| Exam Board | Edexcel |
|---|---|
| Module | C12 (Core Mathematics 1 & 2) |
| Year | 2014 |
| Session | January |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Trigonometric equations in context |
| Type | Sketch transformed/compound trig graph and identify features |
| Difficulty | Moderate -0.8 This is a straightforward trigonometric question requiring a sketch of a horizontally shifted cosine curve, finding intercepts by substitution, and solving a basic trig equation using standard angles. All parts use routine techniques with no problem-solving insight needed, making it easier than average for A-level. |
| Spec | 1.05f Trigonometric function graphs: symmetries and periodicities1.05g Exact trigonometric values: for standard angles1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| Sketch of harmonic curve (sine or cosine); correct section and position relative to axes | M1, A1 | Must be only one cycle; positive \(y\)-intercept; positive gradient at start and finish; solely \(x \geq 0\) |
| Answer | Marks | Guidance |
|---|---|---|
| \((0, \frac{1}{2})\) | B1 | |
| \(\left(\frac{5\pi}{6}, 0\right)\) or \((150°, 0)\) and \(\left(\frac{11\pi}{6}, 0\right)\) or \((330°, 0)\) | B1, B1 | Each answer cao; extra answers in range lose last B1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(\left(x - \frac{\pi}{3}\right) = \frac{\pi}{4}\) or \(-\frac{\pi}{4}\) | M1 | Uses inverse cos correctly to obtain at least one correct answer |
| \(x = \frac{7\pi}{12}\) or \(x = \frac{\pi}{12}\) | M1 A1 A1 | Second M1: adds \(\frac{\pi}{3}\) to previous answer; one correct answer A1; both correct A1; extra answers in range lose final A1 |
## Question 10:
### Part (a):
| Sketch of harmonic curve (sine or cosine); correct section and position relative to axes | M1, A1 | Must be only one cycle; positive $y$-intercept; positive gradient at start and finish; solely $x \geq 0$ |
### Part (b):
| $(0, \frac{1}{2})$ | B1 | |
| $\left(\frac{5\pi}{6}, 0\right)$ or $(150°, 0)$ and $\left(\frac{11\pi}{6}, 0\right)$ or $(330°, 0)$ | B1, B1 | Each answer cao; extra answers in range lose last B1 |
### Part (c):
| $\left(x - \frac{\pi}{3}\right) = \frac{\pi}{4}$ or $-\frac{\pi}{4}$ | M1 | Uses inverse cos correctly to obtain at least one correct answer |
| $x = \frac{7\pi}{12}$ or $x = \frac{\pi}{12}$ | M1 A1 A1 | Second M1: adds $\frac{\pi}{3}$ to previous answer; one correct answer A1; both correct A1; extra answers in range lose final A1 |
---10. The curve $C$ has equation $y = \cos \left( x - \frac { \pi } { 3 } \right) , 0 \leqslant x \leqslant 2 \pi$
\begin{enumerate}[label=(\alph*)]
\item In the space below, sketch the curve $C$.
\item Write down the exact coordinates of the points at which $C$ meets the coordinate axes.
\item Solve, for $x$ in the interval $0 \leqslant x \leqslant 2 \pi$,
$$\cos \left( x - \frac { \pi } { 3 } \right) = \frac { 1 } { \sqrt { 2 } }$$
giving your answers in the form $k \pi$, where $k$ is a rational number.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C12 2014 Q10 [9]}}