Commerce (English Medium) कक्षा १२ - CBSE Question Bank Solutions for Mathematics

If A and B are two independent events such that P(A) = `1/3` and P(B) = `1/4`, then `P(B^'/A)` is ______.

Find the coordinates of the foot of the perpendicular drawn from point (5, 7, 3) to the line `(x - 15)/3 = (y - 29)/8 = (z - 5)/-5`.

Find the coordinates of the image of the point (1, 6, 3) with respect to the line `vecr = (hatj + 2hatk) + λ(hati + 2hatj + 3hatk)`; where 'λ' is a scalar. Also, find the distance of the image from the y – axis.

Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are  `hati + 2hatj - hatk` and `-hati + hatj + hatk`  respectively, internally the ratio 2:1.

Compute P(A|B), if P(B) = 0.5 and P (A ∩ B) = 0.32.

Write the antiderivative of `(3sqrtx+1/sqrtx).`

Let A = {1, 2, 3,......, 9} and R be the relation in A × A defined by (a, b) R (c, d) if a + d = b + c for (a, b), (c, d) in A × A. Prove that R is an equivalence relation. Also, obtain the equivalence class [(2, 5)].

Find the intervals in which the function f(x) = 3x⁴ − 4x³ − 12x² + 5 is

(a) strictly increasing

(b) strictly decreasing

Evaluate : `∫(sin^6x+cos^6x)/(sin^2x.cos^2x)dx`

Write the degree of the differential equation `x^3((d^2y)/(dx^2))^2+x(dy/dx)^4=0`

Find the Cartesian equation of the line which passes through the point (−2, 4, −5) and is parallel to the line `(x+3)/3=(4-y)/5=(z+8)/6`

The amount of pollution content added in air in a city due to x-diesel vehicles is given by P(x) = 0.005x³ + 0.02x² + 30x. Find the marginal increase in pollution content when 3 diesel vehicles are added and write which value is indicated in the above question.

Find the coordinates of the point where the line through the points A(3, 4, 1) and B(5, 1, 6) crosses the XZ plane. Also find the angle which this line makes with the XZ plane.

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

Find the vector and Cartesian equations of the line through the point (1, 2, −4) and perpendicular to the two lines. 

`vecr=(8hati-19hatj+10hatk)+lambda(3hati-16hatj+7hatk) " and "vecr=(15hati+29hatj+5hatk)+mu(3hati+8hatj-5hatk)`

Find the intervals in which f(x) = sin 3x – cos 3x, 0 < x < π, is strictly increasing or strictly decreasing.

The side of an equilateral triangle is increasing at the rate of 2 cm/s. At what rate is its area increasing when the side of the triangle is 20 cm ?

Let N denote the set of all natural numbers and R be the relation on N × N defined by (a, b) R (c, d) if ad (b + c) = bc (a + d). Show that R is an equivalence relation.

If R=[(x, y) : x+2y=8] is a relation on N, write the range of R.

If `f(x) =∫_0^xt sin t dt` , then write the value of f ' (x).