Arts (English Medium) कक्षा १२ - CBSE Question Bank Solutions for Mathematics

Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation:

x + y = tan^–1y   :   y² y′ + y² + 1 = 0

Verify that the given function (explicit or implicit) is a solution of the corresponding differential equation:

`y = sqrt(a^2 - x^2 )  x in (-a,a) : x + y  dy/dx = 0(y != 0)`

The number of arbitrary constants in the general solution of a differential equation of fourth order are ______.

The number of arbitrary constants in the particular solution of a differential equation of third order are ______.

Find the general solution of the differential equation `dy/dx + sqrt((1-y^2)/(1-x^2)) = 0.`

Show that the general solution of the differential equation  `dy/dx + (y^2 + y +1)/(x^2 + x + 1) = 0` is given by (x + y + 1) = A (1 - x - y - 2xy), where A is parameter.

Solve the differential equation `[e^(-2sqrtx)/sqrtx - y/sqrtx] dx/dy = 1 (x != 0).`

Find a particular solution of the differential equation `dy/dx + y cot x = 4xcosec x(x != 0)`, given that y = 0 when `x = pi/2.`

Show that the function given by f(x) = sin x is

1. strictly increasing in `(0, pi/2)`
2. strictly decreasing in `(pi/2, pi)`
3. neither increasing nor decreasing in (0, π)

Find the intervals in which the function f given by f(x) = 2x² − 3x is

1. strictly increasing
2. strictly decreasing

Find the intervals in which the function f given by f(x) = 2x³ − 3x² − 36x + 7 is

1. Strictly increasing
2. Strictly decreasing

Find the intervals in which the following functions are strictly increasing or decreasing:

x² + 2x − 5

Find the intervals in which the following functions are strictly increasing or decreasing:

10 − 6x − 2x²

Find the intervals in which the following functions are strictly increasing or decreasing:

−2x³ − 9x² − 12x + 1

Find the intervals in which the following functions are strictly increasing or decreasing:

 (x + 1)³ (x − 3)³

Find the intervals in which the following functions are strictly increasing or decreasing:

6 − 9x − x²

Show that y = `log(1+x) - (2x)/(2+x), x> -  1`, is an increasing function of x throughout its domain.

Find the values of x for  `y = [x(x - 2)]^2` is an increasing function.