$\frac{(5-\sqrt{3})}{(2+\sqrt{3})} \times \frac{(2-\sqrt{3})}{(2-\sqrt{3})}$ | M1 | Multiplying top and bottom by $(2-\sqrt{3})$. (As shown above is sufficient.)

$= \frac{10 - 2\sqrt{3} - 5\sqrt{3} + (\sqrt{3})^2}{...}$ (or $= \frac{10 - 7\sqrt{3} + 3}{...}$) | M1 | Attempt to multiply out numerator $(5-\sqrt{3})(2-\sqrt{3})$. Must have at least 3 terms correct.

$= 13 - 7\sqrt{3}$ (Allow $\frac{13-7\sqrt{3}}{1}$) | A1 | Although 'denominator = 1' may be implied, the $13 - 7\sqrt{3}$ must obviously be the final answer (not an intermediate step), to score full marks. (Also M0 M1 A1 A1 is not an option.)

$a = 13$ | A1 | The A marks cannot be scored unless the 1st M mark has been scored, but this 1st M mark could be implied by correct expansions of both numerator and denominator.

$b = -7$ | A1 |

**Special case:** If numerator is multiplied by $(2+\sqrt{3})$ instead of $(2-\sqrt{3})$, the 2nd M can still be scored for at least 3 of these terms correct: $10 - 2\sqrt{3} + 5\sqrt{3} - (\sqrt{3})^2$. The maximum score in the special case is 1 mark: M0 M1 A0 A0.

**Answer only:** Scores no marks.

**Alternative method:**
$5 - \sqrt{3} = (a + b\sqrt{3})(2 + \sqrt{3})$ | M1 | At least 3 terms correct.

$(a + b\sqrt{3})(2 + \sqrt{3}) = 2a + a\sqrt{3} + 2b\sqrt{3} + 3$ | M1 | Form and attempt to solve simultaneous equations.

$5 = 2a + 3b$
$-1 = a + 2b$

$a = 13, b = -7$ | A1, A1 |

**Total: 4 marks**

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