| Exam Board | Edexcel |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2018 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Harmonic Form |
| Type | Find value where max/min occurs |
| Difficulty | Standard +0.3 This is a standard C3 harmonic form question with routine application of the R sin(θ - α) technique. Part (a) is direct textbook application, parts (b) and (c) require recognizing that squaring creates predictable max/min values. The multi-part structure and double angle in (c) add slight complexity, but all steps follow standard procedures without requiring novel insight. |
| Spec | 1.05n Harmonic form: a sin(x)+b cos(x) = R sin(x+alpha) etc1.05o Trigonometric equations: solve in given intervals |
| END |
\begin{enumerate}
\item (a) Express $\sin \theta - 2 \cos \theta$ in the form $R \sin ( \theta - \alpha )$, where $R > 0$ and $0 < \alpha < \frac { \pi } { 2 }$
\end{enumerate}
Give the exact value of $R$ and the value of $\alpha$, in radians, to 3 decimal places.
$$\mathrm { M } ( \theta ) = 40 + ( 3 \sin \theta - 6 \cos \theta ) ^ { 2 }$$
(b) Find\\
(i) the maximum value of $\mathrm { M } ( \theta )$,\\
(ii) the smallest value of $\theta$, in the range $0 < \theta \leqslant 2 \pi$, at which the maximum value of $\mathrm { M } ( \theta )$ occurs.
$$N ( \theta ) = \frac { 30 } { 5 + 2 ( \sin 2 \theta - 2 \cos 2 \theta ) ^ { 2 } }$$
(c) Find\\
(i) the maximum value of $\mathrm { N } ( \theta )$,\\
(ii) the largest value of $\theta$, in the range $0 < \theta \leqslant 2 \pi$, at which the maximum value of $\mathrm { N } ( \theta )$ occurs.\\
(Solutions based entirely on graphical or numerical methods are not acceptable.)
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\hfill \mbox{\textit{Edexcel C3 2018 Q9 [9]}}