| Exam Board | Edexcel |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2013 |
| Session | June |
| Marks | 10 |
| 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) = a cos x + b sin x formula where R = √(a²+b²) and tan α = b/a, followed by straightforward equation solving. Part (c) requires recognizing maximum/minimum values, which is a direct consequence of the harmonic form. Slightly above average difficulty due to multiple parts, but all techniques are standard C3 material with no novel insight required. |
| 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 | Mark | Guidance |
| \(R = \sqrt{7^2 + 1^2} = \sqrt{50} = 5\sqrt{2}\) | B1 | Accept \(\pm\sqrt{50}\). Do not accept decimal \(7.07...\) unless \(\sqrt{50}\) or \(5\sqrt{2}\) also seen. |
| \(\alpha = \arctan\left(\frac{1}{7}\right)\), method shown | M1 | For \(\tan\alpha = \pm\frac{1}{7}\) or \(\pm\frac{7}{1}\). If \(R\) used, accept \(\sin\alpha = \pm\frac{1}{R}\) or \(\cos\alpha = \pm\frac{7}{R}\) |
| \(\alpha = 8.13...°\) awrt \(8.1°\) | A1 | Note \(\tan\alpha = 7 \Rightarrow \alpha = 81.9°\) is a common error. Radian answer awrt \(0.1418\) is A0. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\sqrt{50}\cos(x - 8.1) = 5 \Rightarrow \cos(x-8.1) = \frac{5}{\sqrt{50}}\) | M1 | For using part (a) answers and moving from \(R\cos(x\pm\alpha)=5\) to \(\cos(x\pm\alpha)=\frac{5}{R}\). May be implied by sight of 53.1 if \(R\) and \(\alpha\) correct. |
| \(x - 8.1 = 45 \Rightarrow x = 53.1°\) | M1, A1 | For achieving \(x \pm \alpha =\) awrt \(45°\) or \(315°\) leading to one value of \(x\) in range. One correct answer awrt \(53.1°\) or \(323.1°\) |
| \(x - 8.1 = 315 \Rightarrow x = 323.1°\) | M1, A1 | Attempt at secondary value in range. Both values correct awrt \(53.1°\) and \(323.1°\). Withhold if extra values in range. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| One solution if \(\frac{k}{\sqrt{50}} = \pm 1 \Rightarrow k = \pm\sqrt{50}\) | M1, A1ft | For stating \(\frac{k}{\textit{their } R} = 1\) or \(\frac{k}{\textit{their } R} = -1\). Both values \(k = \pm\sqrt{50}\). Follow through on numerical \(R\). |
## Question 3:
**Part (a):**
| Answer/Working | Mark | Guidance |
|---|---|---|
| $R = \sqrt{7^2 + 1^2} = \sqrt{50} = 5\sqrt{2}$ | B1 | Accept $\pm\sqrt{50}$. Do not accept decimal $7.07...$ unless $\sqrt{50}$ or $5\sqrt{2}$ also seen. |
| $\alpha = \arctan\left(\frac{1}{7}\right)$, method shown | M1 | For $\tan\alpha = \pm\frac{1}{7}$ or $\pm\frac{7}{1}$. If $R$ used, accept $\sin\alpha = \pm\frac{1}{R}$ or $\cos\alpha = \pm\frac{7}{R}$ |
| $\alpha = 8.13...°$ awrt $8.1°$ | A1 | Note $\tan\alpha = 7 \Rightarrow \alpha = 81.9°$ is a common error. Radian answer awrt $0.1418$ is A0. |
**Part (b):**
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\sqrt{50}\cos(x - 8.1) = 5 \Rightarrow \cos(x-8.1) = \frac{5}{\sqrt{50}}$ | M1 | For using part (a) answers and moving from $R\cos(x\pm\alpha)=5$ to $\cos(x\pm\alpha)=\frac{5}{R}$. May be implied by sight of 53.1 if $R$ and $\alpha$ correct. |
| $x - 8.1 = 45 \Rightarrow x = 53.1°$ | M1, A1 | For achieving $x \pm \alpha =$ awrt $45°$ or $315°$ leading to one value of $x$ in range. One correct answer awrt $53.1°$ or $323.1°$ |
| $x - 8.1 = 315 \Rightarrow x = 323.1°$ | M1, A1 | Attempt at secondary value in range. Both values correct awrt $53.1°$ and $323.1°$. Withhold if extra values in range. |
**Part (c):**
| Answer/Working | Mark | Guidance |
|---|---|---|
| One solution if $\frac{k}{\sqrt{50}} = \pm 1 \Rightarrow k = \pm\sqrt{50}$ | M1, A1ft | For stating $\frac{k}{\textit{their } R} = 1$ or $\frac{k}{\textit{their } R} = -1$. Both values $k = \pm\sqrt{50}$. Follow through on numerical $R$. |
---3.
$$f ( x ) = 7 \cos x + \sin x$$
Given that $\mathrm { f } ( x ) = R \cos ( x - \alpha )$, where $R > 0$ and $0 < \alpha < 90 ^ { \circ }$,
\begin{enumerate}[label=(\alph*)]
\item find the exact value of $R$ and the value of $\alpha$ to one decimal place.
\item Hence solve the equation
$$7 \cos x + \sin x = 5$$
for $0 \leqslant x < 360 ^ { \circ }$, giving your answers to one decimal place.
\item State the values of $k$ for which the equation
$$7 \cos x + \sin x = k$$
has only one solution in the interval $0 \leqslant x < 360 ^ { \circ }$
\end{enumerate}
\hfill \mbox{\textit{Edexcel C3 2013 Q3 [10]}}