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

Find `|veca| and |vecb|`, if `(veca + vecb).(veca -vecb) = 8 and |veca| = 8|vecb|.`

Find the magnitude of two vectors `veca and vecb`, having the same magnitude and such that the angle between them is 60° and their scalar product is `1/2`.

If `veca` is a nonzero vector of magnitude 'a' and λ a nonzero scalar, then λ`veca` is unit vector if ______.

Find a vector of magnitude 5 units, and parallel to the resultant of the vectors `veca = 2i + 3hatj - hatk` and `vecb = hati - 2hatj + hatk`.

If `veca, vecb, vecc` are mutually perpendicular vectors of equal magnitudes, show that the vector `veca +  vecb+ vecc` is equally inclined to `veca, vecb` and `vecc`.

Evaluate the following definite integrals as limit of sums.

`int_a^b x dx`

Evaluate the following definite integrals as limit of sums.

`int_0^5 (x+1) dx`

Evaluate the following definite integrals as limit of sums. 

`int_2^3 x^2 dx`

Evaluate the following definite integrals as limit of sums.

`int_1^4 (x^2 - x) dx`

Evaluate the following definite integrals as limit of sums `int_(-1)^1 e^x dx`

Evaluate the following definite integrals as limit of sums.

`int_0^4 (x + e^(2x)) dx`

Evaluate the definite integral:

`int_(pi/2)^pi e^x ((1-sinx)/(1-cos x)) dx`

Evaluate the definite integral:

`int_0^(pi/4) (sinx cos x)/(cos^4 x + sin^4 x)`dx

Evaluate the definite integral:

`int_0^(pi/2) (cos^2 x dx)/(cos^2 x + 4 sin^2 x)`

Evaluate the definite integral:

`int_(pi/6)^(pi/3)  (sin x + cosx)/sqrt(sin 2x) dx`

Evaluate the definite integral:

`int_0^1 dx/(sqrt(1+x) - sqrtx)`

Evaluate the definite integral:

`int_0^(pi/4) (sin x +  cos x)/(9+16sin 2x) dx`

Evaluate the definite integral:

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

Evaluate the definite integral:

`int_1^4 [|x - 1|+ |x - 2| + |x -3|]dx`

Prove the following:

`int_1^3 dx/(x^2(x +1)) = 2/3 + log  2/3`