| Exam Board | Edexcel |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2006 |
| Session | January |
| 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 routine application of R cos(x + α) = R cos α cos x - R sin α sin x, solving R² = 12² + 4² and tan α = 4/12, then solving a basic trigonometric equation and identifying minimum values. All steps are textbook procedures with no novel insight required, making it slightly easier than average. |
| 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 | Guidance |
| \(R\cos\alpha = 12,\ R\sin\alpha = 4\) | ||
| \(R = \sqrt{12^2 + 4^2} = \sqrt{160}\) | M1 A1 | Accept if just written down, awrt 12.6 |
| \(\tan\alpha = \frac{4}{12} \Rightarrow \alpha \approx 18.43°\) | M1, A1 | awrt \(18.4°\) (4) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\cos(x + \text{their } \alpha) = \frac{7}{\text{their } R}\ (\approx 0.5534)\) | M1 | |
| \(x + \text{their } \alpha = 56.4°\) | A1 | awrt \(56°\) |
| \(= \ldots,\ 303.6°\) | M1 | \(360°\) - their principal value |
| \(x = 38.0°,\ 285.2°\) | A1, A1 | Ignore solutions out of range (5) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Minimum value is \(-\sqrt{160}\) | B1ft | ft their \(R\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\cos(x + \text{their } \alpha) = -1\) | M1 | |
| \(x \approx 161.57°\) | A1 | cao (3) [12] |
# Question 6:
## Part (a)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $R\cos\alpha = 12,\ R\sin\alpha = 4$ | | |
| $R = \sqrt{12^2 + 4^2} = \sqrt{160}$ | M1 A1 | Accept if just written down, awrt 12.6 |
| $\tan\alpha = \frac{4}{12} \Rightarrow \alpha \approx 18.43°$ | M1, A1 | awrt $18.4°$ **(4)** |
## Part (b)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\cos(x + \text{their } \alpha) = \frac{7}{\text{their } R}\ (\approx 0.5534)$ | M1 | |
| $x + \text{their } \alpha = 56.4°$ | A1 | awrt $56°$ |
| $= \ldots,\ 303.6°$ | M1 | $360°$ - their principal value |
| $x = 38.0°,\ 285.2°$ | A1, A1 | Ignore solutions out of range **(5)** |
## Part (c)(i)
| Answer/Working | Marks | Guidance |
|---|---|---|
| Minimum value is $-\sqrt{160}$ | B1ft | ft their $R$ |
## Part (c)(ii)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\cos(x + \text{their } \alpha) = -1$ | M1 | |
| $x \approx 161.57°$ | A1 | cao **(3) [12]** |
---6.
$$f ( x ) = 12 \cos x - 4 \sin x$$
Given that $\mathrm { f } ( x ) = R \cos ( x + \alpha )$, where $R \geqslant 0$ and $0 \leqslant \alpha \leqslant 90 ^ { \circ }$,
\begin{enumerate}[label=(\alph*)]
\item find the value of $R$ and the value of $\alpha$.
\item Hence solve the equation
$$12 \cos x - 4 \sin x = 7$$
for $0 \leqslant x < 360 ^ { \circ }$, giving your answers to one decimal place.
\item \begin{enumerate}[label=(\roman*)]
\item Write down the minimum value of $12 \cos x - 4 \sin x$.
\item Find, to 2 decimal places, the smallest positive value of $x$ for which this minimum value occurs.\\
\includegraphics[max width=\textwidth, alt={}, center]{5cd53af1-bac9-4ed9-ac45-59ad2e372423-09_60_35_2669_1853}
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{Edexcel C3 2006 Q6 [12]}}