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

Find the equation of the plane passing through the intersection of the planes `vecr . (hati + hatj + hatk)` and `vecr.(2hati + 3hatj - hatk) + 4 = 0` and parallel to the x-axis. Hence, find the distance of the plane from the x-axis.

Find the angle between the vectors `vec"a" + vec"b" and  vec"a" -vec"b" if  vec"a" = 2hat"i"-hat"j"+3hat"k" and vec"b" = 3hat"i" + hat"j"-2hat"k", and"hence find a vector perpendicular to both"  vec"a" + vec"b" and vec"a" - vec"b"`.

Show that the lines `("x"-1)/(3) = ("y"-1)/(-1) = ("z"+1)/(0) = λ and  ("x"-4)/(2) = ("y")/(0) = ("z"+1)/(3)` intersect. Find their point of intersection. 

The negative of a matrix is obtained by multiplying it by ______.

Evaluate the following:

`int sqrt(1 + x^2)/x^4 "d"x`

Evaluate the following:

`int ("d"x)/sqrt(16 - 9x^2)`

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 ______.