EIT Mathematics & Engineering Fundamentals 5 — Questions and Answers
Question 1: Which interpolation method fits a polynomial of degree n-1 exactly through n data points?
- Lagrange interpolation (Correct answer)
- Linear interpolation
- Cubic spline interpolation
- Newton's divided difference (linear only)
Correct answer: Lagrange interpolation
Lagrange interpolation constructs a unique polynomial of degree at most n-1 that passes exactly through all n given data points.
Question 2: If A and B are mutually exclusive events, then P(A ∪ B) equals:
- P(A) + P(B) (Correct answer)
- P(A)·P(B)
- P(A) + P(B) - P(A)·P(B)
- P(A) - P(B)
Correct answer: P(A) + P(B)
Mutually exclusive events cannot occur simultaneously, so P(A ∩ B) = 0 and the addition rule simplifies to P(A) + P(B).
Question 3: The determinant of a 2×2 matrix [[a, b],[c, d]] is:
- ad - bc (Correct answer)
- ab - cd
- ac - bd
- ad + bc
Correct answer: ad - bc
The 2×2 determinant is computed as the product of the main diagonal minus the product of the anti-diagonal: ad - bc.
Question 4: The solution to the first-order linear ODE dy/dt + 2y = 0 with y(0) = 5 is:
- y = 5e^(-2t) (Correct answer)
- y = 5e^(2t)
- y = 5 - 2t
- y = e^(-2t) + 5
Correct answer: y = 5e^(-2t)
Separating variables gives dy/y = -2dt, integrating yields ln|y| = -2t + C, and applying y(0) = 5 gives y = 5e^(-2t).
Question 5: The trapezoidal rule approximates a definite integral by treating the integrand as:
- A straight line between adjacent points (Correct answer)
- A parabola through three points
- A constant over each interval
- An exponential function
Correct answer: A straight line between adjacent points
The trapezoidal rule connects adjacent function values with straight lines, computing the area of resulting trapezoids.
Question 6: For a complex number z = a + bi, the modulus |z| is:
- √(a² + b²) (Correct answer)
- a + b
- a² + b²
- √(a - b)
Correct answer: √(a² + b²)
The modulus is the distance from the origin in the complex plane, given by √(a² + b²) by the Pythagorean theorem.
Question 7: Which property distinguishes an exact differential equation M(x,y)dx + N(x,y)dy = 0?
- ∂M/∂y = ∂N/∂x (Correct answer)
- M + N = 0
- ∂M/∂x = ∂N/∂y
- M·N = constant
Correct answer: ∂M/∂y = ∂N/∂x
An equation is exact when the cross-partial derivatives are equal: ∂M/∂y = ∂N/∂x, guaranteeing existence of a potential function F where dF = Mdx + Ndy.
EIT Mathematics & Engineering Fundamentals 5
Which interpolation method fits a polynomial of degree n-1 exactly through n data points?