| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2017 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Quadratic trigonometric equations |
| Type | Two-part show and solve with cos²θ quartic form |
| Difficulty | Standard +0.8 This question requires multiple trigonometric identities (converting cot and tan to sin/cos, using sin 2θ = 2sin θ cos θ, and the Pythagorean identity) followed by algebraic manipulation to reach the required quartic form, then solving a quadratic in cos²θ with domain restrictions. While systematic, it demands careful algebraic handling and is more involved than standard trigonometric equation questions, placing it moderately above average difficulty. |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05l Double angle formulae: and compound angle formulae1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| 3(i) | Use correct formulae to express the equation in terms of cos θ and sin θ | M1 |
| Use Pythagoras and express the equation in terms of cos θ only | M1 | |
| Obtain correct 3-term equation, e.g.2cos 4 θ+cos 2 θ−2=0 | A1 | |
| Total: | 3 |
| Answer | Marks |
|---|---|
| 3(ii) | 2 |
| Solve a 3-term quadratic in cos θfor cos θ | M1 |
| Obtain answer θ = 152.1°only | A1 |
| Total: | 2 |
Question 3:
--- 3(i) ---
3(i) | Use correct formulae to express the equation in terms of cos θ and sin θ | M1
Use Pythagoras and express the equation in terms of cos θ only | M1
Obtain correct 3-term equation, e.g.2cos 4 θ+cos 2 θ−2=0 | A1
Total: | 3
--- 3(ii) ---
3(ii) | 2
Solve a 3-term quadratic in cos θfor cos θ | M1
Obtain answer θ = 152.1°only | A1
Total: | 2\begin{enumerate}[label=(\roman*)]
\item Express the equation $\cot \theta - 2 \tan \theta = \sin 2\theta$ in the form $a \cos^4 \theta + b \cos^2 \theta + c = 0$, where $a$, $b$ and $c$ are constants to be determined. [3]
\item Hence solve the equation $\cot \theta - 2 \tan \theta = \sin 2\theta$ for $90° < \theta < 180°$. [2]
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2017 Q3 [5]}}