Arts (English Medium) Class 12 - CBSE Question Bank Solutions

If y = 5 cos x – 3 sin x, prove that `(d^2y)/(dx^2) + y = 0`.

If y = cos^–1 x, find `(d^2y)/dx^2` in terms of y alone.

If y = 3 cos (log x) + 4 sin (log x), show that x²y₂ + xy₁ + y = 0.

If y = Ae^mx + Be^nx, show that `(d^2y)/dx^2 - (m+ n) (dy)/dx + mny = 0`.

If y = 500e^7x + 600e^–7x, show that `(d^2y)/(dx^2)` = 49y.

If e^y (x + 1) = 1, show that `(d^2y)/(dx^2) = (dy/dx)^2`.

If y = (tan^–1 x)², show that (x² + 1)² y₂ + 2x (x² + 1) y₁ = 2

Show that the points A (1, –2, –8), B (5, 0, –2) and C (11, 3, 7) are collinear, and find the ratio in which B divides AC.

Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are `P(2veca + vecb)` and `Q(veca - 3vecb)` externally in the ratio 1: 2. Also, show that P is the mid point of the line segment RQ.

Evaluate the integral by using substitution.

`int_0^1 x/(x^2 +1)`dx

Evaluate the integral by using substitution.

`int_0^(pi/2) sqrt(sin phi) cos^5 phidphi`

Evaluate the integral by using substitution.

`int_0^1 sin^(-1) ((2x)/(1+ x^2)) dx`

Evaluate the integral by using substitution.

`int_0^2 xsqrt(x+2)`  (Put x + 2 = `t^2`)

Evaluate the integral by using substitution.

`int_0^(pi/2) (sin x)/(1+ cos^2 x) dx`

Evaluate the integral by using substitution.

`int_0^2 dx/(x + 4 - x^2)`

Evaluate the integral by using substitution.

`int_(-1)^1 dx/(x^2 + 2x  + 5)`

Evaluate the integral by using substitution.

`int_1^2 (1/x- 1/(2x^2))e^(2x) dx`

The value of the integral `int_(1/3)^4 ((x- x^3)^(1/3))/x^4` dx is ______.

If `f(x) = int_0^pi t sin  t  dt`, then f' (x) is ______.

Evaluate `int_0^(pi/4) (sinx + cosx)/(16 + 9sin2x) dx`