| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2020 |
| Session | March |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Complete the square |
| Difficulty | Standard +0.3 This is a standard multi-part question covering completing the square, finding an inverse function with domain restriction, solving a composite function equation, and understanding domain restrictions for compositions. All parts use routine techniques taught in P1 with no novel insight required, making it slightly easier than average. |
| 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 |
|---|---|---|
| \(\left[2(x+3)^2\right]\left[-7\right]\) | B1B1 | Stating \(a=3, b=-7\) gets B1B1 |
| Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| \(y = 2(x+3)^2-7 \to 2(x+3)^2 = y+7 \to (x+3)^2 = \frac{y+7}{2}\) | M1 | First 2 operations correct. Condone sign error or with \(x/y\) interchange |
| \(x+3 = (\pm)\sqrt{\frac{y+7}{2}} \to x = (\pm)\sqrt{\frac{y+7}{2}}-3 \to f^{-1}(x) = -\sqrt{\frac{x+7}{2}}-3\) | A1FT | FT on *their* \(a\) and \(b\). Allow \(y = \ldots\) |
| Domain: \(x \geqslant -5\) or \(\geqslant -5\) or \([-5,\infty)\) | B1 | Do not accept \(y=\ldots,\, f(x)=\ldots,\, f^{-1}(x)=\ldots\) |
| Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| \(fg(x) = 8x^2-7\) | B1FT | SOI. FT on *their* \(-7\) from part (a) |
| \(8x^2-7 = 193 \to x^2 = 25 \to x = -5\) only | B1 | |
| Alternative method: | ||
| \(g(x) = f^{-1}(193) \to 2x-3 = -\sqrt{100}-3\) | M1 | FT on *their* \(f^{-1}(x)\) |
| \(x = -5\) only | A1 | |
| Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| (Largest \(k\) is) \(-\frac{1}{2}\) | B1 | Accept \(-\frac{1}{2}\) or \(k \leqslant -\frac{1}{2}\) |
| Total: 1 |
## Question 9(a):
| $\left[2(x+3)^2\right]\left[-7\right]$ | B1B1 | Stating $a=3, b=-7$ gets B1B1 |
|---|---|---|
| **Total: 2** | | |
---
## Question 9(b):
| $y = 2(x+3)^2-7 \to 2(x+3)^2 = y+7 \to (x+3)^2 = \frac{y+7}{2}$ | M1 | First 2 operations correct. Condone sign error or with $x/y$ interchange |
|---|---|---|
| $x+3 = (\pm)\sqrt{\frac{y+7}{2}} \to x = (\pm)\sqrt{\frac{y+7}{2}}-3 \to f^{-1}(x) = -\sqrt{\frac{x+7}{2}}-3$ | A1FT | FT on *their* $a$ and $b$. Allow $y = \ldots$ |
| Domain: $x \geqslant -5$ or $\geqslant -5$ or $[-5,\infty)$ | B1 | Do not accept $y=\ldots,\, f(x)=\ldots,\, f^{-1}(x)=\ldots$ |
| **Total: 3** | | |
---
## Question 9(c):
| $fg(x) = 8x^2-7$ | B1FT | SOI. FT on *their* $-7$ from part (a) |
|---|---|---|
| $8x^2-7 = 193 \to x^2 = 25 \to x = -5$ only | B1 | |
| **Alternative method:** | | |
| $g(x) = f^{-1}(193) \to 2x-3 = -\sqrt{100}-3$ | M1 | FT on *their* $f^{-1}(x)$ |
| $x = -5$ only | A1 | |
| **Total: 2** | | |
---
## Question 9(d):
| (Largest $k$ is) $-\frac{1}{2}$ | B1 | Accept $-\frac{1}{2}$ or $k \leqslant -\frac{1}{2}$ |
|---|---|---|
| **Total: 1** | | |
---9
\begin{enumerate}[label=(\alph*)]
\item Express $2 x ^ { 2 } + 12 x + 11$ in the form $2 ( x + a ) ^ { 2 } + b$, where $a$ and $b$ are constants.\\
The function f is defined by $\mathrm { f } ( x ) = 2 x ^ { 2 } + 12 x + 11$ for $x \leqslant - 4$.
\item Find an expression for $\mathrm { f } ^ { - 1 } ( x )$ and state the domain of $\mathrm { f } ^ { - 1 }$.\\
The function g is defined by $\mathrm { g } ( x ) = 2 x - 3$ for $x \leqslant k$.
\item For the case where $k = - 1$, solve the equation $\operatorname { fg } ( x ) = 193$.
\item State the largest value of $k$ possible for the composition fg to be defined.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2020 Q9 [8]}}