| Exam Board | SPS |
|---|---|
| Module | SPS SM Pure (SPS SM Pure) |
| Year | 2024 |
| Session | June |
| Marks | 6 |
| Topic | Proof |
| Type | Contradiction proof about integers |
| Difficulty | Standard +0.3 This is a straightforward proof question testing standard techniques (contradiction, counter-example, exhaustion) with relatively simple algebra. Part (a) factors to (x-y)(x+y)=1 requiring minimal insight, part (b) needs only one example like logâ‚‚4, and part (c) involves routine case-checking of n mod 2. These are textbook-level proof exercises slightly easier than average A-level questions. |
| Spec | 1.01a Proof: structure of mathematical proof and logical steps1.01c Disproof by counter example1.01d Proof by contradiction |
\begin{enumerate}
\item a) Use proof by contradiction to prove that there are no positive integers, $x$ and $y$, such that
\end{enumerate}
$$x ^ { 2 } - y ^ { 2 } = 1$$
b) Prove, by counter-example, that the statement
\section*{" if $a$ is rational and $b$ is irrational then $\log _ { a } b$ is irrational"}
is false.\\
c) Use algebra to prove by exhaustion that for all $n \in \mathbb { N }$
$$\text { " } n ^ { 2 } - 2 \text { is not a multiple of } 4 "$$
\section*{(Total for Question 10 is 6 marks)}
\hfill \mbox{\textit{SPS SPS SM Pure 2024 Q10 [6]}}