d"y)/("d"x)` = ______

Addition of matrices is defined if order of the matrices is ______.

If possible, find the sum of the matrices A and B, where A = `[(sqrt(3), 1),(2, 3)]`, and B = `[(x, y, z),(a, "b", 6)]`

If A = `[(1, 2),(-2, 1)]`, B = `[(2, 3),(3, -4)]` and C = `[(1, 0),(-1, 0)]`, verify: A(B + C) = AB + AC

If A = `[(2, 1)]`, B = `[(5, 3, 4),(8, 7, 6)]` and C = `[(-1, 2, 1),(1, 0, 2)]`, verify that A(B + C) = (AB + AC).

If A = `[(1, 0, -1),(2, 1, 3 ),(0, 1, 1)]`, then verify that A² + A = A(A + I), where I is 3 × 3 unit matrix.

If A = `[(1, 2),(4, 1),(5, 6)]` B = `[(1, 2),(6, 4),(7, 3)]`, then verify that: (2A + B)′ = 2A′ + B′

Let A = `[(1, 2),(-1, 3)]`, B = `[(4, 0),(1, 5)]`, C = `[(2, 0),(1, -2)]` and a = 4, b = –2. Show that: A + (B + C) = (A + B) + C

Let A = `[(1, 2),(-1, 3)]`, B = `[(4, 0),(1, 5)]`, C = `[(2, 0),(1, -2)]` and a = 4, b = –2. Show that: (a + b)B = aB + bB

If A = `[(0, -x),(x, 0)]`, B = `[(0, 1),(1, 0)]` and x² = –1, then show that (A + B)² = A² + B². 

If A = `[(1, 2),(4, 1)]`, find A² + 2A + 7I.

Matrix multiplication is ______ over addition.

Matrices of any order can be added.

If `|(2x, 5),(8, x)| = |(6, 5),(8, 3)|`, then find x

Prove that (A^–1)′ = (A′)^–1, where A is an invertible matrix.