SCE Engineering Mathematics 1

Question 1

What is the Laplace transform of the function f(t) = e^(-at)?

Accepted Answer

1/(s+a)

Answer

The Laplace transform of e^(-at) is 1/(s+a) for s > -a, which is a fundamental transform used extensively in control systems and circuit analysis.

Question 2

A matrix A has eigenvalues 2 and 5. What is the determinant of A?

Accepted Answer

10

Answer

The determinant of a matrix equals the product of its eigenvalues. Therefore det(A) = 2 × 5 = 10.

Question 3

Which numerical method uses the formula x_{n+1} = x_n - f(x_n)/f'(x_n)?

Accepted Answer

Newton-Raphson method

Answer

The Newton-Raphson method iteratively refines a root estimate using the function value and its derivative, converging quadratically near simple roots.

Question 4

What is the divergence of the vector field F = (x², xy, xz)?

Accepted Answer

2x + x + x = 4x

Answer

Divergence is ∂F₁/∂x + ∂F₂/∂y + ∂F₃/∂z = 2x + x + x = 4x. This measures the rate of flux expansion at a point.

Question 5

For the differential equation dy/dx + 2y = e^(-x), what is the integrating factor?

Accepted Answer

e^(2x)

Answer

For a first-order linear ODE dy/dx + P(x)y = Q(x), the integrating factor is e^(∫P(x)dx) = e^(∫2dx) = e^(2x).

SCE Engineering Practice Test

SCE Engineering Practice Test

SCE Engineering Mathematics 1