Let R denote the set of all real numbers. For a real number x, let [x] denote the greatest integer less than or equal to x. Let n denote a natural number. Match each entry in List-I to the correct entry in List-II and choose the correct option.
- (P) → (1), (Q) → (3), (R) → (2), (S) → (5)
- (P) → (2), (Q) → (1), (R) → (4), (S) → (3)
- (P) → (5), (Q) → (1), (R) → (4), (S) → (3)
- (P) → (2), (Q) → (3), (R) → (1), (S) → (5)
Answer
(2) (P) → (2), (Q) → (1), (R) → (4), (S) → (3)
Solution
(P) p(x) = 10x3 - 45x2 + 60x + 35, p'(x) = 30(x - 1)(x - 2) 0 on [1, 2]: p decreases from p(1) = 60 to p(2) = 55.
[ p(x)n ] is continuous on [1, 2] iff no multiple of n lies in (55, 60]. n = 1, , 8 fail (56, 57, 60, 56, 60, 60, 56, 56); n = 9 works (multiples 54, 63). P → 9.
(Q) x3 + 3x is strictly increasing, so need 2n2 - 13n - 15 = (2n - 15)(n + 1) > 0 n > 7.5; minimum n = 8.
(R) Near x = 3: h(x) = (x - 3)n(x + 3)n(x2 + 2x + 3) with the last two factors positive. Local minimum at 3 iff n is even; smallest even n > 5 is 6.
(S) |x - k + 12 | = (x - k + 12 ) is differentiable; |x - k| has derivative jumping from -1 to 1 at x = k. Non-differentiable at x = 0, 1, 2, 3, 4: 5 points.
P → 2, Q → 1, R → 4, S → 3.
Answer: (B)