| Exam Board | Edexcel |
|---|---|
| Module | C12 (Core Mathematics 1 & 2) |
| Year | 2016 |
| Session | June |
| 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 C2 trigonometry question requiring a phase-shifted sine sketch, finding intercepts using standard values, and solving a basic trig equation with a known exact value. All parts use routine techniques with no problem-solving insight needed, making it easier than average but not trivial due to the phase shift and exact value requirements. |
| Spec | 1.05f Trigonometric function graphs: symmetries and periodicities1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Harmonic shape sketch | M1 | Graph is harmonic; could be part of cycle; needs at least one max and one min in range \(0, \theta, 2\pi\) |
| Correct position | A1 | One cycle between \(0\) and \(2\pi\); positive \(y\)-intercept; positive \(y\) value at \(2\pi\); max positive, min negative |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\left(0,\frac{\sqrt{2}}{2}\right), \left(\frac{3\pi}{4},0\right), \left(\frac{7\pi}{4},0\right)\) | B1 B1 B1 | Score 1,1,0 for any 2 out of 3; 1,0,0 for any 1 out of 3. No decimals unless exact value also given. Extra answers in range: withhold one mark |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\sin\left(x+\frac{\pi}{4}\right)=\frac{\sqrt{3}}{2} \Rightarrow x=\arcsin\left(\frac{\sqrt{3}}{2}\right)-\frac{\pi}{4}=\frac{\pi}{3}-\frac{\pi}{4}=\frac{\pi}{12}\) | M1A1 | M1: takes arcsin and subtracts \(\frac{\pi}{4}\); accept degrees \((x=60-45)\); do not accept mixed degrees/radians. A1: one of \(\frac{\pi}{12}\) or \(\frac{5\pi}{12}\) |
| \(x=\pi-\frac{\pi}{3}-\frac{\pi}{4}=\frac{5\pi}{12}\) | M1A1 | M1: correct attempt at second value, accept \(x+\frac{\pi}{4}=\pi-\text{their }\frac{\pi}{3}\). A1: both \(\frac{\pi}{12}\) and \(\frac{5\pi}{12}\), no other values in range |
## Question 10:
### Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Harmonic shape sketch | M1 | Graph is harmonic; could be part of cycle; needs at least one max and one min in range $0, \theta, 2\pi$ |
| Correct position | A1 | One cycle between $0$ and $2\pi$; positive $y$-intercept; positive $y$ value at $2\pi$; max positive, min negative |
### Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\left(0,\frac{\sqrt{2}}{2}\right), \left(\frac{3\pi}{4},0\right), \left(\frac{7\pi}{4},0\right)$ | B1 B1 B1 | Score 1,1,0 for any 2 out of 3; 1,0,0 for any 1 out of 3. No decimals unless exact value also given. Extra answers in range: withhold one mark |
### Part (c):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\sin\left(x+\frac{\pi}{4}\right)=\frac{\sqrt{3}}{2} \Rightarrow x=\arcsin\left(\frac{\sqrt{3}}{2}\right)-\frac{\pi}{4}=\frac{\pi}{3}-\frac{\pi}{4}=\frac{\pi}{12}$ | M1A1 | M1: takes arcsin and subtracts $\frac{\pi}{4}$; accept degrees $(x=60-45)$; do not accept mixed degrees/radians. A1: one of $\frac{\pi}{12}$ or $\frac{5\pi}{12}$ |
| $x=\pi-\frac{\pi}{3}-\frac{\pi}{4}=\frac{5\pi}{12}$ | M1A1 | M1: correct attempt at second value, accept $x+\frac{\pi}{4}=\pi-\text{their }\frac{\pi}{3}$. A1: both $\frac{\pi}{12}$ and $\frac{5\pi}{12}$, no other values in range |
---10. The curve $C$ has equation $y = \sin \left( x + \frac { \pi } { 4 } \right) , \quad 0 \leqslant x \leqslant 2 \pi$
\begin{enumerate}[label=(\alph*)]
\item On the axes below, sketch the curve $C$.
\item Write down the exact coordinates of all the points at which the curve $C$ meets or intersects the $x$-axis and the $y$-axis.
\item Solve, for $0 \leqslant x \leqslant 2 \pi$, the equation
$$\sin \left( x + \frac { \pi } { 4 } \right) = \frac { \sqrt { 3 } } { 2 }$$
giving your answers in the form $k \pi$, where $k$ is a rational number.\\
\includegraphics[max width=\textwidth, alt={}, center]{aa75f1c1-ee97-4fee-af98-957e6a3fbba1-14_677_1031_1446_445}
\end{enumerate}
\hfill \mbox{\textit{Edexcel C12 2016 Q10 [9]}}