Every year Paper 2 shows a student's argument and asks for the first wrong line. The spec names the classics: '\(x^2 = y^2\) so \(x = y\)', '\(\sin\alpha = \sin\beta\) so \(\alpha = \beta\)'. Add: dividing by something that may be zero, multiplying an inequality by a quantity of unknown sign, squaring without checking, assuming the conclusion, and applying a theorem outside its conditions.
Contents: The catalogue of invalid steps (Err2)How to attack an 'identify the error' question (Err1)Must-knowTrapsReal questions
The catalogue of invalid steps (Err2)
- \(x^2 = y^2 \Rightarrow x = \pm y\), not \(x = y\). \(x^2 > 9 \Rightarrow |x| > 3\).
- \(\sin\alpha = \sin\beta \Rightarrow \alpha = \beta\) or \(\alpha = 180^\circ - \beta\) (plus periods). Same for \(\cos\) and \(\tan\).
- Dividing by \(x\), \(a - b\) or \(\cos x\) loses the case where it is zero (the '\(1 = 2\)' proof; \(x^2 = 4x\); \(2\cos^2 x = \cos x\)).
- Multiplying an inequality by \(x - 2\) or \(ab\) without knowing the sign.
- Squaring both sides can create extraneous roots: substitute back (\(\sqrt{x + 3} = x - 3\) gives only \(x = 6\)).
- Starting from the statement to be proved and reaching a true statement (needs reversibility and reversed presentation).
- Applying the FTC or an integral across a point where the function is undefined (\(\int_{-1}^1 x^{-2}\,dx\)).
- Proving the converse instead of the statement; checking cases instead of proving generally; using a result that is exactly the claim (circularity).
How to attack an 'identify the error' question (Err1)
- Read each line as a claimed deduction from the previous ones; ask 'is this always valid?' not 'is it true here?'.
- Test the claim itself with a quick counterexample first (\(n = 2\) for '\(4 \mid n^2 \Rightarrow 4 \mid n\)'); if the claim is false, the error is at the step that is not a valid deduction.
- The first error is usually where a hidden assumption enters (sign, non-zero, domain).
- 'Completely correct' is a genuine option: valid inductions and completed-square arguments do appear.
Must-know (no formula booklet)
- \(x^2 = y^2 \Rightarrow x = \pm y\)
- never divide by a possible zero
- inequality \(\times\) negative flips
- squaring: check solutions
- backwards proof needs reversibility
Classic traps
- '\(f'(x) \geq 0\) with equality at one point' still gives an increasing function; the error is in claiming \(f' > 0\) everywhere, not in the conclusion.
- Two even primes? No - but '\(2\) is even and prime' breaks 'every prime is odd'.
- \(f'(a) = f''(a) = 0\) does not force an inflexion (\(x^4\)).
Worked example
Q. Where is the first error in: \(a = b\); \(a^2 = ab\); \(a^2 - b^2 = ab - b^2\); divide by \(a - b\): \(a + b = b\)?
A. Dividing by \(a - b = 0\).
How it has been examined
2023 · Paper 2Question 10solving x^4 - 2x^2 - 3 < 0 by completing the squarequestion papersit online