University Mathematics practice

What is on the worksheet

1. A linear map from R^4 to R^4 is injective. What is its rank?

2. Find det A for A = [[7,10,12],[0,5,22],[0,0,4]]. Round the final result to 4 decimal places if needed.

3. For f(x,y) = 2x²y + 9y³, find ∂f/∂x at (-3, 6). Round the final result to 4 decimal places if needed.

4. Solve 8x + y = 32 and x − 7y = 4. Enter x only. Round the final result to 4 decimal places if needed.

5. Let X be binomial with n = 101 and success probability p = 0.2. Find Var(X). Round the final result to 4 decimal places if needed.

6. Let u = 1x² − y² and v = 5xy. Calculate the signed Jacobian determinant ∂(u,v)/∂(x,y) at (8,7). Round the final result to 4 decimal places if needed.

7. Let F = ∇φ with φ(x,y) = 12x² + 4y². Find the line integral of F from (0,0) to (3,6). Round the final result to 4 decimal places if needed.

8. Apply one Newton iteration to f(x) = x² − 59, starting at x₀ = 19. Enter x₁. Round the final result to 4 decimal places if needed.

9. In an abstract classifier, P(A)=0.5, P(B|A)=0.5, and P(B|not A)=0.4. Calculate P(A|B). Round the final result to 4 decimal places if needed.

10. Solve y′ = −0.8y with y(0) = 41, then calculate y(2). Round the final result to 4 decimal places if needed.

11. Let X be binomial with n = 149 and success probability p = 0.8. Find Var(X). Round the final result to 4 decimal places if needed.

12. Integrate 9xy over the rectangle 0 ≤ x ≤ 2, 0 ≤ y ≤ 6. Round the final result to 4 decimal places if needed.

13. For F = (7y², 9x², 0), calculate the z-component of curl F at (-4,4,0). Round the final result to 4 decimal places if needed.

14. For A = [[3,3],[3,3]], find the eigenvalue belonging to v = (1,1). Round the final result to 4 decimal places if needed.

15. For f(x,y) = 9x²y + 3y³, find ∂f/∂x at (-6, 9). Round the final result to 4 decimal places if needed.