| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2024 |
| Session | March |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Harmonic Form |
| Type | Express and solve equation |
| Difficulty | Challenging +1.2 This is a standard harmonic form question requiring compound angle expansion, R-formula conversion, and solving a transformed equation. While it involves multiple steps and the coefficient √2 adds minor algebraic complexity, the techniques are routine for P3/Further Pure students with no novel problem-solving required. The extended range (-4π < θ < 4π) increases computational work but not conceptual difficulty. |
| Spec | 1.05n Harmonic form: a sin(x)+b cos(x) = R sin(x+alpha) etc1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Use the correct expansion of \(\cos\left(x + \frac{1}{4}\pi\right)\) to obtain \(\sin x + 2\cos x\) | B1 | \(3\sin x + 2\sqrt{2}\left(\frac{1}{\sqrt{2}}\cos x - \frac{1}{\sqrt{2}}\sin x\right)\) |
| State \(R = \sqrt{5}\) | B1 FT | ISW. FT their \(a\sin x + b\cos x\) provided this expression obtained by correct method. |
| Use correct trig formulae to find \(\alpha\) | M1 | \(\alpha = \tan^{-1}(b/a)\) from their \(a\sin x + b\cos x\), or \(\sin^{-1}\) or \(\cos^{-1}\) provided this expression obtained by correct method. NB If \(\cos\alpha = 1\) and \(\sin\alpha = 2\) then M0 A0. |
| Obtain \(\alpha = 1.107\) | A1 | 3 d.p. CAO. Treat answer in degrees as a misread \((63.435°)\). |
| Total | 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\sin^{-1}\left(\frac{1.5}{R}\right)\) | B1 FT | Follow their \(R\). |
| Use a correct method to obtain an un-simplified value of \(\theta\) with their \(\alpha\) | M1 | \(2\left(\sin^{-1}\left(\frac{1.5}{R}\right) - \alpha\right)\) or \(2\left(\pi - \sin^{-1}\left(\frac{1.5}{R}\right) - \alpha\right)\) |
| Obtain one correct answer e.g. \(-0.74\) in the interval | A1 | |
| Obtain second correct answer e.g. \(2.60\) \((2.5986)\) or \(4\pi - 0.74 = 11.8\) or \(2.60 - 4\pi = -9.97\) in the interval | A1 | If uses \(1.11°\) withhold first accuracy mark gained, but allow rest of accuracy marks. Allow \(2.6(0)\). |
| Obtain two more correct answers e.g. \(-9.97\) and \(11.8\) and no others in the interval | A1 | Ignore answers outside the interval. Treat answers in degrees as a misread. \((-571.1°, -42.6°, 148.9°, 677.2°)\). |
| Total | 5 |
## Question 8:
### Part 8(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Use the correct expansion of $\cos\left(x + \frac{1}{4}\pi\right)$ to obtain $\sin x + 2\cos x$ | B1 | $3\sin x + 2\sqrt{2}\left(\frac{1}{\sqrt{2}}\cos x - \frac{1}{\sqrt{2}}\sin x\right)$ |
| State $R = \sqrt{5}$ | B1 FT | ISW. FT their $a\sin x + b\cos x$ provided this expression obtained by correct method. |
| Use correct trig formulae to find $\alpha$ | M1 | $\alpha = \tan^{-1}(b/a)$ from their $a\sin x + b\cos x$, or $\sin^{-1}$ or $\cos^{-1}$ provided this expression obtained by correct method. NB If $\cos\alpha = 1$ and $\sin\alpha = 2$ then M0 A0. |
| Obtain $\alpha = 1.107$ | A1 | 3 d.p. CAO. Treat answer in degrees as a misread $(63.435°)$. |
| **Total** | **4** | |
### Part 8(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\sin^{-1}\left(\frac{1.5}{R}\right)$ | B1 FT | Follow their $R$. |
| Use a correct method to obtain an un-simplified value of $\theta$ with their $\alpha$ | M1 | $2\left(\sin^{-1}\left(\frac{1.5}{R}\right) - \alpha\right)$ or $2\left(\pi - \sin^{-1}\left(\frac{1.5}{R}\right) - \alpha\right)$ |
| Obtain one correct answer e.g. $-0.74$ in the interval | A1 | |
| Obtain second correct answer e.g. $2.60$ $(2.5986)$ or $4\pi - 0.74 = 11.8$ or $2.60 - 4\pi = -9.97$ in the interval | A1 | If uses $1.11°$ withhold first accuracy mark gained, but allow rest of accuracy marks. Allow $2.6(0)$. |
| Obtain two more correct answers e.g. $-9.97$ and $11.8$ and no others in the interval | A1 | Ignore answers outside the interval. Treat answers in degrees as a misread. $(-571.1°, -42.6°, 148.9°, 677.2°)$. |
| **Total** | **5** | |
---8
\begin{enumerate}[label=(\alph*)]
\item Express $3 \sin x + 2 \sqrt { 2 } \cos \left( x + \frac { 1 } { 4 } \pi \right)$ in the form $\mathrm { R } \sin ( \mathrm { x } + \alpha )$, where $R > 0$ and $0 < \alpha < \frac { 1 } { 2 } \pi$. State the exact value of $R$ and give $\alpha$ correct to 3 decimal places.
\item Hence solve the equation
$$6 \sin \frac { 1 } { 2 } \theta + 4 \sqrt { 2 } \cos \left( \frac { 1 } { 2 } \theta + \frac { 1 } { 4 } \pi \right) = 3$$
for $- 4 \pi < \theta < 4 \pi$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2024 Q8 [9]}}