| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2010 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Addition & Double Angle Formulae |
| Type | Prove identity then solve equation |
| Difficulty | Standard +0.3 Part (i) requires routine application of addition formulae with standard angles (30°, 60°) and algebraic simplification—straightforward for P2 level. Part (ii) uses the proven identity to create a trigonometric equation involving sec x, requiring substitution and solving a quadratic in sin x, which is standard technique. The question is slightly easier than average due to the structured 'hence' guidance and use of common angles with known exact values. |
| Spec | 1.05h Reciprocal trig functions: sec, cosec, cot definitions and graphs1.05l Double angle formulae: and compound angle formulae1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Use correct \(\sin(A - B)\) and \(\cos(A - B)\) formulae | M1 | |
| Substitute exact values for sin \(30°\) etc. | M1 | |
| Obtain given answer correctly | A1 | [3] |
| (ii) State \(\sqrt{3} \sin x = \frac{1}{2} \sec x\) | B1 | |
| Rearrange to \(\sin 2x = k\), where \(k\) is a non-zero constant | M1 | |
| Carry out evaluation of \(\frac{1}{2} \sin^{-1}\left(\frac{1}{\sqrt{3}}\right)\) | M1 | |
| Obtain answer \(17.6°\) | A1 | |
| Carry out correct method for second answer | M1 | |
| Obtain remaining 3 answers from \(17.6°, 72.4°, 197.6°, 252.4°\) and no others in the range | A1 | |
| [Ignore answers outside the given range] | [6] |
**(i)** Use correct $\sin(A - B)$ and $\cos(A - B)$ formulae | M1 |
Substitute exact values for sin $30°$ etc. | M1 |
Obtain given answer correctly | A1 | [3]
**(ii)** State $\sqrt{3} \sin x = \frac{1}{2} \sec x$ | B1 |
Rearrange to $\sin 2x = k$, where $k$ is a non-zero constant | M1 |
Carry out evaluation of $\frac{1}{2} \sin^{-1}\left(\frac{1}{\sqrt{3}}\right)$ | M1 |
Obtain answer $17.6°$ | A1 |
Carry out correct method for second answer | M1 |
Obtain remaining 3 answers from $17.6°, 72.4°, 197.6°, 252.4°$ and no others in the range | A1 |
[Ignore answers outside the given range] | [6]8 (i) Prove the identity
$$\sin \left( x - 30 ^ { \circ } \right) + \cos \left( x - 60 ^ { \circ } \right) \equiv ( \sqrt { } 3 ) \sin x$$
(ii) Hence solve the equation
$$\sin \left( x - 30 ^ { \circ } \right) + \cos \left( x - 60 ^ { \circ } \right) = \frac { 1 } { 2 } \sec x$$
for $0 ^ { \circ } < x < 360 ^ { \circ }$.
\hfill \mbox{\textit{CAIE P2 2010 Q8 [9]}}