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Q1
short answer
2 marks
Find : ∫ dx / (x√(x⁴ - 4))
Solution
Integral I = ∫ dx / (x√(x⁴ - 4)). Let x² = t, then 2x dx = dt or dx/x = dt/(2x²). Thus, I = 1/2 ∫ dt / (t√(t² - 4)). Using the substitution t = 2secθ or directly using the integral formula ∫ dx / (x√(x² - a²)) = (1/a)sec⁻¹(|x|/a) + C, we get I = 1/2 * (1/2)sec⁻¹(t/2) + C = (1/4)sec⁻¹(x²/2) + C.
Q2
short answer
2 marks
Find the general solution of the following differential equation: dy/dx - e^(x-y) - x²e^(-y) = 0
Solution
Given dy/dx = e^(x-y) + x²e^(-y) = e^(-y)(e^x + x²). Separating variables: e^y dy = (e^x + x²) dx. Integrating both sides: ∫ e^y dy = ∫ (e^x + x²) dx, which gives e^y = e^x + x³/3 + C.
Q3
short answer
2 marks
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.
Solution
Let 2P(X=x₁) = 3P(X=x₂) = P(X=x₃) = 5P(X=x₄) = k. Then P(X=x₁) = k/2, P(X=x₂) = k/3, P(X=x₃) = k, P(X=x₄) = k/5. Since ΣP(X) = 1, k/2 + k/3 + k + k/5 = 1 => k(15+10+30+6)/30 = 1 => 61k/30 = 1, so k = 30/61. The distribution is P(x₁)=15/61, P(x₂)=10/61, P(x₃)=30/61, P(x₄)=6/61.
Q4
short answer
2 marks
If a = î + ĵ + k̂, a·b = 1 and a×b = ĵ - k̂, then find |b|.
Solution
We know |a×b|² + |a·b|² = |a|²|b|². Given a = î + ĵ + k̂, |a|² = 1²+1²+1² = 3. a·b = 1. a×b = ĵ - k̂, so |a×b|² = 1² + (-1)² = 2. Substituting these into the identity: 2 + (1)² = 3|b|² => 3 = 3|b|² => |b|² = 1. Thus, |b| = 1.
Q5
short answer
2 marks
If a line makes an angle α, β, γ with the coordinate axes, then find the value of cos2α + cos2β + cos2γ.
Solution
For a line with direction cosines l, m, n, we know l²+m²+n² = 1, where l=cosα, m=cosβ, n=cosγ. We need cos2α + cos2β + cos2γ. Using the identity cos2θ = 2cos²θ - 1: (2cos²α - 1) + (2cos²β - 1) + (2cos²γ - 1) = 2(cos²α + cos²β + cos²γ) - 3 = 2(1) - 3 = -1.
Q6
short answer
2 marks
(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.
Q7
short answer
3 marks
Evaluate: ∫₀^(π/4) dx / (1 + tan x)
Q8
short answer
3 marks
(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|.
Q9
short answer
3 marks
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).
Q10
short answer
3 marks
(a) Find: ∫ e^x · sin2x dx
OR
(b) Find: ∫ (2x) / ((x² + 1)(x² + 2)) dx
Q11
long answer
4 marks
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.
Q12
long answer
4 marks
Find the area bounded by the curves y = |x – 1| and y = 1, using integration.
Q13
long answer
4 marks
(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
Q14
long answer
4 marks
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.
