HSC Maths Extension 2 free practice questions with answers

Correct answer: B — For all real numbers x, x² < x

• The negation of ‘for all x, P(x)’ is ‘there exists x such that not P(x)’, and the negation of ‘there exists x such that P(x)’ is ‘for all x, not P(x)’.
• Negate the condition itself: not (x² ≥ x) is x² < x.
• So the negation is: For all real numbers x, x² < x
• Why the other options are wrong:
• A (For all real numbers x, x² ≥ x): The condition must be negated as well as the quantifier changed; this is the original claim for all x.
• C: The quantifier must change from ‘there exists’ to ‘for all’; this keeps ‘there exists’.
• D: ‘It is not true that for all x, P’ means some x fails P, which is not the negation of ‘there exists x with

P’.

To negate a statement: ‘for all’ becomes ‘there exists’ (and vice versa) and the condition is negated; ‘if P then Q’ becomes ‘P and not Q’.