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.
Answer & explanation
Answer: (d) Neither 1 nor 2 nor 3
y is always 8 less than x. When x is positive, y can be positive or negative (x = 10, y = 2; x = 2, y = −6), and when x is negative, y is even more negative. None of the three statements must hold.
- Rewrite the condition as y = x − 8.
- x = 2 gives y = −6: both need not be positive.
- x = 10 gives y = 2: y need not be negative when x is positive.
- If x is negative, y = x − 8 is below −8, so y can never be positive.
- ✗ 1. x = 2, y = −6 satisfies x − y = 8 with y negative.
- ✗ 2. x = 10, y = 2 satisfies the equation with both positive.
- ✗ 3. A negative x makes y = x − 8 even more negative, never positive.
Remember · For 'must be true' statements, one counter-example disproves each; express one variable in terms of the other first.
Question and answer: UPSC's official GS Paper II (2018, Series A) — paper ↗ · answer key ↗. Explanation: Minimalist IAS, checked 30 Sept 2026 (how we verify). ·