CBSE 12 MATHS 2022 SET-1

Find : ∫ dx / (x√(x⁴ - 4))

Find the general solution of the following differential equation: dy/dx - e^(x-y) - x²e^(-y) = 0

Let X be a random variable which assumes values x₁, x₂, x₃, x₄ such that 2P(X = x₁) = 3P(X = x₂) = P(X = x₃) = 5P(X = x₄). Find the probability distribution of X.

If a = î + ĵ + k̂, a·b = 1 and a×b = ĵ - k̂, then find |b|.

If a line makes an angle α, β, γ with the coordinate axes, then find the value of cos2α + cos2β + cos2γ.

(a) Events A and B are such that P(A) = 1/2, P(B) = 7/12 and P(A' ∪ B') = 1/4. Find whether the events A and B are independent or not. OR (b) A box B₁ contains 1 white ball and 3 red balls. Another box B₂ contains 2 white balls and 3 red balls. If one ball is drawn at random from each of the boxes B₁ and B₂, then find the probability that the two balls drawn are of the same colour.

Evaluate: ∫₀^(π/4) dx / (1 + tan x)

(a) If a and b are two vectors such that |a + b| = |b|, then prove that (a + 2b) is perpendicular to a. OR (b) If a and b are unit vectors and θ is the angle between them, then prove that sin(θ/2) = (1/2)|a - b|.

Find the equation of the plane passing through the line of intersection of the planes r·(î + ĵ + k̂) = 1 and r·(2î + 3ĵ - k̂) + 4 = 0 and passing through the point (–2, 3, 1).

(a) Find: ∫ e^x · sin2x dx OR (b) Find: ∫ (2x) / ((x² + 1)(x² + 2)) dx

Three persons A, B and C apply for a job of manager in a private company. Chances of their selection are in the ratio 1 : 2 : 4. The probability that A, B and C can introduce changes to increase the profits of a company are 0.8, 0.5 and 0.3 respectively. If increase in the profit does not take place, find the probability that it is due to the appointment of A.

Find the area bounded by the curves y = |x – 1| and y = 1, using integration.

(a) Solve the following differential equation: (y – sin²x) dx + tan x dy = 0 OR (b) Find the general solution of the differential equation: (x³ + y³) dy = x²y dx

Two motorcycles A and B are running at the speed more than the allowed speed on the roads represented by the lines r = λ(i + 2j - k) and r = (3i + j) + µ(i + j + k) respectively. Based on the above information, answer the following questions: (a) Find the shortest distance between the given lines. (b) Find the point at which the motorcycles may collide.