| Exam Board | OCR |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2008 |
| Session | January |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Trig Graphs & Exact Values |
| Type | Find coordinates of turning points |
| Difficulty | Moderate -0.8 This is a multi-part question testing basic properties of sine graphs and a routine trigonometric equation. Parts (i) and (ii) require only reading from a graph and understanding symmetry of sine functions. Part (iii) involves a standard technique of using sin²x + cos²x = 1 to convert to a quadratic, then solving—a common C2 exercise with no novel insight required. Easier than average for A-level. |
| Spec | 1.05f Trigonometric function graphs: symmetries and periodicities1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| \((90°, 2)\), \((-90°, -2)\) | B1, B1 [2] | State at least 2 correct values; State all 4 correct values (radians is B1 B0) |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(180 - \alpha\) | B1 [1] | State \(180 - \alpha\) |
| (b) \(-\alpha\) or \(\alpha - 180\) | B1 [1] | State \(-\alpha\) or \(\alpha - 180\) (radians or unsimplified is B1B0) |
| Answer | Marks | Guidance |
|---|---|---|
| \(2\sin x = 2 - 3\cos^2 x\) | M1 | Attempt use of \(\cos^2 x = 1 - \sin^2 x\) |
| \(2\sin x = 2 - 3(1 - \sin^2 x)\) | ||
| \(3\sin^2 x - 2\sin x - 1 = 0\) | A1 | Obtain \(3\sin^2 x - 2\sin x - 1 = 0\) aef with no brackets |
| \((3\sin x + 1)(\sin x - 1) = 0\) | M1 | Attempt to solve 3 term quadratic in \(\sin x\) |
| \(\sin x = -\frac{1}{3}\), \(\sin x = 1\) | A1 | Obtain \(x = -19.5°\) |
| \(x = -19.5°\), \(-161°\), \(90°\) | A1\(\sqrt{}\), A1 [6] | Obtain second correct answer in range following their \(x\); Obtain \(90°\) (radians or extra answers is max 5 out of 6). SR: answer only (and no extras) is B1 B1\(\sqrt{}\) B1 |
# Question 9:
## Part (i)
| $(90°, 2)$, $(-90°, -2)$ | B1, B1 **[2]** | State at least 2 correct values; State all 4 correct values (radians is B1 B0) |
## Part (ii)
| **(a)** $180 - \alpha$ | B1 **[1]** | State $180 - \alpha$ |
| **(b)** $-\alpha$ or $\alpha - 180$ | B1 **[1]** | State $-\alpha$ or $\alpha - 180$ (radians or unsimplified is B1B0) |
## Part (iii)
| $2\sin x = 2 - 3\cos^2 x$ | M1 | Attempt use of $\cos^2 x = 1 - \sin^2 x$ |
| $2\sin x = 2 - 3(1 - \sin^2 x)$ | | |
| $3\sin^2 x - 2\sin x - 1 = 0$ | A1 | Obtain $3\sin^2 x - 2\sin x - 1 = 0$ aef with no brackets |
| $(3\sin x + 1)(\sin x - 1) = 0$ | M1 | Attempt to solve 3 term quadratic in $\sin x$ |
| $\sin x = -\frac{1}{3}$, $\sin x = 1$ | A1 | Obtain $x = -19.5°$ |
| $x = -19.5°$, $-161°$, $90°$ | A1$\sqrt{}$, A1 **[6]** | Obtain second correct answer in range following their $x$; Obtain $90°$ (radians or extra answers is max 5 out of 6). SR: answer only (and no extras) is B1 B1$\sqrt{}$ B1 |
---9 (i)
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{2ae05b46-6c9f-4aaa-9cba-1116c0ec27d4-4_376_764_276_733}
\captionsetup{labelformat=empty}
\caption{Fig. 1}
\end{center}
\end{figure}
Fig. 1 shows the curve $y = 2 \sin x$ for values of $x$ such that $- 180 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }$. State the coordinates of the maximum and minimum points on this part of the curve.\\
(ii)
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{2ae05b46-6c9f-4aaa-9cba-1116c0ec27d4-4_371_766_959_731}
\captionsetup{labelformat=empty}
\caption{Fig. 2}
\end{center}
\end{figure}
Fig. 2 shows the curve $y = 2 \sin x$ and the line $y = k$. The smallest positive solution of the equation $2 \sin x = k$ is denoted by $\alpha$. State, in terms of $\alpha$, and in the range $- 180 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }$,
\begin{enumerate}[label=(\alph*)]
\item another solution of the equation $2 \sin x = k$,
\item one solution of the equation $2 \sin x = - k$.\\
(iii) Find the $x$-coordinates of the points where the curve $y = 2 \sin x$ intersects the curve $y = 2 - 3 \cos ^ { 2 } x$, for values of $x$ such that $- 180 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }$.
\end{enumerate}
\hfill \mbox{\textit{OCR C2 2008 Q9 [9]}}