| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2006 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Addition & Double Angle Formulae |
| Type | Prove identity then solve equation |
| Difficulty | Moderate -0.3 Part (i) requires straightforward application of addition formulae with exact values (sin 30°, cos 30°, etc.) and algebraic simplification—standard bookwork. Part (ii) is a routine equation solving √3 cos x = 1 in a restricted domain. This is slightly easier than average as it's a direct application of standard formulae with no problem-solving insight required. |
| Spec | 1.05l Double angle formulae: and compound angle formulae1.05o Trigonometric equations: solve in given intervals1.05p Proof involving trig: functions and identities |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Use trig formulae to express LHS in terms of \(\cos x\) and \(\sin x\) | M1 | |
| Use correct exact values of \(\cos 60°\), \(\sin 60°\), etc. | M1 | |
| Obtain given answer | A1 | Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| State or imply answer is \(\cos^{-1}(1/\sqrt{3})\) | M1 | |
| Obtain answer \(54.7°\) | A1 | Total: 2 |
## Question 2:
### Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Use trig formulae to express LHS in terms of $\cos x$ and $\sin x$ | M1 | |
| Use correct exact values of $\cos 60°$, $\sin 60°$, etc. | M1 | |
| Obtain given answer | A1 | Total: 3 |
### Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| State or imply answer is $\cos^{-1}(1/\sqrt{3})$ | M1 | |
| Obtain answer $54.7°$ | A1 | Total: 2 |
---2 (i) Prove the identity
$$\cos \left( x + 30 ^ { \circ } \right) + \sin \left( x + 60 ^ { \circ } \right) \equiv ( \sqrt { } 3 ) \cos x$$
(ii) Hence solve the equation
$$\cos \left( x + 30 ^ { \circ } \right) + \sin \left( x + 60 ^ { \circ } \right) = 1$$
for $0 ^ { \circ } < x < 90 ^ { \circ }$.
\hfill \mbox{\textit{CAIE P2 2006 Q2 [5]}}