NCEES Mathematics 5 — Questions and Answers
Question 1: Using Simpson's 1/3 rule with n = 2 intervals to approximate ∫₀² x² dx gives:
- 2.667 (Correct answer)
- 2.5
- 3.0
- 2.0
Correct answer: 2.667
Simpson's rule: (h/3)[f(0)+4f(1)+f(2)] = (1/3)[0+4+4] = 8/3 ≈ 2.667, matching the exact value.
Question 2: What is the rank of the matrix [[1, 2, 3], [4, 5, 6], [7, 8, 9]]?
- 2 (Correct answer)
- 3
- 1
- 0
Correct answer: 2
Row reduction shows only 2 linearly independent rows (the third is a linear combination of the first two), so rank = 2.
Question 3: In numerical integration, increasing the number of subintervals generally:
- Decreases the truncation error (Correct answer)
- Increases the truncation error
- Has no effect on accuracy
- Increases round-off error only
Correct answer: Decreases the truncation error
More subintervals means smaller step sizes, reducing truncation (approximation) error.
Question 4: The solution to the linear system Ax = b using Gaussian elimination requires the matrix A to be:
- Non-singular (invertible) (Correct answer)
- Symmetric
- Diagonal
- Positive definite
Correct answer: Non-singular (invertible)
Gaussian elimination finds a unique solution only when A is non-singular (det ≠ 0).
Question 5: The Poisson distribution is most appropriate for modeling:
- Number of events in a fixed interval of time or space (Correct answer)
- Continuous random outcomes
- Binary success/failure outcomes
- Proportions of a population
Correct answer: Number of events in a fixed interval of time or space
The Poisson distribution models the count of rare, independent events occurring in a fixed interval.
Question 6: Stokes' Theorem relates a surface integral of curl to:
- A line integral around the boundary curve (Correct answer)
- A volume integral of divergence
- Another surface integral
- A point evaluation of the function
Correct answer: A line integral around the boundary curve
Stokes' Theorem states ∬(∇×F)·dS = ∮F·dr around the bounding curve.
Question 7: What is the inverse Laplace transform of F(s) = 1/s²?
- t (Correct answer)
- 1
- e^t
- t²/2
Correct answer: t
L⁻¹{1/s²} = t, since L{t} = 1/s².
NCEES Mathematics 5
Using Simpson's 1/3 rule with n = 2 intervals to approximate ∫₀² x² dx gives: