Match the expressions/statements in Column I with expressions/statements in Column II and indicate your answer by darkening the appropriate bubbles in the 4×4 matrix given in the ORS.
Column I
Column II
(A) If −1<x<1, then f(x) satisfies
(p) 0<f(x)<1
(B) If 1<x<2, then f(x) satisfies
(q) f(x)<0
(C) If 3<x<5, then f(x) satisfies
(r) f(x)>0
(D) If x>5, then f(x) satisfies
(s) f(x)<1
Answer: A → I, A → III, A → IV, B → II, B → IV, C → II, C → IV, D → I, D → III, D → IV
Solution
We have f(x)=(x−2)(x−3)(x−1)(x−5).
Analyzing the sign and magnitude in each interval:
(A) For −1<x<1: Both numerator factors are negative, denominator factors are negative, so f(x)>0. Also, 0<f(x)<1. Matches: (p), (r), (s).
(B) For 1<x<2: (x−1)>0, (x−5)<0, (x−2)<0, (x−3)<0, so f(x)<0. Also f(x)<1. Matches: (q), (s).
(C) For 3<x<5: (x−1)>0, (x−5)<0, (x−2)>0, (x−3)>0, so f(x)<0. Also f(x)<1. Matches: (q), (s).
(D) For x>5: All factors positive, f(x)>0 and 0<f(x)<1. Matches: (p), (r), (s).
JEE Advanced · 2007 · 08 Apr 2007 · Paper 2 · Match the columns
