| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2009 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Curve Sketching |
| Type | Range from trigonometric functions |
| Difficulty | Moderate -0.8 This is a straightforward question on trigonometric functions requiring knowledge that sin ranges from -1 to 1, basic graph sketching of a transformed sine curve, and understanding of the horizontal line test for inverses. All parts are routine applications of standard techniques with no problem-solving insight required. |
| Spec | 1.02u Functions: definition and vocabulary (domain, range, mapping)1.02v Inverse and composite functions: graphs and conditions for existence |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(2 \leq f(x) \leq 8\) | B1, B1 [2] | B1 for 2, B1 for 8. Must be stated, not on graph. |
| (ii) \(x \mapsto 5 - 3\sin 2x\) | B1, DB1, B1 [3] | 1 complete oscillation. 1st quadrant, not touching x-axis. Needs to be "down" first and curves. nb If no labels, assume 0 to \(\pi\). |
| (iii) No inverse – not 1 : 1. | B1 [1] | co. Independent of graph. |
**(i)** $2 \leq f(x) \leq 8$ | B1, B1 [2] | B1 for 2, B1 for 8. Must be stated, not on graph.
**(ii)** $x \mapsto 5 - 3\sin 2x$ | B1, DB1, B1 [3] | 1 complete oscillation. 1st quadrant, not touching x-axis. Needs to be "down" first and curves. nb If no labels, assume 0 to $\pi$.
**(iii)** No inverse – not 1 : 1. | B1 [1] | co. Independent of graph.
**Total: [6]**
---4 The function f is defined by f : $x \mapsto 5 - 3 \sin 2 x$ for $0 \leqslant x \leqslant \pi$.\\
(i) Find the range of f .\\
(ii) Sketch the graph of $y = \mathrm { f } ( x )$.\\
(iii) State, with a reason, whether f has an inverse.
\hfill \mbox{\textit{CAIE P1 2009 Q4 [6]}}