| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2010 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Trig Graphs & Exact Values |
| Type | Find function constants from given conditions |
| Difficulty | Moderate -0.8 This is a straightforward question requiring substitution of given values into a cosine function to form simultaneous equations, then using exact trig values. All steps are routine: finding constants from conditions, stating range from amplitude, and evaluating at a standard angle. Below average difficulty as it's purely procedural with no problem-solving or insight required. |
| Spec | 1.05g Exact trigonometric values: for standard angles1.05k Further identities: sec^2=1+tan^2 and cosec^2=1+cot^2 |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(f(0) = 10, a + b = 10\) | B1 | EITHER OF THESE |
| \(f(\frac{2}{3}\pi) = 1, a - \frac{b}{2} = 1\) | B1 | both co |
| \(\to a = 4, b = 6\) | [2] | √ for his "\(a - b\)" to "\(a + b\)" |
| (ii) Range is \(-2\) to \(10\). | B1√ | √ for his "\(a - b\)" to "\(a + b\)" |
| [1] | ||
| (iii) \(\cos\left(\frac{5}{6}\pi\right) = -\cos\left(\frac{1}{6}\pi\right) = -\frac{\sqrt{3}}{2}\) | B1 | For \(\cos 30° = \frac{1}{2}\sqrt{3}\) used somewhere. |
| \(\to 4 - 3\sqrt{3}\) | B1 | co |
| [2] |
$f: x \mapsto a + b\cos x$
(i) $f(0) = 10, a + b = 10$ | B1 | EITHER OF THESE
$f(\frac{2}{3}\pi) = 1, a - \frac{b}{2} = 1$ | B1 | both co
$\to a = 4, b = 6$ | [2] | √ for his "$a - b$" to "$a + b$"
(ii) Range is $-2$ to $10$. | B1√ | √ for his "$a - b$" to "$a + b$"
| [1] |
(iii) $\cos\left(\frac{5}{6}\pi\right) = -\cos\left(\frac{1}{6}\pi\right) = -\frac{\sqrt{3}}{2}$ | B1 | For $\cos 30° = \frac{1}{2}\sqrt{3}$ used somewhere.
$\to 4 - 3\sqrt{3}$ | B1 | co
| [2] |
---3 The function $\mathrm { f } : x \mapsto a + b \cos x$ is defined for $0 \leqslant x \leqslant 2 \pi$. Given that $\mathrm { f } ( 0 ) = 10$ and that $\mathrm { f } \left( \frac { 2 } { 3 } \pi \right) = 1$, find\\
(i) the values of $a$ and $b$,\\
(ii) the range of $f$,\\
(iii) the exact value of $\mathrm { f } \left( \frac { 5 } { 6 } \pi \right)$.
\hfill \mbox{\textit{CAIE P1 2010 Q3 [5]}}