June 2025 Paper 1 Q21
21 Show that \(\quad \dfrac{20\sin^2 x}{3 + \cos^2 x - 3\sin^2 x} \quad\) can be simplified to \(\ k\tan^2 x\ \) where \(k\) is an integer. [2 marks]
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1 | ||
| Denominator \(3 + \cos^2 x - 3(1 - \cos^2 x)\) | M1 | oe after identity used eg \(\cos^2 x + 3\cos^2 x\) |
| \(\dfrac{20\sin^2 x}{4\cos^2 x} = 5\tan^2 x\) | A1 | |
| Alternative method 2 | ||
| Denominator \(4 - 4\sin^2 x\) | M1 | from \(\cos^2 x = 1 - \sin^2 x\) |
| \(\dfrac{20\sin^2 x}{4\cos^2 x} = 5\tan^2 x\) | A1 | |
| Alternative method 3 | ||
| Denominator \(3(\cos^2 x + \sin^2 x) + \cos^2 x - 3\sin^2 x\) | M1 | oe after identity used eg \(\cos^2 x + 3\cos^2 x\) |
| \(\dfrac{20\sin^2 x}{4\cos^2 x} = 5\tan^2 x\) | A1 | |
Additional guidance
| Only mark using one of the alts – once the candidate starts to treat the solution as an equation by moving terms around from one side of the identity to the other then stop awarding marks | |
| Correct notation for \(5\tan^2 x\) needed to score A mark | |
| M mark can be awarded for denominator worked out separately but not the A mark (unless recovered) |