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

Find the values of x for which the function f(x) = x³ – 6x² – 36x + 7 is strictly increasing

Find the values of x, for which the function f(x) = x³ + 12x² + 36𝑥 + 6 is monotonically decreasing

Find the values of x for which f(x) = 2x³ – 15x² – 144x – 7 is

1. Strictly increasing
2. strictly decreasing

Show that function f(x) = tan x is increasing in `(0, π/2)`.

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

Lines `overliner = (hati + hatj - hatk) + λ(2hati - 2hatj + hatk)` and `overliner = (4hati - 3hatj + 2hatk) + μ(hati - 2hatj + 2hatk)` are coplanar. Find the equation of the plane determined by them.

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

Find `dy/dx` at x = 0.

Find the particular solution of the differential equation `dy/dx` = e^2y cos x, when x = `π/6`, y = 0

Let f(x) = x³ – 6x² + 9x + 18, then f(x) is strictly increasing in ______.

The function f(x) = sin⁴x + cos⁴x is an increasing function if ______.

Verify that y sec x = tan x + c is a solution of the differential equation `dy/dx + y tan x` = sec x.

Solution of differential equation `e^(x - 2y) = dy/dx` is ______.

Solution of the differential equation `dy/dx = (xy^2 + x)/(x^2y + y)` is ______.

Find the general solution of the differential equation `tan y * dy/dx = sin(x + y) - sin(x - y)`.

Find the values of x for which the function f(x) = `x/(x^2 + 1)` is strictly decreasing.

Solution of the differential equation `(x + y dy/dx)(x^2 + y^2)` = 1, is ______.

If y = `(x + sqrt(x^2 - 1))^m`, show that `(x^2 - 1)(d^2y)/(dx^2) + xdy/dx` = m²y

Solve:

`1 + (dy)/(dx) = cosec (x + y)`; put x + y = u.

Examine the maxima and minima of the function f(x) = 2x³ - 21x² + 36x - 20 . Also, find the maximum and minimum values of f(x). 

Show that the points (1, 1, 1) and (-3, 0, 1) are equidistant from the plane `bar r (3bari+4barj-12bark)+13=0`