| Exam Board | Edexcel |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2024 |
| Session | January |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Proof |
| Type | Counter example to disprove statement |
| Difficulty | Moderate -0.8 Part (i) requires only testing small natural numbers until finding one where n²+3n+1 is composite (e.g., n=4 gives 29, n=5 gives 41, n=6 gives 55=5×11), which is straightforward trial. Part (ii) is a standard proof by exhaustion considering n as even/odd, showing n²-2 leaves remainder 2 or 3 when divided by 4—a routine modular arithmetic argument commonly seen in A-level proof questions. Both parts are accessible with basic techniques and no novel insight required. |
| Spec | 1.01a Proof: structure of mathematical proof and logical steps1.01c Disproof by counter example |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Substitutes a value e.g. \(n=6\) into \(n^2+3n+1\) where \(n^2+3n+1\) is not prime | M1 | Possible values \(n=6,11,13,16\) etc. \(n=0\) not acceptable |
| Correct calculation e.g. \(n^2+3n+1=55\) AND conclusion "which is not prime" | A1 | Must reference not being prime or show divisibility e.g. \(6^2+3\times6+1=55\) which is not prime |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Attempts to find \(n^2-2\) for either odds or evens e.g. \((2p+1)^2-2\) or \((2p)^2-2\) | M1 | Allow \(n=2k+2\) or \(2n\pm5\); condone \(n=2n\) and \(n=2n\pm1\) |
| Achieves \((2p+1)^2-2=4p^2+4p-1\) or \((2p)^2-2=4p^2-2\) AND shows why expression is not a multiple of 4 | A1 | e.g. \(4p^2+4p-1=4(p^2+p)-1\) is A1; do not accept "cannot take 4 as common factor" without showing |
| Attempts to find \(n^2-2\) for both odds and evens | dM1 | Dependent on previous M1 |
| Achieves both \((2p+1)^2-2=4p^2+4p-1\) and \((2p)^2-2=4p^2-2\), shows neither is a multiple of 4, with concluding statement | A1* | Must have conclusion e.g. "hence not a multiple of 4 for all \(n\)"; withhold if \(n\) used instead of \(k\) |
# Question 8:
## Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Substitutes a value e.g. $n=6$ into $n^2+3n+1$ where $n^2+3n+1$ is **not prime** | M1 | Possible values $n=6,11,13,16$ etc. $n=0$ not acceptable |
| Correct calculation e.g. $n^2+3n+1=55$ AND conclusion "which is not prime" | A1 | Must reference not being prime or show divisibility e.g. $6^2+3\times6+1=55$ which is not prime |
## Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Attempts to find $n^2-2$ for either odds or evens e.g. $(2p+1)^2-2$ or $(2p)^2-2$ | M1 | Allow $n=2k+2$ or $2n\pm5$; condone $n=2n$ and $n=2n\pm1$ |
| Achieves $(2p+1)^2-2=4p^2+4p-1$ or $(2p)^2-2=4p^2-2$ AND shows why expression is not a multiple of 4 | A1 | e.g. $4p^2+4p-1=4(p^2+p)-1$ is A1; do not accept "cannot take 4 as common factor" without showing |
| Attempts to find $n^2-2$ for **both** odds and evens | dM1 | Dependent on previous M1 |
| Achieves both $(2p+1)^2-2=4p^2+4p-1$ and $(2p)^2-2=4p^2-2$, shows neither is a multiple of 4, with concluding statement | A1* | Must have conclusion e.g. "hence not a multiple of 4 for all $n$"; withhold if $n$ used instead of $k$ |
---\begin{enumerate}
\item (i) Use a counter example to show that the following statement is false
\end{enumerate}
$$\text { " } n ^ { 2 } + 3 n + 1 \text { is prime for all } n \in \mathbb { N } \text { " }$$
(ii) Use algebra to prove by exhaustion that for all $n \in \mathbb { N }$
$$\text { " } n ^ { 2 } - 2 \text { is not a multiple of } 4 \text { " }$$
\hfill \mbox{\textit{Edexcel P2 2024 Q8 [6]}}