| Exam Board | Edexcel |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2022 |
| Session | October |
| Marks | 3 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Proof |
| Type | Proof by exhaustion with table |
| Difficulty | Easy -1.2 This is a straightforward proof by exhaustion requiring only substitution of the constraint c=b+2 into a+b+c=10 to get a=8-2b, then testing small integer values of b. The table structure guides students through the process, and recognizing that products are even requires only basic parity knowledge. Significantly easier than average A-level proof questions. |
| Spec | 1.01a Proof: structure of mathematical proof and logical steps |
| \(a\) | \(b\) | \(c\) |
| 1 | ||
| 2 | ||
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Any one correct row: \((a,b,c) \in \{(6,1,3),(4,2,4),(2,3,5)\}\). Products do not need to be found. | B1 | e.g. \(a=6, b=1, c=3 \Rightarrow abc=18\); or \(a=4, b=2, c=4 \Rightarrow abc=32\); or \(a=2, b=3, c=5 \Rightarrow abc=30\) |
| Attempts the product \(abc\) for at least 2 valid combinations | M1 | Note M1 usually follows B1 |
| Finds all three valid combinations with correct products AND shows why this is exhaustive, with conclusion | A1* | Requires: all three correct products; no other combinations unless crossed out/discounted; minimal conclusion e.g. "product of \(a\), \(b\) and \(c\) is even, hence proven" |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Uses information to obtain correct equation e.g. \(a+2b=8\) or \(a=8-2b\) | B1 | |
| States \(a\) must be even and considers product \(abc\) in some way | M1 | |
| States \(abc\) is even with a reason e.g. "even \(\times\) anything is even" | A1* |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Correct equation e.g. \(a+2b=8\), \(a=8-2b\) | B1 | |
| \(abc=(8-2b)b(b+2)\); attempts product of \(a\), \(b\) and \(c\) in terms of \(b\) | M1 | |
| \(abc=2(4-b)b(b+2)\) which is even, hence proven | A1* | Must conclude \(abc\) is even with minimal conclusion; no algebraic errors; note \(abc=-2b^3+4b^2+16b\) is insufficient unless 2 is factored out |
# Question 1:
| Answer/Working | Mark | Guidance |
|---|---|---|
| Any one correct row: $(a,b,c) \in \{(6,1,3),(4,2,4),(2,3,5)\}$. Products do not need to be found. | B1 | e.g. $a=6, b=1, c=3 \Rightarrow abc=18$; or $a=4, b=2, c=4 \Rightarrow abc=32$; or $a=2, b=3, c=5 \Rightarrow abc=30$ |
| Attempts the product $abc$ for at least 2 valid combinations | M1 | Note M1 usually follows B1 |
| Finds all three valid combinations with correct products AND shows why this is exhaustive, with conclusion | A1* | Requires: all three correct products; no other combinations unless crossed out/discounted; minimal conclusion e.g. "product of $a$, $b$ and $c$ is even, hence proven" |
**Algebraic/logic approach:**
| Answer/Working | Mark | Guidance |
|---|---|---|
| Uses information to obtain correct equation e.g. $a+2b=8$ or $a=8-2b$ | B1 | |
| States $a$ must be even and considers product $abc$ in some way | M1 | |
| States $abc$ is even with a reason e.g. "even $\times$ anything is even" | A1* | |
**Pure Algebraic approach:**
| Answer/Working | Mark | Guidance |
|---|---|---|
| Correct equation e.g. $a+2b=8$, $a=8-2b$ | B1 | |
| $abc=(8-2b)b(b+2)$; attempts product of $a$, $b$ and $c$ in terms of $b$ | M1 | |
| $abc=2(4-b)b(b+2)$ which is even, hence proven | A1* | Must conclude $abc$ is even with minimal conclusion; no algebraic errors; note $abc=-2b^3+4b^2+16b$ is insufficient unless 2 is factored out |
---\begin{enumerate}
\item Given that $a , b$ and $c$ are integers greater than 0 such that
\end{enumerate}
\begin{itemize}
\item $c = b + 2$
\item $a + b + c = 10$
\end{itemize}
Prove, by exhaustion, that the product of $a , b$ and $c$ is always even.\\
You may use the table below to illustrate your answer.
You may not need to use all rows of this table.
\begin{center}
\begin{tabular}{|l|l|l|}
\hline
$a$ & $b$ & $c$ \\
\hline
& 1 & \\
\hline
& 2 & \\
\hline
& & \\
\hline
& & \\
\hline
& & \\
\hline
& & \\
\hline
\end{tabular}
\end{center}
\hfill \mbox{\textit{Edexcel P2 2022 Q1 [3]}}