| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2003 |
| Session | June |
| 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 -0.3 This question involves sketching a standard sine curve with amplitude 3, finding a gradient through a known maximum point (straightforward calculation k=6/π), and using symmetry to identify the intersection at (-π/2, -3). While multi-part, each step uses routine techniques with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.02m Graphs of functions: difference between plotting and sketching1.02q Use intersection points: of graphs to solve equations1.05f Trigonometric function graphs: symmetries and periodicities |
| Answer | Marks | Guidance |
|---|---|---|
| (i) [Graph showing complete cycle with curves, not lines, -3 to +3 shown or implied, from \(-\pi\) to \(\pi\). Degrees ok] | B2, 1 [2] | For complete cycle, shape including curves, not lines, -3 to +3 shown or implied, for \(-\pi\) to \(\pi\). Degrees ok |
| (ii) \(x = \pi/2\), \(y = 3\) (allow if 90°) \(\rightarrow k = 6/\pi\) co. | M1, A1 [2] | Realising maximum is \((\pi/2, 3)\) + sub. Co (even if no graph) |
| (iii) \((-\pi/2, -3)\) – must be radians | B1 [1] | Co (could come from incorrect graph) |
**(i)** [Graph showing complete cycle with curves, not lines, -3 to +3 shown or implied, from $-\pi$ to $\pi$. Degrees ok] | B2, 1 [2] | For complete cycle, shape including curves, not lines, -3 to +3 shown or implied, for $-\pi$ to $\pi$. Degrees ok
**(ii)** $x = \pi/2$, $y = 3$ (allow if 90°) $\rightarrow k = 6/\pi$ co. | M1, A1 [2] | Realising maximum is $(\pi/2, 3)$ + sub. Co (even if no graph)
**(iii)** $(-\pi/2, -3)$ – must be radians | B1 [1] | Co (could come from incorrect graph)6 (i) Sketch the graph of the curve $y = 3 \sin x$, for $- \pi \leqslant x \leqslant \pi$.
The straight line $y = k x$, where $k$ is a constant, passes through the maximum point of this curve for $- \pi \leqslant x \leqslant \pi$.\\
(ii) Find the value of $k$ in terms of $\pi$.\\
(iii) State the coordinates of the other point, apart from the origin, where the line and the curve intersect.
\hfill \mbox{\textit{CAIE P1 2003 Q6 [5]}}