| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2024 |
| Session | November |
| Marks | 9 |
| 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 product-to-sum formula application followed by a routine equation solve. Part (a) uses the product formula 2sin A sin B = cos(A-B) - cos(A+B), then converts to R sin(θ-α) form using standard techniques. Part (b) is straightforward substitution and inverse trig. While multi-step, all techniques are textbook exercises 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) etc1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Expand to obtain \(2\sin^2\theta + 2\sqrt{3}\sin\theta\cos\theta\) | B1 | Needs to be simplified but may be implied by later work. |
| Attempt to express in terms of \(\sin 2\theta\) and \(\cos 2\theta\) | *M1 | Needs to be in the form \(k + p\cos 2\theta + q\sin 2\theta\). |
| Obtain \(1 - \cos 2\theta + \sqrt{3}\sin 2\theta\) | A1 | |
| Obtain \(R = 2\) | A1 FT | Following *their* \(p\sin 2\theta + q\cos 2\theta\) form. |
| Use appropriate trigonometry to find \(\alpha\) | DM1 | For *their* \(p\sin 2\theta + q\cos 2\theta\) form. Allow for \(60°\). |
| Obtain \(1 + 2\sin(2\theta - 30)\) | A1 | |
| Alternative Method: | ||
| Expand to obtain \(2\sin^2\theta + 2\sqrt{3}\sin\theta\cos\theta\) | B1 | Needs to be simplified. |
| For expansion of \(a + R\sin(2\theta - \alpha)\) and comparison with \(2\sin^2\theta + 2\sqrt{3}\sin\theta\cos\theta\) | *M1 | Need comparison of at least 2 like terms: \(R\sin\alpha = 1\), \(R\cos\alpha = \sqrt{3}\), \(a - R\sin\alpha = 0\). |
| \(R\sin\alpha = 1\) and \(R\cos\alpha = \sqrt{3}\) leading to values of \(R\) and \(\alpha\) | DM1 | |
| Obtain \(R = 2\) | A1 FT | |
| Obtain \(\alpha = 30°\) and \(a = 1\) | A1 | |
| Obtain \(1 + 2\sin(2\theta - 30)\) | A1 | |
| Total | 6 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| State \(\sin(2\theta - 30) = -\dfrac{6}{10}\) | \*B1 FT | Following *their* answer to part (a). Must be using \(\sin(2\theta - \alpha) = -\dfrac{1}{5} - their\) (a). Must have a value for \(a\). |
| Attempt complete method to find smallest positive value | DM1 | |
| Obtain \(123.4\) only | A1 | Or greater accuracy. |
| 3 |
## Question 7(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Expand to obtain $2\sin^2\theta + 2\sqrt{3}\sin\theta\cos\theta$ | B1 | Needs to be simplified but may be implied by later work. |
| Attempt to express in terms of $\sin 2\theta$ and $\cos 2\theta$ | *M1 | Needs to be in the form $k + p\cos 2\theta + q\sin 2\theta$. |
| Obtain $1 - \cos 2\theta + \sqrt{3}\sin 2\theta$ | A1 | |
| Obtain $R = 2$ | A1 FT | Following *their* $p\sin 2\theta + q\cos 2\theta$ form. |
| Use appropriate trigonometry to find $\alpha$ | DM1 | For *their* $p\sin 2\theta + q\cos 2\theta$ form. Allow for $60°$. |
| Obtain $1 + 2\sin(2\theta - 30)$ | A1 | |
| **Alternative Method:** | | |
| Expand to obtain $2\sin^2\theta + 2\sqrt{3}\sin\theta\cos\theta$ | B1 | Needs to be simplified. |
| For expansion of $a + R\sin(2\theta - \alpha)$ and comparison with $2\sin^2\theta + 2\sqrt{3}\sin\theta\cos\theta$ | *M1 | Need comparison of at least 2 like terms: $R\sin\alpha = 1$, $R\cos\alpha = \sqrt{3}$, $a - R\sin\alpha = 0$. |
| $R\sin\alpha = 1$ and $R\cos\alpha = \sqrt{3}$ leading to values of $R$ and $\alpha$ | DM1 | |
| Obtain $R = 2$ | A1 FT | |
| Obtain $\alpha = 30°$ and $a = 1$ | A1 | |
| Obtain $1 + 2\sin(2\theta - 30)$ | A1 | |
| **Total** | **6** | |
## Question 7(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| State $\sin(2\theta - 30) = -\dfrac{6}{10}$ | **\*B1 FT** | Following *their* answer to part **(a)**. Must be using $\sin(2\theta - \alpha) = -\dfrac{1}{5} - their$ **(a)**. Must have a value for $a$. |
| Attempt complete method to find smallest positive value | **DM1** | |
| Obtain $123.4$ only | **A1** | Or greater accuracy. |
| | **3** | |7
\begin{enumerate}[label=(\alph*)]
\item Express $4 \sin \theta \sin \left( \theta + 60 ^ { \circ } \right)$ in the form
$$a + R \sin ( 2 \theta - \alpha ) ,$$
where $a$ and $R$ are positive integers and $0 ^ { \circ } < \alpha < 90 ^ { \circ }$.\\
\includegraphics[max width=\textwidth, alt={}, center]{18aea465-b5b0-48f0-970a-e9ede1dc9370-13_2723_33_99_21}
\item Hence find the smallest positive value of $\theta$ satisfying the equation
$$\frac { 1 } { 5 } + 4 \sin \theta \sin \left( \theta + 60 ^ { \circ } \right) = 0 .$$
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.\\
\includegraphics[max width=\textwidth, alt={}, center]{18aea465-b5b0-48f0-970a-e9ede1dc9370-14_2714_38_109_2010}
\begin{center}
\end{center}
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2024 Q7 [9]}}