1964 AMC 12 Problem 12

Attempt Problem 12 of the 1964 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1964 AMC 12 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

12.

Which of the following is the negation of the statement: For all

x

of a certain set,

x^2\gt0?

For all

x,

x^2\lt0

For all

x,

x^2\leq0

For no

x,

x^2\gt0

For some

x,

x^2\gt0

For some

x,

x^2\leq0

The negation of “for all” begins with “for some”

Negating

x^2\gt0

gives

x^2\leq0

The negation of a universal statement is an existential statement, and the negation of

x^2\gt0

x^2\leq0.

Thus the negation is: for some

x,

x^2\leq0.

Therefore, the correct answer is E.

Problem 12 in other years