| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2017 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Trig Proofs |
| Type | Solve equation using proven identity |
| Difficulty | Moderate -0.3 This is a straightforward two-part trigonometric identity and equation question. Part (i) requires routine algebraic manipulation (cross-multiplying, expanding, using tan = sin/cos) to reach the given form. Part (ii) uses the Pythagorean identity to solve a simple equation. While it requires multiple steps, the techniques are standard P1 fare with no novel insight needed, 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 |
|---|---|
| 5(i) | 2sinθ+cosθ 2sinθ |
| Answer | Marks | Guidance |
|---|---|---|
| sinθ+cosθ cosθ | M1 | Replace tanθ by sinθ/cosθ |
| 2sinθcosθ+cos2θ=2sin2θ+2sinθcosθ⇒c2 =2s2 | M1 A1 | Mult by c(s + c) or making this a common denom.. For A1 simplification to |
| Answer | Marks |
|---|---|
| Total: | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| 5(ii) | tan2θ=1/2 or cos2θ=2/3 or sin2θ=1/3 | B1 |
| θ=35.3° or 1 44.7° | B1 B1 FT | FT for 180 ‒ other solution. SR B1 for radians 0.615, 2.53 (0.196π, 0.804π) |
| Answer | Marks | Guidance |
|---|---|---|
| Total: | 3 | |
| Question | Answer | Marks |
Question 5:
--- 5(i) ---
5(i) | 2sinθ+cosθ 2sinθ
=
sinθ+cosθ cosθ | M1 | Replace tanθ by sinθ/cosθ
2sinθcosθ+cos2θ=2sin2θ+2sinθcosθ⇒c2 =2s2 | M1 A1 | Mult by c(s + c) or making this a common denom.. For A1 simplification to
AG without error or omission must be seen.
Total: | 3
--- 5(ii) ---
5(ii) | tan2θ=1/2 or cos2θ=2/3 or sin2θ=1/3 | B1 | Use tanθ=s/c or c2 +s2 =1 and simplify to one of these results
θ=35.3° or 1 44.7° | B1 B1 FT | FT for 180 ‒ other solution. SR B1 for radians 0.615, 2.53 (0.196π, 0.804π)
Extra solutions in range amongst solutions of which 2 are correct gets B1B0
Total: | 3
Question | Answer | Marks | Guidance\begin{enumerate}[label=(\roman*)]
\item Show that the equation $\frac{2 \sin \theta + \cos \theta}{\sin \theta + \cos \theta} = 2 \tan \theta$ may be expressed as $\cos^2 \theta = 2 \sin^2 \theta$. [3]
\item Hence solve the equation $\frac{2 \sin \theta + \cos \theta}{\sin \theta + \cos \theta} = 2 \tan \theta$ for $0° < \theta < 180°$. [3]
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2017 Q5 [6]}}