GATE Engineering Mathematics & Analytical Skills 2 — Questions and Answers
Question 1: The eigenvalues of a real symmetric matrix are always:
- Complex conjugates
- Real (Correct answer)
- Purely imaginary
- Zero
Correct answer: Real
Real symmetric matrices always have real eigenvalues, a fundamental result in linear algebra.
Question 2: If f(x) = x³ - 3x² + 2x, what are the critical points of f?
- x = 0 and x = 2
- x = 1 and x = 1/3
- x = 1 and x = 2/3 (Correct answer)
- x = 0 and x = 1
Correct answer: x = 1 and x = 2/3
Setting f'(x) = 3x² - 6x + 2 = 0 gives x = (6 ± √(36-24))/6 = 1 ± 1/√3, which simplifies to x = 1 and x = 2/3 approximately.
Question 3: The probability that a fair die shows an even number given it shows a number greater than 3 is:
- 1/2
- 2/3 (Correct answer)
- 1/3
- 3/4
Correct answer: 2/3
Numbers greater than 3 are {4,5,6}; even ones are {4,6}, so P = 2/3.
Question 4: The Laplace transform of t·e^(at) is:
- 1/(s-a)² (Correct answer)
- 1/(s+a)
- s/(s-a)²
- 1/(s-a)
Correct answer: 1/(s-a)²
Using the first shifting theorem and L{t} = 1/s², L{t·e^(at)} = 1/(s-a)².
Question 5: Which numerical method has the highest order of convergence for finding roots?
- Bisection method
- Secant method
- Newton-Raphson method (Correct answer)
- Regula Falsi method
Correct answer: Newton-Raphson method
Newton-Raphson has quadratic (second-order) convergence, faster than the others listed.
Question 6: For a standard normal distribution, approximately what percentage of data lies within two standard deviations of the mean?
- 68%
- 95% (Correct answer)
- 99.7%
- 90%
Correct answer: 95%
The empirical rule states approximately 95% of data lies within ±2σ of the mean.
Question 7: The curl of a gradient field ∇f is:
- ∇²f
- A non-zero vector
- Zero vector (Correct answer)
- ∇f itself
Correct answer: Zero vector
For any scalar field f with continuous second-order partial derivatives, ∇×(∇f) = 0.
GATE Engineering Mathematics & Analytical Skills 2
The eigenvalues of a real symmetric matrix are always: