| Exam Board | SPS |
|---|---|
| Module | SPS SM Pure (SPS SM Pure) |
| Year | 2021 |
| Session | June |
| Marks | 3 |
| Topic | Proof |
| Type | Algebraic inequality proof |
| Difficulty | Moderate -0.8 Part (a) is a standard AM-GM inequality proof requiring algebraic manipulation of $(\sqrt{x}-\sqrt{y})^2 \geq 0$, which is a common textbook exercise. Part (b) simply requires providing any counter example with negative numbers (e.g., x=y=-1), which is trivial. The question tests basic proof techniques but requires minimal problem-solving or insight. |
| Spec | 1.01a Proof: structure of mathematical proof and logical steps1.01c Disproof by counter example |
\begin{enumerate}[label=(\alph*)]
\item Prove that for all positive values of $x$ and $y$
$$\sqrt{xy} \leq \frac{x + y}{2}$$
[2]
\item Prove by counter example that this is not true when $x$ and $y$ are both negative. [1]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM Pure 2021 Q13 [3]}}