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

The solution of the differential equation (x² + 1) \[\frac{dy}{dx}\] + (y² + 1) = 0, is

The solution of the differential equation \[\frac{dy}{dx} - ky = 0, y\left( 0 \right) = 1\] approaches to zero when x → ∞, if

The solution of the differential equation \[\left( 1 + x^2 \right)\frac{dy}{dx} + 1 + y^2 = 0\], is

The solution of the differential equation \[\frac{dy}{dx} = \frac{x^2 + xy + y^2}{x^2}\], is

The number of arbitrary constants in the general solution of differential equation of fourth order is

The number of arbitrary constants in the particular solution of a differential equation of third order is

Which of the following differential equations has y = x as one of its particular solution?

The general solution of the differential equation \[\frac{dy}{dx} = e^{x + y}\], is

The general solution of the differential equation \[\frac{y dx - x dy}{y} = 0\], is

The general solution of a differential equation of the type \[\frac{dx}{dy} + P_1 x = Q_1\] is

The general solution of the differential equation e^x dy + (y e^x + 2x) dx = 0 is

Prove that the function f(x) = log_a x is increasing on (0, ∞) if a > 1 and decreasing on (0, ∞), if 0 < a < 1 ?

Show that f(x) = \[\frac{1}{x}\] is a decreasing function on (0, ∞) ?

Show that f(x) = \[\frac{1}{1 + x^2}\] decreases in the interval [0, ∞) and increases in the interval (−∞, 0] ?

Show that f(x) = \[\frac{1}{1 + x^2}\] is neither increasing nor decreasing on R ?

Without using the derivative, show that the function f (x) = | x | is.
(a) strictly increasing in (0, ∞)
(b) strictly decreasing in (−∞, 0) .

Without using the derivative show that the function f (x) = 7x − 3 is strictly increasing function on R ?