EIT Mathematics & Engineering Fundamentals 2 — Questions and Answers
Question 1: The Laplace transform of f(t) = t·e^(2t) is:
- 1/(s-2)^2 (Correct answer)
- 1/(s+2)^2
- 2/(s-2)^2
- 1/(s-2)
Correct answer: 1/(s-2)^2
Using the shifting theorem, L{t·e^(at)} = 1/(s-a)^2, so with a=2 the result is 1/(s-2)^2.
Question 2: A matrix A is singular if and only if:
- det(A) = 0 (Correct answer)
- trace(A) = 0
- A = A^T
- rank(A) = n
Correct answer: det(A) = 0
A singular matrix has a zero determinant, meaning it has no inverse and its rows/columns are linearly dependent.
Question 3: Which numerical method has the fastest convergence rate for finding roots?
- Newton-Raphson (Correct answer)
- Bisection
- False Position
- Fixed-Point Iteration
Correct answer: Newton-Raphson
Newton-Raphson converges quadratically (order 2), making it faster than bisection and false position near the root.
Question 4: The curl of a conservative vector field F is:
- Zero everywhere (Correct answer)
- A non-zero constant
- Equal to the gradient of F
- Undefined
Correct answer: Zero everywhere
A conservative field can be expressed as the gradient of a scalar potential, and the curl of any gradient is identically zero.
Question 5: For a second-order linear ODE with constant coefficients, if the characteristic equation has repeated real roots r, the general solution is:
- (C1 + C2·t)·e^(rt) (Correct answer)
- C1·e^(rt) + C2·e^(rt)
- C1·cos(rt) + C2·sin(rt)
- C1·e^(rt) + C2·t
Correct answer: (C1 + C2·t)·e^(rt)
Repeated roots require the second independent solution to be multiplied by t to avoid linear dependence.
Question 6: In probability, if P(A) = 0.4 and P(B) = 0.3 and events A and B are independent, what is P(A ∩ B)?
- 0.12 (Correct answer)
- 0.70
- 0.58
- 0.10
Correct answer: 0.12
For independent events, P(A ∩ B) = P(A)·P(B) = 0.4 × 0.3 = 0.12.
Question 7: The Taylor series expansion of e^x about x = 0 truncated after the cubic term is:
- 1 + x + x²/2 + x³/6 (Correct answer)
- 1 + x + x² + x³
- x + x²/2 + x³/6
- 1 + x + x²/2! + x³/3
Correct answer: 1 + x + x²/2 + x³/6
The Maclaurin series for e^x is Σ(x^n/n!), so the first four terms are 1 + x + x²/2! + x³/3! = 1 + x + x²/2 + x³/6.
EIT Mathematics & Engineering Fundamentals 2
The Laplace transform of f(t) = t·e^(2t) is: