CBSE · Maths
CBSE 12 MATHS 2022 SET-1
2022
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14 questions
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Q1
short answer
2 marks
Find : ∫ dx / (x√(x⁴ - 4))
Step-by-Step AI Explanation
Verified Method
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
Step-by-Step AI Explanation
Verified Method
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.
Step-by-Step AI Explanation
Verified Method
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|.
Step-by-Step AI Explanation
Verified Method
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.
Step-by-Step AI Explanation
Verified Method
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.
Step-by-Step AI Explanation
Verified Method
(a) P(A)=1/2, P(B)=7/12. P(A' ∪ B') = P((A∩B)') = 1 - P(A∩B) = 1/4 => P(A∩B) = 3/4. Since P(A)P(B) = (1/2)*(7/12) = 7/24 and 7/24 ≠ 3/4, the events are not independent. (b) B₁: 1W, 3R; B₂: 2W, 3R. P(Same) = P(WW) + P(RR) = (1/4 * 2/5) + (3/4 * 3/5) = 2/20 + 9/20 = 11/20.
Q7
short answer
3 marks
Evaluate: ∫₀^(π/4) dx / (1 + tan x)
Step-by-Step AI Explanation
Verified Method
Let I = ∫₀^(π/4) dx / (1 + tan x) = ∫₀^(π/4) cos x / (cos x + sin x) dx. Using the property ∫₀^a f(x) dx = ∫₀^a f(a-x) dx: I = ∫₀^(π/4) sin x / (sin x + cos x) dx. Adding both: 2I = ∫₀^(π/4) (cos x + sin x) / (cos x + sin x) dx = ∫₀^(π/4) 1 dx = π/4. Thus, I = π/8.
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|.
Step-by-Step AI Explanation
Verified Method
(a) |a+b|=|b| => |a+b|²=|b|² => |a|²+|b|²+2a·b=|b|² => |a|²+2a·b=0. We want to show (a+2b)·a = 0. (a+2b)·a = a·a + 2b·a = |a|²+2a·b. From the first step, this equals 0. Hence, perpendicular. (b) |a-b|² = |a|²+|b|²-2a·b = 1+1-2cosθ = 2(1-cosθ) = 2(2sin²(θ/2)) = 4sin²(θ/2). Taking the square root, |a-b| = 2sin(θ/2), so 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).
Step-by-Step AI Explanation
Verified Method
The equation of a plane passing through the intersection of planes P1: r·(i + j + k) - 1 = 0 and P2: r·(2i + 3j - k) + 4 = 0 is given by (P1) + λ(P2) = 0. Substituting the forms, [r·(i + j + k) - 1] + λ[r·(2i + 3j - k) + 4] = 0. Grouping terms: r·((1+2λ)i + (1+3λ)j + (1-λ)k) = 1 - 4λ. Since the plane passes through (-2, 3, 1), we substitute x=-2, y=3, z=1: -2(1+2λ) + 3(1+3λ) + 1(1-λ) = 1 - 4λ. Simplifying: -2 - 4λ + 3 + 9λ + 1 - λ = 1 - 4λ, which gives 2 + 4λ = 1 - 4λ, so 8λ = -1, λ = -1/8. Substituting λ back: [r·(i+j+k) - 1] - 1/8[r·(2i+3j-k) + 4] = 0. Multiplying by 8: 8r·(i+j+k) - 8 - r·(2i+3j-k) - 4 = 0. Thus, r·(6i + 5j + 9k) = 12.
Q10
short answer
3 marks
(a) Find: ∫ e^x · sin2x dx
OR
(b) Find: ∫ (2x) / ((x² + 1)(x² + 2)) dx
Step-by-Step AI Explanation
Verified Method
Solving part (b): Let I = ∫ (2x) / ((x² + 1)(x² + 2)) dx. Let t = x², then dt = 2x dx. The integral becomes ∫ dt / ((t+1)(t+2)). Using partial fractions: 1/((t+1)(t+2)) = A/(t+1) + B/(t+2). We find A=1, B=-1. So, ∫ [1/(t+1) - 1/(t+2)] dt = log|t+1| - log|t+2| + C = log|(x²+1)/(x²+2)| + C.
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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.
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Q12
long answer
4 marks
Find the area bounded by the curves y = |x – 1| and y = 1, using integration.
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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
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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.
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