Higher June 2025 Paper 1 Q18
18 Show that \(\quad \dfrac{\sin 30^\circ \times \cos 45^\circ}{\tan 60^\circ} \quad\) can be written in the form \(\dfrac{\sqrt{a}}{b}\)
where \(a\) and \(b\) are integers. [3 marks]
| Answer | Mark | Comments |
|---|---|---|
| \(\sin 30 = \dfrac{1}{2}\) and \(\cos 45 = \dfrac{1}{\sqrt{2}}\) or \(\dfrac{\sqrt{2}}{2}\) and \(\tan 60 = \sqrt{3}\) | M2 | oe eg \(\dfrac{\frac{1}{2} \times \frac{1}{\sqrt{2}}}{\sqrt{3}}\) or \(\dfrac{\frac{1}{2} \times \frac{\sqrt{2}}{2}}{\sqrt{3}}\) or \(\dfrac{1}{2\sqrt{6}}\) may be seen in a table implied by position in a calculation eg \(\dfrac{1}{2} \times \dfrac{1}{\sqrt{2}} \times \dfrac{1}{\sqrt{3}}\) is correct for M2 M1 1 or 2 correct values |
| \(\dfrac{\sqrt{6}}{12}\) with all three correct values seen | A1 | oe in the form \(\dfrac{\sqrt{a}}{b}\) eg \(\dfrac{\sqrt{24}}{24}\) condone \(a = 6\) and \(b = 12\) with all three correct values seen |
Additional guidance
| Allow \(\sqrt{1}\) for 1 throughout | |
| Allow, eg \(\dfrac{\sqrt{3}/2}{1/2}\) for tan 60 | |
| Correct answer from \(\dfrac{\frac{1}{\sqrt{2}} \times \frac{1}{2}}{\sqrt{3}}\) scores only M1 for tan 60 unless the correct values are attributed to sin 30 and cos 45 elsewhere in the working | |
| Do not allow further work eg \(\dfrac{\sqrt{6}}{12} = \dfrac{\sqrt{1}}{2}\) (with all three correct values seen) | M2A0 |