## Japan-IMF Scholarship Program for Asia 2025-26 — Basic Mathematics Aptitude Test (Test A)

## Source details

**Canonical URL:** [Japan-IMF Scholarship Program for Asia 2025-26 — Basic Mathematics Aptitude Test (Test A)](https://www.imf.org/-/media/files/oap/jispa/2025/jispa-math-test-2025.pdf)

## Other formats

- [Markdown version](/-/media/files/oap/jispa/2025/jispa-math-test-2025.pdf.md)
- [Structured JSON version](/-/media/files/oap/jispa/2025/jispa-math-test-2025.pdf.json)

---

### Instructions
- You have 60 minutes to complete
- Calculators are not allowed
- Please show all your work and write your answers in the designated space
- Fields: Country; Reference Number; Name

### Problems
- Problem 1. Compute the following
  - 100 / 2 + 5
  - Answer:

- Problem 2. Suppose that Y = K^α L^(1−α). Which of the following is true?
  - Answer:

- Problem 3. Suppose that y = x^2 − 2ax. Which of the following is equivalent to this equation:
  - Answer:

- Problem 4. Solve for x:
  - 1 / x = 3 / (1 − x)
  - Answer:

- Problem 5. Note that e is Euler’s constant. Solve for x:
  - e^(6x−10) / e^x = 1
  - Answer:

- Problem 6. Solve for x:
  - (1 + x)^(2/(1 + x)) = 3
  - Answer:

- Problem 7. Solve for x:
  - 2 ln (1 + x) − ln (1 + x) = ln (3)
  - Answer:

- Problem 8. Find a first order Taylor expansion (nearby x = 0) of f(x) = ln(1 + x). Using this, approximate ln(1 + .02)
  - Answer:

- Problem 9. Suppose there is a box containing 50 red balls and 50 green balls. A ball is removed and then set aside (and not returned to the box). Which of the following is true?
  - Answer:

- Problem 10. Suppose that we have the set of numbers:
  - X = [87, 63, 40, 76, 2, 28, 6, 30, 25, 49]
  - The average value of this set of numbers is x̄ = 40.6. Compute the following, where x_i ∈ X, Σ_{i=1}^{10} x_i.
  - Answer:

- Problem 11. Suppose that we have the set of numbers:
  - X = [87, 63, 40, 76, 2, 28, 6, 30, 25, 49]
  - The average value of this set of numbers is x̄ = 40.6. Compute the following, where x_i ∈ X, Σ_{i=1}^{10} (x_i − x̄)
  - Answer:

- Problem 12. Compute the derivative of the function
  - f(x) = x ln (x) + 1
  - Answer:

- Problem 13. Suppose that y(t) = x(t) / z(t) where x(t) and z(t) are functions of time
  - x(t) = x_0 e^{g_x t}, z(t) = z_0 e^{g_z t} where g_x and g_z are constants. Compute g_y ≡ d ln y(t) / dt
  - Answer:

- Problem 14. Suppose that
  - ∫_{−∞}^{∞} f(x) dx = 1
  - ∫_{−∞}^{0} f(x) dx = 0.2
  - Compute ∫_{0}^{∞} f(x) dx
  - Answer:

- Problem 15. Note that Φ(X) ≡ ∫_{−∞}^{X} (1 / √(2π)) e^{−1/2 (z^2)} dz. Evaluate the following integral:
  - Y ≡ ∫_{−∞}^{X} (1 / √(2π)) (1 / σ) e^{−1/2 ((x − μ)/σ)^2} dx (1)
  - Answer:

- Problem 16. Find the value of x in which
  - φ(x) = (1 / √(2π)) (1 / σ) e^{−1/2 ((x − μ)/σ)^2}, σ > 0
  - is maximized:
  - Answer:

- Problem 17. Suppose that A = [1   2   3]. Compute A A^T:
  - Answer:

- Problem 18. Suppose that
  - [y_1 y_2]^T = [1   2; 3   5] [x_1 x_2]^T.
  - Write y_1 as a function of x_1 and x_2:
  - Answer:

- Problem 19. If possible, solve for the values of x_1, x_2, and x_3 in which
  - [ [1    5    10]; [2   10   20]; [3    01] ] [x_1 x_2 x_3]^T = [1 3 5]^T
  - Answer:

- Problem 20. Evaluate the limit (use L’Hôpital’s rule)
  - lim_{x→∞} x ln (1 + 1/x)
  - Answer:

---


_Source: https://www.imf.org/-/media/files/oap/jispa/2025/jispa-math-test-2025.pdf_