| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/2 (Pre-U Mathematics Paper 2) |
| Session | Specimen |
| Marks | 12 |
| Topic | Harmonic Form |
| Type | Find maximum or minimum value |
| Difficulty | Moderate -0.3 This question tests standard trigonometric techniques: Pythagorean identity with quadrant consideration, half-angle formulas, and R-cos(x+α) form with range finding. All parts follow routine procedures taught in A-level/Pre-U courses with no novel problem-solving required. The multi-part structure and 12 total marks suggest moderate length, but each component is straightforward application of learned formulas, making it slightly easier than average. |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05l Double angle formulae: and compound angle formulae1.05n Harmonic form: a sin(x)+b cos(x) = R sin(x+alpha) etc |
| Answer | Marks | Guidance |
|---|---|---|
| (i) (a) \(\sin \theta = \pm\sqrt{1 - \cos^2 \theta} = \pm\sqrt{1 - \left(\frac{7}{25}\right)^2}\) | M1 | \(= \pm\frac{24}{25}\) |
| (b) Use of \(\cos \theta = 1 - 2\sin^2(\frac{\theta}{2})\) | M1 | Obtain \(\sin(\frac{\theta}{2}) = \pm\frac{4}{5}\) |
| (c) \(\sec(\frac{\theta}{2}) = -\frac{5}{4}\) | B1 | 1 mark |
| (ii) (a) \(4 \cos x - 3 \sin x = 5\left(\frac{4}{5}\cos x - \frac{3}{5}\sin x\right) = 5\cos\left(x + \tan^{-1}\left(\frac{3}{4}\right)\right)\) | B1B1 | 2 marks |
| (b) Graphically indicate the function on its domain, or attempt a justification for at least one extreme value | M1 | Maximum value \(5\cos\left(\tan^{-1}\left(\frac{3}{4}\right)\right) = 4\) |
(i) (a) $\sin \theta = \pm\sqrt{1 - \cos^2 \theta} = \pm\sqrt{1 - \left(\frac{7}{25}\right)^2}$ | M1 | $= \pm\frac{24}{25}$ | A1 | Choose negative alternative | A1 | 3 marks
(b) Use of $\cos \theta = 1 - 2\sin^2(\frac{\theta}{2})$ | M1 | Obtain $\sin(\frac{\theta}{2}) = \pm\frac{4}{5}$ | A1 | Choose positive alternative | A1 | 3 marks
(c) $\sec(\frac{\theta}{2}) = -\frac{5}{4}$ | B1 | 1 mark
(ii) (a) $4 \cos x - 3 \sin x = 5\left(\frac{4}{5}\cos x - \frac{3}{5}\sin x\right) = 5\cos\left(x + \tan^{-1}\left(\frac{3}{4}\right)\right)$ | B1B1 | 2 marks
(b) Graphically indicate the function on its domain, or attempt a justification for at least one extreme value | M1 | Maximum value $5\cos\left(\tan^{-1}\left(\frac{3}{4}\right)\right) = 4$ | A1 | Minimum value $-5$ | A1 | 3 marks\begin{enumerate}[label=(\roman*)]
\item Given that $\cos \theta = \frac{7}{25}$, where $\frac{3}{2}\pi < \theta < 2\pi$, determine the exact values of
\begin{enumerate}[label=(\alph*)]
\item $\sin \theta$, [3]
\item $\sin(\frac{1}{2}\theta)$, [3]
\item $\sec(\frac{1}{2}\theta)$. [1]
\end{enumerate}
\item \begin{enumerate}[label=(\alph*)]
\item Express $4 \cos x - 3 \sin x$ in the form $A \cos(x + \alpha)$, where $A > 0$ and $0 < \alpha < \frac{1}{2}\pi$. [2]
\item Hence find the greatest and least values of $4 \cos x - 3 \sin x$ for $0 \leqslant x \leqslant \pi$. [3]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9794/2 Q7 [12]}}