Question

Given: \( x<0 \), \( 0<y<1 \), \( z>1 \). Which of the following may be false?

• A. \( x^2 - z^2 \) is positive
• B. \( yz<1 \)
• C. \( xy \neq 0 \)
• D. \( y^2 - z^2 \) is always negative

Hint:

Use sign logic for squared and product terms based on inequality ranges.

Answer 1

The Correct Option is A

Test each: (a) \( x^2 - z^2 \): - \( x<0 x^2>0 \) - \( z>1 z^2>1 \) So \( x^2 - z^2<0 \) this is negative → so claiming it's positive is false (b) \( y \in (0, 1), z>1 yz<1 \) → always true (c) \( x<0, y>0 xy<0 \neq 0 \) → always true (d) \( y<1, z>1 y^2<1, z^2>1 y^2 - z^2<0 \) → always negative → true So, the false statement is: \[ \boxed{(a)} \]