HSC Arts (English Medium) 12th Standard Board Exam - Maharashtra State Board Important Questions for Mathematics and Statistics

If f'(4) = 5, f(4) = 3, g'(6) = 7 and R(x) = g[3 + f(x)] then R'(4) = ______

If x = cos⁻¹(t), y = `sqrt(1 - "t"^2)` then `("d"y)/("d"x)` = ______

If y = `"e"^(1 + logx)` then find `("d"y)/("d"x)` 

If y = log [cos(x⁵)] then find `("d"y)/("d"x)`

Differentiate sin² (sin⁻¹(x²)) w.r. to x

Differentiate `cot^-1((cos x)/(1 + sinx))` w.r. to x

Differentiate `sin^-1((2cosx + 3sinx)/sqrt(13))` w.r. to x

If log₅ `((x^4 + "y"^4)/(x^4 - "y"^4))` = 2, show that `("dy")/("d"x) = (12x^3)/(13"y"^2)`

If y = `sqrt(cos x + sqrt(cos x + sqrt(cos x + ...... ∞)`, show that `("d"y)/("d"x) = (sin x)/(1 - 2y)`

If x sin(a + y) + sin a cos(a + y) = 0 then show that `("d"y)/("d"x) = (sin^2("a" + y))/(sin"a")`

If x = f(t) and y = g(t) are differentiable functions of t so that y is a differentiable function of x and `(dx)/(dt)` ≠ 0 then `(dy)/(dx) = ((dy)/(dt))/((dx)/(d"))`.
Hence find `(dy)/(dx)` if x = sin t and y = cost

If `int (dx)/(4x^2 - 1)` = A log `((2x - 1)/(2x + 1))` + c, then A = ______.

If y = `e^(m tan^-1x)` then show that `(1 + x^2) (d^2y)/(dx^2) + (2x - m) (dy)/(dx)` = 0

Solve `x + y (dy)/(dx) = sec(x^2 + y^2)`

Find `(dy)/(dx)`, if x³ + x2y + xy² + y³ = 81

If y = `sqrt(tan x + sqrt(tanx + sqrt(tanx + .... +  ∞)`, then show that `dy/dx = (sec^2x)/(2y - 1)`.

Find `dy/dx` at x = 0.

If y = sin^–1x, then show that `(1 - x^2) (d^2y)/(dx^2) - x * dy/dx` = 0

Find `dy/dx`, if y = (log x)^x.

Evaluate:

`int log x dx`

If x = f(t) and y = g(t) are differentiable functions of t, so that y is function of x and `(dx)/dt ≠ 0` then prove that `dy/(dx) = (dy/dt)/((dx)/dt)`. Hence find `dy/(dx)`, if x = at², y = 2at.