Class 12 Mathematics and Statistics Part 2 – Daily Revision MCQ (English Medium)
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If y = xˣ (x > 0), then dy/dx = ?
Chapter 1: Differentiation
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Answer: C. xˣ(1 + log x)
Take log: log y = x log x; differentiate: (1/y) dy/dx = log x + 1, so dy/dx = xˣ(1 + log x).If y = e²ˣ, then d²y/dx² = ?
Chapter 1: Differentiation
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Answer: B. 4e²ˣ
dy/dx = 2e²ˣ and differentiating again gives d²y/dx² = 4e²ˣ.For f(x) = x² − 4x + 3 on [1, 3], what value of c satisfies Rolle's theorem?
Chapter 2: Applications of Derivatives
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Answer: A. 2
f(1) = f(3) = 0, and f′(c) = 2c − 4 = 0 gives c = 2, which lies in (1, 3).On which interval is f(x) = 2x³ − 3x² decreasing?
Chapter 2: Applications of Derivatives
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Answer: D. (0, 1)
f′(x) = 6x² − 6x = 6x(x − 1), which is negative only for 0 < x < 1.∫ log x dx = ?
Chapter 3: Indefinite Integration
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Answer: C. x log x − x + c
Integrate by parts taking 1 as the second function: x log x − ∫ x·(1/x) dx = x log x − x + c.Using the properties of definite integrals, ∫ from 0 to π/2 of sin x/(sin x + cos x) dx = ?
Chapter 4: Definite Integration
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Answer: B. π/4
Replacing x by π/2 − x gives the same integral with cos x on top; adding both, 2I = ∫ from 0 to π/2 of 1 dx = π/2, so I = π/4.What is the area of the region enclosed between the line y = x and the parabola y = x²?
Chapter 5: Application of Definite Integration
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Answer: D. 1/6 sq. unit
They meet at x = 0 and x = 1; area = ∫ from 0 to 1 of (x − x²) dx = 1/2 − 1/3 = 1/6.What is the general solution of the differential equation dy/dx = eˣ⁻ʸ?
Chapter 6: Differential Equations
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Answer: A. eʸ = eˣ + c
Separate the variables: eʸ dy = eˣ dx; integrating both sides gives eʸ = eˣ + c.A random variable X takes the values 0, 1 and 2 with probabilities 0.2, 0.5 and 0.3. What is E(X)?
Chapter 7: Probability Distributions
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Answer: B. 1.1
E(X) = Σ x·P(x) = 0(0.2) + 1(0.5) + 2(0.3) = 0.5 + 0.6 = 1.1.A binomial variable X has mean 6 and variance 2. What are n and p?
Chapter 8: Binomial Distribution
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Answer: A. n = 9, p = 2/3
np = 6 and npq = 2 give q = 2/6 = 1/3, so p = 2/3 and n = 6 ÷ (2/3) = 9.
Class 12: MCQ in other subjects
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All questions are written by PadhaiNotes in our own words from the chapters of the official Maharashtra State Board Balbharati textbook (ebalbharati.in). Each question shows its chapter.