| Answer/Scheme | Marks | Guidance |
|---|---|---|
| $\frac{8-\sqrt{15}}{2\sqrt{3}+\sqrt{5}} \times \frac{2\sqrt{3}-\sqrt{5}}{2\sqrt{3}-\sqrt{5}} = \frac{16\sqrt{3}-8\sqrt{5}-2\sqrt{45}+\sqrt{75}}{12-5}$ | M1 | Attempts to rationalise the denominator by multiplying both numerator and denominator by $k(2\sqrt{3}-\sqrt{5})$, where $k$ is an integer usually 1, and proceeds to a fraction such as $\frac{...√3 ± ...√5 ± ...√45 ± ...√75}{...}$ or $\frac{...}{12-5}$ or $\frac{...}{7}$. Allow $\sqrt{45}$ and $\sqrt{75}$ to be written in terms of $\sqrt{3}$ and $\sqrt{5}$ as well. Condone slips in multiplying out as well as miscopying errors. Note $\frac{21\sqrt{3}-14\sqrt{5}}{7}$ with no intermediate working can still score M1. Attempting to multiply by a multiple of $\frac{2\sqrt{3}+\sqrt{5}}{2\sqrt{3}+\sqrt{5}}$ is M0dM0A0. |
| e.g. $\frac{21\sqrt{3}-14\sqrt{5}}{7}$ | dM1 | Attempts to simplify surds and may collect terms to achieve a fraction where: the numerator is in terms of $\sqrt{3}$ and $\sqrt{5}$ only, the denominator is a multiple of 7 (which may be unsimplified). It is dependent on the previous method mark. If they have not fully multiplied out the numerator (or implied) then this mark cannot be scored. e.g. $\frac{(8-\sqrt{15})(2\sqrt{3}-\sqrt{5})}{...} = \frac{16\sqrt{3}+8\sqrt{5}-2\sqrt{45}+\sqrt{75}}{...}$ scores maximum M1dM0A0. Note $\frac{21\sqrt{3}-14\sqrt{5}}{7}$ with **no previous working seen** scores M1dM0A0 because they have not shown any multiplying out of the brackets on the numerator. Note $\frac{8-\sqrt{15}}{2\sqrt{3}+\sqrt{5}} = \frac{16\sqrt{3}-8\sqrt{5}-2\sqrt{45}+\sqrt{75}}{12-25}$ is M1dM0A0 because they have not collected terms on the numerator (or changed them all to be in terms of $\sqrt{3}$ and $\sqrt{5}$ before simplifying the final answer). |
| $3\sqrt{3}-2\sqrt{5}$ | A1 | $3\sqrt{3}-2\sqrt{5}$. The answer does not imply the method marks. Do not withhold this mark for slips in working such as invisible brackets provided they are recovered/implied by further work. |
| **(a) Total: 3 marks** | | |
| $(x+5\sqrt{3})\sqrt{5} = 40 - 2x\sqrt{3} \Rightarrow x\sqrt{5}+2x\sqrt{3} = 40-5\sqrt{15}$ | M1 | Multiplies out the brackets and isolates the $x$ terms on one side. Condone for this mark if the surds are converted to rounded decimals. e.g. $(\sqrt{5}+2\sqrt{3})x = 5(8-\sqrt{15})$ scores M1. Alternatively divides both sides by $\sqrt{5}$ and isolates the $x$ terms on one side. e.g. $x+2x\frac{\sqrt{3}}{\sqrt{5}} = \frac{40}{\sqrt{5}}-\sqrt{3}$. Condone slips in their working and invisible brackets, but there must be two terms on each side of the equation. |
| $(x=)\frac{40-5\sqrt{15}}{2\sqrt{3}+\sqrt{5}}$ | A1 | $x = \frac{40-5\sqrt{15}}{2\sqrt{3}+\sqrt{5}}$ or exact equivalent (which cannot be $15\sqrt{3}-10\sqrt{5}$). |
| $(x=)15\sqrt{3}-10\sqrt{5}$ | A1ft | $x = 15\sqrt{3}-10\sqrt{5}$ only (or exact simplified equivalent) e.g. $x = 5(3\sqrt{3}-2\sqrt{5})$. This mark cannot be awarded without the previous A mark being scored. Follow through on their part (a) answer of the form $a\sqrt{3}+b\sqrt{5}$ so award for $5a\sqrt{3}+5b\sqrt{5}$ one including $5(a\sqrt{3}+b\sqrt{5})$ where $a$ and $b$ may be fractions. Isw once they have achieved the correct answer. Note: If candidate has unsimplified answer in part (a) e.g. $\frac{21\sqrt{3}-14\sqrt{5}}{7}$ then do not withhold this mark in part (b) if they proceed to e.g. $\frac{105\sqrt{3}-70\sqrt{5}}{7}$ as they have already been penalised once for not giving answer in simplest form. |
| **(b) Total: 3 marks** | | |
| **Total for Question 3: 6 marks** | | |

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