BMath Bachelor of Mathematics Bachelor of Mathematics: Applied Mathematics Problem Solving 2 — Questions and Answers
Question 1: A mass-spring system obeys m·x'' + kx = 0. If m = 4 kg and k = 16 N/m, what is the angular frequency ω?
- 2 rad/s (Correct answer)
- 4 rad/s
- 8 rad/s
- 1 rad/s
Correct answer: 2 rad/s
ω = √(k/m) = √(16/4) = √4 = 2 rad/s.
Question 2: Using Euler's method with step size h = 0.1 to solve y' = y, y(0) = 1, what is the approximation for y(0.1)?
- 1.1 (Correct answer)
- 1.0
- 1.2
- 1.105
Correct answer: 1.1
Euler's step: y₁ = y₀ + h·f(x₀,y₀) = 1 + 0.1·1 = 1.1.
Question 3: What is the Fourier series representation of an odd function on [-π, π]?
- It contains only sine terms (Correct answer)
- It contains only cosine terms
- It contains both sine and cosine terms
- It contains only constant terms
Correct answer: It contains only sine terms
An odd function satisfies f(-x) = -f(x), so all cosine (even) coefficients vanish, leaving only sine terms.
Question 4: The Laplace transform of e^(at) is:
- 1/(s - a) (Correct answer)
- 1/(s + a)
- a/(s² + a²)
- s/(s² - a²)
Correct answer: 1/(s - a)
L{e^(at)} = ∫₀^∞ e^(at)e^(-st)dt = 1/(s-a) for s > a.
Question 5: In linear programming, the optimal solution of a bounded feasible region always occurs at:
- A vertex (corner point) of the feasible region (Correct answer)
- The centroid of the feasible region
- Any point on the boundary
- The origin
Correct answer: A vertex (corner point) of the feasible region
The fundamental theorem of linear programming states the optimum is attained at a vertex of the convex feasible region.
Question 6: A random variable X has PDF f(x) = 2x on [0,1]. What is E[X]?
- 2/3 (Correct answer)
- 1/2
- 1/3
- 3/4
Correct answer: 2/3
E[X] = ∫₀¹ x·2x dx = 2∫₀¹ x² dx = 2·(1/3) = 2/3.
Question 7: The gradient of f(x,y) = x²y + y³ at the point (1, 2) is:
- (4, 13) (Correct answer)
- (4, 1)
- (2, 13)
- (1, 4)
Correct answer: (4, 13)
∇f = (2xy, x² + 3y²); at (1,2): (2·1·2, 1 + 3·4) = (4, 13).
BMath Bachelor of Mathematics Bachelor of Mathematics: Applied Mathematics Problem Solving 2
A mass-spring system obeys m·x'' + kx = 0.
If m = 4 kg and k = 16 N/m, what is the angular frequency ω?