## Japan-IMF Scholarship Program for Asia 2022-2023 Basic Mathematics Aptitude Test Test A

## Source details

**Canonical URL:** [Japan-IMF Scholarship Program for Asia 2022-2023 Basic Mathematics Aptitude Test Test A](https://www.imf.org/-/media/files/oap/jispa/2022/samplemath2022.pdf)

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### Test instructions and metadata
- Full score: 40
- You have 60 minutes to complete the test
- Calculators are not allowed

### Arithmetic and Elementary Algebra (Questions 1–8)
- 1. (1 point) Calculate 7÷2×(−4) +12.
- 2. (1 point) Find the largest number out of √(3/2), √(6/3), and 2√5.
- 3. (2 points) Calculate √12 / (√3−1) + √12 / (√3+1).
- 4. (2 points) Calculate 49^(1 / log_8 7).
- 5. (2 points) Calculate x^3 −1 / x^2 + x + 1 for x = 101.
- 6. (2 points) Solve the following system of linear equations in x and y:
  - 5x + 2y = 12
  - 3x − 2y = 4
- 7. (2 points) Solve for x from the following equation b/(x−a) + a/(x−b) = 2, where a and b are parameters such that a̸=0, b̸=0, and a̸=b.
- 8. (2 points) Find the first-order derivative of the function f(x) = x^2 + 2/(x+1).

### Domain, Sequences, and Multivariable Calculus (Questions 9–12)
- 9. (2 points) For which x function f(x) = √(2 + x − x^2) is well-defined?
- 10. (2 points) Calculate the sum ∑_{d=0}^{19} (3 + d).
- 11. (2 points) Let f(x) = a^x, where a > 0 is a parameter. For which a > 0 function f(x) is decreasing in x, and for which a > 0 this function is increasing?
- 12. (2 points) Find the partial derivative ∂^2 f(x,y) / ∂x ∂y for f(x,y) = xy + x / y.

### Integration, Extrema, and Limits (Questions 13–16)
- 13. (2 points) Evaluate the integral 8 ∫_1^{x} 1/3 dx.
- 14. (2 points) Find all local extrema of function f(x) = x^3 − 6x^2 + 9x − 4.
- 15. (2 points) Assuming that n takes positive integer values, find lim_{n→∞} 10000 n / (n^2 + 1).
- 16. (2 points) Evaluate the limit lim_{x→∞} x / (x + ln x).

### Linear Algebra and Probability (Questions 17–19)
- 17. (2 points) Find the determinant of matrix
  - ( 4 −3 5
      3 −2 8
      1 −7 −5 )
- 18. (2 points) For matrix A = ( 1 2 ; 3 4 ) find A^{−1}.
- 19. (2 points) There are 5 oranges and 4 apples in a box. Without looking inside the box, you take 2 random fruits from it. What is the probability that both fruits are oranges?

### Counting and Geometry in R^3 (Questions 20–21)
- 20. (2 points) Consider all 4-digit numbers consisting only of digits 1, 2, and 3. What is the proportion of such numbers that start with digit 2?
- 21. (2 points) Coordinates of points A and B in the three-dimensional space are given by A = (8, 0, 1) and B = (1, 1, −1). Find the distance between A and B.

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_Source: https://www.imf.org/-/media/files/oap/jispa/2022/samplemath2022.pdf_