| Exam Board | OCR |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Session | Specimen |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Quadratic trigonometric equations |
| Type | Show then solve: sin²/cos² substitution |
| Difficulty | Moderate -0.3 This is a standard trigonometric equation requiring the identity cos²θ = 1 - sin²θ to convert to quadratic form, then solving a straightforward quadratic and finding angles in the given range. While it involves multiple steps (8 marks total), each step follows routine procedures taught in C2 with no novel insight required, making it slightly easier than average. |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| LHS is \(15(1 - \sin^2 \theta^{\circ})\) | M1 | For using the relevant trig identity |
| A1 | 2 | For correct 3-term quadratic |
| Hence equation is \(15\sin^2 \theta + \sin \theta - 2 = 0\) |
| Answer | Marks | Guidance |
|---|---|---|
| \((5\sin \theta + 2)(3\sin \theta - 1) = 0\) | M1 | For factoring, or other solution method |
| A1 | For both correct values | |
| Hence \(\sin \theta = -\frac{2}{5}\) or \(\frac{1}{3}\) | ||
| So \(\theta = 19.5, 160.5, 203.6, 336.4\) | M1 | For any relevant inverse sine operation |
| A1 | For any one correct value | |
| A1′ | For corresponding second value | |
| A1′ | 6 | For both remaining values |
| Total: 8 |
## Part (i)
LHS is $15(1 - \sin^2 \theta^{\circ})$ | M1 | For using the relevant trig identity
| A1 | 2 | For correct 3-term quadratic
Hence equation is $15\sin^2 \theta + \sin \theta - 2 = 0$ | |
## Part (ii)
$(5\sin \theta + 2)(3\sin \theta - 1) = 0$ | M1 | For factoring, or other solution method
| A1 | | For both correct values
Hence $\sin \theta = -\frac{2}{5}$ or $\frac{1}{3}$ | |
So $\theta = 19.5, 160.5, 203.6, 336.4$ | M1 | For any relevant inverse sine operation
| A1 | | For any one correct value
| A1′ | | For corresponding second value
| A1′ | 6 | For both remaining values
| **Total: 8** | |\begin{enumerate}[label=(\roman*)]
\item Show that the equation $15\cos^2\theta = 13 + \sin\theta$ may be written as a quadratic equation in $\sin\theta$. [2]
\item Hence solve the equation, giving all values of $\theta$ such that $0 \leq \theta \leq 360$. [6]
\end{enumerate}
\hfill \mbox{\textit{OCR C2 Q5 [8]}}