| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2012 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Reciprocal Trig & Identities |
| Type | Convert equation to quadratic form |
| Difficulty | Moderate -0.3 This is a slightly below-average A-level question. Part (i) requires routine manipulation using tan x = sin x/cos x and the Pythagorean identity to form a quadratic in sin x—a standard textbook exercise. Part (ii) applies the same method with a compound angle substitution and requires solving within a given range, adding minimal complexity. The techniques are well-practiced and the question follows a predictable structure with no novel insight required. |
| Spec | 1.05j Trigonometric identities: tan=sin/cos and sin^2+cos^2=11.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Replaces \(\tan x\) by \(\sin x \div \cos x\) | M1 | Uses \(t = s \div c\) |
| \(\rightarrow 2c^2 = 3s \rightarrow 2s^2 + 3s - 2 = 0\) | M1 A1 | Uses \(s^2 + c^2 = 1\); correct eqn |
| [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Solution of quadratic | M1 | Method for quadratic \(= 0\) and \(\div 2\) |
| \(\rightarrow y = 15°\) | A1 | co; works with \(2y\) first before \(\div 2\) |
| \(2y\) can also be \(180 - 30\) | DM1 A1\(\checkmark\) | for \(90° - 1^{st}\) answer |
| \(\rightarrow y = 75°\) | (loses \(\checkmark\) mark if extra solution in range) | |
| [4] |
## Question 6:
### Part (i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Replaces $\tan x$ by $\sin x \div \cos x$ | M1 | Uses $t = s \div c$ |
| $\rightarrow 2c^2 = 3s \rightarrow 2s^2 + 3s - 2 = 0$ | M1 A1 | Uses $s^2 + c^2 = 1$; correct eqn |
| **[3]** | | |
### Part (ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Solution of quadratic | M1 | Method for quadratic $= 0$ and $\div 2$ |
| $\rightarrow y = 15°$ | A1 | co; works with $2y$ first before $\div 2$ |
| $2y$ can also be $180 - 30$ | DM1 A1$\checkmark$ | for $90° - 1^{st}$ answer |
| $\rightarrow y = 75°$ | | (loses $\checkmark$ mark if extra solution in range) |
| **[4]** | | |
---6 (i) Show that the equation $2 \cos x = 3 \tan x$ can be written as a quadratic equation in $\sin x$.\\
(ii) Solve the equation $2 \cos 2 y = 3 \tan 2 y$, for $0 ^ { \circ } \leqslant y \leqslant 180 ^ { \circ }$.
\hfill \mbox{\textit{CAIE P1 2012 Q6 [7]}}