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Paper 2 reasoning · Updated 11 November 2026

TMUA Paper 2 practice

Paper 2 tests reasoning more than knowledge. If any of these trip you up, read our free lesson on the logic of arguments first.

Try each question without a calculator, allowing about 4 minutes, then open the solution.

Question 1. Which statement is the contrapositive of “if n2 is even, then n is even”?

  1. A  If n is even, then n2 is even.
  2. B  If n is odd, then n2 is odd.
  3. C  If n2 is odd, then n is odd.
  4. D  n2 is even only if n is odd.
  5. E  If n is odd, then n2 is even.
Show answer and worked solution

Answer: B. The contrapositive of “if P then Q” is “if not Q then not P”. Here: if n is not even, then n2 is not even. Trap: option A is the converse, which is a different statement.

Question 2. For real x, the condition x > 2 is ___ for x2 > 4.

  1. A  necessary but not sufficient
  2. B  sufficient but not necessary
  3. C  necessary and sufficient
  4. D  neither necessary nor sufficient
Show answer and worked solution

Answer: B. If x > 2 then certainly x2 > 4, so it is sufficient. But x = −3 gives x2 > 4 without x > 2, so it is not necessary.

Question 3. Which is the negation of “every prime number greater than 2 is odd”?

  1. A  Every prime number greater than 2 is even.
  2. B  No prime number greater than 2 is odd.
  3. C  There is a prime number greater than 2 that is even.
  4. D  There is a prime number greater than 2 that is odd.
  5. E  Every even number greater than 2 is not prime.
Show answer and worked solution

Answer: C. The negation of “for all, P” is “there exists one for which P fails”. So: some prime greater than 2 is not odd, i.e. is even. Trap: “every prime is even” is far stronger than the negation.

Question 4. A “proof” that 1 = 2: (I) Let a = b. (II) Then a2 = ab. (III) So a2 − b2 = ab − b2. (IV) Factorise: (a + b)(a − b) = b(a − b). (V) So a + b = b, i.e. 2b = b, and 1 = 2. Which step is invalid?

  1. A  II
  2. B  III
  3. C  IV
  4. D  V
  5. E  none: the argument is valid
Show answer and worked solution

Answer: D. Step V divides both sides by a − b, which is 0 because a = b. Dividing by zero is not allowed. Steps II to IV are all correct.

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