| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2022 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Complete the square |
| Difficulty | Moderate -0.8 This is a straightforward multi-part question testing standard techniques: completing the square (routine algebraic manipulation), finding range from completed square form (direct reading), and finding inverse of a quadratic with restricted domain (standard procedure). All parts are textbook exercises requiring no problem-solving insight, making it easier than average. |
| Spec | 1.02e Complete the square: quadratic polynomials and turning points1.02v Inverse and composite functions: graphs and conditions for existence |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \([f(x)] = \{-2(x+2)^2\} - \{5\}\) | B1 B1 | |
| Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \([f(x)] < -7\) | B1 | Allow \(y < -7\), \(<-7\), \((-\infty, -7)\) or less than \(-7\); \(-\infty\langle f(x)\langle -7,-7\rangle f(x)\rangle -\infty\), \(f < -7\) |
| Total: 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(y = -2(x+2)^2 - 5 \rightarrow (x+2)^2 = \dfrac{-(y+5)}{2}\) | M1 | Operations correct. Allow sign errors. FT *their* quadratic from (a) |
| \(x = [\pm]\sqrt{\dfrac{-(y+5)}{2}} - 2\) | M1 | Operations correct. Allow sign errors. FT *their* quadratic from (a) |
| \([\text{f}^{-1}(x)] = -2 - \sqrt{\dfrac{-(x+5)}{2}}\) or \(-2 - \sqrt{\dfrac{(x+5)}{-2}}\) | A1 | Allow \([\text{f}^{-1}(x)] = -2 - \sqrt{\dfrac{x+5}{-2}}\) |
| Total: 3 |
## Question 2(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $[f(x)] = \{-2(x+2)^2\} - \{5\}$ | B1 B1 | |
| **Total: 2** | | |
## Question 2(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $[f(x)] < -7$ | B1 | Allow $y < -7$, $<-7$, $(-\infty, -7)$ or less than $-7$; $-\infty\langle f(x)\langle -7,-7\rangle f(x)\rangle -\infty$, $f < -7$ |
| **Total: 1** | | |
## Question 2(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $y = -2(x+2)^2 - 5 \rightarrow (x+2)^2 = \dfrac{-(y+5)}{2}$ | M1 | Operations correct. Allow sign errors. FT *their* quadratic from **(a)** |
| $x = [\pm]\sqrt{\dfrac{-(y+5)}{2}} - 2$ | M1 | Operations correct. Allow sign errors. FT *their* quadratic from **(a)** |
| $[\text{f}^{-1}(x)] = -2 - \sqrt{\dfrac{-(x+5)}{2}}$ or $-2 - \sqrt{\dfrac{(x+5)}{-2}}$ | A1 | Allow $[\text{f}^{-1}(x)] = -2 - \sqrt{\dfrac{x+5}{-2}}$ |
| **Total: 3** | | |2 The function f is defined by $\mathrm { f } ( x ) = - 2 x ^ { 2 } - 8 x - 13$ for $x < - 3$.
\begin{enumerate}[label=(\alph*)]
\item Express $\mathrm { f } ( x )$ in the form $- 2 ( x + a ) ^ { 2 } + b$, where $a$ and $b$ are integers.
\item Find the range of f.
\item Find an expression for $\mathrm { f } ^ { - 1 } ( x )$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2022 Q2 [6]}}