| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2022 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Harmonic Form |
| Type | Express and solve equation |
| Difficulty | Challenging +1.2 This is a multi-step harmonic form question requiring expansion of compound angles, algebraic manipulation to collect sin/cos terms, and solving a transformed equation. While it involves several techniques (double angle formulas, sec conversion, compound angle expansion, R-α form), these are standard P2 procedures. The substitution θ=2β in part (b) is straightforward once part (a) is complete. More challenging than routine exercises but follows predictable patterns without requiring novel insight. |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05k Further identities: sec^2=1+tan^2 and cosec^2=1+cot^21.05l Double angle formulae: and compound angle formulae1.05n Harmonic form: a sin(x)+b cos(x) = R sin(x+alpha) etc |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use \(\sin 2\theta = 2\sin\theta\cos\theta\) and \(\sec\theta = \frac{1}{\cos\theta}\) to obtain \(6\sin\theta\) | B1 | |
| Expand second term to obtain \(5\sqrt{3}\cos\theta + 5\sin\theta\) | B1 | |
| Simplify to obtain \(11\sin\theta + 5\sqrt{3}\cos\theta\) | B1 | |
| State \(R = 14\) | B1 FT | FT *their* \(k_1\cos\theta + k_2\sin\theta\) |
| Use appropriate trigonometry to find \(\alpha\) | M1 | |
| Obtain \(\alpha = 38.21\) | A1 | AWRT |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State or imply \(14\sin(2\beta + 38.21) = 2\) | B1 FT | FT *their* \(R\) and \(\alpha\) |
| Carry out correct process to find value of \(\beta\) between \(0°\) and \(90°\) | M1 | |
| Obtain \(66.8\) | A1 | AWRT |
## Question 8(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use $\sin 2\theta = 2\sin\theta\cos\theta$ and $\sec\theta = \frac{1}{\cos\theta}$ to obtain $6\sin\theta$ | B1 | |
| Expand second term to obtain $5\sqrt{3}\cos\theta + 5\sin\theta$ | B1 | |
| Simplify to obtain $11\sin\theta + 5\sqrt{3}\cos\theta$ | B1 | |
| State $R = 14$ | B1 FT | FT *their* $k_1\cos\theta + k_2\sin\theta$ |
| Use appropriate trigonometry to find $\alpha$ | M1 | |
| Obtain $\alpha = 38.21$ | A1 | AWRT |
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## Question 8(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| State or imply $14\sin(2\beta + 38.21) = 2$ | B1 FT | FT *their* $R$ and $\alpha$ |
| Carry out correct process to find value of $\beta$ between $0°$ and $90°$ | M1 | |
| Obtain $66.8$ | A1 | AWRT |8
\begin{enumerate}[label=(\alph*)]
\item Express $3 \sin 2 \theta \sec \theta + 10 \cos \left( \theta - 30 ^ { \circ } \right)$ in the form $R \sin ( \theta + \alpha )$ where $R > 0$ and $0 ^ { \circ } < \alpha < 90 ^ { \circ }$. Give the value of $\alpha$ correct to 2 decimal places.
\item Hence solve the equation $3 \sin 4 \beta \sec 2 \beta + 10 \cos \left( 2 \beta - 30 ^ { \circ } \right) = 2$ for $0 ^ { \circ } < \beta < 90 ^ { \circ }$.\\
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2022 Q8 [9]}}