The statements (S1), (S2), and (S3) pertain to the scores obtained by students in an exam. The maximum possible marks in the exam is 150.
(S1) The highest score is 100.
(S2) The fourth highest score is 76.
(S3) There are at least four students whose scores are within 25 of each other.
Which one of the following options is necessarily correct?
(S1) and (S2) together imply (S3)
(S1) and (S3) together imply (S2)
(S2) and (S3) together imply (S1)
(S1) implies (S3)
ANSWER
(A) (S1) and (S2) together imply (S3): the top four scores all lie between 76 and 100, so they are within 24 of each other.
Official answer: ASource: Official answer key, GATE 2026 EC (IIT Guwahati)
Test each option by asking whether the two given statements force the third. A single set of scores that satisfies the given statements but not the third is enough to rule an implication out. "Within 25 of each other" means every pair of the four scores differs by at most 25.
(A) Suppose (S1) and (S2) hold. Since a fourth highest score exists, there are at least four students. The top four scores are no higher than the highest score (100) and no lower than the fourth highest (76), so all four lie in .
Any two of these four scores therefore differ by at most . So at least four students have scores within 25 of each other, which is (S3). (S1) and (S2) always force (S3), so (A) is correct.
(B) Counterexample: four students scoring 100, 95, 92 and 90. The highest score is 100, so (S1) holds, and all four lie within of each other, so (S3) holds. But the fourth highest score is 90, not 76, so (S2) fails. (B) is not necessarily correct.
(C) Counterexample: four students scoring 90, 80, 78 and 76. The fourth highest score is 76, so (S2) holds, and all four lie within of each other, so (S3) holds. But the highest score is 90, not 100, so (S1) fails. (C) is not necessarily correct.
(D) Counterexample: four students scoring 100, 70, 40 and 10. The highest score is 100, so (S1) holds. With only these four students, the four scores span , so no four students are within 25 of each other and (S3) fails. (D) is not necessarily correct.
Only (A) holds for every possible set of scores. The maximum of 150 marks plays no part beyond keeping every score used here valid.
KEY IDEA
Knowing the highest and the fourth highest score boxes the top four students into an interval whose width is their difference. Here that width is , so (S3) is forced. Each of the other implications falls to a single counterexample.
WATCH OUT FOR
Treating an implication as true because it holds in one convenient example (for example, choosing (C) or (D) because some score list fits), instead of asking whether it holds in every case. The other slip is reading (S3) as a claim about the highest score. (S3) fixes no particular value, so on its own it can never tell you the highest or the fourth highest score.
From the GATE 2026 EC paper. Every question of this paper · Download the full paper · All 41 Reasoning questions, with notes
