| Exam Board | WJEC |
|---|---|
| Module | Unit 1 (Unit 1) |
| Year | 2022 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Proof |
| Type | Counter example to disprove statement |
| Difficulty | Standard +0.3 This question tests understanding of algebraic inequalities and proof techniques at a basic level. Statement A is a standard inequality provable by rearranging to (x-y)² ≥ 0, while Statement B fails for negative values (an easy counter-example). The question explicitly tells students one is true and one is false, removing the need for independent analysis. The proof required is straightforward algebraic manipulation, and finding a counter-example requires only testing simple cases like negative numbers. This is slightly easier than average as it's highly scaffolded and tests fundamental AS-level proof skills. |
| Spec | 1.01a Proof: structure of mathematical proof and logical steps1.01c Disproof by counter example |
In each of the two statements below, $x$ and $y$ are real numbers. One of the statements is true while the other is false.
A: $x^2 + y^2 \geqslant 2xy$, for all real values of $x$ and $y$.
B: $x + y \geqslant 2\sqrt{xy}$, for all real values of $x$ and $y$.
\begin{enumerate}[label=(\alph*)]
\item Identify the statement which is false. Find a counter example to show that this statement is in fact false. [3]
\item Identify the statement which is true. Give a proof to show that this statement is in fact true. [2]
\end{enumerate}
\hfill \mbox{\textit{WJEC Unit 1 2022 Q6 [5]}}