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

Evaluate `cos[sin^-1  1/4 + sec^-1  4/3]`

Prove that `2sin^-1  3/5 - tan^-1  17/31 = pi/4`

Prove that cot^–17 + cot^–18 + cot^–118 = cot^–13

Solve the equation `sin^-1 6x + sin^-1 6sqrt(3)x = - pi/2`

Show that `2tan^-1 {tan  alpha/2 * tan(pi/4 - beta/2)} = tan^-1  (sin alpha cos beta)/(cosalpha + sinbeta)`

If `tan^-1x = pi/10` for some x ∈ R, then the value of cot^–1x is ______.

If α ≤ 2 sin^–1x + cos^–1x ≤ β, then ______.

Evaluate `cos[cos^-1 ((-sqrt(3))/2) + pi/6]`

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 < ______.