| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2012 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Standard trigonometric equations |
| Type | Simple double angle equations (direct substitution) |
| Difficulty | Moderate -0.8 This is a straightforward trigonometric equation requiring the standard technique of dividing by cos(2x) to get tan(2x) = -3, then solving in the given range. Part (i) is routine bookwork with 4 solutions in the doubled angle range. Part (ii) simply requires recognizing the period extends the pattern, making this easier than average A-level content. |
| Spec | 1.05l Double angle formulae: and compound angle formulae1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(\to \tan 2x = -3\); \(2x = 180 - 71.6\) or \(360 - 71.6\); \(x = 54.2°\) or \(144.2°\); Also \(234.2°\) and \(324.2°\) | M1, M1, A1, A1 | Uses \(\tan 2x = k\) and works with "2x"; Finds "2x" before \(\div 2\); co; co (both of these need 2nd M) for \(180° +\) his answer(s) |
| (ii) 12 answers | B1 | for 3 times the number of solns to (i) |
$\sin 2x + 3\cos 2x = 0$
**(i)** $\to \tan 2x = -3$; $2x = 180 - 71.6$ or $360 - 71.6$; $x = 54.2°$ or $144.2°$; Also $234.2°$ and $324.2°$ | M1, M1, A1, A1 | Uses $\tan 2x = k$ and works with "2x"; Finds "2x" before $\div 2$; co; co (both of these need 2nd M) for $180° +$ his answer(s)
**(ii)** 12 answers | B1 | for 3 times the number of solns to (i)\begin{enumerate}[label=(\roman*)]
\item Solve the equation $\sin 2x + 3 \cos 2x = 0$ for $0° \leqslant x \leqslant 360°$. [5]
\item How many solutions has the equation $\sin 2x + 3 \cos 2x = 0$ for $0° \leqslant x \leqslant 1080°$? [1]
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2012 Q4 [6]}}