Arts (English Medium) कक्षा १२ - CBSE Important Questions

The derivative of x^2x w.r.t. x is ______.

If f(x) = `{{:(x^2"," if x ≥ 1),(x"," if x < 1):}`, then show that f is not differentiable at x = 1.

If y = tan x + sec x then prove that `(d^2y)/(dx^2) = cosx/(1 - sinx)^2`.

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 value of c in Rolle's theorem for the function `f(x) = x^3 - 3x " in " (-sqrt3, 0)`

Show that the function `f(x) = x^3 - 3x^2 + 6x - 100` is increasing on R

Show that the surface area of a closed cuboid with square base and given volume is minimum, when it is a cube.

The volume of a sphere is increasing at the rate of 3 cubic centimeter per second. Find the rate of increase of its surface area, when the radius is 2 cm

Prove that the semi-vertical angle of the right circular cone of given volume and least curved surface is \[\cot^{- 1} \left( \sqrt{2} \right)\] .

Prove that the function f : N → N, defined by f(x) = x² + x + 1 is one-one but not onto. Find the inverse of f: N → S, where S is range of f.

Evaluate : `intsin(x-a)/sin(x+a)dx`

Evaluate : ` int x^2/((x^2+4)(x^2+9))dx`

Evaluate : `∫_0^(π/2)(sin^2 x)/(sinx+cosx)dx`

Evaluate `∫_0^(3/2)|x cosπx|dx`

Evaluate :

`int(sqrt(cotx)+sqrt(tanx))dx`

Evaluate the definite integrals `int_0^pi (x tan x)/(sec x + tan x)dx`

Evaluate: `int_1^4 {|x -1|+|x - 2|+|x - 4|}dx`

Find : ` int  (sin 2x ) /((sin^2 x + 1) ( sin^2 x + 3 ) ) dx`

Prove that `int_0^"a" "f" ("x") "dx" = int_0^"a" "f" ("a" - "x") "d x",` hence evaluate `int_0^pi ("x" sin "x")/(1 + cos^2 "x") "dx"`

Find the general solution of the differential equation: `e^((dy)/(dx)) = x^2`.