## Japan-IMF Scholarship Program for Asia 2026-2027 — Basic Mathematics Aptitude Test (Test A, Answer sheet)

## Source details

**Canonical URL:** [Japan-IMF Scholarship Program for Asia 2026-2027 — Basic Mathematics Aptitude Test (Test A, Answer sheet)](https://www.imf.org/-/media/files/oap/jispa/2026/revamp/math-test-2026.pdf)

## Other formats

- [Markdown version](/-/media/files/oap/jispa/2026/revamp/math-test-2026.pdf.md)
- [Structured JSON version](/-/media/files/oap/jispa/2026/revamp/math-test-2026.pdf.json)

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### Instructions
- You have 60 minutes to complete
- Calculators are not allowed

### Problems and answer fields
- Country:
- Reference Number:
- Name:

- Problem 1. Compute the following
  - 100
    4
    + 5
  - Answer:

- Problem 2. Suppose that
  - Y = K^α L^{1−α}
  - Answer:

- Problem 3. Suppose that
  - y = x^2 + a x
  - Answer:

- Problem 4. Solve for x:
  - x^{n−1} / x^n = 2 / (1−x)
  - Answer:

- Problem 5. Note that e is Euler’s constant. Solve for x:
  - e^{5x−10} = 1
  - Answer:

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

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

- Problem 8. Find a first order Taylor expansion (nearby x = 36) for the function
  - f(x) = 6 √x
  - Using this, approximate 6 √38
  - Answer:

- Problem 9. Suppose that two cyclists are biking towards each other. Both cyclists maintain a constant and equal speed of 10 kilometers per hour. The cyclists begin 100 kilometers apart. An insect flies between the cyclists at a constant speed of 40 kilometers per hour. Assume that as the insect reaches one of the cyclist, it immediately turns around (with no change in speed). It keeps flying until the cyclists meet. How far does the insect fly?
  - 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 ̄x = 40.6. Compute the following
  - ∑_{i=1}^{10} 2 x_i, x_i ∈ X
  - 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 ̄x = 40.6. Compute the following
  - ∑_{i=1}^{10} (2 x_i − ̄x), x_i ∈ X
  - Answer:

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

- Problem 13. Suppose that
  - y(t) = (x(t))^2 where x(t) = x_0 e^{g_x t}
  - Compute g_y ≡ d ln y(t) / dt
  - Answer:

- Problem 14. Assume that a, b, c are positive constants. Evaluate f′(0) where,
  - f(x) = c ln a x^2 + b x + c
  - Answer:

- Problem 15. Assume that κ > 1 and b > 0. Define and solve for
  - F(x) = ∫_{x}^{x B κ x κ B x^{κ+1}} dx
  - (as printed)
  - Answer:

- Problem 16. Assume that κ > 1 and b > 0. Define,
  - F(x) = ∫_{x}^{x B κ x κ B x^{κ+1}} dx.
  - Answer:

- Problem 17. Suppose that A = [0 2 3]. Compute A A^T:
  - Answer:

- Problem 18. Suppose that matrix A is
  - A = [ a−λ   b
          c     d−λ ]
  - Answer:

- Problem 19. If possible, solve for the values of x_1, x_2, and x_3 in which
  - [ 1   5  10
      1   5  10
      3   0   1 ] [ x_1
                   x_2
                   x_3 ] = [ 1
                              2
                              5 ]
  - Answer:

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

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_Source: https://www.imf.org/-/media/files/oap/jispa/2026/revamp/math-test-2026.pdf_