## JAPAN-IMF SCHOLARSHIP PROGRAM FOR ASIA 2023-2024 — Basic Mathematics Aptitude Test (Full score: 40)

## Source details

**Canonical URL:** [JAPAN-IMF SCHOLARSHIP PROGRAM FOR ASIA 2023-2024 — Basic Mathematics Aptitude Test (Full score: 40)](https://www.imf.org/-/media/files/oap/jispa/2023/imf-math-test-2023-for-brochure-website.pdf)

## Other formats

- [Markdown version](/-/media/files/oap/jispa/2023/imf-math-test-2023-for-brochure-website.pdf.md)
- [Structured JSON version](/-/media/files/oap/jispa/2023/imf-math-test-2023-for-brochure-website.pdf.json)

---

### Instructions and test metadata
- Full score: 40
- Time allowed: 60 minutes
- Calculators: No calculators are allowed.
- Candidates must show all work and write answers in the designated space.

### Arithmetic and algebra
- Q1. Calculate 11 − (−6)^3 ÷ (−2)^2.  
- Q2. Calculate (x − 2)^5 (3x − 6)^4 for x = 2756.  
- Q3. Calculate (√3)^3 + 2√2 + (√3)^3 − 2√2.  

### Linear equations and basic algebraic manipulation
- Q4. Solve for x and y in the system:
  - −0.5x + 0.4y = −7/15
  - 1/3 x + 0.25y = 1
- Q5. Solve for x in the equation: 4^x − 0.25 = 12.

### Logarithms, inequalities, and series
- Q6. Solve for x in: 2 log_3 (x + 13) − log_3 (−2x + 1) = 2.
- Q7. Find the largest integer x that satisfies |2 − log_3 x| < 3.
- Q8. Evaluate the infinite sum: Σ_{t=1 to ∞} t (0.5)^t.

### Calculus — differentiation and tangent lines
- Q9. Find the equation of a line that is tangential to y = −2 (x − 1)^3 + 1 at (x, y) = (2, −1).
- Q10. Find the first derivative of f(x) = (2x − 1)(3x^2 + x − 2).
- Q11. Evaluate the second derivative of f(x) = 2 e^{−2x+2} + ln(3x + 1) at x = 1.

### Multivariable calculus and integrals
- Q12. Find the cross partial derivative ∂^2 f(x, y) / ∂x ∂y of f(x, y) = x ln(2x + y).
- Q13. Evaluate the integral ∫_{−1}^{5} (x^2 − 2x + 3) dx.
- Q14. Evaluate the integral ∫_{−2}^{0} e^{−2x} dx, where e denotes Euler’s constant.

### Linear algebra — matrices and linear systems
- Q15. Given matrices A and B below, find the matrix X that satisfies A X = B.
  - A = [ [2 −1] [−2 3] ], B = [ [5 4] [−3 −8] ].
- Q16. Solve for x1, x2, and x3 where
  - [ [2 −3 −1] [−1 5 2] [1 −1 3] ] [ x1 x2 x3 ]^T = [ −4 3 8 ]^T.

### Optimization with constraints (Lagrange multipliers / constrained maxima)
- Q17. Find the values of x and y that solve:
  - maximize ln(x + 8) + 2 ln(2y − 5)
  - subject to x + 4y = 8
- Q18. Find the values of x and y that solve:
  - maximize 3 ln(x − 2) + y
  - subject to 2x + y = 10, x ≥ 4, y ≥ −6

### Probability and combinatorics
- Q19. Suppose you roll two fair dice. What is the probability that the sum of two numbers is at least 8?
- Q20. Suppose you randomly pick an integer from 500 to 999. What is the probability that the integer you picked is divisible by both 4 and 6?

---


_Source: https://www.imf.org/-/media/files/oap/jispa/2023/imf-math-test-2023-for-brochure-website.pdf_