SCE Engineering Mathematics 4 — Questions and Answers
Question 1: Green's theorem relates a line integral around a closed curve to:
- A double integral over the enclosed region (Correct answer)
- A triple integral over a volume
- A surface integral
- Another line integral
Correct answer: A double integral over the enclosed region
Green's theorem converts a line integral around a simple closed curve C to a double integral over the plane region D bounded by C, connecting circulation and flux to area integrals.
Question 2: What is the condition number of a matrix used for?
- Measuring sensitivity of the solution to input perturbations (Correct answer)
- Finding eigenvalues
- Computing the determinant
- Performing LU decomposition
Correct answer: Measuring sensitivity of the solution to input perturbations
The condition number measures how much the output changes for a small change in input. A large condition number indicates an ill-conditioned matrix where solutions are numerically unstable.
Question 3: The Gauss-Seidel iterative method differs from Jacobi's method in that it:
- Uses updated values as soon as they are computed (Correct answer)
- Always converges faster
- Requires a symmetric matrix
- Uses matrix inversion
Correct answer: Uses updated values as soon as they are computed
Gauss-Seidel immediately uses newly computed values in subsequent equations within the same iteration, often leading to faster convergence than Jacobi's method.
Question 4: What is the Wronskian used for in differential equations?
- Testing linear independence of solutions (Correct answer)
- Finding particular solutions
- Computing eigenvalues
- Solving boundary value problems
Correct answer: Testing linear independence of solutions
The Wronskian determinant tests whether a set of functions are linearly independent. If the Wronskian is non-zero, the functions form a fundamental set of solutions.
Question 5: A Poisson distribution is most appropriate for modeling:
- The number of events in a fixed interval when events occur independently at a constant rate (Correct answer)
- Continuous measurements with symmetric distribution
- The time between successive events
- Binary outcomes in repeated trials
Correct answer: The number of events in a fixed interval when events occur independently at a constant rate
The Poisson distribution models the count of independent events occurring at a constant average rate in a fixed interval of time or space, such as defects per unit or arrivals per hour.
Question 6: The convolution integral f*g is commutative, meaning:
- f*g = g*f (Correct answer)
- f*g = f·g
- f*g = f + g
- f*g ≠ g*f in general
Correct answer: f*g = g*f
Convolution is commutative: the order of the two functions does not matter. This property is important in signal processing and systems analysis.
SCE Engineering Mathematics 4
Green's theorem relates a line integral around a closed curve to: