Engineering Mathematics GATE Practice Set
Mixed GATE-style questions on matrices, linear systems, eigenvalues, limits, maxima, integration, probability, distributions and statistics.
These questions mix every lesson of the Engineering Mathematics track, in the three GATE formats. Every numerical answer is recomputed by a program when the site is built. Round only at the end, to the number of decimal places asked for.
Linear algebra
[Determinant] For which value of k is the matrix [[1, 2, 3], [0, k, 4], [2, 1, 5]] singular?
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Answer: 12Expanding along the first row: 1(5k − 4) − 2(0 − 8) + 3(0 − 2k) = 5k − 4 + 16 − 6k = 12 − k. It is 0 when k = 12.
[Eigenvalues] A = [[2, 1], [1, 2]]. What is the sum of the eigenvalues of A³?
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Answer: 28A has trace 4 and determinant 3, so its eigenvalues are 1 and 3. A³ has eigenvalues 1³ = 1 and 3³ = 27, summing to 28 (which is also the trace of A³).
[Linear systems] A 4 × 4 system Ax = b has det(A) = 0 and is consistent. How many solutions does it have?
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Answer: Cdet(A) = 0 means rank(A) < 4. A consistent system with rank below the number of unknowns has free variables, hence infinitely many solutions.
Calculus
[Limits] What is limx→0 (e^(3x) − 1 − 3x) / x²?
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Answer: 4.5e^(3x) = 1 + 3x + 9x²/2 + …, so the numerator is 9x²/2 + higher terms and the limit is 9/2 = 4.5. (L'Hôpital twice gives 9e^(3x)/2 → 9/2.)
[Maxima] What is the maximum value of f(x) = x e^(−x) for x ≥ 0? Give the answer to two decimal places.
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Answer: 0.37f′(x) = e^(−x)(1 − x) = 0 at x = 1, where f″(1) = −e⁻¹ < 0. The maximum is f(1) = 1/e ≈ 0.37.
[Integration] What is ∫₀¹ x² eˣ dx? Give the answer to three decimal places.
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Answer: 0.718By parts twice: ∫ x²eˣ dx = eˣ(x² − 2x + 2). From 0 to 1: e(1 − 2 + 2) − 2 = e − 2 ≈ 0.718.
Probability and statistics
[Probability] Two cards are drawn without replacement from a standard deck. What is the probability that both are hearts? Give the answer to four decimal places.
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Answer: 0.058813/52 × 12/51 = 1/4 × 4/17 = 1/17 ≈ 0.0588.
[Bayes] Machines M1, M2 and M3 make 50%, 30% and 20% of a factory's output, with defect rates 1%, 2% and 3%. A randomly chosen item is defective. What is the probability that M3 made it? Give the answer to four decimal places.
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Answer: 0.3529P(defective) = 0.005 + 0.006 + 0.006 = 0.017. P(M3 | defective) = 0.006 / 0.017 ≈ 0.3529.
[Poisson] Calls arrive at an average rate of 2 per minute (Poisson). What is the probability of exactly one call in a given minute? Give the answer to four decimal places.
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Answer: 0.2707P(X = 1) = e⁻² × 2¹ / 1! = 2e⁻² ≈ 0.2707.
[Exponential] The lifetime of a component is exponentially distributed with mean 4 years. What is the probability that it lasts more than 4 years? Give the answer to four decimal places.
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Answer: 0.3679λ = 1/4, so P(X > 4) = e^(−4/4) = e⁻¹ ≈ 0.3679, whatever the component's age so far (memorylessness).
[Expectation] What is the variance of the number shown by a fair six-sided die? Give the answer to two decimal places.
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Answer: 2.92E[X] = 3.5 and E[X²] = 91/6, so Var(X) = 91/6 − 12.25 = 35/12 ≈ 2.92.
[Statistics] One class of 20 students has a mean score of 50 and another class of 30 students has a mean of 60. What is the mean score of all 50 students?
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Answer: 56The combined mean weights each class by its size: (20 × 50 + 30 × 60) / 50 = 2800 / 50 = 56.