| Exam Board | OCR MEI |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2008 |
| Session | January |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Proof |
| Type | Divisibility proof for all integers |
| Difficulty | Moderate -0.3 This is a straightforward divisibility proof requiring factorization and case analysis. Part (i) involves factoring to 3n(n+2) and recognizing that for even n, one of n or n+2 is divisible by 4. Part (ii) requires testing odd n (counterexample: n=1 gives 9). While it requires algebraic manipulation and understanding of even/odd properties, it's a standard C1 proof question with no novel insight needed—slightly easier than average due to its routine nature. |
| Spec | 1.01a Proof: structure of mathematical proof and logical steps |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(n = 2m\) | Marks: M1 | Guidance: or any attempt at generalising; M0 for just trying numbers |
| \(3n^2 + 6n = 12n^2 + 12n\) or \(= 12m(n + 1)\) | Marks: M2 | Guidance: or M1 for \(3n^2 + 6n = 3n(n + 2) = 3 \times\) even \(\times\) even and M1 for explaining that 4 is a factor of even \(\times\) even; or M1 for 12 is a factor of \(6n\) when \(n\) is even and M1 for 4 is a factor of \(n^2\) so 12 is a factor of \(3n^2\) |
| (ii) showing false when \(n\) is odd e.g. \(3n^2 + 6n = \) odd \(+\) even \(=\) odd | Marks: B2 | Guidance: or \(3n(n + 2) = 3 \times\) odd \(\times\) odd \(=\) odd or counterexample showing not always true; M1 for false with partial explanation or incorrect calculation |
**(i)** $n = 2m$ | **Marks:** M1 | **Guidance:** or any attempt at generalising; M0 for just trying numbers | **Total:** 1
$3n^2 + 6n = 12n^2 + 12n$ or $= 12m(n + 1)$ | **Marks:** M2 | **Guidance:** or M1 for $3n^2 + 6n = 3n(n + 2) = 3 \times$ even $\times$ even and M1 for explaining that 4 is a factor of even $\times$ even; or M1 for 12 is a factor of $6n$ when $n$ is even and M1 for 4 is a factor of $n^2$ so 12 is a factor of $3n^2$ | **Total:** 2
**(ii)** showing false when $n$ is odd e.g. $3n^2 + 6n = $ odd $+$ even $=$ odd | **Marks:** B2 | **Guidance:** or $3n(n + 2) = 3 \times$ odd $\times$ odd $=$ odd or counterexample showing not always true; M1 for false with partial explanation or incorrect calculation | **Total:** 3
**Total for Question 9:** 59 (i) Prove that 12 is a factor of $3 n ^ { 2 } + 6 n$ for all even positive integers $n$.\\
(ii) Determine whether 12 is a factor of $3 n ^ { 2 } + 6 n$ for all positive integers $n$.
\hfill \mbox{\textit{OCR MEI C1 2008 Q9 [5]}}