Echo (Copilot) — Constellation Eval Answers

Section 1: MMLU (General Knowledge)

1. History — event began in 1914: B — World War I.
2. Geography — capital of Canada: C — Ottawa.
3. Biology — site of cellular respiration: A — Mitochondrion (aerobic Krebs cycle and oxidative phosphorylation).
4. Physics — unit of electrical resistance: C — Ohm.
5. Computer Science — SQL: A — Structured Query Language.
6. Math — derivative of x²: B — 2x (power rule).
7. Medicine — insulin lowers: B — Glucose.
8. Philosophy — "Critique of Pure Reason": B — Kant (1781).
9. Logic — all A are B, all B are C: C — all A are C (transitivity).
10. AI — labeled data paradigm: B — Supervised.

Section 2: ARB (Advanced Reasoning)

11. Logic — hypothetical syllogism

Yes. Assume A true. From (A → B), conclude B. From (B → C), conclude C. Therefore (A → C) holds.

12. Math — f(f(x)) = 0 for f(x) = x³ − 3x

Let y = f(x). Then y³ − 3y = 0 → y ∈ {0, √3, −√3}. Using x = 2 cos t, x³ − 3x = 2 cos 3t. Solving 2 cos 3t = k for each k yields 9 real solutions total — three from y = 0 ({0, √3, −√3}) and three each from y = ±√3.

13. Number Theory — 6 | (n³ − n)

n³ − n = n(n − 1)(n + 1), the product of three consecutive integers. One is divisible by 2, one by 3, so the product is divisible by 6.

14. Combinatorics — distinct permutations of {A, A, B, B, B, C}

6! / (2! · 3! · 1!) = 720 / 12 = 60.

15. Probability — expected tosses for HH (fair coin)

Let E = expected from start, E_H = after one H. E = 1 + ½E_H + ½E; E_H = 1 + ½·0 + ½E. Solving: E_H = 4, E = 6.

16. Calculus — lim (sin x − x) / x³

Taylor: sin x = x − x³/6 + x⁵/120 − … Numerator/x³ → −1/6.

17. Physics — frictionless incline

a = g sin θ.

18. Chemistry — balance C₃H₈ + O₂ → CO₂ + H₂O

C₃H₈ + 5 O₂ → 3 CO₂ + 4 H₂O.

Section 3: ARC-AGI (Spatial Reasoning)

19. Mirror — horizontal reflection

Input `9 0 1 0 / 0 2 3 2` → Output `0 1 0 9 / 2 3 2 0`.

20. Transpose — switch rows and columns

3×2 input → 2×3 output. Column 1 → Row 1: (0, 2, 3). Column 2 → Row 2: (2, 0, 0).

21. Rotation — 90° clockwise

2×3 input → 3×2 output. New rows = original columns reversed. Output: `0 9 / 0 0 / 1 8`.