Commerce (English Medium) इयत्ता १२ - CBSE Question Bank Solutions

If the distance of the point (1, 1, 1) from the plane x – y + z + λ = 0 is `5/sqrt(3)`, find the value(s) of λ.

Evaluate: `int_0^(2π) (1)/(1 + e^(sin x)`dx

The two adjacent sides of a parallelogram are represented by vectors `2hati - 4hatj + 5hatk` and `hati - 2hatj - 3hatk`. Find the unit vector parallel to one of its diagonals, Also, find the area of the parallelogram.

Find the distance of the point (1, –2, 0) from the point of the line `vecr = 4hati + 2hatj + 7hatk + λ(3hati + 4hatj + 2hatk)` and the point `vecr.(hati - hatj + hatk)` = 10.

Evaluate: `int_(-1)^3 |x^3 - x|dx`

Evaluate: `int_((-π)/2)^(π/2) (sin|x| + cos|x|)dx`

Find the coordinates of points on line `x/1 = (y - 1)/2 = (z + 1)/2` which are at a distance of `sqrt(11)` units from origin.

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

Evaluate `int_-1^1 |x^4 - x|dx`.

If the angle between `veca` and `vecb` is `π/3` and `|veca xx vecb| = 3sqrt(3)`, then the value of `veca.vecb` is ______.

If `int_0^(2π) cos^2 x  dx = k int_0^(π/2) cos^2 x  dx`, then the value of k is ______.

If `|veca xx vecb| = sqrt(3)` and `veca.vecb` = – 3, then angle between `veca` and `vecb` is ______.

`int_-1^1 |x - 2|/(x - 2) dx`, x ≠ 2 is equal to ______.

Assertion (A): `int_2^8 sqrt(10 - x)/(sqrt(x) + sqrt(10 - x))dx` = 3.

Reason (R): `int_a^b f(x) dx = int_a^b f(a + b - x) dx`.

Find the area of a parallelogram whose adjacent sides are determined by the vectors `veca = hati - hatj + 3hatk` and `vecb = 2hati - 7hatj + hatk`.

The value of `int_0^(π/4) (sin 2x)dx` is ______.

Evaluate: `int_(-π//4)^(π//4) (cos 2x)/(1 + cos 2x)dx`.

Evaluate: `int_0^π x/(1 + sinx)dx`.

If `veca = hati + hatj + hatk` and `vecb = hati + 2hatj + 3hatk` then find a unit vector perpendicular to both `veca + vecb` and `veca - vecb`.

For any integer n, the value of `int_-π^π e^(cos^2x) sin^3 (2n + 1)x  dx` is ______.