Moderate -0.8 This is a straightforward disproof by counterexample requiring only substitution of small values of n until finding one where 4n² + 3 is composite. Most students will quickly find n=2 gives 19 (prime) then n=3 gives 39 = 3×13 (composite). Requires minimal calculation and no sophisticated mathematical insight, making it easier than average.
A student suggests that for any positive integer \(n\) the value of the expression
$$4n^2 + 3$$
is always a prime number.
Prove that the student's statement is false by finding a counter example.
Fully justify your answer.
[3 marks]
Question 5:
5 | Evaluates 4n2 + 3 for at least
one value of n | 3.1a | M1 | n = 1 gives 7
n = 2 gives 19
n = 3 gives 39
39 = 3 × 13 so not prime
Obtains one value that is not
prime | 2.3 | A1
Shows explicitly that their value
factorises. | 2.4 | R1
Question 5 Total | 3
Q | Marking instructions | AO | Marks | Typical solution
A student suggests that for any positive integer $n$ the value of the expression
$$4n^2 + 3$$
is always a prime number.
Prove that the student's statement is false by finding a counter example.
Fully justify your answer.
[3 marks]
\hfill \mbox{\textit{AQA AS Paper 2 2024 Q5 [3]}}