| Exam Board | OCR |
|---|---|
| Module | H240/02 (Pure Mathematics and Statistics) |
| Year | 2023 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Topic | Proof |
| Type | Contradiction proof about integers |
| Difficulty | Standard +0.8 This is a proof by contradiction requiring modular arithmetic insight (recognizing that if a² - 4b = 2, then a² ≡ 2 (mod 4), which is impossible since squares are only 0 or 1 mod 4). While the individual steps are accessible, the problem requires students to recognize the appropriate proof technique and work with congruences, which goes beyond routine A-level proof questions. The 'hence' structure provides scaffolding, but students must still connect the parts logically. |
| Spec | 2.02f Measures of average and spread2.02g Calculate mean and standard deviation2.02h Recognize outliers |
A student wishes to prove that, for all positive integers $a$ and $b$, $a^2 - 4b \neq 2$.
\begin{enumerate}[label=(\alph*)]
\item Prove that $a^2 - 4b = 2 \Rightarrow a$ is even. [2]
\item Hence or otherwise prove that, for all positive integers $a$ and $b$, $a^2 - 4b \neq 2$. [3]
\end{enumerate}
\hfill \mbox{\textit{OCR H240/02 2023 Q7 [5]}}