| Exam Board | Edexcel |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2012 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Harmonic Form |
| Type | Express double angle or product |
| Difficulty | Standard +0.3 This is a standard harmonic form question with routine techniques: finding R and α using Pythagoras and tan, solving a transformed equation, using double angle identities, and finding a maximum. All steps are textbook procedures requiring no novel insight, though the multi-part structure and algebraic manipulation make it slightly above average difficulty. |
| Spec | 1.05l Double angle formulae: and compound angle formulae1.05n Harmonic form: a sin(x)+b cos(x) = R sin(x+alpha) etc1.05o Trigonometric equations: solve in given intervals |
$$f ( x ) = 7 \cos 2 x - 24 \sin 2 x$$
Given that $\mathrm { f } ( x ) = R \cos ( 2 x + \alpha )$, where $R > 0$ and $0 < \alpha < 90 ^ { \circ }$,
\begin{enumerate}[label=(\alph*)]
\item find the value of $R$ and the value of $\alpha$.
\item Hence solve the equation
$$7 \cos 2 x - 24 \sin 2 x = 12.5$$
for $0 \leqslant x < 180 ^ { \circ }$, giving your answers to 1 decimal place.
\item Express $14 \cos ^ { 2 } x - 48 \sin x \cos x$ in the form $a \cos 2 x + b \sin 2 x + c$, where $a , b$, and $c$ are constants to be found.
\item Hence, using your answers to parts (a) and (c), deduce the maximum value of
$$14 \cos ^ { 2 } x - 48 \sin x \cos x$$
\end{enumerate}
\hfill \mbox{\textit{Edexcel C3 2012 Q8 [12]}}