Let P, Q, R, S and T be five statements such that : I. If P is true, then both Q and S are true. II. If R and S are true, then T is false. Which of the following can be concluded ? 1. If T is true, then at least one of P and R must be false. 2. If Q is true, then P is true. Select the correct answer using the code given below :

(a) 1 only ✓ UPSC's answer
(b) 2 only
(c) Both 1 and 2
(d) Neither 1 nor 2

Why the answer is (a)

• Statement I is P → (Q and S), so P true forces S true.

• Statement II is (R and S) → not T, so if T is true, R and S cannot both be true.

• If T is true and both P and R were true, P true would make S true, so R and S would both be true.

• That would force T false by Statement II, contradicting the assumption that T is true; hence T true implies P and R cannot both be true.

• Therefore statement 1, "If T is true, then at least one of P and R must be false," is valid.

• Statement 2 is the converse of P → Q and is not valid; for example, P false, Q true, R false, S false, T true satisfies both given statements but makes Q true while P false, so only statement 1 follows and option (a) is correct.

Why the other options are wrong

(b) 2 only: Option (b) is wrong because statement 2 is not a valid conclusion, while statement 1 is valid.
(c) Both 1 and 2: Option (c) is wrong because statement 2 does not follow from the given implications.
(d) Neither 1 nor 2: Option (d) is wrong because statement 1 does follow from the given implications.

Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2023, held on 28 May 2023. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.