Arts (English Medium) Class 12 - CBSE Important Questions for Mathematics

If y = x^x, prove that \[\frac{d^2 y}{d x^2} - \frac{1}{y} \left( \frac{dy}{dx} \right)^2 - \frac{y}{x} = 0 .\]

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.

Find the equation of tangent to the curve `y = sqrt(3x -2)` which is parallel to the line 4x − 2y + 5 = 0. Also, write the equation of normal to the curve at the point of contact.

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`

If `tan^(-1)  (x- 3)/(x - 4) + tan^(-1)  (x +3)/(x + 4) = pi/4`, then find the value of x.

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

Evaluate:

\[\int \cos^{-1} \left(\sin x \right) \text{dx}\]

Evaluate : \[\int\frac{\cos 2x + 2 \sin^2 x}{\cos^2 x}dx\] .

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`.

Find: `int x^2/((x^2 + 1)(3x^2 + 4))dx`

Evaluate: `int_-2^1 sqrt(5 - 4x - x^2)dx`

Evaluate `int_0^(π//4) log (1 + tanx)dx`.

Find `int dx/sqrt(sin^3x cos(x - α))`.