| Exam Board | Edexcel |
|---|---|
| Module | Paper 1 (Paper 1) |
| Year | 2022 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Proof |
| Type | Algebraic inequality proof |
| Difficulty | Standard +0.8 This is a two-part proof question requiring (i) proof by contradiction about parity of integers and (ii) algebraic manipulation of an inequality with integer constraints. Part (i) is a standard proof technique but requires careful logical reasoning. Part (ii) requires expanding, rearranging, and factoring the inequality while respecting the constraint x<0, which adds a subtle layer of difficulty when dividing by negative values. This is moderately challenging for A-level, requiring both proof technique and careful algebraic manipulation beyond routine exercises. |
| Spec | 1.01d Proof by contradiction1.02g Inequalities: linear and quadratic in single variable |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Assume there exist integers \(p\) and \(q\) such that \(pq\) is even and both \(p\) and \(q\) are odd | B1 | Must use "assume"/"let"/"there is" or similar; must state "\(pq\) is even" and "\(p\) and \(q\) are (both) odd" |
| Sets \(p = 2m+1\) and \(q = 2n+1\) (different variables) and attempts \(pq = (2m+1)(2n+1) = \ldots\) | M1 | Different variables required; \(p=2n+1\) and \(q=2n-1\) is M0 |
| \(pq = (2m+1)(2n+1) = 4mn+2m+2n+1 = 2(2mn+m+n)+1\); states this is odd, giving a contradiction, so "if \(pq\) is even, then at least one of \(p\) and \(q\) is even" | A1* | Requires: correct calculation, correct reason it is odd, minimal conclusion statement |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \((x+y)^2 < 9x^2 + y^2 \Rightarrow 2xy < 8x^2\) | M1 | Multiply out and cancel terms to reach correct intermediate line e.g. \(2x(4x-y)>0\) |
| As \(x < 0\), \(2y > 8x \Rightarrow y > 4x\) | A1* | Full rigorous proof; point at which inequality reverses must be correct with correct reason given; no incorrect lines permitted |
# Question 7(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Assume there exist integers $p$ and $q$ such that $pq$ is even and both $p$ and $q$ are odd | B1 | Must use "assume"/"let"/"there is" or similar; must state "$pq$ is even" and "$p$ and $q$ are (both) odd" |
| Sets $p = 2m+1$ and $q = 2n+1$ (different variables) and attempts $pq = (2m+1)(2n+1) = \ldots$ | M1 | Different variables required; $p=2n+1$ and $q=2n-1$ is M0 |
| $pq = (2m+1)(2n+1) = 4mn+2m+2n+1 = 2(2mn+m+n)+1$; states this is odd, giving a contradiction, so "if $pq$ is even, then at least one of $p$ and $q$ is even" | A1* | Requires: correct calculation, correct reason it is odd, minimal conclusion statement |
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# Question 7(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $(x+y)^2 < 9x^2 + y^2 \Rightarrow 2xy < 8x^2$ | M1 | Multiply out and cancel terms to reach correct intermediate line e.g. $2x(4x-y)>0$ |
| As $x < 0$, $2y > 8x \Rightarrow y > 4x$ | A1* | Full rigorous proof; point at which inequality reverses must be correct with correct reason given; no incorrect lines permitted |\begin{enumerate}
\item (i) Given that $p$ and $q$ are integers such that
\end{enumerate}
use algebra to prove by contradiction that at least one of $p$ or $q$ is even.\\
(ii) Given that $x$ and $y$ are integers such that
\begin{itemize}
\item $x < 0$
\item $( x + y ) ^ { 2 } < 9 x ^ { 2 } + y ^ { 2 }$\\
show that $y > 4 x$
\end{itemize}
\hfill \mbox{\textit{Edexcel Paper 1 2022 Q7 [5]}}