| Exam Board | AQA |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2009 |
| Session | January |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Trigonometric equations in context |
| Type | Sketch basic trig graph and solve |
| Difficulty | Moderate -0.8 This is a routine C2 trigonometry question requiring standard techniques: solving a basic trig equation with calculator, reading symmetry properties from a sine graph, and sketching a transformed sine curve. All parts are textbook exercises with no problem-solving or novel insight required, making it easier than average A-level questions. |
| 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 |
|---|---|---|
| \(\{x\} = \sin^{-1}(0.8) = 0.927(29\ldots) \{=\beta\}\) | M1 | \(\sin^{-1}(0.8)\) PI |
| \(\{x\} = \pi - \beta\) | m1 | |
| \(x = 0.927(29\ldots), \quad 2.21(42\ldots)\) | A1 | Both. Ignore values outside interval 0−2π but A0 if 'extra' values inside the given interval. 3 marks total |
| Answer | Marks | Guidance |
|---|---|---|
| \(\left(\frac{3\pi}{2}, -1\right)\) | B2,1 | B1 if one coordinate correct or \(\left(-1, \frac{3\pi}{2}\right)\). 2 marks total |
| Answer | Marks | Guidance |
|---|---|---|
| \(\pi - \alpha\) | B1 | 1 mark total |
| Answer | Marks | Guidance |
|---|---|---|
| \(RS = (2\pi - \alpha) - (\pi + \alpha) = \pi - 2\alpha\) | M1 | OE eg \(RS = PQ = (\pi - \omega) - \alpha\) |
| \(= \pi - 2\alpha\) | A1 | Must be simplified. 2 marks total |
| Answer | Marks | Guidance |
|---|---|---|
| B1 | Since curve with positive gradient at O with at least 3 stationary points between 0 and 2π. | |
| B1 | Correct shaped curve with 2 max and 2 min between 0 and 2π. | |
| B1 | All 5 correct points of intersection with x-axis with \(\frac{\pi}{2}, \pi\) and \(\frac{3\pi}{2}\) clearly shown. | |
| Maximum points \(\left(\frac{\pi}{4}, 1\right)\) and \(\left(\frac{5\pi}{4}, 1\right)\) stated or clearly shown on the sketch | B2,1 | B1 for either: 1 as the y-coordinate of max pt(s) or: two max pts between 0 and 2π with correct x-coordinates. 5 marks total |
**7(a)**
$\{x\} = \sin^{-1}(0.8) = 0.927(29\ldots) \{=\beta\}$ | M1 | $\sin^{-1}(0.8)$ PI
$\{x\} = \pi - \beta$ | m1 |
$x = 0.927(29\ldots), \quad 2.21(42\ldots)$ | A1 | Both. Ignore values outside interval 0−2π but A0 if 'extra' values inside the given interval. 3 marks total
**7(b)(i)**
$\left(\frac{3\pi}{2}, -1\right)$ | B2,1 | B1 if one coordinate correct or $\left(-1, \frac{3\pi}{2}\right)$. 2 marks total
**7(b)(ii)**
$\pi - \alpha$ | B1 | 1 mark total
**7(b)(iii)**
$RS = (2\pi - \alpha) - (\pi + \alpha) = \pi - 2\alpha$ | M1 | OE eg $RS = PQ = (\pi - \omega) - \alpha$
$= \pi - 2\alpha$ | A1 | Must be simplified. 2 marks total
**7(c)**
| B1 | Since curve with positive gradient at O with at least 3 stationary points between 0 and 2π.
| B1 | Correct shaped curve with 2 max and 2 min between 0 and 2π.
| B1 | All 5 correct points of intersection with x-axis with $\frac{\pi}{2}, \pi$ and $\frac{3\pi}{2}$ clearly shown.
Maximum points $\left(\frac{\pi}{4}, 1\right)$ and $\left(\frac{5\pi}{4}, 1\right)$ stated or clearly shown on the sketch | B2,1 | B1 for either: 1 as the y-coordinate of max pt(s) or: two max pts between 0 and 2π with correct x-coordinates. 5 marks total
**Total for Q7: 13 marks**
---7
\begin{enumerate}[label=(\alph*)]
\item Solve the equation $\sin x = 0.8$ 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 the curve $y = \sin x , 0 \leqslant x \leqslant 2 \pi$ and the lines $y = k$ and $y = - k$.\\
\includegraphics[max width=\textwidth, alt={}, center]{0e19665b-5ee5-49e4-8de2-6c8dd17f61eb-5_497_780_552_689}
The line $y = k$ intersects the curve at the points $P$ and $Q$, and the line $y = - k$ intersects the curve at the points $R$ and $S$.
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 the point $Q$ in terms of $\pi$ and $\alpha$.
\item Find the length of $R S$ in terms of $\pi$ and $\alpha$, giving your answer in its simplest form.
\end{enumerate}\item Sketch the graph of $y = \sin 2 x$ for $0 \leqslant x \leqslant 2 \pi$, indicating the coordinates of points where the graph intersects the $x$-axis and the coordinates of any maximum points.
\end{enumerate}
\hfill \mbox{\textit{AQA C2 2009 Q7 [13]}}