If x - y = 8, then which of the following must be true? 1. Both x and y must be positive for any value of x and y. 2. If x is positive, y must be negative for any value of x and y. 3. If x is negative, y must be positive for any value of x and y. 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 nor 3 ✓ UPSC's answer

Why the answer is (d)

• Statement 1 is false because x and y can both be negative (e.g., x = -1, y = -9) or both positive (e.g., x = 10, y = 2), so they are not required to be positive.

• Statement 2 is false because if x is positive, y can also be positive (e.g., x = 10, y = 2) or negative (e.g., x = 1, y = -7), so y is not necessarily negative.

• Statement 3 is false because if x is negative, y must be even more negative to satisfy x - y = 8 (e.g., x = -1, y = -9), so y cannot be positive.

• Since all three statements are false, the correct code is 'Neither 1 nor 2 nor 3'.

Why the other options are wrong

(a) 1 only: Statement 1 is false because x and y can both be negative, such as x = -1 and y = -9.
(b) 2 only: Statement 2 is false because if x is positive, y can also be positive, such as x = 10 and y = 2.
(c) Both 1 and 2: Both Statement 1 and Statement 2 are false, as demonstrated by counterexamples where signs vary independently.

Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2018, held on 3 June 2018. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.