Higher November 2020 Unit 1 Q18
18. A circle has radius \(r\) cm, where \(r\) is an integer.
The side of a square is of length \(x\) cm.
If the circle and square have the same area, explain why \(x\) cannot be an integer.
You should consider algebraic expressions in your answer. [2]
| Scheme | Marks | Comments |
|---|---|---|
| Sight of \(x = (\sqrt{\pi}) \times r\) OR \(x = \sqrt{(\pi r^2)}\) or equivalent | B1 | Allow an equivalent expression, e.g. \(r = x / (\sqrt{\pi})\) or \(r = \sqrt{(x^2 / \pi)}\). Allow use of 3.14 for \(\pi\). |
| Convincing concluding argument e.g. \(x\) is irrational since \(\pi\) (and therefore \(\sqrt{\pi}\)) is irrational. | E1 | E1 depends on B1. Accept e.g. multiplying an integer by \(\sqrt{\pi}\) will not produce another integer; multiplying an integer by \(\sqrt{\pi}\) will produce an infinite decimal. Do not accept a reason based on \(\sqrt{\pi}\) not being a whole number. Consideration of a specific numerical case gains no credit. |
Allow an alternative method
| Scheme | Marks | Comments |
|---|---|---|
\(x^2\) and \(\pi r^2\) both seen WITH a related statement about
\(\pi r^2\) (or \(3.14 r^2\)) cannot be a square number; multiplying an integer by \(\pi\) (or 3.14) cannot produce a square number; \(\pi r^2\) is irrational; \(\pi\) times an integer (squared) is a decimal (or cannot be an integer). | E1 | For \(x^2 = \pi r^2\), allow an equivalent equation, e.g. \(r^2 = x^2 / \pi\). Allow use of 3.14 for \(\pi\). Do not accept a statement that \(3.14 r^2\) is not an integer or that \(3.14 r^2\) is irrational. |
| Convincing concluding argument leading to \(x\) (not \(x^2\)) being a non-integer e.g. \(x\) is irrational since \(x^2\) is irrational; \(x\) is not an integer since \(x^2\) is a decimal. | E1 | Depends on previous E1 Consideration of a specific numerical case gains no credit. |