Higher November 2022 Paper 1 Q21
21. Write \(7 - \sqrt{27} + \dfrac{44}{5 + \sqrt{3}}\) in the form \(p + q\sqrt{3}\). [5]
| Scheme | Marks | Comments |
|---|---|---|
| Sight of \(\sqrt{27} = 3\sqrt{3}\) | B1 | |
| \(\dfrac{44}{5 + \sqrt{3}} \times \dfrac{5 - \sqrt{3}}{5 - \sqrt{3}}\) oe | M1 | |
| \(44 \times \dfrac{5 - \sqrt{3}}{22}\) oe | B1 | For a correct denominator of 22 |
| \(7 - 3\sqrt{3} + 10 - 2\sqrt{3}\) oe | M1 | FT for final M1 A0 provided B1 M1 previously awarded |
| \(17 - 5\sqrt{3}\) | A1 | CAO |
Alternative method
| Scheme | Marks | Comments |
|---|---|---|
| Sight of \(\sqrt{27} = 3\sqrt{3}\) | B1 | May be seen at any stage |
| \(\dfrac{(7 - \sqrt{27})(5 + \sqrt{3}) + 44}{5 + \sqrt{3}}\) \(\left(= \dfrac{35 + 7\sqrt{3} - 5\sqrt{27} - \sqrt{27 \times 3}}{5 + \sqrt{3}}\right)\) oe \(\left(= \dfrac{70 - 8\sqrt{3}}{5 + \sqrt{3}}\right)\) oe si | M1 | |
| \(= \dfrac{(70 - 8\sqrt{3})(5 - \sqrt{3})}{(5 + \sqrt{3})(5 - \sqrt{3})}\) oe si | M1 | FT ‘their \(\dfrac{70 - 8\sqrt{3}}{5 + \sqrt{3}}\)’ provided B1 M1 previously awarded |
| \(= \dfrac{(350 - 70\sqrt{3} - 40\sqrt{3} + 24)}{22}\) oe si | B1 | For a correct denominator of 22 |
| \(= 17 - 5\sqrt{3}\) | A1 | CAO |
| (5) |