EIT Mathematics and Statistics 5 — Questions and Answers
Question 1: The Fourier series of a periodic function requires which condition on the function over its period?
- The function must be continuous everywhere
- The function must satisfy Dirichlet conditions (bounded, finite extrema, finite discontinuities) (Correct answer)
- The function must be differentiable everywhere
- The function must have zero mean
Correct answer: The function must satisfy Dirichlet conditions (bounded, finite extrema, finite discontinuities)
Dirichlet conditions — bounded variation, finite number of maxima/minima and discontinuities — are sufficient for Fourier series convergence.
Question 2: For a binomial distribution with n = 10 trials and p = 0.3, the mean number of successes is:
- 3.0 (Correct answer)
- 2.1
- 7.0
- 0.3
Correct answer: 3.0
The mean of a binomial distribution is μ = np = 10 × 0.3 = 3.0.
Question 3: What is the rank of the matrix [[1,2,3],[2,4,6],[0,0,0]]?
- 1 (Correct answer)
- 2
- 3
- 0
Correct answer: 1
Row 2 is twice row 1, and row 3 is zero, so only one row is linearly independent, giving rank = 1.
Question 4: The trapezoidal rule for numerical integration approximates the area under a curve as a series of:
- Rectangles using midpoint heights
- Trapezoids connecting consecutive data points (Correct answer)
- Parabolic arcs through three points
- Cubic splines
Correct answer: Trapezoids connecting consecutive data points
The trapezoidal rule connects consecutive function values with straight lines, forming trapezoids whose areas are summed.
Question 5: The gradient of f(x,y) = x²y + 3y² evaluated at (1, 2) is:
- (4, 13) (Correct answer)
- (4, 1)
- (2, 13)
- (1, 7)
Correct answer: (4, 13)
∂f/∂x = 2xy = 4, ∂f/∂y = x² + 6y = 1 + 12 = 13, so ∇f(1,2) = (4, 13).
Question 6: A chi-square goodness-of-fit test is used to determine whether:
- Two means are equal
- Observed frequencies match expected frequencies from a theoretical distribution (Correct answer)
- Two variances are equal
- A dataset is normally distributed using the z-test
Correct answer: Observed frequencies match expected frequencies from a theoretical distribution
The chi-square test compares observed counts to expected counts to assess how well data fits a hypothesized distribution.
Question 7: What is the solution to the system of equations: 2x + y = 5 and x - y = 1?
- x = 2, y = 1 (Correct answer)
- x = 3, y = -1
- x = 1, y = 3
- x = 2, y = -1
Correct answer: x = 2, y = 1
Adding the equations: 3x = 6, so x = 2; substituting back: y = 5 - 2(2) = 1.
EIT Mathematics and Statistics 5
The Fourier series of a periodic function requires which condition on the function over its period?