| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2022 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Harmonic Form |
| Type | Express double angle or product |
| Difficulty | Standard +0.3 This is a standard harmonic form question requiring double angle identities (sin2θ, cos2θ), then R-α form conversion using standard techniques, followed by routine equation solving and integration. While multi-part, each step follows textbook procedures with no novel insight required, making it slightly easier than average. |
| Spec | 1.05l Double angle formulae: and compound angle formulae1.05n Harmonic form: a sin(x)+b cos(x) = R sin(x+alpha) etc1.08c Integrate e^(kx), 1/x, sin(kx), cos(kx) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use correct identity for \(\sin 2\theta\) or \(\cos 2\theta\) (or both) | M1 | |
| Obtain \(6\sin 2\theta + 8\cos 2\theta\ (+8)\) | A1 | |
| State \(R = 10\) | B1 FT | FT *their* \(a\sin 2\theta + b\cos 2\theta\) form |
| Use appropriate trigonometry to find \(\alpha\) using *their* \(a\sin 2\theta + b\cos 2\theta\) form | M1 | Allow \(0.927\) |
| Obtain \(10\cos(2\theta - 0.6435) + 8\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| State or imply \(\cos(2\theta - 0.6435) = 0.9\) | B1 FT | FT *their* \(R\), \(\alpha\) and \(k\) provided RHS less than 1. |
| Carry out correct process to find any positive value of \(\theta\) | M1 | \(0.547\) |
| Obtain \(0.0962\) | A1 | AWRT |
| 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Integrate trigonometry term from part (a) to obtain \(k_1\sin(2\theta - \text{their } 0.6435)\) | M1 | any non-zero constant \(k_1\). |
| Obtain \(5\sin(2\theta - 0.6435) + 8\theta\) | A1 | condone absence of \(\ldots + c\). |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Integrate to obtain at least form \(k_2\cos 2\theta + k_3\sin 2\theta\) | M1 | any non-zero constants \(k_2, k_3\). |
| Obtain \(-3\cos 2\theta + 4\sin 2\theta + 8\theta\) | A1 | condone absence of \(\ldots + c\). |
| 2 |
## Question 8(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use correct identity for $\sin 2\theta$ or $\cos 2\theta$ (or both) | M1 | |
| Obtain $6\sin 2\theta + 8\cos 2\theta\ (+8)$ | A1 | |
| State $R = 10$ | B1 FT | FT *their* $a\sin 2\theta + b\cos 2\theta$ form |
| Use appropriate trigonometry to find $\alpha$ using *their* $a\sin 2\theta + b\cos 2\theta$ form | M1 | Allow $0.927$ |
| Obtain $10\cos(2\theta - 0.6435) + 8$ | A1 | |
## Question 8(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| State or imply $\cos(2\theta - 0.6435) = 0.9$ | **B1 FT** | FT *their* $R$, $\alpha$ and $k$ provided RHS less than 1. |
| Carry out correct process to find any positive value of $\theta$ | **M1** | $0.547$ |
| Obtain $0.0962$ | **A1** | AWRT |
| | **3** | |
## Question 8(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Integrate trigonometry term from part **(a)** to obtain $k_1\sin(2\theta - \text{their } 0.6435)$ | **M1** | any non-zero constant $k_1$. |
| Obtain $5\sin(2\theta - 0.6435) + 8\theta$ | **A1** | condone absence of $\ldots + c$. |
**Alternative method for question 8(c):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| Integrate to obtain at least form $k_2\cos 2\theta + k_3\sin 2\theta$ | **M1** | any non-zero constants $k_2, k_3$. |
| Obtain $-3\cos 2\theta + 4\sin 2\theta + 8\theta$ | **A1** | condone absence of $\ldots + c$. |
| | **2** | |8 The expression $\mathrm { f } ( \theta )$ is defined by $\mathrm { f } ( \theta ) = 12 \sin \theta \cos \theta + 16 \cos ^ { 2 } \theta$.
\begin{enumerate}[label=(\alph*)]
\item Express $\mathrm { f } ( \theta )$ in the form $R \cos ( 2 \theta - \alpha ) + k$, where $R > 0,0 < \alpha < \frac { 1 } { 2 } \pi$ and $k$ is a constant. State the values of $R$ and $k$, and give the value of $\alpha$ correct to 4 significant figures.
\item Find the smallest positive value of $\theta$ satisfying the equation $\mathrm { f } ( \theta ) = 17$.
\item Find $\int f ( \theta ) d \theta$.\\
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 [10]}}