Edexcel IAL P4 Proof Questions
Proof by contradiction (e.g. irrationality of √2, infinitely many primes).
- P4 · Pure Mathematics 4
- Calculator allowed
- 5 practice questions
What the questions test
The 5 P4 proof practice questions cover:
Worked example 1
Prove by contradiction that there are infinitely many prime numbers.
Mark scheme
- M1 for assuming that there is a finite number of prime numbers, listing them as $p_1, p_2, \dots, p_n$.
- M1 for constructing a new number $N = (p_1 \times p_2 \times \dots \times p_n) + 1$.
- M1 for arguing that dividing $N$ by any of the primes $p_i$ leaves a remainder of $1$, meaning $N$ has no prime factors from the finite list.
- A1 for concluding that $N$ must either be a new prime itself or have a prime factor not in the list, contradicting the assumption that the list contains all primes.
Worked example 2
Prove by contradiction that if $a$ and $b$ are integers, then $a^2 - 4b \ne 2$.
Mark scheme
- M1 for assuming that there exist integers $a$ and $b$ such that $a^2 - 4b = 2$, which rearranges to $a^2 = 4b + 2 = 2(2b + 1)$.
- M1 for deducing that since $a^2$ is an even number, $a$ must also be an even number.
- M1 for letting $a = 2k$, substituting to get $(2k)^2 = 4b + 2 \implies 4k^2 = 4b + 2$, and dividing by $2$ to get $2k^2 = 2b + 1 \implies 2(k^2 - b) = 1$.
- A1 for observing that the left-hand side $2(k^2 - b)$ is even while the right-hand side $1$ is odd, which is a contradiction, so $a^2 - 4b \ne 2$.
FAQ
What proof is in Edexcel IAL P4?
Proof by contradiction (e.g. irrationality of √2, infinitely many primes).
Is P4 a calculator paper?
Yes, a calculator is allowed in P4, but method marks are only given for working you write down.
How are P4 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.