SCE Engineering Mathematics 3 — Questions and Answers
Question 1: The probability density function of a normal distribution is characterized by which two parameters?
- Mean (μ) and standard deviation (σ) (Correct answer)
- Mean and mode
- Median and range
- Variance and skewness
Correct answer: Mean (μ) and standard deviation (σ)
The normal (Gaussian) distribution is fully described by its mean μ (center) and standard deviation σ (spread). This distribution is fundamental in engineering statistics and quality control.
Question 2: What is the curl of a gradient field?
- Always zero (Correct answer)
- The divergence
- The Laplacian
- A scalar field
Correct answer: Always zero
The curl of any gradient field is always zero (∇ × ∇φ = 0). This is a fundamental identity in vector calculus, meaning conservative fields are irrotational.
Question 3: In the z-transform, what does z⁻¹ represent?
- A unit delay (Correct answer)
- A unit advance
- Integration
- Differentiation
Correct answer: A unit delay
In discrete-time signal processing and control, z⁻¹ represents a unit delay operator, shifting a sequence by one sample period.
Question 4: What condition must be satisfied for a function to have a convergent Taylor series?
- The function must be infinitely differentiable and the remainder must approach zero (Correct answer)
- The function must be continuous only
- The function must be periodic
- The function must be bounded
Correct answer: The function must be infinitely differentiable and the remainder must approach zero
A function needs to be infinitely differentiable (smooth) at the point, and the Taylor remainder term must converge to zero for the series to converge to the function value.
Question 5: The method of Lagrange multipliers is used to:
- Optimize a function subject to equality constraints (Correct answer)
- Solve systems of linear equations
- Find eigenvalues of a matrix
- Compute definite integrals
Correct answer: Optimize a function subject to equality constraints
Lagrange multipliers introduce auxiliary variables to convert a constrained optimization problem into an unconstrained system, finding extrema on constraint surfaces.
Question 6: What is the trace of a 3×3 identity matrix?
- 3 (Correct answer)
- 1
- 0
- 9
Correct answer: 3
The trace is the sum of diagonal elements. The 3×3 identity matrix has 1s on the diagonal, so trace = 1 + 1 + 1 = 3.
SCE Engineering Mathematics 3
The probability density function of a normal distribution is characterized by which two parameters?