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

Evaluate the following:

`int (3x - 1)/sqrt(x^2 + 9) "d"x`

Evaluate the following:

`int sqrt(5 - 2x + x^2) "d"x`

Evaluate the following:

`int x/(x^4 - 1) "d"x`

Evaluate the following:

`int sqrt(2"a"x - x^2)  "d"x`

Evaluate the following:

`int sqrt(x)/(sqrt("a"^3 - x^3)) "d"x`

Evaluate the following:

`int ("d"x)/(xsqrt(x^4 - 1))`  (Hint: Put x² = sec θ)

Evaluate the following:

`int_1^2 ("d"x)/sqrt((x - 1)(2 - x))`

Find the angle between the vectors `2hat"i" - hat"j" + hat"k"` and `3hat"i" + 4hat"j" - hat"k"`.

If `vec"a" + vec"b" + vec"c"` = 0, show that `vec"a" xx vec"b" = vec"b" xx vec"c" = vec"c" xx vec"a"`. Interpret the result geometrically?

Using vectors, find the area of the triangle ABC with vertices A(1, 2, 3), B(2, – 1, 4) and C(4, 5, – 1).

Using vectors, prove that the parallelogram on the same base and between the same parallels are equal in area.

Show that area of the parallelogram whose diagonals are given by `vec"a"` and `vec"b"` is `(|vec"a" xx vec"b"|)/2`. Also find the area of the parallelogram whose diagonals are `2hat"i" - hat"j" + hat"k"` and `hat"i" + 3hat"j" - hat"k"`.

If `vec"a" = hat"i" + hat"j" + hat"k"` and `vec"b" = hat"j" - hat"k"`, find a vector `vec"c"` such that `vec"a" xx vec"c" = vec"b"` and `vec"a"*vec"c"` = 3.

The value of λ for which the vectors `3hat"i" - 6hat"j" + hat"k"` and `2hat"i" - 4hat"j" + lambdahat"k"` are parallel is ______.

The vectors from origin to the points A and B are `vec"a" = 2hat"i" - 3hat"j" + 2hat"k"` and `vec"b" = 2hat"i" + 3hat"j" + hat"k"`, respectively, then the area of triangle OAB is ______.

For any vector `vec"a"`, the value of `(vec"a" xx hat"i")^2 + (vec"a" xx hat"j")^2 + (vec"a" xx hat"k")^2` is equal to ______.

If `|vec"a"|` = 10, `|vec"b"|` = 2 and `vec"a".vec"b"` = 12, then value of `|vec"a" xx vec"b"|` is ______.

The vectors `lambdahat"i" + hat"j" + 2hat"k", hat"i" + lambdahat"j" - hat"k"` and `2hat"i" - hat"j" + lambdahat"k"` are coplanar if ______.

If `|vec"a"|` = 4 and −3 ≤ λ ≤ 2, then the range of `|lambdavec"a"|` is ______.

The value of the expression `|vec"a" xx vec"b"|^2 + (vec"a".vec"b")^2` is ______.