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

In the given cases, find the distance of each of the given points from the corresponding given plane.

Point                 Plane

(2, 3, – 5)           x + 2y – 2z = 9

In the given cases, find the distance of each of the given points from the corresponding given plane.

Point              Plane

(– 6, 0, 0)        2x – 3y + 6z – 2 = 0

Find the distance of the point (−1, −5, −­10) from the point of intersection of the line `vecr = 2hati -hatj + 2hatk + lambda(3hati + 4hatj + 2hatk)` and the plane `vecr.(hati -hatj + hatk) = 5`.

Distance between the two planes: 2x + 3y + 4z = 4 and 4x + 6y + 8z = 12 is

(A) 2 units

(B) 4 units

(C) 8 units

(D)`2/sqrt29 "units"`

If `veca = 2hati + 2hatj + 3hatk,  vecb = -veci + 2hatj + hatk and vecc = 3hati + hatj` are such that `veca + lambdavecb`  is perpendicular to `vecc`, then find the value of λ.

Find `|veca × vecb|`, if `veca = hati - 7hatj + 7hatk` and `vecb = 3hati - 2hatj + 2hatk`.

Find a unit vector perpendicular to each of the vector  `veca  + vecb` and `veca - vecb`, where `veca = 3hati + 2hatj + 2hatk` and `vecb = hati + 2hatj  - 2hatk`.

If a unit vector `veca` makes an angles `pi/3` with `hati, pi/4` with `hatj` and an acute angle θ with `hatk`, then find θ and, hence the compounds of `veca`.

Show that `(veca - vecb) xx (veca + vecb) = 2(veca xx vecb)`.

Find λ and μ if  `(2hati + 6hatj + 27hatk) xx (hati + lambdahatj + muhatk) = vec0`.

Given that `veca.vecb = 0` and `veca xx vecb = 0` What can you conclude about the vectors `veca and vecb`?

Let the vectors `veca, vecb, vecc` given as `a_1hati + a_2hatj + a_3hatk, b_1hati + b_2hatj + b_3hatk, c_1hati + c_2hatj + c_3hatk` Then show that = `veca xx (vecb+ vecc) = veca xx vecb + veca xx vecc.`

If either `veca = vec0`  or `vecb = vec0`, then `veca xxvecb = vec0`. Is the converse true? Justify your answer with an example.

Find the area of the triangle with vertices A (1, 1, 2), B (2, 3, 5) and C (1, 5, 5).

Let the vectors `veca` and `vecb` be such that `|veca| = 3` and `|vecb| = sqrt2/3`, then `veca xx vecb` is a unit vector, if the angle between `veca` and `vecb` is ______.

Area of a rectangle having vertices A, B, C, and D with position vectors `-hati + 1/2 hatj + 4hatk, hati + 1/2 hatj + 4hatk, and -hati - 1/2j + 4hatk,` respectively is ______.

By using the properties of the definite integral, evaluate the integral:

`int_0^(pi/2) cos^2 x dx`

By using the properties of the definite integral, evaluate the integral:

`int_0^(pi/2)  sqrt(sinx)/(sqrt(sinx) + sqrt(cos x)) dx` 

By using the properties of the definite integral, evaluate the integral:

`int_0^(pi/2) sin^(3/2)x/(sin^(3/2)x + cos^(3/2) x) dx`