| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2016 |
| Session | November |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Addition & Double Angle Formulae |
| Type | Show equation reduces to tan form |
| Difficulty | Moderate -0.8 This is a straightforward algebraic manipulation using double angle formulae followed by routine solving. Part (i) requires simple rearrangement to isolate tan 2x (collecting sin 2x and cos 2x terms, then dividing), and part (ii) involves standard inverse tan calculation with range consideration. No conceptual difficulty or novel insight required—purely procedural application of basic trigonometric identities. |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(2\sin 2x = 6\cos 2x \rightarrow \tan 2x = k\) | M1 | Expand and collect as far as \(\tan 2x =\) a constant from \(\sin \div \cos\), soi, cwo |
| \(\tan 2x = 3\) or \(k = 3\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(x = (\tan^{-1}(\text{their } k)) \div 2\) | M1 | Inverse then \(\div 2\), soi |
| \((71.6°\) or \(-108.4°) \div 2\) | ||
| \(x = 35.8°, -54.2°\) | A1 A1\(\checkmark\) | \(\checkmark\) on 1st answer \(+/- 90°\) if in given range but no extra solutions in the given range. Both SR A1A0 |
| \(x = 0.624^c, -0.946^c\) | ||
| \(x = 0.198\pi^c, -0.301\pi^c\) |
## Question 2(i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $2\sin 2x = 6\cos 2x \rightarrow \tan 2x = k$ | M1 | Expand and collect as far as $\tan 2x =$ a constant from $\sin \div \cos$, soi, cwo |
| $\tan 2x = 3$ or $k = 3$ | A1 | |
## Question 2(ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $x = (\tan^{-1}(\text{their } k)) \div 2$ | M1 | Inverse then $\div 2$, soi |
| $(71.6°$ or $-108.4°) \div 2$ | | |
| $x = 35.8°, -54.2°$ | A1 A1$\checkmark$ | $\checkmark$ on 1st answer $+/- 90°$ if in given range but no extra solutions in the given range. Both SR A1A0 |
| $x = 0.624^c, -0.946^c$ | | |
| $x = 0.198\pi^c, -0.301\pi^c$ | | |
---2 (i) Express the equation $\sin 2 x + 3 \cos 2 x = 3 ( \sin 2 x - \cos 2 x )$ in the form $\tan 2 x = k$, where $k$ is a constant.\\
(ii) Hence solve the equation for $- 90 ^ { \circ } \leqslant x \leqslant 90 ^ { \circ }$.
\hfill \mbox{\textit{CAIE P1 2016 Q2 [5]}}