Commerce (English Medium) कक्षा १२ - CBSE Question Bank Solutions for Mathematics

By using the properties of the definite integral, evaluate the integral:

`int_0^(2x) cos^5 xdx`

By using the properties of the definite integral, evaluate the integral:

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

By using the properties of the definite integral, evaluate the integral:

`int_0^pi log(1+ cos x) dx`

By using the properties of the definite integral, evaluate the integral:

`int_0^a  sqrtx/(sqrtx + sqrt(a-x))   dx`

By using the properties of the definite integral, evaluate the integral:

`int_0^4 |x - 1| dx`

Show that `int_0^a f(x)g (x)dx = 2 int_0^a f(x) dx`  if f and g are defined as f(x) = f(a-x) and g(x) + g(a-x) = 4.

`int_(-pi/2)^(pi/2) (x^3 + x cos x + tan^5 x + 1) dx ` is ______.

The value of `int_0^(pi/2) log  ((4+ 3sinx)/(4+3cosx))` dx is ______.

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`

If θ is the angle between two vectors `hati - 2hatj + 3hatk and 3hati - 2hatj + hatk` find `sin theta`

Let `veca = 4hati + 5hatj - hatk`, `vecb  = hati - 4hatj + 5hatk` and `vecc = 3hati + hatj - hatk`. Find a vector `vecd` which is perpendicular to both `vecc` and `vecb and vecd.veca = 21`

\[\int\limits_0^k \frac{1}{2 + 8 x^2} dx = \frac{\pi}{16},\] find the value of k.

\[\int\limits_0^a 3 x^2 dx = 8,\] find the value of a.

If \[f\left( a + b - x \right) = f\left( x \right)\] , then prove that

Write the equation of a plane which is at a distance of \[5\sqrt{3}\] units from origin and the normal to which is equally inclined to coordinate axes.

Find the distance of the point  \[2 \hat{i} - \hat{j} - 4 \hat{k}\]  from the plane  \[\vec{r} \cdot \left( 3 \hat{i}  - 4 \hat{j}  + 12 \hat{k}  \right) - 9 = 0 .\]

Show that the points \[\hat{i}  - \hat{j}  + 3 \hat{k}  \text{ and }  3 \hat{i}  + 3 \hat{j}  + 3 \hat{k} \] are equidistant from the plane \[\vec{r} \cdot \left( 5 \hat{i}  + 2 \hat{j}  - 7 \hat{k}  \right) + 9 = 0 .\]