| Exam Board | Edexcel |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Harmonic Form |
| Type | Express and solve equation |
| Difficulty | Standard +0.3 This is a standard harmonic form question requiring conversion of a sin θ - b cos θ to R sin(θ - α), followed by routine equation solving and finding maximum values. All steps follow textbook procedures with no novel insight required, making it slightly easier than average for C3 level. |
| 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/Working | Marks | Notes |
| \(4\sin\theta - 3\cos\theta = R\sin\theta\cos\alpha - R\cos\theta\sin\alpha\) | ||
| \(\sin\theta\) terms: \(4 = R\cos\alpha\) | ||
| \(\cos\theta\) terms: \(3 = R\sin\alpha\) | ||
| \(\tan\alpha = 0.75\) | M1 | |
| \(\alpha = 36.9°\) | A1 | |
| \(R^2 = 4^2 + 3^2 = 25 \Rightarrow R = 5\) | M1 A1 | (4 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Notes |
| \(5\sin(\theta - 36.9°) = 3\) | ||
| \(\sin(\theta - 36.9°) = 0.6\) | M1 | |
| \(\theta - 36.9° = 36.9°,\ 143.1°\) | A1 M1 | |
| \(\theta = 73.7°,\ 180°\) | A1 A1 | awrt 74° (5 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Notes |
| Maximum value \(= 5\) | B1 | (1 mark) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Notes |
| \(\sin(\theta - 36.9°) = 1\) | M1 | |
| \(\theta - 36.9° = 90°\) | ||
| \(\theta = 90° + 36.9° = 126.9°\) | A1 | (2 marks) |
# Question 8:
## Part (a)
| Answer/Working | Marks | Notes |
|---|---|---|
| $4\sin\theta - 3\cos\theta = R\sin\theta\cos\alpha - R\cos\theta\sin\alpha$ | | |
| $\sin\theta$ terms: $4 = R\cos\alpha$ | | |
| $\cos\theta$ terms: $3 = R\sin\alpha$ | | |
| $\tan\alpha = 0.75$ | M1 | |
| $\alpha = 36.9°$ | A1 | |
| $R^2 = 4^2 + 3^2 = 25 \Rightarrow R = 5$ | M1 A1 | (4 marks) |
## Part (b)
| Answer/Working | Marks | Notes |
|---|---|---|
| $5\sin(\theta - 36.9°) = 3$ | | |
| $\sin(\theta - 36.9°) = 0.6$ | M1 | |
| $\theta - 36.9° = 36.9°,\ 143.1°$ | A1 M1 | |
| $\theta = 73.7°,\ 180°$ | A1 A1 | awrt 74° (5 marks) |
## Part (c)
| Answer/Working | Marks | Notes |
|---|---|---|
| Maximum value $= 5$ | B1 | (1 mark) |
## Part (d)
| Answer/Working | Marks | Notes |
|---|---|---|
| $\sin(\theta - 36.9°) = 1$ | M1 | |
| $\theta - 36.9° = 90°$ | | |
| $\theta = 90° + 36.9° = 126.9°$ | A1 | (2 marks) |
**Total: 12 marks**
I can see this image only shows the mark allocation summary table (AO1, AO2, AO3, AO4, AO5 breakdown by question), not the actual mark scheme content with worked solutions, mark allocations (M1, A1, B1, etc.), and guidance notes.
To extract the detailed mark scheme content you're asking for, I would need to see the subsequent pages of the mark scheme document that contain the actual question-by-question worked solutions and marking criteria.
Could you please share the remaining pages of the mark scheme? They would typically follow this summary table and contain entries like:\begin{enumerate}
\item In a particular circuit the current, $I$ amperes, is given by
\end{enumerate}
$$I = 4 \sin \theta - 3 \cos \theta , \quad \theta > 0$$
where $\theta$ is an angle related to the voltage.
Given that $I = R \sin ( \theta - \alpha )$, where $R > 0$ and $0 \leqslant \alpha < 360 ^ { \circ }$,\\
(a) find the value of $R$, and the value of $\alpha$ to 1 decimal place.\\
(b) Hence solve the equation $4 \sin \theta - 3 \cos \theta = 3$ to find the values of $\theta$ between 0 and $360 ^ { \circ }$.\\
(c) Write down the greatest value for $I$.\\
(d) Find the value of $\theta$ between 0 and $360 ^ { \circ }$ at which the greatest value of $I$ occurs.\\
8. continued
\hfill \mbox{\textit{Edexcel C3 Q8 [12]}}