| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2013 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Find minimum domain for inverse |
| Difficulty | Moderate -0.3 This is a straightforward composite/inverse function question involving trigonometry. Parts (i)-(iii) are routine: solving a simple trig equation, finding range from amplitude, and sketching a cosine graph. Part (iv) requires understanding that cosine must be one-to-one (answer: k=π), and part (v) is standard inverse function manipulation using arccos. While multi-part, each step uses direct application of standard techniques without requiring problem-solving insight, making it slightly easier than average. |
| Spec | 1.02n Sketch curves: simple equations including polynomials1.02u Functions: definition and vocabulary (domain, range, mapping)1.02v Inverse and composite functions: graphs and conditions for existence1.05o Trigonometric equations: solve in given intervals |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(3\cos x - 2 = 0 \to \cos x = \frac{2}{3} \to x = 0.841\) or 5.44 | M1, A1 A1✓ [3] | Makes cos subject, then \(\cos^{-1}\). ✓ for \(2\pi\) – 1st answer. |
| (ii) range is \(-5 \le f(x) \le 1\) | B2, 1 [2] | B1 for \(\ge -5\). B1 for \(\le 1\). |
| (iii) [Graph showing one complete cycle from 0 to \(2\pi\), starting and ending at same point, decreasing, with one minimum] | B1, B1, B1 [2] | B1 starts and ends at same point. Starts decreasing. One cycle only. B1 for shape, not '∨' or '∪'. B1 [1] |
| (iv) max value of \(k = \pi\) or \(180°\). | M1, A1 [2] | Make \(x\) the subject, copes with 'cos'. Needs to be in terms of \(x\). |
| (v) \(g^{-1}(x) = \cos^{-1}\left(\frac{x + 2}{3}\right)\) |
$f : x \mapsto 3\cos x - 2$ for $0 \le x \le 2\pi$.
(i) $3\cos x - 2 = 0 \to \cos x = \frac{2}{3} \to x = 0.841$ or 5.44 | M1, A1 A1✓ [3] | Makes cos subject, then $\cos^{-1}$. ✓ for $2\pi$ – 1st answer.
(ii) range is $-5 \le f(x) \le 1$ | B2, 1 [2] | B1 for $\ge -5$. B1 for $\le 1$.
(iii) [Graph showing one complete cycle from 0 to $2\pi$, starting and ending at same point, decreasing, with one minimum] | B1, B1, B1 [2] | B1 starts and ends at same point. Starts decreasing. One cycle only. B1 for shape, not '∨' or '∪'. B1 [1]
(iv) max value of $k = \pi$ or $180°$. | M1, A1 [2] | Make $x$ the subject, copes with 'cos'. Needs to be in terms of $x$.
(v) $g^{-1}(x) = \cos^{-1}\left(\frac{x + 2}{3}\right)$ | | |8 A function f is defined by $\mathrm { f } : x \mapsto 3 \cos x - 2$ for $0 \leqslant x \leqslant 2 \pi$.\\
(i) Solve the equation $\mathrm { f } ( x ) = 0$.\\
(ii) Find the range of f .\\
(iii) Sketch the graph of $y = \mathrm { f } ( x )$.
A function g is defined by $\mathrm { g } : x \mapsto 3 \cos x - 2$ for $0 \leqslant x \leqslant k$.\\
(iv) State the maximum value of $k$ for which g has an inverse.\\
(v) Obtain an expression for $\mathrm { g } ^ { - 1 } ( x )$.
\hfill \mbox{\textit{CAIE P1 2013 Q8 [10]}}