GATE Engineering Mathematics & Analytical Skills 3 — Questions and Answers
Question 1: The rank of a matrix A is defined as:
- The number of rows in A
- The maximum number of linearly independent rows (or columns) (Correct answer)
- The determinant of A
- The trace of A
Correct answer: The maximum number of linearly independent rows (or columns)
Rank is the dimension of the column space (or row space) of a matrix, equal to the max number of linearly independent rows or columns.
Question 2: If ∫₀^∞ e^(-st) f(t) dt = F(s), then ∫₀^∞ e^(-st) f'(t) dt equals:
- sF(s) - f(0) (Correct answer)
- sF(s) + f(0)
- F(s)/s
- F'(s)
Correct answer: sF(s) - f(0)
Using integration by parts, the Laplace transform of a derivative is L{f'(t)} = sF(s) - f(0).
Question 3: A bag contains 3 red and 5 blue balls. Two balls are drawn without replacement. What is the probability both are red?
- 9/64
- 3/28 (Correct answer)
- 6/56
- 1/8
Correct answer: 3/28
P = (3/8) × (2/7) = 6/56 = 3/28.
Question 4: The solution to the differential equation dy/dx + y = e^(-x) with y(0)=0 is:
- y = xe^(-x) (Correct answer)
- y = e^(-x) - e^(-2x)
- y = x·e^x
- y = (1-x)e^(-x)
Correct answer: y = xe^(-x)
The integrating factor is e^x; solving gives y = xe^(-x) after applying the initial condition.
Question 5: Which of the following is NOT a property of a Poisson distribution?
- Mean equals variance
- Events occur independently
- The distribution is symmetric about the mean (Correct answer)
- Describes rare event counts in a fixed interval
Correct answer: The distribution is symmetric about the mean
Poisson distributions are right-skewed (not symmetric), though they approach symmetry as λ → ∞.
Question 6: Using Simpson's 1/3 rule, ∫₀^2 x² dx with n=2 subintervals gives:
- 2.667 (Correct answer)
- 2.333
- 3.000
- 2.500
Correct answer: 2.667
Simpson's 1/3: (h/3)[f(0)+4f(1)+f(2)] = (1/3)[0+4+4] = 8/3 ≈ 2.667, matching the exact value.
Question 7: The divergence theorem relates a surface integral to:
- A line integral over the boundary curve
- A volume integral over the enclosed region (Correct answer)
- Another surface integral
- A path-independent scalar field
Correct answer: A volume integral over the enclosed region
Gauss's divergence theorem states ∯F·dA = ∭(∇·F)dV, converting a closed surface integral to a volume integral.
GATE Engineering Mathematics & Analytical Skills 3
The rank of a matrix A is defined as: