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

Using properties of determinants, prove that

`|(sin alpha, cos alpha, cos(alpha+ delta)),(sin beta, cos beta, cos (beta + delta)),(sin gamma, cos gamma, cos (gamma+ delta))| = 0`

Differentiate the following w.r.t. x:

`e^x/sinx`

Differentiate the following w.r.t. x: 

`e^(sin^(-1) x)`

Differentiate the following w.r.t. x:

`e^(x^3)`

Differentiate the following w.r.t. x: 

sin (tan^–1 e^–x)

Differentiate the following w.r.t. x:

log (cos e^x)

Differentiate the following w.r.t. x:

`e^x + e^(x^2) + "..." + e^(x^5)`

Differentiate the following w.r.t. x:

`sqrt(e^(sqrtx))`, x > 0

Differentiate the following w.r.t. x:

log (log x), x > 1

Differentiate the following w.r.t. x: 

`cos x/log x`, x > 0

Differentiate the following w.r.t. x:

cos (log x + e^x), x > 0

Differentiate the function with respect to x:

(log x)^{log x}, x > 1

Differentiate the function with respect to x:

cos (a cos x + b sin x), for some constant a and b.

Using the fact that sin (A + B) = sin A cos B + cos A sin B and the differentiation, obtain the sum formula for cosines.

Compute the magnitude of the following vector:

`veca = hati + hatj + hatk;` `vecb = 2hati - 7hatj - 3hatk`;  `vecc = 1/sqrt3 hati + 1/sqrt3 hatj - 1/sqrt3 hatk`

The value of is `hati.(hatj xx hatk)+hatj.(hatixxhatk)+hatk.(hatixxhatj)` is ______.

If θ is the angle between any two vectors `veca` and `vecb,` then `|veca.vecb| = |veca xx vecb|` when θ is equal to ______.

For the differential equation, find the general solution:

`dy/dx = (1 - cos x)/(1+cos x)`