| Exam Board | OCR MEI |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Marks | 5 |
| 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 straightforward C2 trigonometric equation requiring the standard identity cos²θ = 1 - sin²θ to convert to quadratic form (already shown in part i), then simple factorisation and solving. The restricted domain and routine nature of both steps make this slightly easier than average, though it does require knowing the key identity and basic solving technique. |
| Spec | 1.02f Solve quadratic equations: including in a function of unknown1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| (i) \(2(-\sin^2\theta) + 7\sin\theta = 5\) | 1 | for \(\cos^2\theta + \sin^2\theta = 1\) o.e. used |
| (ii) \((2\sin\theta - 1)(\sin\theta - 3)\) | M1 | \(1^{\text{st}}\) and \(3^{\text{rd}}\) terms in expansion correct |
| \(\sin\theta = \frac{1}{2}\) | DM1 | f.t. facto |
| \(30°\) and \(150°\) | A1 | B1,B1 for each solution obtained by any valid method, ignore extra solns outside range |
| A1 | \(30°\), \(150°\) plus extra soln(s) scores 1 | 5 |
## Question 10:
| Answer/Working | Mark | Guidance |
|---|---|---|
| (i) $2(-\sin^2\theta) + 7\sin\theta = 5$ | 1 | for $\cos^2\theta + \sin^2\theta = 1$ o.e. used |
| (ii) $(2\sin\theta - 1)(\sin\theta - 3)$ | M1 | $1^{\text{st}}$ and $3^{\text{rd}}$ terms in expansion correct |
| $\sin\theta = \frac{1}{2}$ | DM1 | f.t. facto |
| $30°$ and $150°$ | A1 | B1,B1 for each solution obtained by any valid method, ignore extra solns outside range |
| | A1 | $30°$, $150°$ plus extra soln(s) scores 1 | 5 |10 (i) Show that the equation $2 \cos ^ { 2 } \theta + 7 \sin \theta = 5$ may be written in the form
$$2 \sin ^ { 2 } \theta - 7 \sin \theta + 3 = 0$$
(ii) By factorising this quadratic equation, solve the equation for values of $\theta$ between $0 ^ { \circ }$ and $180 ^ { \circ }$. [4]
\hfill \mbox{\textit{OCR MEI C2 Q10 [5]}}