Edexcel IAL P2 Proof Questions
Proof by deduction, proof by exhaustion and disproof by counter-example, in number, algebra, inequalities and coordinate geometry (proof by contradiction is P4).
- P2 · Pure Mathematics 2
- Calculator allowed
- 12 practice questions
What the questions test
The 12 P2 proof practice questions cover:
Worked example 1
(a) Prove by exhaustion that the square of any integer leaves a remainder of 0, 1 or 4 when it is divided by 5. [4]
(b) Hence show that the last digit of a square number is never 2, 3, 7 or 8. [1]
Mark scheme
(a) Considers every case: \(n=5k,\ 5k\pm1,\ 5k\pm2\) (or \(5k,\ 5k+1,\ 5k+2,\ 5k+3,\ 5k+4\)), \(k\) an integer M1
Squares at least two cases correctly, e.g. \((5k\pm1)^2=25k^2\pm10k+1\) M1
All cases correct: \((5k)^2=5(5k^2)\); \((5k\pm1)^2=5(5k^2\pm2k)+1\); \((5k\pm2)^2=5(5k^2\pm4k)+4\) A1
Conclusion: every integer is in one of the cases, so the remainder is always 0, 1 or 4 A1 (cso)
(b) A whole number ending in 2 or 7 leaves remainder 2, and one ending in 3 or 8 leaves remainder 3, on division by 5; by (a) a square leaves remainder 0, 1 or 4, so it cannot end in 2, 3, 7 or 8 B1
Worked example 2
Prove that \(n^4-n^2\) is a multiple of 12 for every integer \(n\).
Mark scheme
Factorises: \(n^4-n^2=n^2(n-1)(n+1)\) M1
\((n-1)n(n+1)\) is a product of three consecutive integers, so one factor is a multiple of 3 B1
Considers \(n\) even and \(n\) odd M1
\(n\) even: \(n=2k\), so \(n^2=4k^2\) is a multiple of 4 A1
\(n\) odd: \(n-1\) and \(n+1\) are both even, so \((n-1)(n+1)\) is a multiple of 4 A1
The number is a multiple of 3 and of 4, which have no common factor, so it is a multiple of 12 A1 (cso)
FAQ
What proof is in Edexcel IAL P2?
Proof by deduction, proof by exhaustion and disproof by counter-example, in number, algebra, inequalities and coordinate geometry (proof by contradiction is P4).
Is P2 a calculator paper?
Yes, a calculator is allowed in P2, but method marks are only given for working you write down.
How are P2 proof questions marked?
With M marks for a correct method, A marks for accurate answers that depend on the M mark before them, and B marks for independent results, as in the worked examples on this page.