GATE ECE 2026 | GA | Question: 2

Question

The statements (S$1$), (S$2$), and (S$3$) pertain to the scores obtained by students in an exam. The maximum possible marks in the exam is $150$.

(S$1$) The highest score is $100$.
(S$2$) The fourth highest score is $76$.
(S$3$) There are at least four students whose scores are within $25$ of each other.

Which one of the following options is necessarily correct?

• A. (S$1$) and (S$2$) together imply (S$3$)
• B. (S$1$) and (S$3$) together imply (S$2$)
• C. (S$2$) and (S$3$) together imply (S$1$)
• D. (S$1$) implies (S$3$)

Answer 1

S1 & S2 together imply S3 bcoz assuming S1 & S2 to be true, S3 is definitely True

S1 and S3 cannot necessarily imply S2 to be true

Answer 2

It is given that,

$\text{Range} = \text{Maximum Score} - \text{Minimum Score} \le 25$

Let the student scores arranged in descending order be: $s_1 \ge s_2 \ge s_3 \ge s_4 \ge \dots$

From (S1): The highest score is $100 \implies s_1 = 100$.

From (S2): The fourth highest score is $76 \implies s_4 = 76$

Because the scores are ordered, the second and third highest scores ($s_2$ and $s_3$) must fall between $s_4$ and $s_1$: $76 \le s_4 \le s_3 \le s_2 \le s_1 = 100$

$\text{Max Difference} = s_1 - s_4 = 100 - 76 = 24$

Since $24 \le 25$, the scores of these four students ($s_1, s_2, s_3, s_4$) are guaranteed to be within $25$ of each other.

So answer is Option A