| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2018 |
| 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 question tests standard inverse function concepts (finding domain restrictions for one-one functions, determining range, finding inverse expressions) and composite functions with algebraic manipulation. While it requires multiple steps, all techniques are routine for P1 level—identifying the vertex of a parabola for part (a)(i), applying inverse function methods, and systematic substitution/expansion for part (b). No novel problem-solving insight is needed, making it slightly easier than average. |
| Spec | 1.02u Functions: definition and vocabulary (domain, range, mapping)1.02v Inverse and composite functions: graphs and conditions for existence1.02w Graph transformations: simple transformations of f(x) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Greatest value of \(a\) is \(3\) | B1 | Must be in terms of \(a\). Allow \(a < 3\). Allow \(a \leqslant 3\) |
| 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Range is \(y > -1\) | B1 | Ft on their \(a\). Accept any equivalent notation |
| \(y = (x-3)^2 - 1 \to (x-3)^2 = 1+y \to x = 3(\pm)\sqrt{1+y}\) | M1 | Order of operations correct. Allow sign errors |
| \(f^{-1}(x) = 3 - \sqrt{1+x}\) cao | A1 | |
| 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(gg(2x) = \left[(2x-3)^2 - 3\right]^2\) | B1 | |
| \((2x-3)^4 - 6(2x-3)^2 + 9\) | B1 | |
| 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\left[16x^4 - 96x^3 + 216x^2 - 216x + 81\right] + \left[(-24x^2 + 72x - 54) + 9\right]\) | B4,3,2,1,0 | |
| \(16x^4 - 96x^3 + 192x^2 - 144x + 36\) | ||
| 4 |
## Question 11(a)(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Greatest value of $a$ is $3$ | **B1** | Must be in terms of $a$. Allow $a < 3$. Allow $a \leqslant 3$ |
| | **1** | |
---
## Question 11(a)(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Range is $y > -1$ | **B1** | Ft on their $a$. Accept any equivalent notation |
| $y = (x-3)^2 - 1 \to (x-3)^2 = 1+y \to x = 3(\pm)\sqrt{1+y}$ | **M1** | Order of operations correct. Allow sign errors |
| $f^{-1}(x) = 3 - \sqrt{1+x}$ cao | **A1** | |
| | **3** | |
---
## Question 11(b)(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $gg(2x) = \left[(2x-3)^2 - 3\right]^2$ | **B1** | |
| $(2x-3)^4 - 6(2x-3)^2 + 9$ | **B1** | |
| | **2** | |
---
## Question 11(b)(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\left[16x^4 - 96x^3 + 216x^2 - 216x + 81\right] + \left[(-24x^2 + 72x - 54) + 9\right]$ | **B4,3,2,1,0** | |
| $16x^4 - 96x^3 + 192x^2 - 144x + 36$ | | |
| | **4** | |11
\begin{enumerate}[label=(\alph*)]
\item The one-one function f is defined by $\mathrm { f } ( x ) = ( x - 3 ) ^ { 2 } - 1$ for $x < a$, where $a$ is a constant.
\begin{enumerate}[label=(\roman*)]
\item State the greatest possible value of $a$.
\item It is given that $a$ takes this greatest possible value. State the range of f and find an expression for $\mathrm { f } ^ { - 1 } ( x )$.
\end{enumerate}\item The function g is defined by $\mathrm { g } ( x ) = ( x - 3 ) ^ { 2 }$ for $x \geqslant 0$.
\begin{enumerate}[label=(\roman*)]
\item Show that $\operatorname { gg } ( 2 x )$ can be expressed in the form $( 2 x - 3 ) ^ { 4 } + b ( 2 x - 3 ) ^ { 2 } + c$, where $b$ and $c$ are constants to be found.
\item Hence expand $\operatorname { gg } ( 2 x )$ completely, simplifying your answer.\\
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2018 Q11 [10]}}