| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/2 (Pre-U Mathematics Paper 2) |
| Year | 2010 |
| Session | June |
| Marks | 6 |
| Topic | Standard trigonometric equations |
| Type | Solve using given identity |
| Difficulty | Standard +0.3 Part (i) is a straightforward algebraic identity using difference of squares factorization and the Pythagorean identity. Part (ii) requires substituting the result, using the double angle formula to get 2cos²x - 1 = cos(2x), then solving cos(2x) = cos(x), which involves standard trigonometric equation techniques but with a minor twist of solving for multiple angles. Overall slightly easier than average due to the guided structure and routine techniques. |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| Using difference of squares (or attempting to remove the sine or cosine): \(\cos^4 x - \sin^4 x = (\cos^2 x + \sin^2 x)(\cos^2 x - \sin^2 x)\) | M1 | |
| \(\equiv 2\cos^2 x - 1\) using trig identities | A1 | [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Required to solve \(2\cos^2 x - 1 = \cos x\) which factorises as \((2\cos x + 1)(\cos x - 1) = 0\), or use of the quadratic formula | M1 | A1 |
| Solutions are: \(120°, 240°\) | A1 | |
| \(0°, 360°\) | A1 | [4] |
### (i)
Using difference of squares (or attempting to remove the sine or cosine): $\cos^4 x - \sin^4 x = (\cos^2 x + \sin^2 x)(\cos^2 x - \sin^2 x)$ | M1 |
$\equiv 2\cos^2 x - 1$ using trig identities | A1 | [2]
### (ii)
Required to solve $2\cos^2 x - 1 = \cos x$ which factorises as $(2\cos x + 1)(\cos x - 1) = 0$, or use of the quadratic formula | M1 | A1 |
Solutions are: $120°, 240°$ | A1 |
$0°, 360°$ | A1 | [4]
*Note: A1 for any two correct, A1A1 for all correct.*\begin{enumerate}[label=(\roman*)]
\item Show that
$$\cos^4 x - \sin^4 x = 2\cos^2 x - 1.$$ [2]
\item Hence find the solutions of
$$\cos^4 x - \sin^4 x = \cos x,$$
where $0° \leqslant x \leqslant 360°$. [4]
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9794/2 2010 Q4 [6]}}