| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2017 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Reciprocal Trig & Identities |
| Type | Prove identity then solve equation |
| Difficulty | Standard +0.3 Part (i) is a standard trigonometric identity proof requiring manipulation of sec and tan into sin/cos form, then algebraic simplification—routine for P1 level. Part (ii) is a straightforward equation solve using the proven identity. This is a typical textbook-style question testing standard techniques without requiring novel insight, making it slightly easier than average. |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05k Further identities: sec^2=1+tan^2 and cosec^2=1+cot^21.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\text{LHS} = \left(\frac{1}{c} - \frac{s}{c}\right)^2\) | M1 | Eliminates tan by replacing with \(\frac{\sin}{\cos}\) leading to a function of sin and/or cos only. |
| \(= \frac{(1-s)^2}{1-s^2}\) | M1 | Uses \(s^2 + c^2 = 1\) leading to a function of sin only. |
| \(= \frac{(1-s)(1-s)}{(1-s)(1+s)} = \frac{1-\sin\theta}{1+\sin\theta}\) | A1 | AG. Must show use of factors for A1. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Uses part (i) \(\rightarrow 2 - 2s = 1 + s\) | ||
| \(\rightarrow s = \frac{1}{3}\) | M1 | Uses part (i) to obtain \(s = k\) |
| \(\theta = 19.5°\) or \(160.5°\) | A1A1 FT | FT from error in \(19.5°\). Allow \(0.340^c\) (\(0.3398^c\)) & \(2.80(2)\) or \(0.108\pi^c\) & \(0.892\pi^c\) for A1 only. Extra answers in the range lose the second A1 if gained for \(160.5°\). |
## Question 3:
**Part (i):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\text{LHS} = \left(\frac{1}{c} - \frac{s}{c}\right)^2$ | **M1** | Eliminates tan by replacing with $\frac{\sin}{\cos}$ leading to a function of sin and/or cos only. |
| $= \frac{(1-s)^2}{1-s^2}$ | **M1** | Uses $s^2 + c^2 = 1$ leading to a function of sin only. |
| $= \frac{(1-s)(1-s)}{(1-s)(1+s)} = \frac{1-\sin\theta}{1+\sin\theta}$ | **A1** | AG. Must show use of factors for **A1**. |
**Part (ii):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| Uses part (i) $\rightarrow 2 - 2s = 1 + s$ | | |
| $\rightarrow s = \frac{1}{3}$ | **M1** | Uses part (i) to obtain $s = k$ |
| $\theta = 19.5°$ or $160.5°$ | **A1A1 FT** | FT from error in $19.5°$. Allow $0.340^c$ ($0.3398^c$) & $2.80(2)$ or $0.108\pi^c$ & $0.892\pi^c$ for **A1** only. Extra answers in the range lose the second **A1** if gained for $160.5°$. |
---3 (i) Prove the identity $\left( \frac { 1 } { \cos \theta } - \tan \theta \right) ^ { 2 } \equiv \frac { 1 - \sin \theta } { 1 + \sin \theta }$.\\
(ii) Hence solve the equation $\left( \frac { 1 } { \cos \theta } - \tan \theta \right) ^ { 2 } = \frac { 1 } { 2 }$, for $0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }$.\\
\includegraphics[max width=\textwidth, alt={}, center]{8a3f8707-67a4-4069-aba5-7e9496cb1748-06_572_460_258_845}
The diagram shows a circle with radius $r \mathrm {~cm}$ and centre $O$. Points $A$ and $B$ lie on the circle and $A B C D$ is a rectangle. Angle $A O B = 2 \theta$ radians and $A D = r \mathrm {~cm}$.\\
\hfill \mbox{\textit{CAIE P1 2017 Q3 [6]}}