Q. If P means ‘greater than (>)’; Q means less than (<)’; R means ‘not greater than (>)’; S means ‘not less than (<)’ and T means ‘equal to (=)’, then consider the following statements:
1. If 2x(S)3y and 3x(T)4z, then 9y(P)8z.
2. If x(Q)2y and y(R)z, then x(R)z.
Which of the statements given above is/are correct?
a) 1 only
b) 2 only
c) Both 1 and 2
d) Neither 1 nor 2
Correct answer: d) Neither 1 nor 2
Question from UPSC Prelims 2024 CSAT Paper
Explanation:
Let’s analyze each statement step by step:
Statement 1: If 2x(S)3y and 3x(T)4z, then 9y(P)8z.
1. 2x(S)3y means 2x is not less than 3y, or 2x ≥ 3y
2. 3x(T)4z means 3x is equal to 4z, or 3x = 4z
3. From (2), we can say x = 4z/3
4. Substituting this in (1): 2(4z/3) ≥ 3y
5. Simplifying: 8z/3 ≥ 3y
6. Multiplying both sides by 3: 8z ≥ 9y
7. This is equivalent to 9y ≤ 8z, which is the opposite of 9y(P)8z (9y > 8z)
Therefore, statement 1 is incorrect.
Statement 2: If x(Q)2y and y(R)z, then x(R)z.
1. x(Q)2y means x < 2y
2. y(R)z means y is not greater than z, or y ≤ z
3. From (1) and (2), we can say x < 2y and y ≤ z
4. This means x < 2z
5. However, x(R)z means x is not greater than z, or x ≤ z
6. While x < 2z is true, we cannot conclude that x ≤ z is always true
Therefore, statement 2 is also incorrect.
Since both statements are incorrect, the answer is: d) Neither 1 nor 2