| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2024 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Find intersection points |
| Difficulty | Challenging +1.2 This question requires completing the square (routine), finding where g^{-1}f(x) = g(x), which leads to a quadratic equation that must have a repeated root (discriminant = 0). While it involves composition and the condition for a single intersection point, these are standard A-level techniques with clear signposting across multiple parts. |
| Spec | 1.02e Complete the square: quadratic polynomials and turning points1.02u Functions: definition and vocabulary (domain, range, mapping)1.02v Inverse and composite functions: graphs and conditions for existence |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Express \(f(x)\) as \(a-(x-3)^2\) or \(a-(3-x)^2\) where \(a = \pm 19\) or \(\pm 1\) | M1 | OE. If the form \(-f(x) = (x^2-6x-10)\) is used the form must be returned to \(f(x) = \ldots\) Completed square form must give \(-x^2\). Answers must come from completion of the square (not calculus or graphs) |
| \(19-(3-x)^2\) or \(19-(x-3)^2\) | A1 | OE |
| \(f(x) \leqslant 19\) or \(y \leqslant 19\) with \(\leqslant\), not \(<\); or \(-\infty < f(x) \leqslant 19\) or \(-\infty \leqslant f(x) \leqslant 19\); or \((-\infty, 19]\) or \([-\infty, 19]\) | A1FT | Using *their* constant following the award of M1. SC B1 answer only or answer from a method not involving completion of the square |
| Total | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(g^{-1}(x) = \frac{1}{4}(x-k)\) | B1 | |
| \(g^{-1}f(x) = \frac{1}{4}(10 + 6x - x^2 - k) = 4x + k\) | M1 | OE. May use *their* completed square form for \(f(x)\). |
| Simplify the quadratic equation obtained from \(g^{-1}f(x) = g(x)\) provided \(k\) is present and apply \(b^2 - 4ac = 0\) to this quadratic equation | \*M1 | Expect \(x^2 + 10x - 10 + 5k = 0\) |
| Obtain \(100 - 4(5k-10) = 0\) and hence \(k = 7\) | A1 | |
| Use *their* \(k\) to form and solve a quadratic in \(x\) | DM1 | Allow if *their* quadratic has two solutions. |
| \((-5, -13)\) only | A1 | SC B1 if no method seen. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State \(f(x) = gg(x)\) | (B1) | |
| \(gg(x) = 16x + 5k\) | (M1) | |
| Apply \(b^2 - 4ac = 0\) to quadratic equation obtained from \(f(x) = gg(x)\) | (\*M1) | Provided \(k\) is present. |
| \(100 - 4(5k-10) = 0\) and hence \(k = 7\) | (A1) | |
| 6 |
## Question 11(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Express $f(x)$ as $a-(x-3)^2$ or $a-(3-x)^2$ where $a = \pm 19$ or $\pm 1$ | M1 | OE. If the form $-f(x) = (x^2-6x-10)$ is used the form must be returned to $f(x) = \ldots$ Completed square form must give $-x^2$. Answers must come from completion of the square (not calculus or graphs) |
| $19-(3-x)^2$ or $19-(x-3)^2$ | A1 | OE |
| $f(x) \leqslant 19$ or $y \leqslant 19$ with $\leqslant$, not $<$; or $-\infty < f(x) \leqslant 19$ or $-\infty \leqslant f(x) \leqslant 19$; or $(-\infty, 19]$ or $[-\infty, 19]$ | A1FT | Using *their* constant following the award of M1. **SC B1** answer only or answer from a method not involving completion of the square |
| **Total** | **3** | |
## Question 11(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| $g^{-1}(x) = \frac{1}{4}(x-k)$ | **B1** | |
| $g^{-1}f(x) = \frac{1}{4}(10 + 6x - x^2 - k) = 4x + k$ | **M1** | OE. May use *their* completed square form for $f(x)$. |
| Simplify the quadratic equation obtained from $g^{-1}f(x) = g(x)$ provided $k$ is present and apply $b^2 - 4ac = 0$ to this quadratic equation | **\*M1** | Expect $x^2 + 10x - 10 + 5k = 0$ |
| Obtain $100 - 4(5k-10) = 0$ and hence $k = 7$ | **A1** | |
| Use *their* $k$ to form and solve a quadratic in $x$ | **DM1** | Allow if *their* quadratic has two solutions. |
| $(-5, -13)$ only | **A1** | **SC B1** if no method seen. |
**Alternative Method for first 4 marks:**
| Answer | Mark | Guidance |
|--------|------|----------|
| State $f(x) = gg(x)$ | **(B1)** | |
| $gg(x) = 16x + 5k$ | **(M1)** | |
| Apply $b^2 - 4ac = 0$ to quadratic equation obtained from $f(x) = gg(x)$ | **(\*M1)** | Provided $k$ is present. |
| $100 - 4(5k-10) = 0$ and hence $k = 7$ | **(A1)** | |
| | **6** | |11 The function f is defined by $\mathrm { f } ( x ) = 10 + 6 x - x ^ { 2 }$ for $x \in \mathbb { R }$.
\begin{enumerate}[label=(\alph*)]
\item By completing the square, find the range of f .\\
\includegraphics[max width=\textwidth, alt={}, center]{d6976a4b-aecf-43f1-a3f2-bcad37d03585-16_2715_37_143_2010}
The function g is defined by $\mathrm { g } ( x ) = 4 x + k$ for $x \in \mathbb { R }$ where $k$ is a constant.
\item It is given that the graph of $y = \mathrm { g } ^ { - 1 } \mathrm { f } ( x )$ meets the graph of $y = \mathrm { g } ( x )$ at a single point $P$.
Determine the coordinates of $P$.\\
If you use the following page to complete the answer to any question, the question number must be clearly shown.\\
\includegraphics[max width=\textwidth, alt={}, center]{d6976a4b-aecf-43f1-a3f2-bcad37d03585-18_2715_35_143_2012}
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2024 Q11 [9]}}