SCE Engineering Mathematics 6

Question 1

Stokes' theorem relates a surface integral to:

Accepted Answer

A line integral around the boundary of the surface

Answer

Stokes' theorem equates the surface integral of the curl of a vector field to the line integral of the field around the boundary curve, generalizing Green's theorem to 3D.

Question 2

What is the primary purpose of singular value decomposition (SVD)?

Accepted Answer

Decomposing any matrix into orthogonal and diagonal components for analysis

Answer

SVD decomposes any m×n matrix into UΣV^T where U and V are orthogonal and Σ is diagonal. It is used in data compression, pseudoinverse computation, and principal component analysis.

Question 3

The error function erf(x) is commonly used in engineering for:

Accepted Answer

Calculating probabilities related to normal distributions

Answer

The error function erf(x) = (2/√π)∫₀ˣ e^(-t²)dt is directly related to the cumulative normal distribution and appears frequently in heat transfer and diffusion problems.

Question 4

In regression analysis, the coefficient of determination R² represents:

Accepted Answer

The proportion of variance in the dependent variable explained by the model

Answer

R² ranges from 0 to 1 and indicates how well the regression model explains the variability of the response data. An R² of 0.85 means 85% of variance is explained by the model.

Question 5

The trapezoidal rule approximation error is proportional to:

Accepted Answer

h² (the square of the step size)

Answer

The trapezoidal rule has a local error of O(h³) per interval and a global error of O(h²), making it a second-order method. Halving h reduces the global error by a factor of 4.

SCE Engineering Practice Test

SCE Engineering Practice Test

SCE Engineering Mathematics 6