Identifying Errors in Proofs
Locate the first invalid step in purported proofs and diagnose hidden restrictions, reversed logic and unjustified algebra or trigonometry.
How to prepare for the current TMUA
Secure the shared mathematical specification, practise application in unfamiliar settings, then make every Paper 2 implication and proof step explicit.
Core concepts
Concept 1
Check each proof transition for its assumptions, domain and reversibility.
Exam cue: Find the earliest step that is not guaranteed by the previous line.
Concept 2
Detect division by a possibly zero expression, invalid cancellation and lost or introduced solutions.
Exam cue: Record any expression divided by or square-rooted.
Concept 3
Distinguish a true conclusion from a valid proof of that conclusion.
Exam cue: Test the suspect implication with a counterexample.
Risk pitfalls and guardrails
Accepting a proof because its final statement is true.
Guardrail: Do not treat examples as proof, reverse an implication, cancel a possible zero or rely on calculator-only intuition.
Cancelling a factor that may equal zero.
Guardrail: Do not treat examples as proof, reverse an implication, cancel a possible zero or rely on calculator-only intuition.
Assuming equal sine values force equal angles.
Guardrail: Do not treat examples as proof, reverse an implication, cancel a possible zero or rely on calculator-only intuition.
Memory anchors
First bad step
Diagnose the earliest unjustified transition, not the final symptom.
Zero divisor
Before cancelling a factor, establish that it is non-zero.
Squaring
Squaring can introduce solutions and needs a final check.
Square root
The principal square root is non-negative.
Same sine
Equal sine values need not mean equal angles.
True conclusion
A true result can still follow from an invalid argument.
Checkpoint rule
Do the check-up only after you can summarize each concept in one sentence and identify one dangerous pitfall from memory.
Knowledge Check (after reading)
Short check-up to confirm understanding of this module.
Check-up Questions
A proof says: let a=b≠0; then a²=ab, so a²−b²=ab−b²; divide by a−b to get a+b=b; hence 2b=b. What is the error?
A student solves x²=9 by taking square roots and writes x=3. What was lost?
Answer all questions to submit.
Next step personalized recommendations
Continue learning
Move forward only after this module is stable.
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