| Exam Board | OCR MEI |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Proof |
| Type | Counter example to disprove statement |
| Difficulty | Moderate -0.3 Part (i) requires finding a single counterexample (e.g., n=2 gives 11, n=3 gives 29, n=4 gives 83, n=5 gives 245=5×49), which is straightforward trial. Part (ii) is a simple proof by induction or modular arithmetic showing 3^n ≡ 3,9,7,1 (mod 10) cycles without reaching 5. Both parts are routine proof techniques with minimal steps, making this slightly easier than a typical C3 question. |
| Spec | 1.01a Proof: structure of mathematical proof and logical steps1.01c Disproof by counter example |
| Answer | Marks | Guidance |
|---|---|---|
| 2 | (i) | 35 + 2 = 245 [which is not prime] |
| Answer | Marks |
|---|---|
| [2] | Attempt to find counter- |
| Answer | Marks |
|---|---|
| identified | If A0, allow M1 for 3n + 2 correctly |
| Answer | Marks | Guidance |
|---|---|---|
| 2 | (ii) | (30 = 1), 31 = 3, 32 = 9, 33 = 27, 34 = 81, … |
| Answer | Marks |
|---|---|
| so its last digit cannot be a ‘5’ | M1 |
| Answer | Marks |
|---|---|
| B1 | Evaluate 3n for n = 0 to 4 or 1 |
| Answer | Marks |
|---|---|
| must state conclusion for B2 | allow just final digit written |
Question 2:
2 | (i) | 35 + 2 = 245 [which is not prime] | M1
A1
[2] | Attempt to find counter-
example
correct counter-example
identified | If A0, allow M1 for 3n + 2 correctly
evaluated for 3 values of n
2 | (ii) | (30 = 1), 31 = 3, 32 = 9, 33 = 27, 34 = 81, …
so units digits cycle through 1, 3, 9, 7, 1, 3,
…
so cannot be a ‘5’.
OR
3n is not divisible by 5
all numbers ending in ‘5’ are divisible by 5.
so its last digit cannot be a ‘5’ | M1
A1
B1
B1 | Evaluate 3n for n = 0 to 4 or 1
to 5
must state conclusion for B2 | allow just final digit written
[2]\begin{enumerate}[label=(\roman*)]
\item Disprove the following statement:
$$3^n + 2 \text{ is prime for all integers } n \geqslant 0.$$ [2]
\item Prove that no number of the form $3^n$ (where $n$ is a positive integer) has 5 as its final digit. [2]
\end{enumerate}
\hfill \mbox{\textit{OCR MEI C3 Q2 [4]}}