| Exam Board | OCR MEI |
|---|---|
| Module | C1 (Core Mathematics 1) |
| 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 algebraic proof question requiring factorization and divisibility arguments. Part (i) involves factoring to 3n(n+2) and noting that for even n, n(n+2) is the product of consecutive even numbers (divisible by 4), making the expression divisible by 12. Part (ii) requires testing an odd value to show it fails. While it requires proof technique, the algebra is simple and the reasoning is direct, making it slightly easier than average for A-level. |
| Spec | 1.01a Proof: structure of mathematical proof and logical steps1.01b Logical connectives: congruence, if-then, if and only if |
| Answer | Marks |
|---|---|
| 4 | (i) n = 2m |
| Answer | Marks |
|---|---|
| 3n2 + 6n = odd + even = odd | M1 |
| Answer | Marks |
|---|---|
| B2 | or any attempt at generalising; M0 for |
| Answer | Marks |
|---|---|
| explanation or incorrect calculation | 5 |
Question 4:
4 | (i) n = 2m
3n2 + 6n = 12m2 + 12m or
= 12m(m + 1)
(iiii)) showing false w n is odd e.g.
3n2 + 6n = odd + even = odd | M1
M2
B2 | or any attempt at generalising; M0 for
just trying numbers
or M1 for 3n2 + 6n = 3n (n + 2) = 3 ×
even × even and M1 for explaining that
4 is a factor of even × even
or M1 for 12 is a factor of 6n when n is
even and M1 for 4 is a factor of n2 so 12
is a factor of 3n2
or 3n (n + 2) =3 × odd × odd = odd or
counterexample showing not always
true; M1 for false with partial
explanation or incorrect calculation | 5\begin{enumerate}[label=(\roman*)]
\item Prove that 12 is a factor of $3n^2 + 6n$ for all even positive integers $n$. [3]
\item Determine whether 12 is a factor of $3n^2 + 6n$ for all positive integers $n$. [2]
\end{enumerate}
\hfill \mbox{\textit{OCR MEI C1 Q4 [5]}}