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

Show that the lines: `(1 - x)/2 = (y - 3)/4 = z/(-1)` and `(x - 4)/3 = (2y - 2)/(-4) = z - 1` are coplanar.

If y = `sqrt(ax + b)`, prove that `y((d^2y)/dx^2) + (dy/dx)^2` = 0.

The position vectors of three consecutive vertices of a parallelogram ABCD are `A(4hati + 2hatj - 6hatk), B(5hati - 3hatj + hatk)`, and `C(12hati + 4hatj + 5hatk)`. The position vector of D is given by ______.

If x = A cos 4t + B sin 4t, then `(d^2x)/(dt^2)` is equal to ______.

If y = tan x + sec x then prove that `(d^2y)/(dx^2) = cosx/(1 - sinx)^2`.

Evaluate: `int_0^(π/2) sin 2x tan^-1 (sin x) dx`.

If x = a cos t and y = b sin t, then find `(d^2y)/(dx^2)`.

Read the following passage and answer the questions given below:

1. Find the rate of growth of the plant with respect to the number of days exposed to the sunlight.
2. Does the rate of growth of the plant increase or decrease in the first three days? What will be the height of the plant after 2 days?

Find the distance between the point (7, 2, 4) and the plane determined by the points A(2, 5, −3), B(−2, −3, 5) and C(5, 3, −3).

Differentiate x^sinx+(sinx)^cosx with respect to x.

Evaluate `int_(-2)^2x^2/(1+5^x)dx`

Find the distance of a point (2, 5, −3) from the plane `vec r.(6hati-3hatj+2 hatk)=4`

Find the equation of the plane through the line of intersection of the planes x + y + z = 1 and 2x + 3y + 4z = 5 which is perpendicular to the plane x − y + z = 0. Also find the distance of the plane, obtained above, from the origin.

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

Point                    Plane
(0, 0, 0)           3x – 4y + 12 z = 3

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

Point                   Plane

(3, – 2, 1)             2x – y + 2z + 3 = 0

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.