SCE Engineering Mathematics 2

Question 1

What is the rank of a 3×3 matrix with two identical rows?

Accepted Answer

At most 2

Answer

Two identical rows are linearly dependent, reducing the maximum number of linearly independent rows to at most 2. The rank could be lower if additional dependencies exist.

Question 2

Simpson's 1/3 rule requires how many equally spaced points minimum?

Accepted Answer

3 points (2 intervals)

Answer

Simpson's 1/3 rule fits a quadratic polynomial through 3 points, requiring a minimum of 2 equal subintervals. It provides exact results for polynomials up to degree 3.

Question 3

What is the gradient of the scalar field φ = x²y + yz²?

Accepted Answer

(2xy, x² + z², 2yz)

Answer

The gradient is (∂φ/∂x, ∂φ/∂y, ∂φ/∂z) = (2xy, x² + z², 2yz). The gradient points in the direction of steepest increase.

Question 4

A system of linear equations Ax = b has no solution when:

Accepted Answer

rank(A) < rank([A|b])

Answer

By the Rouché–Capelli theorem, a system is inconsistent (no solution) when the rank of the coefficient matrix is less than the rank of the augmented matrix.

Question 5

The second derivative test classifies a critical point as a saddle point when:

Accepted Answer

D = f_xx·f_yy - (f_xy)² < 0

Answer

When the discriminant D is negative, the critical point is a saddle point regardless of the sign of f_xx, indicating the surface curves up in one direction and down in another.

SCE Engineering Practice Test

SCE Engineering Practice Test

SCE Engineering Mathematics 2