Correct Option (D)
The problem presents three statements concerning the selection of five candidates (P, Q, R, S, T), where two statements are true and one is false. We must deduce conclusions based on these statements.
- The first true statement is: "One of P and Q was selected for the job." This implies that exactly one candidate from the set {P, Q} was selected.
- The false statement is: "At least one of R and S was selected for the job." Since this statement is false, its logical negation must be true. The negation of "at least one of R and S was selected" is "Neither R nor S was selected for the job." Therefore, it is definitively concluded that R was not selected, and S was not selected.
- The second true statement is: "At most two of R, S and T were selected for the job." Given the conclusion from the false statement that R and S were not selected, this implies that from the set {R, S, T}, only T could potentially be selected. The condition "at most two" is satisfied whether T is selected (resulting in 1 selected candidate from {R, S, T}) or T is not selected (resulting in 0 selected candidates from {R, S, T}).
Synthesizing these deductions:
- Exactly one candidate from {P, Q} was selected.
- R was not selected.
- S was not selected.
- T may or may not have been selected.
Now, let us evaluate the given conclusions:
Conclusion 1: "At least four were selected for the job."
- If T was not selected: The total number of selected candidates would be 1 (from P/Q) + 0 (from R, S, T) = 1.
- If T was selected: The total number of selected candidates would be 1 (from P/Q) + 1 (T) = 2.
In both possible scenarios, the total number of selected candidates is either 1 or 2. Therefore, the conclusion "At least four were selected for the job" is false.
Conclusion 2: "S was selected for the job."
- From the negation of the false statement, it was established that S was not selected for the job.
Therefore, Conclusion 2 is false.
Since both Conclusion 1 and Conclusion 2 are false, neither conclusion can be drawn as true. Hence, option (D) is the correct answer.
Incorrect Options:
Options (A), (B), and (C) are incorrect because the analysis of the given statements demonstrates that both Conclusion 1 and Conclusion 2 are false. Conclusion 1 is false because the maximum number of selected candidates is 2. Conclusion 2 is false because the negation of the false statement explicitly indicates that S was not selected.