Age 16 to 18
Challenge Level

For any real numbers $a$, $b$, and $c$ where $a \ge b$, consider these statements:

  1. $-b \ge -a$
  2. $a^2 + b^2 \ge 2ab$
  3. $ac \ge bc$

Which of the statements 1, 2, and 3 must be true?

(A) None of them

(B) Statement 1 only

(C) Statement 2 only

(D) Statement 3 only

(E) Statements 1 & 2 only

(F) Statements 1 & 3 only

(G) Statements 2 & 3 only

(H) Statements 1, 2 and 3 

This question comes from TMUA Specimen paper 2.

The Test of Mathematics for University Admission (or TMUA) is designed to give you the opportunity to demonstrate that you have the essential mathematical thinking and reasoning skills needed for a demanding undergraduate Mathematics or Mathematics-related course. There are several UK universities which encourage their applicants to sit TMUA, and a good performance may result in a reduced offer.

For more information about TMUA visit