| Exam Board | SPS |
|---|---|
| Module | SPS SM Pure (SPS SM Pure) |
| Year | 2022 |
| Session | June |
| Marks | 6 |
| Topic | Proof |
| Type | Contradiction proof of irrationality |
| Difficulty | Standard +0.8 Part (a) requires understanding divisibility proofs and algebraic manipulation (factoring n³-1 and using n=3k+1), which is moderately challenging. Part (b) combines logarithm manipulation with proof by contradiction using fundamental theorem of arithmetic—a technique less routine than standard A-level proofs. Both parts demand mathematical maturity beyond typical exercises, but the steps are accessible with careful reasoning. |
| Spec | 1.01a Proof: structure of mathematical proof and logical steps1.01d Proof by contradiction1.06c Logarithm definition: log_a(x) as inverse of a^x1.06e Logarithm as inverse: ln(x) inverse of e^x |
\begin{enumerate}[label=\alph*)]
\item Prove that
$$n - 1 \text{ is divisible by } 3 \Rightarrow n^3 - 1 \text{ is divisible by } 9$$ [3 marks]
\item Show that if $\log_2 3 = \frac{p}{q}$, then
$$2^p = 3^q.$$
Use proof by contradiction to prove that $\log_2 3$ is irrational. [3 marks]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM Pure 2022 Q15 [6]}}