GATE 2026 CS (CS1) – Question 5
‘When the teacher is in the room, all students stand silently.’
If the above statement is true, which one of the following statements is not necessarily true?
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Correct answer: (C) If all students are standing, then the teacher is in the room.
Explanation
Let $P$: 'The teacher is in the room' and $Q$: 'All students stand silently'.
The given statement is $P \implies Q$.
- Option (A): 'If any student is not standing silently $\implies$ teacher is not in the room' is $\neg Q \implies \neg P$, which is the contrapositive. It is logically equivalent to $P \implies Q$, hence necessarily true.
- Option (B): 'When teacher is in room, all students are silent' directly follows from $Q$, hence true.
- Option (C): 'If all students are standing, then the teacher is in the room' is the converse statement $Q \implies P$. The truth of $P \implies Q$ does not guarantee the truth of its converse $Q \implies P$. (Students could be standing for another reason even if the teacher is absent). Hence it is NOT necessarily true.
- Option (D): 'When teacher is in room, all students are standing' directly follows from $Q$, hence true.
Therefore, option (C) is the correct answer.