| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2005 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Find minimum domain for inverse |
| Difficulty | Standard +0.3 This question involves standard techniques for finding range of sine functions, sketching transformations, and determining domains for inverse functions. The key insight that the function must be one-to-one (monotonic) is straightforward, requiring A=90° where sine reaches its maximum. Finding the inverse involves routine algebraic manipulation and applying arcsin. While multi-part, each step uses well-practiced A-level techniques without requiring novel problem-solving. |
| Spec | 1.02u Functions: definition and vocabulary (domain, range, mapping)1.02v Inverse and composite functions: graphs and conditions for existence1.05f Trigonometric function graphs: symmetries and periodicities1.05i Inverse trig functions: arcsin, arccos, arctan domains and graphs |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Range \(1 \leq f(x) \leq 5\) | B2,1,0 [2] | Needs 1, 5, \(\leq\). One off for each error. |
| (ii) | B2,1,0 [2] | Must be exactly 1 full oscillation - this overrides the rest. Starts and ends at 3. Correct shape needed. Curves, not blatant lines. |
| Answer | Marks | Guidance |
|---|---|---|
| (iii) Maximum value of \(A = 90\) or \(\frac{\pi}{2}\) | B1 [1] | CAO |
| (iv) \(y = 3 - 2\sin x\) | M1 | Attempt to make \(x\) the subject and then to replace \(x\) by \(y\). Needs \(\sin^{-1}()\). |
| \(g^{-1}(x) = \sin^{-1}\left(\frac{3-x}{2}\right)\) | A1 [2] | Everything correct inc \(\sin^{-1}\). Allow these marks anywhere. |
$f : x \to 3 - 2\sin x$ for $0° \leq x \leq 360°$
(i) Range $1 \leq f(x) \leq 5$ | B2,1,0 [2] | Needs 1, 5, $\leq$. One off for each error.
(ii) | B2,1,0 [2] | Must be exactly 1 full oscillation - this overrides the rest. Starts and ends at 3. Correct shape needed. Curves, not blatant lines.
$g: x \to 3 - 2\sin x$ for $0° \leq x \leq A°$
(iii) Maximum value of $A = 90$ or $\frac{\pi}{2}$ | B1 [1] | CAO
(iv) $y = 3 - 2\sin x$ | M1 | Attempt to make $x$ the subject and then to replace $x$ by $y$. Needs $\sin^{-1}()$.
$g^{-1}(x) = \sin^{-1}\left(\frac{3-x}{2}\right)$ | A1 [2] | Everything correct inc $\sin^{-1}$. Allow these marks anywhere.7 A function f is defined by f : $x \mapsto 3 - 2 \sin x$, for $0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }$.\\
(i) Find the range of f .\\
(ii) Sketch the graph of $y = \mathrm { f } ( x )$.
A function g is defined by $\mathrm { g } : x \mapsto 3 - 2 \sin x$, for $0 ^ { \circ } \leqslant x \leqslant A ^ { \circ }$, where $A$ is a constant.\\
(iii) State the largest value of $A$ for which g has an inverse.\\
(iv) When $A$ has this value, obtain an expression, in terms of $x$, for $\mathrm { g } ^ { - 1 } ( x )$.
\hfill \mbox{\textit{CAIE P1 2005 Q7 [7]}}