| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2018 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Trigonometric equations in context |
| Type | Convert sin/cos ratio to tan |
| Difficulty | Moderate -0.3 This is a standard A-level trig equation question requiring routine techniques: rearranging to tan x form, sketching basic trig curves, and using graphical intersection to solve an inequality. While multi-part, each step follows textbook methods with no novel insight required, making it slightly easier than average. |
| Spec | 1.02g Inequalities: linear and quadratic in single variable1.05f Trigonometric function graphs: symmetries and periodicities1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(2\cos x = -3\sin x \rightarrow \tan x = -\dfrac{2}{3}\) | M1 | Use of \(\tan = \sin/\cos\) to get \(\tan =\), or other valid method to find \(\sin\) or \(\cos =\). M0 for \(\tan x = +/-\dfrac{3}{2}\) |
| \(\rightarrow x = 146.3°\) or \(326.3°\) awrt | A1 A1FT | FT for \(180°\) added to an incorrect first answer in the given range. The second A1 is withheld if any further values in the range \(0° \leqslant x \leqslant 360°\) are given. Answers in radians score A0, A0 |
| Total | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Sketch of \(y = 2\cos x\): one complete cycle; start and finish at top of curve at roughly the same positive \(y\) value and go below the \(x\) axis by roughly the same distance | B1 | Can be a poor curve but not straight lines |
| Sketch of \(y = -3\sin x\): one complete cycle; start and finish on the \(x\) axis, must be inverted and go below and then above the \(x\) axis by roughly the same distance | B1 | Can be a poor curve but not straight lines |
| Fully correct answer including the sine curve with clearly larger amplitude than cosine curve. Must now be reasonable curves | B1 | Note: Separate diagrams can score 2/3 |
| Total | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(x < 146.3°,\ x > 326.3°\) | B1FT B1FT | Does not need to include \(0°\), \(360°\). \(\sqrt{}\) from their answers in (i). Allow combined statement as long as correct inequalities if taken separately. SC: For two correct values including ft but with \(\leqslant\) and \(\geqslant\): B1 |
| Total | 2 |
## Question 10:
### Part (i):
| Answer | Mark | Guidance |
|--------|------|----------|
| $2\cos x = -3\sin x \rightarrow \tan x = -\dfrac{2}{3}$ | M1 | Use of $\tan = \sin/\cos$ to get $\tan =$, or other valid method to find $\sin$ or $\cos =$. M0 for $\tan x = +/-\dfrac{3}{2}$ |
| $\rightarrow x = 146.3°$ or $326.3°$ awrt | A1 A1FT | FT for $180°$ added to an incorrect first answer in the given range. The second A1 is withheld if any further values in the range $0° \leqslant x \leqslant 360°$ are given. Answers in radians score A0, A0 |
| **Total** | **3** | |
### Part (ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Sketch of $y = 2\cos x$: one complete cycle; start and finish at top of curve at roughly the same positive $y$ value and go below the $x$ axis by roughly the same distance | B1 | Can be a poor curve but not straight lines |
| Sketch of $y = -3\sin x$: one complete cycle; start and finish on the $x$ axis, must be inverted and go below and then above the $x$ axis by roughly the same distance | B1 | Can be a poor curve but not straight lines |
| Fully correct answer including the sine curve with clearly larger amplitude than cosine curve. Must now be reasonable curves | B1 | Note: Separate diagrams can score 2/3 |
| **Total** | **3** | |
### Part (iii):
| Answer | Mark | Guidance |
|--------|------|----------|
| $x < 146.3°,\ x > 326.3°$ | B1FT B1FT | Does not need to include $0°$, $360°$. $\sqrt{}$ from their answers in (i). Allow combined statement as long as correct inequalities if taken separately. SC: For two correct values including ft but with $\leqslant$ and $\geqslant$: B1 |
| **Total** | **2** | |
---10 (i) Solve the equation $2 \cos x + 3 \sin x = 0$, for $0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }$.\\
(ii) Sketch, on the same diagram, the graphs of $y = 2 \cos x$ and $y = - 3 \sin x$ for $0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }$.\\
(iii) Use your answers to parts (i) and (ii) to find the set of values of $x$ for $0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }$ for which $2 \cos x + 3 \sin x > 0$.\\
\hfill \mbox{\textit{CAIE P1 2018 Q10 [8]}}