# Q. 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

(b) 2 only

(c) Both 1 and 2

(d) Neither 1 nor 2

Correct Answer: (a) 1 only

#### Question from UPSC Prelims 2023 CSAT

**Explanation : **

Let’s analyze each statement:

1. If T is true, then at least one of P and R must be false.

- The contrapositive of 1 is: “If both P and R are true, then T is false”. This statement can be derived from the given statements as follows:
- If P is true, then Q and S are true (by I).
- If R and S are true, then T is false (by II).
- Therefore, if P and R are true, then Q, S and T are false (by
**transitivity**).

- Since the contrapositive of 1 is true, 1 is also true.

2. If Q is true, then P is true.

According to statement I, if P is true, then both Q and S are true. However, this does not imply that if Q is true, then P must be true. It is possible for Q to be true while P is false. Therefore, this statement is false.

So, the correct answer is (a) 1 only.