| Exam Board | AQA |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2007 |
| Session | January |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Trigonometric equations in context |
| Type | Solve double/multiple angle equation |
| Difficulty | Moderate -0.8 This is a straightforward C2 trigonometry question testing standard techniques: using inverse cosine with a calculator, reading graph features, applying symmetry of cosine, identifying a horizontal stretch transformation, and solving a double-angle equation. All parts are routine textbook exercises requiring recall and basic application rather than problem-solving or insight, making it easier than average. |
| Spec | 1.02w Graph transformations: simple transformations of f(x)1.05f Trigonometric function graphs: symmetries and periodicities1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| (a) | \(\{x =\} \cos^{-1}(0.3) = 1.266....\) \(\{= \beta\}\) | M1 |
| \(\{x = \} 2\pi - \beta\) | m1 | Condone degrees or mix. |
| \(x = 1.27\), \(5.02\) | A1 | Accept 1.26 to 1.27 with 5.01 to 5.02 inclusive |
| Total: 3 | ||
| (b)(i) | \(M(\pi, -1)\) | B1;B1 |
| (ii) | \(\{x_Q\} = 2\pi - \alpha\) | B1 |
| (c) | Stretch (I) in x-direction (II) scale factor \(\frac{1}{2}\) (III) | M1, A1 |
| Total: 2 | ||
| (d) | \(\cos 2x = \cos\frac{4\pi}{5} \Rightarrow 2x = \frac{4\pi}{5}\) | B1 |
| \(\Rightarrow x = \frac{2\pi}{5}(= \alpha)\) | Condone decimals/degrees | |
| \(x = \pi - \alpha\); OE | M1 | OE eg \(2x = 2\pi - \frac{4\pi}{5}\) Correct quadrant; condone degrees/decimals/mix |
| \(x = \pi + \alpha\); \(x = 2\pi - \alpha\); OE | m1 | Need both (OE for \(2x =\)) with no extras (quadrants) within the given interval. Condone degrees/decimals/mix |
| \(x = \frac{2\pi}{5}, \frac{3\pi}{5}, \frac{7\pi}{5}, \frac{8\pi}{5}\) | A1 | Need all 4 solutions for \(x\) but condone unsimplified provided in terms of \(\pi\) Ignore extra values outside the given interval. |
| Total: 12 |
(a) | $\{x =\} \cos^{-1}(0.3) = 1.266....$ $\{= \beta\}$ | M1 | $\cos^{-1}(0.3)$ PI by eg $72°$ or $73°$ |
| $\{x = \} 2\pi - \beta$ | m1 | Condone degrees or mix. |
| $x = 1.27$, $5.02$ | A1 | Accept 1.26 to 1.27 with 5.01 to 5.02 inclusive |
| | **Total: 3** | |
(b)(i) | $M(\pi, -1)$ | B1;B1 | B1 for each coordinate |
(ii) | $\{x_Q\} = 2\pi - \alpha$ | B1 | OE (unsimplified) |
(c) | Stretch (I) in x-direction (II) scale factor $\frac{1}{2}$ (III) | M1, A1 | Need(I) & one of (II),(III) |
| | **Total: 2** | |
(d) | $\cos 2x = \cos\frac{4\pi}{5} \Rightarrow 2x = \frac{4\pi}{5}$ | B1 | OE. (From correct work) |
| $\Rightarrow x = \frac{2\pi}{5}(= \alpha)$ | | Condone decimals/degrees |
| | | |
| $x = \pi - \alpha$; OE | M1 | OE eg $2x = 2\pi - \frac{4\pi}{5}$ Correct quadrant; condone degrees/decimals/mix |
| | | |
| $x = \pi + \alpha$; $x = 2\pi - \alpha$; OE | m1 | Need both (OE for $2x =$) with no extras (quadrants) within the given interval. Condone degrees/decimals/mix |
| | | |
| $x = \frac{2\pi}{5}, \frac{3\pi}{5}, \frac{7\pi}{5}, \frac{8\pi}{5}$ | A1 | Need all 4 solutions for $x$ but condone unsimplified provided in terms of $\pi$ Ignore extra values outside the given interval. |
| | **Total: 12** | |
---8
\begin{enumerate}[label=(\alph*)]
\item Solve the equation $\cos x = 0.3$ in the interval $0 \leqslant x \leqslant 2 \pi$, giving your answers in radians to three significant figures.
\item The diagram shows the graph of $y = \cos x$ for $0 \leqslant x \leqslant 2 \pi$ and the line $y = k$.\\
\includegraphics[max width=\textwidth, alt={}, center]{c16d94a6-52f2-4bf3-acee-0b227ae55a1a-5_524_805_559_648}
The line $y = k$ intersects the curve $y = \cos x , 0 \leqslant x \leqslant 2 \pi$, at the points $P$ and $Q$. The point $M$ is the minimum point of the curve.
\begin{enumerate}[label=(\roman*)]
\item Write down the coordinates of the point $M$.
\item The $x$-coordinate of $P$ is $\alpha$.
Write down the $x$-coordinate of $Q$ in terms of $\pi$ and $\alpha$.
\end{enumerate}\item Describe the geometrical transformation that maps the graph of $y = \cos x$ onto the graph of $y = \cos 2 x$.
\item Solve the equation $\cos 2 x = \cos \frac { 4 \pi } { 5 }$ in the interval $0 \leqslant x \leqslant 2 \pi$, giving the values of $x$ in terms of $\pi$.\\
(4 marks)
\end{enumerate}
\hfill \mbox{\textit{AQA C2 2007 Q8 [12]}}