| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2004 |
| Session | November |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Trig Graphs & Exact Values |
| Type | Sketch two trig curves and count intersections/solutions |
| Difficulty | Moderate -0.8 This is a straightforward graph-sketching question requiring knowledge of standard trig transformations (amplitude and period) and the ability to count intersections visually. Both graphs are standard A-level curves with no complex features, and the interval is simple. The question requires only routine application of basic trig graph knowledge with no problem-solving or algebraic manipulation needed. |
| 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 |
| Graph of \( | \sin x | \) (half cycle above axis for \(0\) to \(\pi\), max 2, \(\frac{1}{2}\) cycle only) |
| One whole cycle for \(0\) to \(\pi\) | B1 | |
| \(-1\) to \(1\) shown with one cycle only | B1 | |
| Providing 2 trig graphs used | B1 | [4] Ignore other half if \(0\) to \(2\pi\) used |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\rightarrow 2\) points of intersection | \(\text{B1}\sqrt{}\) | [1] |
## Question 4(i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Graph of $|\sin x|$ (half cycle above axis for $0$ to $\pi$, max 2, $\frac{1}{2}$ cycle only) | B1 | Half a cycle – all above axis for $0$ to $\pi$. 2 shown as max with $\frac{1}{2}$ cycle only |
| One whole cycle for $0$ to $\pi$ | B1 | |
| $-1$ to $1$ shown with one cycle only | B1 | |
| Providing 2 trig graphs used | B1 | [4] Ignore other half if $0$ to $2\pi$ used |
## Question 4(ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\rightarrow 2$ points of intersection | $\text{B1}\sqrt{}$ | [1] |
---4 (i) Sketch and label, on the same diagram, the graphs of $y = 2 \sin x$ and $y = \cos 2 x$, for the interval $0 \leqslant x \leqslant \pi$.\\
(ii) Hence state the number of solutions of the equation $2 \sin x = \cos 2 x$ in the interval $0 \leqslant x \leqslant \pi$.
\hfill \mbox{\textit{CAIE P1 2004 Q4 [5]}}