SCE Engineering Mathematics 2 — Questions and Answers
Question 1: What is the rank of a 3×3 matrix with two identical rows?
- At most 2 (Correct answer)
- Always 3
- Always 1
- Cannot be determined
Correct answer: At most 2
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?
- 3 points (2 intervals) (Correct answer)
- 2 points (1 interval)
- 4 points (3 intervals)
- 5 points (4 intervals)
Correct answer: 3 points (2 intervals)
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?
- (2xy, x² + z², 2yz) (Correct answer)
- (x²y, yz², 0)
- (2x, 1, 2z)
- (2xy, x², 2yz)
Correct answer: (2xy, x² + z², 2yz)
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:
- rank(A) < rank([A|b]) (Correct answer)
- rank(A) = rank([A|b])
- det(A) ≠ 0
- A is square
Correct answer: rank(A) < rank([A|b])
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:
- D = f_xx·f_yy - (f_xy)² < 0 (Correct answer)
- D > 0 and f_xx > 0
- D > 0 and f_xx < 0
- D = 0
Correct answer: D = f_xx·f_yy - (f_xy)² < 0
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.
Question 6: What is the order of the Runge-Kutta method commonly known as RK4?
- Fourth order (Correct answer)
- Second order
- Third order
- First order
Correct answer: Fourth order
RK4 is a fourth-order method, meaning its local truncation error is O(h⁵) and global error is O(h⁴), providing excellent accuracy for solving ordinary differential equations.
SCE Engineering Mathematics 2
What is the rank of a 3×3 matrix with two identical rows?