| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2008 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Trigonometric equations in context |
| Type | Sketch transformed/compound trig graph and identify features |
| Difficulty | Moderate -0.8 This is a straightforward question testing basic understanding of cosine transformations. Part (i) requires simple simultaneous equations using max/min values of cos x. Part (ii) is a routine trig equation. Part (iii) is a standard sketch. All parts are textbook exercises with no problem-solving insight required, making it easier than average. |
| Spec | 1.02n Sketch curves: simple equations including polynomials1.05f Trigonometric function graphs: symmetries and periodicities1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| \(a + b = 10 \text{ and } a - b = -2 \rightarrow a = 4 \text{ and } b = 6\) | M1, A1, A1 | M1 for either correct. A1 both correct. Co |
| Answer | Marks | Guidance |
|---|---|---|
| \(4 - 6\cos x = 0 \rightarrow \cos x = \frac{2}{3} \rightarrow x = 48.2° \text{ or } 311.8°\) | M1, A1, A1√ | Makes \(\cos x\) subject and uses inv cos. For \(1^{\text{st}}\) angle. \(\sqrt{}\) for \(360° - \) "his angle" |
| Answer | Marks | Guidance |
|---|---|---|
| [Graph description: Must be just one cycle. Starts at –2 and ends at –2. Max at 10. "V shapes" lose a mark. Parabolas lose 1 mark.] | B2, 1 | [2] |
**(i)**
$a + b = 10 \text{ and } a - b = -2 \rightarrow a = 4 \text{ and } b = 6$ | M1, A1, A1 | M1 for either correct. A1 both correct. Co | If $a - b = 10, a + b = -2$, treat as MR –1, (i) $a = 4, b = -6$, (ii) 131.8, 228.2, (iii) Sketch is mirror image in $y = 4$
**(ii)**
$4 - 6\cos x = 0 \rightarrow \cos x = \frac{2}{3} \rightarrow x = 48.2° \text{ or } 311.8°$ | M1, A1, A1√ | Makes $\cos x$ subject and uses inv cos. For $1^{\text{st}}$ angle. $\sqrt{}$ for $360° - $ "his angle"
**(iii)**
[Graph description: Must be just one cycle. Starts at –2 and ends at –2. Max at 10. "V shapes" lose a mark. Parabolas lose 1 mark.] | B2, 1 | [2]
---5 The function f is such that $\mathrm { f } ( x ) = a - b \cos x$ for $0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }$, where $a$ and $b$ are positive constants. The maximum value of $\mathrm { f } ( x )$ is 10 and the minimum value is - 2 .\\
(i) Find the values of $a$ and $b$.\\
(ii) Solve the equation $\mathrm { f } ( x ) = 0$.\\
(iii) Sketch the graph of $y = \mathrm { f } ( x )$.
\hfill \mbox{\textit{CAIE P1 2008 Q5 [8]}}