| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2016 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Find inverse function |
| Difficulty | Moderate -0.3 This is a standard multi-part inverse function question covering routine techniques: finding range from a restricted domain, finding intercepts, sketching a transformed sine curve, and finding the inverse by rearranging y = 4sin(x) - 1. All parts are textbook exercises requiring no novel insight, though the multiple parts and careful attention to domain/range make it slightly more substantial than the most basic questions. |
| Spec | 1.02u Functions: definition and vocabulary (domain, range, mapping)1.02v Inverse and composite functions: graphs and conditions for existence1.05i Inverse trig functions: arcsin, arccos, arctan domains and graphs |
| Answer | Marks | Guidance |
|---|---|---|
| \(f: x \rightarrow 4\sin x - 1\) for \(-\frac{\pi}{2} \leq x \leq \frac{\pi}{2}\) | B1 | \(-5\) and \(3\) |
| Range \(-5 \leq f(x) \leq 3\) | B1 [2] | Correct range |
| Answer | Marks | Guidance |
|---|---|---|
| \(4s - 1 = 0 \rightarrow s = \frac{1}{4} \rightarrow x = 0.253\) | M1 A1 | Makes \(\sin x\) subject. Degrees M1 A0, \((14.5°)\) |
| \(x = 0 \rightarrow y = -1\) | B1 [3] |
| Answer | Marks | Guidance |
|---|---|---|
| [Graph showing curve with range from part (i)] | B1\(\checkmark\) B1 [2] | Shape from their range in (i). Flattens, curve. |
| Answer | Marks | Guidance |
|---|---|---|
| range \(-\frac{1}{2}\pi \leq f^{-1}(x) \leq \frac{1}{2}\pi\) | B1 | |
| domain \(-5 \leq x \leq 3\) | B1\(\checkmark\) | \(\checkmark\) on part (i) (only for 2 numerical values) |
| Inverse \(f^{-1}(x) = \sin^{-1}\left(\frac{x+1}{4}\right)\) | M1 A1 [4] | Correct order of operations |
# Question 11:
## Part (i):
$f: x \rightarrow 4\sin x - 1$ for $-\frac{\pi}{2} \leq x \leq \frac{\pi}{2}$ | **B1** | $-5$ and $3$ |
Range $-5 \leq f(x) \leq 3$ | **B1** [2] | Correct range |
## Part (ii):
$4s - 1 = 0 \rightarrow s = \frac{1}{4} \rightarrow x = 0.253$ | **M1 A1** | Makes $\sin x$ subject. Degrees **M1 A0**, $(14.5°)$ |
$x = 0 \rightarrow y = -1$ | **B1** [3] | |
## Part (iii):
[Graph showing curve with range from part (i)] | **B1$\checkmark$ B1** [2] | Shape from their range in (i). Flattens, curve. |
## Part (iv):
range $-\frac{1}{2}\pi \leq f^{-1}(x) \leq \frac{1}{2}\pi$ | **B1** | |
domain $-5 \leq x \leq 3$ | **B1$\checkmark$** | $\checkmark$ on part (i) (only for 2 numerical values) |
Inverse $f^{-1}(x) = \sin^{-1}\left(\frac{x+1}{4}\right)$ | **M1 A1** [4] | Correct order of operations |11 The function f is defined by $\mathrm { f } : x \mapsto 4 \sin x - 1$ for $- \frac { 1 } { 2 } \pi \leqslant x \leqslant \frac { 1 } { 2 } \pi$.\\
(i) State the range of f .\\
(ii) Find the coordinates of the points at which the curve $y = \mathrm { f } ( x )$ intersects the coordinate axes.\\
(iii) Sketch the graph of $y = \mathrm { f } ( x )$.\\
(iv) Obtain an expression for $\mathrm { f } ^ { - 1 } ( x )$, stating both the domain and range of $\mathrm { f } ^ { - 1 }$.
{www.cie.org.uk} after the live examination series.
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\hfill \mbox{\textit{CAIE P1 2016 Q11 [11]}}