| Exam Board | Edexcel |
|---|---|
| Module | Paper 2 (Paper 2) |
| Year | 2020 |
| Session | October |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Proof |
| Type | Proof by exhaustion with cases |
| Difficulty | Moderate -0.3 This is a standard proof by exhaustion using modular arithmetic. Students need to consider three cases (n ≡ 0, 1, 2 mod 3) and show n² is either 0 or 1 mod 3. While it requires understanding of proof structure and modular arithmetic, it's a well-practiced technique with straightforward algebra and only three cases to check, making it slightly easier than average. |
| Spec | 1.01a Proof: structure of mathematical proof and logical steps |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Any natural number expressed as \(3k\), \(3k+1\), \(3k+2\) (or e.g. \(3k-1\), \(3k\), \(3k+1\)); attempts to square any two distinct cases | M1 | Key step of writing naturals in distinct forms and squaring |
| \((3k)^2 = 9k^2 = 3 \times 3k^2\) is a multiple of 3; \((3k+1)^2 = 9k^2+6k+1 = 3(3k^2+2k)+1\) is one more than a multiple of 3; \((3k+2)^2 = 9k^2+12k+4 = 3(3k^2+4k+1)+1\) is one more than a multiple of 3 | A1 (M1 on EPEN) | Successfully shows for 2 cases squares are multiple of 3 or 1 more; must be made explicit algebraically |
| Attempts to square all 3 distinct cases | M1 (A1 on EPEN) | Recognises all 3 forms needed |
| Achieves accurate results for all three cases and gives minimal conclusion | A1 | Allow tick, QED etc. |
| Total: (4) |
## Question 16 (Proof by Exhaustion):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Any natural number expressed as $3k$, $3k+1$, $3k+2$ (or e.g. $3k-1$, $3k$, $3k+1$); attempts to square **any two** distinct cases | M1 | Key step of writing naturals in distinct forms and squaring |
| $(3k)^2 = 9k^2 = 3 \times 3k^2$ is a multiple of 3; $(3k+1)^2 = 9k^2+6k+1 = 3(3k^2+2k)+1$ is one more than a multiple of 3; $(3k+2)^2 = 9k^2+12k+4 = 3(3k^2+4k+1)+1$ is one more than a multiple of 3 | A1 (M1 on EPEN) | Successfully shows for 2 cases squares are multiple of 3 or 1 more; must be made explicit algebraically |
| Attempts to square all 3 distinct cases | M1 (A1 on EPEN) | Recognises all 3 forms needed |
| Achieves accurate results for all three cases and gives minimal conclusion | A1 | Allow tick, QED etc. |
| **Total: (4)** | | |\begin{enumerate}
\item Use algebra to prove that the square of any natural number is either a multiple of 3 or one more than a multiple of 3
\end{enumerate}
\hfill \mbox{\textit{Edexcel Paper 2 2020 Q16 [4]}}