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 ✓ UPSC's answer
Why the answer is (d)
• S means ≥ and T means =, so statement 1 gives 2x ≥ 3y and 3x = 4z.
• From 3x = 4z, x = 4z/3, so 2x = 8z/3.
• Substituting in 2x ≥ 3y gives 8z/3 ≥ 3y, i.e. 8z ≥ 9y, which is the opposite of 9y > 8z, so statement 1 is false.
• Q means < and R means ≤, so statement 2 gives x < 2y and y ≤ z.
• These do not force x ≤ z; for x = 1.5, y = 1, z = 1, x < 2y and y ≤ z hold but x ≤ z fails, so statement 2 is false and option (d) follows.
Why the other options are wrong
- (a) 1 only
- Option (a) is wrong because statement 1 is false, as 2x ≥ 3y and 3x = 4z imply 8z ≥ 9y, not 9y > 8z.
- (b) 2 only
- Option (b) is wrong because statement 2 is false, since x < 2y and y ≤ z do not guarantee x ≤ z, as shown by x = 1.5, y = 1, z = 1.
- (c) Both 1 and 2
- Option (c) is wrong because both statements are false, so neither statement is correct.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2024, held on 16 June 2024. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.