HSC Science (General) १२ वीं कक्षा - Maharashtra State Board Question Bank Solutions for Mathematics and Statistics

Differentiate `sin^-1((2x)/(1 + x^2))w.r.t. cos^-1((1 - x^2)/(1 + x^2))`

Differentiate `tan^-1((x)/(sqrt(1 - x^2))) w.r.t. sec^-1((1)/(2x^2 - 1))`.

Differentiate `cos^-1((1 - x^2)/(1 + x^2)) w.r.t. tan^-1 x.`

Differentiate `tan^-1((cosx)/(1 + sinx)) w.r.t. sec^-1 x.`

Differentiate x^x w.r.t. x^six.

Differentiate `tan^-1((sqrt(1 + x^2) - 1)/(x)) w.r.t  tan^-1((2xsqrt(1 - x^2))/(1 - 2x^2))`.

Find `(d^2y)/(dx^2)` of the following : x = a(θ – sin θ), y = a(1 – cos θ)

Find `(d^2y)/(dx^2)` of the following : x = sinθ, y = sin³θ at θ = `pi/(2)`

Find `(d^2y)/(dx^2)` of the following : x = a cos θ, y = b sin θ at θ = `π/4`.

If x = at² and y = 2at, then show that `xy(d^2y)/(dx^2) + a` = 0.

If y = `e^(mtan^-1x)`, show that `(1 + x^2)(d^2y)/(dx^2) + (2x - m)"dy"/"dx"` = 0.

If x = cos t, y = e^mt, show that `(1 - x^2)(d^2y)/(dx^2) - x"dy"/"dx" - m^2y` = 0.

If y = x + tan x, show that `cos^2x.(d^2y)/(dx^2) - 2y + 2x` = 0.

If y = e^ax.sin(bx), show that y₂ – 2ay₁ + (a² + b²)y = 0.

If `sec^-1((7x^3 - 5y^3)/(7^3 + 5y^3)) = "m", "show"  (d^2y)/(dx^2)` = 0.

If 2y = `sqrt(x + 1) + sqrt(x - 1)`, show that 4(x² – 1)y₂ + 4xy₁ – y = 0.

If y = sin (m cos^–1x), then show that `(1 - x^2)(d^2y)/(dx^2) - x"dy"/"dx" + m^2y` = 0.

If x² + 6xy + y² = 10, show that `(d^2y)/(dx^2) = (80)/(3x + y)^3`.

If x = a sin t – b cos t, y = a cos t + b sin t, show that `(d^2y)/(dx^2) = -(x^2 + y^2)/(y^3)`.

Find the n^th derivative of the following : (ax + b)^m