GATE 2025 CY – Question 7
Weight of a person can be expressed as a function of their age. The function usually varies from person to person. Suppose this function is identical for two brothers, and it monotonically increases till the age of 50 years and then it monotonically decreases. Let a1 and a2 (in years) denote the ages of the brothers and a1 < a2. Which one of the following statements is correct about their age on the day when they attain the same weight?
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Correct answer: (B) $a_1<50<a_2$
Explanation
The weight function is **increasing up to age 50** and **decreasing after 50** (it has its maximum at 50). The brothers have the same function, and at some day they have the same weight, with ages $a_1<a_2$.
**Key idea.** A strictly increasing function takes each value only once, and so does a strictly decreasing one. So two different ages with the same weight cannot both lie in the increasing part, and cannot both lie in the decreasing part.
Therefore the younger brother must be in the increasing phase ($a_1<50$) and the older in the decreasing phase ($a_2>50$):
$$a_1<50<a_2 .$$
Answer **B**. (Neither can be exactly 50: at the maximum there is no other age with the same weight.)