Question 15

**(P)** Let p(x) = 10x³ − 45x² + 60x + 35. Then p'(x) = 30(x−1)(x−2) ≤ 0 on [1,2], so p decreases from p(1) = 60 to p(2) = 55. The floor function f = [p(x)/n] is continuous on [1,2] iff no integer lies in (55/n, 60/n]. - n = 8: (6.875, 7.5] contains 7 → discontinuous. - n = 9: (6.11, 6.67] contains no integer → continuous. - Smaller n (1–7) all give intervals containing an integer. So minimum n = 9 → (2). **(Q)** x³ + 3x is strictly increasing, so g is increasing iff 2n² − 13n − 15 = (2n − 15)(n + 1) > 0, i.e. n > 7.5. Minimum n = 8 → (1). **(R)** Near x = 3, h(x) = (x−3)ⁿ · (x+3)ⁿ(x²+2x+3), where (x+3)ⁿ(x²+2x+3) > 0. Since h(3) = 0, x = 3 is a local minimum iff (x−3)ⁿ ≥ 0 on both sides, i.e. n even. Smallest even n > 5 is 6 → (4). **(S)** cos|x − k + ½| = cos(x − k + ½) is smooth everywhere. sin|x − k| is non-differentiable only at x = k (one-sided derivatives ±1). So l is non-differentiable exactly at x = 0, 1, 2, 3, 4: 5 points → (3). Matching: P→2, Q→1, R→4, S→3, which is option (B).

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