| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2013 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Addition & Double Angle Formulae |
| Type | Prove identity then solve equation |
| Difficulty | Standard +0.8 Part (i) requires applying compound angle formulae to expand sin(x-60°) and cos(x-30°), then recognizing the resulting expression simplifies to sin x (non-trivial algebraic manipulation). Part (ii) uses this result to solve a trigonometric equation involving cosec x and cot² x, requiring substitution and solving a quadratic in sin x. This is more demanding than routine formula application but less challenging than proof-heavy or multi-concept Further Maths questions. |
| 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 |
|---|---|---|
| (i) Use correct \(\sin(A - B)\) and \(\cos(A - B)\) formula | M1 | |
| Substitute exact values for \(\cos 30°\) etc. | M1 | |
| Obtain given answer correctly | A1 | [3] |
| (ii) State \(2\cos ec \, x = 3\cot^2 x - 2\) | B1 | |
| Use \(\cot^2 x = \csc ec^2 x - 1\) | M1 | |
| Attempt solution of quadratic equation in \(\csc ec \, x\) or \(\sin x\) (\(3\csc ec^2 x - 2\csc ec x - 5 = 0\) or \(3\sin^2 x - 2\sin x - 3 = 0\)) | M1 | |
| Obtain \(\sin x = \frac{3}{5}\) or \(-1\) | A1✓ | |
| Obtain one correct answer for \(\sin^{-1}\left(\frac{3}{5}\right)\) | A1 | |
| Obtain remaining 2 answers from \(36.9°\), \(143.1°\), \(270°\) and no others in the range [Ignore answers outside the given range] | A1 | [6] |
| SC If only answer given is \(270°\) | B1 |
**(i)** Use correct $\sin(A - B)$ and $\cos(A - B)$ formula | M1 |
Substitute exact values for $\cos 30°$ etc. | M1 |
Obtain given answer correctly | A1 | [3]
**(ii)** State $2\cos ec \, x = 3\cot^2 x - 2$ | B1 |
Use $\cot^2 x = \csc ec^2 x - 1$ | M1 |
Attempt solution of quadratic equation in $\csc ec \, x$ or $\sin x$ ($3\csc ec^2 x - 2\csc ec x - 5 = 0$ or $3\sin^2 x - 2\sin x - 3 = 0$) | M1 |
Obtain $\sin x = \frac{3}{5}$ or $-1$ | A1✓ |
Obtain one correct answer for $\sin^{-1}\left(\frac{3}{5}\right)$ | A1 |
Obtain remaining 2 answers from $36.9°$, $143.1°$, $270°$ and no others in the range [Ignore answers outside the given range] | A1 | [6]
SC If only answer given is $270°$ | B1 |8 (i) Prove the identity
$$\frac { 1 } { \sin \left( x - 60 ^ { \circ } \right) + \cos \left( x - 30 ^ { \circ } \right) } \equiv \operatorname { cosec } x$$
(ii) Hence solve the equation
$$\frac { 2 } { \sin \left( x - 60 ^ { \circ } \right) + \cos \left( x - 30 ^ { \circ } \right) } = 3 \cot ^ { 2 } x - 2$$
for $0 ^ { \circ } < x < 360 ^ { \circ }$.
\hfill \mbox{\textit{CAIE P2 2013 Q8 [9]}}