OCR MEI C3 2007 June — Question 5 2 marks

Exam BoardOCR MEI
ModuleC3 (Core Mathematics 3)
Year2007
SessionJune
Marks2
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicProof
TypeCounter example to disprove statement
DifficultyEasy -1.2 This is a straightforward disproof by counterexample requiring only substitution of small integer values. Students need only test n=1,2,3,... until finding a composite value (n=4 gives 29, which is prime, but n=5 gives 41, prime, then n=6 gives 55=5×11). The conceptual demand is minimal—no proof technique beyond trial is needed, making this easier than average A-level work.
Spec1.01c Disproof by counter example
5 Prove that the following statement is false.
For all integers \(n\) greater than or equal to \(1 , n ^ { 2 } + 3 n + 1\) is a prime number.
Question 5:
AnswerMarks Guidance
Let \(n=4\): \(16 + 12 + 1 = 29\) — need a non-prime exampleM1 Attempt to find counter-example
\(n=4\): \(n^2+3n+1 = 29\) (prime); try \(n=7\): \(49+21+1=71\) (prime)— —
e.g. \(n=4\): gives \(29\); correct counter-example shown, e.g. \(n\) where result is compositeM1A1 —
Counter-example: e.g. \(n=7\) gives \(71\); or valid composite shownA1 [2] Valid counter-example with value confirmed not prime 
# Question 5:

| Let $n=4$: $16 + 12 + 1 = 29$ — need a non-prime example | M1 | Attempt to find counter-example |
| $n=4$: $n^2+3n+1 = 29$ (prime); try $n=7$: $49+21+1=71$ (prime) | — | — |
| e.g. $n=4$: gives $29$; correct counter-example shown, e.g. $n$ where result is composite | M1A1 | — |
| Counter-example: e.g. $n=7$ gives $71$; or valid composite shown | A1 | [2] Valid counter-example with value confirmed not prime |

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5 Prove that the following statement is false.\\
For all integers $n$ greater than or equal to $1 , n ^ { 2 } + 3 n + 1$ is a prime number.

\hfill \mbox{\textit{OCR MEI C3 2007 Q5 [2]}}
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