| Exam Board | OCR MEI |
|---|---|
| Module | AS Paper 1 (AS Paper 1) |
| Year | 2021 |
| Session | November |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Proof |
| Type | Counter example to disprove statement |
| Difficulty | Moderate -0.3 Part (a) is trivial arithmetic (24 < 256). Part (b) requires finding a counterexample to disprove a statement, which is a basic proof technique, but the counterexamples (n=1 or n=2) are immediately obvious to check. This is easier than average as it requires minimal problem-solving and tests only basic understanding of proof by counterexample. |
| Spec | 1.01c Disproof by counter example |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(4! = 24\) and \(4^4 = 256\) or \(4! = 4 \times 3 \times 2 \times 1\) and \(4^4 = 4 \times 4 \times 4 \times 4\) | M1 | Evaluating both expressions or clearly comparing the factors of each |
| So \(4! < 4^4\) | E1 | Clear conclusion seen |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Using counterexample \(n = 1\); \(1! = 1^1 = 1\) | M1 | Attempt to find a counterexample |
| So the statement is false / Nina is incorrect | E1 | Clear argument about the statement from \(n = 1\) |
## Question 4(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $4! = 24$ and $4^4 = 256$ or $4! = 4 \times 3 \times 2 \times 1$ and $4^4 = 4 \times 4 \times 4 \times 4$ | M1 | Evaluating both expressions or clearly comparing the factors of each |
| So $4! < 4^4$ | E1 | Clear conclusion seen |
**Total: [2]**
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## Question 4(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Using counterexample $n = 1$; $1! = 1^1 = 1$ | M1 | Attempt to find a counterexample |
| So the statement is false / Nina is incorrect | E1 | Clear argument about the statement from $n = 1$ |
**Total: [2]**4
\begin{enumerate}[label=(\alph*)]
\item Show that $4 ! < 4 ^ { 4 }$.
\item Nina believes that the statement $n ! < n ^ { n }$ is true for all positive integers $n$.
Prove that Nina is not correct.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI AS Paper 1 2021 Q4 [4]}}