HSC Science (General) 12th Standard Board Exam - Maharashtra State Board Question Bank Solutions for Mathematics and Statistics

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`

If `f'(x)=k(cosx-sinx), f'(0)=3 " and " f(pi/2)=15`, find f(x).

Find the approximate value of cos (89°, 30'). [Given is: 1° = 0.0175°C]

Show that the height of the cylinder of maximum volume, that can be inscribed in a sphere of radius R is `(2R)/sqrt3.`  Also, find the maximum volume.

Find the equation of the planes parallel to the plane x + 2y+ 2z + 8 =0 which are at the distance of 2  units from the point (1,1, 2)

An open box is to be made out of a piece of a square card board of sides 18 cms by cutting off equal squares from the comers and turning up the sides. Find the maximum volume of the box.

Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius r is `(4r)/3`. Also find maximum volume in terms of volume of the sphere