| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2012 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Trigonometric equations in context |
| Type | Reduce to quadratic in trig |
| Difficulty | Standard +0.3 Part (i) is a standard trigonometric equation requiring the Pythagorean identity to convert to a quadratic in sin θ, then solving—routine A-level technique. Part (ii) requires recognizing that if 10° is the smallest solution for 2cos²(nθ) = 3sin(nθ), then n = 3 (since the smallest solution to part (i) is 30°), followed by finding all solutions in the transformed interval. This adds a layer of reasoning beyond pure recall but remains a straightforward deduction, making it slightly easier than average overall. |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(2(1 - \sin^2\theta) = 3\sin\theta\) | M1 | Use \(c^2 + s^2 = 1\) |
| \((2\sin\theta - 1)(\sin\theta + 2) = 0\) | M1 | Attempt to solve |
| \(\theta = 30°\) or \(150°\) | A1A1 | cao |
| (ii) \(n = \frac{\text{their }30}{30} = 3\) | B1 | If provided \(n\) is an integer |
| \((\text{their }3)\theta = 720 + \text{their}150 = 870\) | M1 | Allow full list up to at least 870 cao |
| \(\theta = 290°\) | A1 | [3] |
**(i)** $2(1 - \sin^2\theta) = 3\sin\theta$ | M1 | Use $c^2 + s^2 = 1$
$(2\sin\theta - 1)(\sin\theta + 2) = 0$ | M1 | Attempt to solve
$\theta = 30°$ or $150°$ | A1A1 | cao | [4]
**(ii)** $n = \frac{\text{their }30}{30} = 3$ | B1 | If provided $n$ is an integer
$(\text{their }3)\theta = 720 + \text{their}150 = 870$ | M1 | Allow full list up to at least 870 cao
$\theta = 290°$ | A1 | [3]7 (i) Solve the equation $2 \cos ^ { 2 } \theta = 3 \sin \theta$, for $0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }$.\\
(ii) The smallest positive solution of the equation $2 \cos ^ { 2 } ( n \theta ) = 3 \sin ( n \theta )$, where $n$ is a positive integer, is $10 ^ { \circ }$. State the value of $n$ and hence find the largest solution of this equation in the interval $0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }$.
\hfill \mbox{\textit{CAIE P1 2012 Q7 [7]}}