Consider the following statements in respect of five candidates P, Q, R, S and T. Two statements are true and one statement is false. True Statement : One of P and Q was selected for the job. False Statement : At least one of R and S was selected for the job. True Statement : At most two of R, S and T were selected for the job. Which of the following conclusions can be drawn ? 1. At least four were selected for the job. 2. S was selected for the job. Select the correct answer using the code given below :
- (a) 1 only
- (b) 2 only
- (c) Both 1 and 2
- (d) Neither 1 nor 2 ✓ UPSC's answer
Why the answer is (d)
• The false statement 'At least one of R and S was selected' implies that neither R nor S was selected for the job.
• Since R and S are not selected, the condition 'At most two of R, S and T were selected' is automatically satisfied regardless of T's status, as the count of selected candidates among them is either 0 or 1.
• The true statement 'One of P and Q was selected' means exactly one of P or Q is selected, while the other is not.
• The total number of selected candidates is therefore either 1 (if T is not selected) or 2 (if T is selected), which contradicts the conclusion that at least four were selected.
• Since S is definitely not selected, the conclusion that S was selected is false.
• Therefore, neither conclusion 1 nor conclusion 2 can be drawn, making option (d) the correct answer.
Why the other options are wrong
- (a) 1 only
- Conclusion 1 is false because the maximum number of selected candidates is two (one from P/Q and possibly T), not at least four.
- (b) 2 only
- Conclusion 2 is false because the falsity of the statement 'At least one of R and S was selected' proves that S was not selected.
- (c) Both 1 and 2
- Both conclusions are false; S was not selected, and fewer than four candidates were selected.
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.