| Exam Board | SPS |
|---|---|
| Module | SPS SM Pure (SPS SM Pure) |
| Year | 2023 |
| Session | September |
| Marks | 6 |
| Topic | Proof |
| Type | Counter example to disprove statement |
| Difficulty | Standard +0.8 This question requires understanding of proof techniques and counterexamples across three distinct statements. Part (a) needs finding a counterexample (e.g., n=5 gives 33=3×11), part (b) requires completing the square to show (x-4)²+1>0, and part (c) needs a counterexample like p=√2, q=√2 giving pq=2. While each individual part is accessible, the variety of proof methods and the need to recognize when to prove vs disprove elevates this above routine exercises, though it remains within standard A-level proof expectations. |
| Spec | 1.01c Disproof by counter example1.02e Complete the square: quadratic polynomials and turning points |
Prove or disprove each of the following statements:
\begin{enumerate}[label=(\alph*)]
\item If $n$ is an integer, then $3n^2 - 11n + 13$ is a prime number. [2]
\item If $x$ is a real number, then $x^2 - 8x + 17$ is positive. [2]
\item If $p$ and $q$ are irrational numbers, then $pq$ is irrational. [2]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM Pure 2023 Q13 [6]}}