| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2011 |
| Session | November |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Trig Graphs & Exact Values |
| Type | Sketch trig curve and straight line, count intersections |
| Difficulty | Moderate -0.8 This is a straightforward question requiring a standard sketch of cos 2θ (double frequency cosine) and a horizontal line, counting intersections visually, then scaling up by recognizing the pattern repeats over intervals of 2π. All steps are routine with no problem-solving insight needed beyond basic periodicity. |
| Spec | 1.05f Trigonometric function graphs: symmetries and periodicities1.05g Exact trigonometric values: for standard angles1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Correct cosine curve for at least 1 oscillation | B1 | Range \(-1 \to 1\). Ignore labels on \(\theta\) axis |
| Exactly 2 complete oscillations in \([0, 2\pi]\) | B1 | |
| Line \(y = \frac{1}{2}\) correct | B1 | [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(4\) | B1\(\sqrt{}\) | [1] Ft *their* graph. Accept \(30°, 150°, 210°, 330°\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(20\) | B1\(\sqrt{}\) | [1] Or \(5 \times\) *their* part (ii) |
## Question 3:
### Part (i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Correct cosine curve for at least 1 oscillation | B1 | Range $-1 \to 1$. Ignore labels on $\theta$ axis |
| Exactly 2 complete oscillations in $[0, 2\pi]$ | B1 | |
| Line $y = \frac{1}{2}$ correct | B1 | [3] |
### Part (ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $4$ | B1$\sqrt{}$ | [1] Ft *their* graph. Accept $30°, 150°, 210°, 330°$ |
### Part (iii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $20$ | B1$\sqrt{}$ | [1] Or $5 \times$ *their* part **(ii)** |
---3 (i) Sketch, on a single diagram, the graphs of $y = \cos 2 \theta$ and $y = \frac { 1 } { 2 }$ for $0 \leqslant \theta \leqslant 2 \pi$.\\
(ii) Write down the number of roots of the equation $2 \cos 2 \theta - 1 = 0$ in the interval $0 \leqslant \theta \leqslant 2 \pi$.\\
(iii) Deduce the number of roots of the equation $2 \cos 2 \theta - 1 = 0$ in the interval $10 \pi \leqslant \theta \leqslant 20 \pi$.
\hfill \mbox{\textit{CAIE P1 2011 Q3 [5]}}