Key points

  • Paper 2 tests whether you can reason with mathematical statements, so reading each statement exactly is part of the maths.
  • 'If A then B' doesn't mean 'if B then A', but it does mean 'if not B then not A'.
  • Review correct answers that took too long as well as wrong ones, and note whether each mistake came from the maths or from reading the statement.

How Paper 2 differs from Paper 1

TMUA has two papers, each with 20 multiple-choice questions in 75 minutes. Paper 1 asks you to apply mathematical knowledge in new situations. Paper 2 is about mathematical reasoning and simple ideas from elementary logic.

Paper 2 questions don't usually need harder algebra than Paper 1. A question might contain a familiar expression, but the words around it often matter more: whether a conclusion follows, whether two statements say the same thing, or which example disproves a claim.

If you only practise calculation-heavy sets, Paper 2 can come as a surprise. Give it its own practice from early on.

A statement, its converse and its contrapositive

Take the statement 'if n is a multiple of 4, then n is even'. It's true.

The converse swaps the two parts: 'if n is even, then n is a multiple of 4'. That's false, and n = 6 shows it. Under time pressure it's easy to treat a statement and its converse as the same, because they sound balanced, and a Paper 2 question may well offer you the converse as one of the options.

The contrapositive swaps the two parts and negates both: 'if n is not even, then n is not a multiple of 4', or more simply, 'if n is odd, then n isn't a multiple of 4'. That's true. A statement and its contrapositive are always both true or both false, because they say the same thing in two ways.

Here's one with algebra. 'If x > 2, then x² > 4' is true. Its converse, 'if x² > 4, then x > 2', is false, because x = −3 gives x² = 9. Its contrapositive, 'if x² ≤ 4, then x ≤ 2', is true.

When you meet a statement in a question, jot the converse and contrapositive next to it in a few words. It takes seconds and stops you mixing them up.

Necessary and sufficient

These two words come up often and are easy to swap. In 'if A then B', A is sufficient for B: knowing A is enough to know B. B is necessary for A: A can't happen without B.

In the example above, being a multiple of 4 is sufficient for being even. Being even is necessary for being a multiple of 4, but it isn't sufficient, as 6 shows.

If a question asks whether a condition is necessary, sufficient, both or neither, test each direction separately. Check 'if A then B', then check 'if B then A'. Each one is true or false on its own.

Reading 'all' and 'some'

Watch the small words. To show that a claim about all numbers is false, you need one exception. To show that a claim about some number is true, you need one example that works. Don't try to check every case when the claim only says some.

Check the conditions on the variables as well. A claim that holds for every positive integer might fail for zero, for negative numbers or for fractions. If the question says n is an integer, use that. If it doesn't say so, don't assume it.

Finding counterexamples quickly

A counterexample only needs to break the claim once, so start with the simplest cases you can think of. Good ones to try first are 0, 1, −1, ½ and 2, two equal values, and simple shapes such as a square or an equilateral triangle.

For example, 'x² ≥ x for every real number x' looks true if you try 2, 3 and 10. Try ½ and you get ¼, which is smaller than ½, so the claim is false.

If building a counterexample is taking you three minutes, it's probably more complicated than the question needs. Go back to your list of small cases. The point of the list is to get you from doubting a claim to disproving it faster.

Reviewing a Paper 2 mock

When you go through a Paper 2 mock, mark each mistake as either maths or reading. 'I didn't know the method' and 'I answered the converse' need different kinds of practice.

Name each mistake precisely. Useful names include 'confused sufficient with necessary', 'ignored the integer condition', 'chose a true statement that didn't answer the question' and 'accepted a pattern from three examples'. Once a mistake has a name, you can find questions that practise it.

Look at correct answers that took a long time too. If you got there by testing lots of cases when one small counterexample would have done, write that down as well.