Higher June 2024 Paper 1 Q21
21. The mean of the three values \(x\), \(\dfrac{12}{\sqrt{3}}\) and \(\left(\sqrt{3}\right)^5\) is \(6\sqrt{3}\).
Calculate the value of \(x\).
Give your answer in the form \(a\sqrt{3}\) where \(a\) is an integer. [4]
| Scheme | Marks | Comments |
|---|---|---|
| (Total of the 3 values =) \(18\sqrt{3}\) | B1 | May be seen in later working |
| \(\left(\dfrac{12}{\sqrt{3}} =\right)\ 4\sqrt{3}\) | B1 | |
| \(\left(\sqrt{3}^{\,5} =\right)\ 9\sqrt{3}\) | B1 | |
| \((x = 18\sqrt{3} - 4\sqrt{3} - 9\sqrt{3} =)\ 5\sqrt{3}\) | B1 | FT provided at least B2 previously awarded, the third value of the form \(a\sqrt{3}\) and \(x \gt 0\) Mark the final answer unless \(x\) clearly identified |
Alternative method 1 (multiplying by \(\sqrt{3}\))
| Scheme | Marks | Comments |
|---|---|---|
| (Total of the 3 values =) \(18\sqrt{3}\) | B1 | May be seen in later working |
| \(\sqrt{3}x + 12 + \sqrt{3}^{\,6} = 54\) | B1 | FT their \(a\sqrt{3}\) Using \(\left(x + \dfrac{12}{\sqrt{3}} + \sqrt{3}^{\,5} = 18\sqrt{3}\right)\times\sqrt{3}\) |
| \(\sqrt{3}^{\,6} = 27\) | B1 | |
| \(x = \left(\dfrac{54 - 12 - 27}{\sqrt{3}} =\right)\ 5\sqrt{3}\) | B1 | FT provided at least B1 B1 previously awarded, \(\sqrt{3}^{\,6}\) is an integer and \(x \gt 0\) Mark the final answer unless \(x\) clearly identified |
Alternative method 2 (multiplying by \(\sqrt{3}\))
| Scheme | Marks | Comments |
|---|---|---|
| (Total of the 3 values =) \(18\sqrt{3}\) | B1 | May be seen in later working |
| \(\left(\sqrt{3}^{\,5} =\right)\ 9\sqrt{3}\) | B1 | |
| \(\sqrt{3}x + 12 + 27 = 54\) or better | B1 | FT their \(a\sqrt{3}\) Using \(\left(x + \dfrac{12}{\sqrt{3}} + 9\sqrt{3} = 18\sqrt{3}\right)\times\sqrt{3}\) |
| \(x = \left(\dfrac{54 - 12 - 27}{\sqrt{3}} =\right)\ 5\sqrt{3}\) | B1 | FT their \(a\sqrt{3}\) provided at least B1 B1 previously awarded and \(x \gt 0\) Mark the final answer unless \(x\) clearly identified |
| (4) |