EIT Mathematics & Engineering Fundamentals 4 — Questions and Answers
Question 1: The partial derivative ∂/∂x of f(x,y) = x²y + sin(xy) is:
- 2xy + y·cos(xy) (Correct answer)
- x² + x·cos(xy)
- 2xy + cos(xy)
- 2x + y·cos(xy)
Correct answer: 2xy + y·cos(xy)
Treating y as constant: ∂/∂x(x²y) = 2xy and ∂/∂x(sin(xy)) = y·cos(xy) by the chain rule.
Question 2: For the Poisson distribution with mean λ, the variance is:
- λ (Correct answer)
- λ²
- √λ
- 1/λ
Correct answer: λ
A unique property of the Poisson distribution is that both its mean and variance equal the parameter λ.
Question 3: The rank of a matrix is defined as the:
- Number of linearly independent rows (or columns) (Correct answer)
- Number of rows minus number of columns
- Value of the determinant
- Number of zero eigenvalues
Correct answer: Number of linearly independent rows (or columns)
Rank equals the maximum number of linearly independent row vectors (or equivalently column vectors) in the matrix.
Question 4: What does a Bode plot display?
- Magnitude and phase of a transfer function vs. log frequency (Correct answer)
- Time-domain step response
- Poles and zeros on the complex plane
- Frequency vs. amplitude in linear scale
Correct answer: Magnitude and phase of a transfer function vs. log frequency
A Bode plot consists of two graphs: magnitude (in dB) and phase (in degrees) plotted against logarithmic frequency.
Question 5: The inverse Laplace transform of 1/(s² + ω²) is:
- sin(ωt)/ω (Correct answer)
- cos(ωt)
- sin(ωt)
- e^(ωt)
Correct answer: sin(ωt)/ω
From standard Laplace transform tables, L{sin(ωt)} = ω/(s²+ω²), so L⁻¹{1/(s²+ω²)} = sin(ωt)/ω.
Question 6: In regression analysis, the coefficient of determination R² represents:
- The proportion of variance in y explained by x (Correct answer)
- The slope of the regression line
- The correlation coefficient squared only when negative
- The sum of squared residuals
Correct answer: The proportion of variance in y explained by x
R² ranges from 0 to 1 and measures how well the independent variable(s) explain the variability of the dependent variable.
Question 7: Stoke's theorem relates the surface integral of the curl of F to:
- The line integral of F around the bounding curve (Correct answer)
- The volume integral of F
- The surface integral of F itself
- The divergence of F over the surface
Correct answer: The line integral of F around the bounding curve
Stokes' theorem states ∫∫(∇×F)·dS = ∮F·dr, linking the curl flux through a surface to circulation around its boundary.
EIT Mathematics & Engineering Fundamentals 4
The partial derivative ∂/∂x of f(x,y) = x²y + sin(xy) is: