| Exam Board | OCR |
|---|---|
| Module | PURE |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Topic | Proof |
| Type | Algebraic inequality proof |
| Difficulty | Standard +0.3 This question tests the QM-AM inequality proof and connection to variance. Part (a)(i) is trivial notation (1 mark). Part (a)(ii) requires expanding and rearranging (a²+b²)/2 ≥ ((a+b)/2)² to show (a-b)²≥0, which is a standard algebraic manipulation (3 marks). Part (b) asks for interpretation that variance is non-negative, requiring only conceptual understanding (1 mark). While it involves proof, the algebra is straightforward and the result is a well-known inequality, making it slightly easier than average. |
| Spec | 1.01a Proof: structure of mathematical proof and logical steps2.02f Measures of average and spread |
\begin{enumerate}[label=(\alph*)]
\item Two real numbers are denoted by $a$ and $b$.
\begin{enumerate}[label=(\roman*)]
\item Write down expressions for the following.
\begin{itemize}
\item The mean of the squares of $a$ and $b$
\item The square of the mean of $a$ and $b$ [1]
\end{itemize}
\item Prove that the mean of the squares of $a$ and $b$ is greater than or equal to the square of their mean. [3]
\end{enumerate}
\item You are given that the result in part (a)(ii) is true for any two or more real numbers.
Explain what this result shows about the variance of a set of data. [1]
\end{enumerate}
\hfill \mbox{\textit{OCR PURE Q7 [5]}}