| Exam Board | Edexcel |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2023 |
| Session | January |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Proof by induction |
| Type | Prove by exhaustion (cases) |
| Difficulty | Moderate -0.8 This is a straightforward proof by exhaustion requiring students to check the remaining two cases (n=3k+1 and n=3k+2) using the same algebraic technique already demonstrated. It's routine modular arithmetic with clear structure provided, making it easier than average but not trivial since it requires understanding the exhaustion method and careful algebra. |
| Spec | 1.01a Proof: structure of mathematical proof and logical steps |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Considers another relevant case for \(n\), e.g. \(n = 3k-1\) or \(3k+1\) or \(3k+2\). For example \(n = 3k+1\): \(2n^2+n+1 = 2(3k+1)^2+(3k+1)+1 = 18k^2+15k+4\) | M1 | Attempts to substitute a second case into expression in \(k\), multiplies out brackets and collects terms. Condone slips. May attempt e.g. \(3k+4\) or any other multiples of those given. |
| \(= 3(6k^2+5k+1)+1\) which is not divisible by 3 | A1 | Completes with no errors to show expression is not divisible by 3, usually by factorising part of the expression, or may identify a particular term not divisible by 3. |
| Considers a third relevant case to complete all 3 cases for \(n\), e.g. \(n = 3k-1\): \(2n^2+n+1 = 2(3k-1)^2+(3k-1)+1 = 18k^2-9k+2\) | dM1 | Third case must be different from the given case and not equivalent to their first relevant case. Attempts to substitute into expression in \(k\), multiplies out and collects terms. Condone slips. Must have been in terms of \(k\) for all cases. |
| \(= 3(6k^2-3k)+2\) which is not divisible by 3. Hence \(2n^2+n+1\) is not divisible by 3. | A1 | Completes with no errors to show 3rd expression is not divisible by 3 (usually by factorising, or identifying a term not divisible by 3) and concludes proof with a statement e.g. hence \(2n^2+n+1\) not divisible by 3. Alternatively may write statement/preamble at beginning and end proof with tick/QED. |
| (4) |
| Answer | Marks | Guidance |
|---|---|---|
| Case | \(2n^2+n+1\) | Possible factorisation |
| \(3k-2\) | \(18k^2-21k+7\) | \(=3(6k^2-7k+2)+1\) |
| \(3k-1\) | \(18k^2-9k+2\) | \(=3(6k^2-3k)+2\) |
| \(3k+1\) | \(18k^2+15k+4\) | \(=3(6k^2+5k+1)+1\) |
| \(3k+2\) | \(18k^2+27k+11\) | \(=3(6k^2+9k+3)+2\) |
| \(3k+4\) | \(18k^2+51k+37\) | \(=3(6k^2+17k+12)+1\) |
## Question 10:
| Answer/Working | Mark | Guidance |
|---|---|---|
| Considers another relevant case for $n$, e.g. $n = 3k-1$ or $3k+1$ or $3k+2$. For example $n = 3k+1$: $2n^2+n+1 = 2(3k+1)^2+(3k+1)+1 = 18k^2+15k+4$ | M1 | Attempts to substitute a second case into expression in $k$, multiplies out brackets and collects terms. Condone slips. May attempt e.g. $3k+4$ or any other multiples of those given. |
| $= 3(6k^2+5k+1)+1$ which is not divisible by 3 | A1 | Completes with no errors to show expression is not divisible by 3, usually by factorising part of the expression, or may identify a particular term not divisible by 3. |
| Considers a third relevant case to complete all 3 cases for $n$, e.g. $n = 3k-1$: $2n^2+n+1 = 2(3k-1)^2+(3k-1)+1 = 18k^2-9k+2$ | dM1 | Third case must be different from the given case and not equivalent to their first relevant case. Attempts to substitute into expression in $k$, multiplies out and collects terms. Condone slips. Must have been in terms of $k$ for all cases. |
| $= 3(6k^2-3k)+2$ which is not divisible by 3. Hence $2n^2+n+1$ is not divisible by 3. | A1 | Completes with no errors to show 3rd expression is not divisible by 3 (usually by factorising, or identifying a term not divisible by 3) and concludes proof with a statement e.g. hence $2n^2+n+1$ not divisible by 3. Alternatively may write statement/preamble at beginning and end proof with tick/QED. |
| | **(4)** | |
**Reference table for checking accuracy of quadratics:**
| Case | $2n^2+n+1$ | Possible factorisation |
|---|---|---|
| $3k-2$ | $18k^2-21k+7$ | $=3(6k^2-7k+2)+1$ |
| $3k-1$ | $18k^2-9k+2$ | $=3(6k^2-3k)+2$ |
| $3k+1$ | $18k^2+15k+4$ | $=3(6k^2+5k+1)+1$ |
| $3k+2$ | $18k^2+27k+11$ | $=3(6k^2+9k+3)+2$ |
| $3k+4$ | $18k^2+51k+37$ | $=3(6k^2+17k+12)+1$ |\begin{enumerate}
\item A student was asked to prove by exhaustion that\\
if $n$ is an integer then $2 n ^ { 2 } + n + 1$ is not divisible by 3
\end{enumerate}
The start of the student's proof is shown in the box below.
Consider the case when $n = 3 k$
$$2 n ^ { 2 } + n + 1 = 18 k ^ { 2 } + 3 k + 1 = 3 \left( 6 k ^ { 2 } + k \right) + 1$$
which is not divisible by 3
Complete this proof.
\hfill \mbox{\textit{Edexcel P2 2023 Q10 [4]}}