| Exam Board | Edexcel |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2009 |
| Session | January |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Harmonic Form |
| Type | Applied context modeling |
| Difficulty | Standard +0.3 This is a standard C3 harmonic form question with straightforward application to a real-world context. Part (a) is routine bookwork using R cos(θ-α) = R cos α cos θ + R sin α sin θ to find R and α. Parts (b-d) apply this directly to find maxima/minima. All steps are algorithmic 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 |
|---|---|---|
| \(R^2 = 3^2 + 4^2\) | M1 | |
| \(R = 5\) | A1 | |
| \(\tan \alpha = \frac{4}{3}\) | M1 | |
| \(\alpha = 53...°\) | awrt 53° | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Maximum value is 5 | B1 ft their R | |
| At the maximum, \(\cos(\theta - \alpha) = 1\) or \(\theta - \alpha = 0\) | M1 | |
| \(\theta = \alpha = 53...°\) | ft their α | A1 ft |
| Answer | Marks | Guidance |
|---|---|---|
| \(f(t) = 10 + 5\cos(15t - \alpha)°\) | Given | |
| Minimum occurs when \(\cos(15t - \alpha)° = -1\) | M1 | |
| The minimum temperature is \((10 - 5)° = 5°\) | A1 ft | (2) |
| Answer | Marks | Guidance |
|---|---|---|
| \(15t - \alpha = 180\) | M1 | |
| \(t = 15.5\) | awrt 15.5 | M1 A1 |
**(a)**
$R^2 = 3^2 + 4^2$ | M1
$R = 5$ | A1
$\tan \alpha = \frac{4}{3}$ | M1
$\alpha = 53...°$ | awrt 53° | A1 | (4)
**(b)**
Maximum value is 5 | B1 ft their R
At the maximum, $\cos(\theta - \alpha) = 1$ or $\theta - \alpha = 0$ | M1
$\theta = \alpha = 53...°$ | ft their α | A1 ft | (3)
**(c)**
$f(t) = 10 + 5\cos(15t - \alpha)°$ | Given
Minimum occurs when $\cos(15t - \alpha)° = -1$ | M1
The minimum temperature is $(10 - 5)° = 5°$ | A1 ft | (2)
**(d)**
$15t - \alpha = 180$ | M1
$t = 15.5$ | awrt 15.5 | M1 A1 | (3) [12]8.\\
\begin{enumerate}[label=(\alph*)]
\item Express $3 \cos \theta + 4 \sin \theta$ in the form $R \cos ( \theta - \alpha )$, where $R$ and $\alpha$ are constants, $R > 0$ and $0 < \alpha < 90 ^ { \circ }$.
\item Hence find the maximum value of $3 \cos \theta + 4 \sin \theta$ and the smallest positive value of $\theta$ for which this maximum occurs.
The temperature, $\mathrm { f } ( t )$, of a warehouse is modelled using the equation
$$f ( t ) = 10 + 3 \cos ( 15 t ) ^ { \circ } + 4 \sin ( 15 t ) ^ { \circ }$$
where $t$ is the time in hours from midday and $0 \leqslant t < 24$.
\item Calculate the minimum temperature of the warehouse as given by this model.
\item Find the value of $t$ when this minimum temperature occurs.\\
\end{enumerate}
\hfill \mbox{\textit{Edexcel C3 2009 Q8 [12]}}