| Exam Board | Edexcel |
|---|---|
| Module | C12 (Core Mathematics 1 & 2) |
| Year | 2018 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Trigonometric equations in context |
| Type | Solve shifted trig equation |
| Difficulty | Standard +0.3 Part (i) is a straightforward phase-shifted sine equation requiring basic rearrangement and calculator work. Part (ii) involves using the Pythagorean identity to convert to a quadratic in cos θ, then solving—a standard C2 technique. Both parts are routine applications of well-practiced methods with no novel insight required, making this slightly easier than the average A-level question. |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\sin(\ldots) = -\frac{2}{5}\) | M1 | Allow \(\text{invsin}\!\left(\frac{2}{5}\right)=23.6°\) or one of the given angles |
| \(\ldots = -23.6°\) (or \(203.6°\) or \(336.4°\)) | A1 | In radians allow awrt 0.41, 2.73 |
| \(x = 138.6°\) or \(271.4°\) (allow awrt) | dM1 A1 | dM1: subtracting 65 from any of their answers. A1: cao, withhold if extra solutions in \([0°, 360°]\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(12(1-\cos^2\theta) + \cos\theta = 6\) | M1 | Attempts to use \(\sin^2\theta = 1-\cos^2\theta\) |
| Solves three-term quadratic \(12\cos^2\theta - \cos\theta - 6 = 0\) | dM1 | Solves 3TQ; implied by sight of \(\pm\) correct values |
| \(\cos\theta = -\frac{2}{3}\) or \(\frac{3}{4}\) | A1 | |
| \(\theta = 2.30, 3.98, 0.723\) or \(5.56\) | M1 A1 A1 | M1: inverse cosine for valid \(\cos\theta\). A1: two angles correct in degrees or radians. A1: all four correct (awrt), no extras. Condone 2.3 for 2.30 |
# Question 12:
## Part (i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\sin(\ldots) = -\frac{2}{5}$ | M1 | Allow $\text{invsin}\!\left(\frac{2}{5}\right)=23.6°$ or one of the given angles |
| $\ldots = -23.6°$ (or $203.6°$ or $336.4°$) | A1 | In radians allow awrt 0.41, 2.73 |
| $x = 138.6°$ or $271.4°$ (allow awrt) | dM1 A1 | dM1: subtracting 65 from any of their answers. A1: cao, withhold if extra solutions in $[0°, 360°]$ |
**[4 marks]**
## Part (ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $12(1-\cos^2\theta) + \cos\theta = 6$ | M1 | Attempts to use $\sin^2\theta = 1-\cos^2\theta$ |
| Solves three-term quadratic $12\cos^2\theta - \cos\theta - 6 = 0$ | dM1 | Solves 3TQ; implied by sight of $\pm$ correct values |
| $\cos\theta = -\frac{2}{3}$ or $\frac{3}{4}$ | A1 | |
| $\theta = 2.30, 3.98, 0.723$ or $5.56$ | M1 A1 A1 | M1: inverse cosine for valid $\cos\theta$. A1: two angles correct in degrees or radians. A1: all four correct (awrt), no extras. Condone 2.3 for 2.30 |
**[6 marks]**
---12. [In this question solutions based entirely on graphical or numerical methods are not acceptable.]\\
(i) Solve for $0 \leqslant x < 360 ^ { \circ }$,
$$5 \sin \left( x + 65 ^ { \circ } \right) + 2 = 0$$
giving your answers in degrees to one decimal place.\\
(ii) Find, for $0 \leqslant \theta < 2 \pi$, all the solutions of
$$12 \sin ^ { 2 } \theta + \cos \theta = 6$$
giving your answers in radians to 3 significant figures.\\
\hfill \mbox{\textit{Edexcel C12 2018 Q12 [10]}}