| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2010 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Find inverse function |
| Difficulty | Standard +0.3 This is a straightforward inverse function question with standard techniques: stating range from domain transformation, evaluating at a specific point, sketching a transformed tan graph, and algebraically finding the inverse. While it involves multiple parts and requires understanding of tan function properties, each step follows routine procedures typical of P1 level with no novel problem-solving required. |
| Spec | 1.02v Inverse and composite functions: graphs and conditions for existence |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Range of \(f \leq 3\) | \(B1\) | co. Allow \(<\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(f\!\left(\frac{2}{3}\pi\right) = 3 - 2\sqrt{3}\) | \(B1\) | co |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Graph | \(B2, 1, 0\) Indep. | Starting at \(y=3\); Shape correct – no turning points; Tending tangentially towards \(x = \pi\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(y = 3 - 2\tan\!\left(\frac{x}{2}\right)\) | \(M1\) | Attempt at making \(x\) the subject |
| \(M1\) | Order of operations all ok | |
| \(\rightarrow f^{-1}(x) = 2\tan^{-1}\!\left(\frac{3-x}{2}\right)\) | \(A1\) | co – but with \(x\), not \(y\) |
## Question 7:
### Part (i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Range of $f \leq 3$ | $B1$ | co. Allow $<$ |
### Part (ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $f\!\left(\frac{2}{3}\pi\right) = 3 - 2\sqrt{3}$ | $B1$ | co |
### Part (iii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Graph | $B2, 1, 0$ Indep. | Starting at $y=3$; Shape correct – no turning points; Tending tangentially towards $x = \pi$ |
### Part (iv):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $y = 3 - 2\tan\!\left(\frac{x}{2}\right)$ | $M1$ | Attempt at making $x$ the subject |
| | $M1$ | Order of operations all ok |
| $\rightarrow f^{-1}(x) = 2\tan^{-1}\!\left(\frac{3-x}{2}\right)$ | $A1$ | co – but with $x$, not $y$ |
---7 A function f is defined by f : $x \mapsto 3 - 2 \tan \left( \frac { 1 } { 2 } x \right)$ for $0 \leqslant x < \pi$.\\
(i) State the range of f .\\
(ii) State the exact value of $\mathrm { f } \left( \frac { 2 } { 3 } \pi \right)$.\\
(iii) Sketch the graph of $y = \mathrm { f } ( x )$.\\
(iv) Obtain an expression, in terms of $x$, for $\mathrm { f } ^ { - 1 } ( x )$.
\hfill \mbox{\textit{CAIE P1 2010 Q7 [7]}}