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

Find: `int logx/(1 + log x)^2 dx`

If `hata` and `hatb` are unit vectors, then prove that `|hata + hatb| = 2 cos  theta/2`, where θ is the angle between them.

Evaluate: `int_(-1)^2 |x^3 - 3x^2 + 2x|dx`

The value of `int_2^3 x/(x^2 + 1)`dx is ______.

If two vectors `veca` and `vecb` are such that `|veca|` = 2, `|vecb|` = 3 and `veca.vecb` = 4, then `|veca - 2vecb|` is equal to ______.

Find the equation of the plane passing through the line of intersection of the planes `vecr(hati + hatj + hatk)` = 10 and `vecr.(2hati + 3hatj - hatk)` + 4 = 0 and passing through (–2, 3, 1).

If `veca, vecb, vecc` are three non-zero unequal vectors such that `veca.vecb = veca.vecc`, then find the angle between `veca` and `vecb - vecc`.

Three vectors `veca, vecb` and `vecc` satisfy the condition `veca + vecb + vecc = vec0`. Evaluate the quantity μ = `veca.vecb + vecb.vecc + vecc.veca`, if `|veca|` = 3, `|vecb|` = 4 and `|vecc|` = 2.

If `veca.hati = veca.(hati + hatj) = veca.(hati + hatj + hatk)` = 1, then `veca` is ______.

In the following cases, determine whether the given planes are parallel or perpendicular, and in case they are neither, find the angles between them.

7x + 5y + 6z + 30 = 0 and 3x – y – 10z + 4 = 0

In the following cases, determine whether the given planes are parallel or perpendicular, and in case they are neither, find the angles between them.

2x + y + 3z – 2 = 0 and x – 2y + 5 = 0

In the following cases, determine whether the given planes are parallel or perpendicular, and in case they are neither, find the angles between them.

2x – 2y + 4z + 5 = 0 and 3x – 3y + 6z – 1 = 0

In the following cases, determine whether the given planes are parallel or perpendicular, and in case they are neither, find the angles between them.

2x – y + 3z – 1 = 0 and 2x – y + 3z + 3 = 0

In the following cases, determine whether the given planes are parallel or perpendicular, and in case they are neither, find the angles between them.

4x + 8y + z – 8 = 0 and y + z – 4 = 0

Find the angle between the vectors `hati - 2hatj + 3hatk` and `3hati - 2hatj + hatk`.

Find the projection of the vector `hati - hatj` on the vector `hati + hatj`.

Find the projection of the vector `hati + 3hatj + 7hatk`  on the vector `7hati - hatj + 8hatk`.

If the vertices A, B, C of a triangle ABC are (1, 2, 3), (–1, 0, 0), (0, 1, 2), respectively, then find ∠ABC. [∠ABC is the angle between the vectors `bar(BA)` and `bar(BC)`].