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

Prove that `tan^-1 ((sqrt(1 + x^2) + sqrt(1 - x^2))/((1 + x^2) - sqrt(1 - x^2))) = pi/2 + 1/2 cos^-1x^2`

Prove that `sin^-1  8/17 + sin^-1  3/5 = sin^-1  7/85`

Show that `tan(1/2 sin^-1  3/4) = (4 - sqrt(7))/3` and justify why the other value `(4 + sqrt(7))/3` is ignored?

If a₁, a₂, a₃,...,a_n is an arithmetic progression with common difference d, then evaluate the following expression.

`tan[tan^-1("d"/(1 + "a"_1 "a"_2)) + tan^-1("d"/(21 + "a"_2 "a"_3)) + tan^-1("d"/(1 + "a"_3 "a"_4)) + ... + tan^-1("d"/(1 + "a"_("n" - 1) "a""n"))]`

If 3 tan^–1x + cot^–1x = π, then x equals ______.

If `sin^-1 ((2"a")/(1 + "a"^2)) + cos^-1 ((1 - "a"^2)/(1 + "a"^2)) = tan^-1 ((2x)/(1 - x^2))`. where a, x ∈ ] 0, 1, then the value of x is ______.

The value of the expression `tan (1/2 cos^-1  2/sqrt(5))` is ______.

If |x| ≤ 1, then `2 tan^-1x + sin^-1 ((2x)/(1 + x^2))` is equal to ______.

If cos^–1α + cos^–1β + cos^–1γ = 3π, then α(β + γ) + β(γ + α) + γ(α + β) equals ______.

The number of real solutions of the equation `sqrt(1 + cos 2x) = sqrt(2) cos^-1 (cos x)` in `[pi/2, pi]` is ______.

If cos^–1x > sin^–1x, then ______.

If y = `2 tan^-1x + sin^-1 ((2x)/(1 + x^2))` for all x, then ______ < y < ______.

The value of cot^–1(–x) for all x ∈ R in terms of cot^–1x is ______.

The matrix A = `[(0, 0, 5),(0, 5, 0),(5, 0, 0)]` is a ______.

If A and B are matrices of same order, then (3A –2B)′ is equal to______.

If two matrices A and B are of the same order, then 2A + B = B + 2A.

For the non singular matrix A, (A′)^–1 = (A^–1)′.

AB = AC ⇒ B = C for any three matrices of same order.

If A = `[(3, -4),(1, 1),(2, 0)]` and B = `[(2, 1, 2),(1, 2, 4)]`, then verify (BA)² ≠ B²A² 

Show by an example that for A ≠ O, B ≠ O, AB = O