| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2010 |
| Session | November |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Trig Graphs & Exact Values |
| Type | Sketch trig curve and straight line, count intersections |
| Difficulty | Standard +0.3 This question requires sketching a standard sine curve and adding a straight line to count intersections. While it involves rearranging the equation and recognizing that the line y = (π - x)/(2π) should be added, the sketching is routine and counting intersections is straightforward visual analysis. Slightly above average due to the algebraic manipulation needed to identify the correct line, but still a standard graphical question type. |
| Spec | 1.02q Use intersection points: of graphs to solve equations1.05f Trigonometric function graphs: symmetries and periodicities |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Correct sine curve | B1 | 2 shown or implied |
| [1] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Required line \(y = 1 - \frac{x}{\pi}\) | B1 | |
| Line through \((0,1)\), \((\pi, 0)\) drawn | B1 | SC B1 for correct graphs without 1 or 2 marked |
| 3 roots | B1\(\sqrt{}\) | ft on trig curve and line |
| [3] |
## Question 4:
### Part (i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Correct sine curve | B1 | 2 shown or implied |
| **[1]** | | |
### Part (ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Required line $y = 1 - \frac{x}{\pi}$ | B1 | |
| Line through $(0,1)$, $(\pi, 0)$ drawn | B1 | SC B1 for correct graphs without 1 or 2 marked |
| 3 roots | B1$\sqrt{}$ | ft on trig curve and line |
| **[3]** | | |
---4 (i) Sketch the curve $y = 2 \sin x$ for $0 \leqslant x \leqslant 2 \pi$.\\
(ii) By adding a suitable straight line to your sketch, determine the number of real roots of the equation
$$2 \pi \sin x = \pi - x$$
State the equation of the straight line.
\hfill \mbox{\textit{CAIE P1 2010 Q4 [4]}}