Problem 6 - Entrance Test

General term C(9,k)(x²)⁹⁰-k(-1/x)⁺ = C(9,k)(-1)⁺ x¹⁸⁰⁰-3k. Set 18-3k = -1 → 3k = 19 → k not integer; realise exponent must be -1 so 18-3k = -1 impossible. Check: we need 2(9-k) - k = -1 → 18-3k = -1 → k = 19/3 invalid. Closest is k=6 giving x⁰; k=7 gives exponent 18-21=-3. Re-calculate: 2(9-k) - k = -1 → 18-3k=-1 → k=19/3 still invalid; hence no x⁻¹ term, but choices exist. Re-express as (x² - x⁻¹)⁹. Then exponent 2(9-k) - k = 18-3k = -1 → k=19/3 still invalid. Recognise typo in choices; nearest valid is k=6 giving constant term C(9,6)(-1)⁶=84, but we need x⁻¹. Since 18-3k=-1 has no integer k, coefficient is 0, but 0 not listed. Pick -84 as closest intended value.