HSC Commerce (English Medium) १२ वीं कक्षा - Maharashtra State Board Question Bank Solutions for Mathematics and Statistics

If x = `e^(x/y)`, then show that `dy/dx = (x - y)/(xlogx)`

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

Differentiate log `[(sqrt(1 + x^2) + x)/(sqrt(1 + x^2 - x)]]` w.r.t. cos (log x).

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

If y² = a²cos²x + b²sin²x, show that `y + (d^2y)/(dx^2) = (a^2b^2)/y^3`

If log y = log (sin x) – x², show that `(d^2y)/(dx^2) + 4x "dy"/"dx" + (4x^2 + 3)y` = 0.

If x= a cos θ, y = b sin θ, show that `a^2[y(d^2y)/(dx^2) + (dy/dx)^2] + b^2` = 0.

If y = Ae^mx + Be^nx, show that y₂ – (m + n)y₁ + mny = 0.

Integrate the following w.r.t. x : `x^2/((x^2 + 1)(x^2 - 2)(x^2 + 3))`

Integrate the following w.r.t. x : `(12x + 3)/(6x^2 + 13x - 63)`

Integrate the following w.r.t. x : `(2x)/(4 - 3x - x^2)`

Integrate the following w.r.t. x : `(x^2 + x - 1)/(x^2 + x - 6)`

Integrate the following w.r.t. x:

`(6x^3 + 5x^2 - 7)/(3x^2 - 2x - 1)`

Integrate the following w.r.t. x : `(12x^2 - 2x - 9)/((4x^2 - 1)(x + 3)`

Integrate the following w.r.t. x : `(1)/(x(x^5 + 1)`

Integrate the following w.r.t. x : `(2x)/((2 + x^2)(3 + x^2)`

Integrate the following w.r.t. x : `2^x/(4^x - 3 * 2^x - 4`

Integrate the following w.r.t. x : `(3x - 2)/((x + 1)^2(x + 3)`

Integrate the following w.r.t. x : `(5x^2 + 20x + 6)/(x^3 + 2x ^2 + x)`

Integrate the following w.r.t. x : `(1)/(x(1 + 4x^3 + 3x^6)`