October 2021 Paper 3 Q6
6 In this question you must show detailed reasoning.
Show that \(\displaystyle\sum_{r=1}^{3} \frac{1}{\sqrt{r+1} + \sqrt{r}} = 1\). [4]
| Scheme | Marks | AO |
|---|---|---|
| DR \(\dfrac{1}{\sqrt{2}+1} + \dfrac{1}{\sqrt{3}+\sqrt{2}} + \dfrac{1}{\sqrt{4}+\sqrt{3}}\) | B1 | 1.1a |
| \(\dfrac{\sqrt{2}-1}{2-1} + \dfrac{\sqrt{3}-\sqrt{2}}{1} + \dfrac{\sqrt{4}-\sqrt{3}}{1}\) | M1 A1 | 3.1a 1.1 |
| \(\sqrt{2} - 1 + \sqrt{3} - \sqrt{2} + 2 - \sqrt{3} = 1\) | A1 | 2.1 |
| [4] |
Notes
DR: This question included the instruction: In this question you must show detailed reasoning.
B1: Substituting values
M1: Attempt to rationalise denominator for one term
Either \(\times\dfrac{\sqrt{2}-1}{\sqrt{2}-1}\) or \(\dfrac{\sqrt{2}-1}{2-1}\) at least once for M1
A1: All correct
A1: Convincing completion (AG)