| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2022 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Find inverse function |
| Difficulty | Moderate -0.3 This is a straightforward multi-part question on inverse functions requiring standard techniques: differentiation to check monotonicity, algebraic manipulation to find the inverse, and comparing forms. Part (c) adds mild interest by requiring recognition that f = f^(-1) when coefficients match, but overall this is slightly easier than average due to its routine nature and clear structure. |
| Spec | 1.02v Inverse and composite functions: graphs and conditions for existence1.07i Differentiate x^n: for rational n and sums |
| Answer | Marks | Guidance |
|---|---|---|
| \(f'(x) = -3(-1)(4)(4x-p)^{-2} \left[= \frac{12}{(4x-p)^2}\right]\) | B2, 1, 0 | |
| \(> 0\) Hence increasing function | B1FT | Correct conclusion from *their* \(f'(x)\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(y = 2 - \frac{3}{4x-p} \Rightarrow (y-2)(4x-p) = -3\) or \(4xy - py = 8x - 2p - 3\) | M1 | OE Form horizontal equation. Sign errors only, no missing terms. May go directly to \(4y = p - \frac{3}{x-2}\) OE M1 M1 |
| \(4xy - 8x = py - 2p - 3 \Rightarrow 4x(y-2) = p(y-2) - 3\) or \(4x = -\frac{3}{x-2} + p\) | M1 | OE Factorise out \([4]x\) or \([4]y\) |
| \(x = \frac{p(y-2)-3}{4(y-2)} \left[\Rightarrow x = \frac{p}{4} - \frac{3}{4y-8}\right]\) or \(\frac{-\frac{3}{x-2}+p}{4}\) | M1 | OE Make \(x\) (or \(y\)) the subject |
| \(\left[f^{-1}(x) =\right] \frac{p}{4} - \frac{3}{4x-8}\) | A1 | OE in correct form (must be in terms of \(x\)) |
| Answer | Marks |
|---|---|
| \([p =]\ 8\) | B1 |
## Question 8:
### Part 8(a):
| $f'(x) = -3(-1)(4)(4x-p)^{-2} \left[= \frac{12}{(4x-p)^2}\right]$ | B2, 1, 0 | |
|---|---|---|
| $> 0$ Hence increasing function | B1FT | Correct conclusion from *their* $f'(x)$ |
### Part 8(b):
| $y = 2 - \frac{3}{4x-p} \Rightarrow (y-2)(4x-p) = -3$ **or** $4xy - py = 8x - 2p - 3$ | M1 | OE Form horizontal equation. Sign errors only, no missing terms. May go directly to $4y = p - \frac{3}{x-2}$ OE M1 M1 |
|---|---|---|
| $4xy - 8x = py - 2p - 3 \Rightarrow 4x(y-2) = p(y-2) - 3$ **or** $4x = -\frac{3}{x-2} + p$ | M1 | OE Factorise out $[4]x$ or $[4]y$ |
| $x = \frac{p(y-2)-3}{4(y-2)} \left[\Rightarrow x = \frac{p}{4} - \frac{3}{4y-8}\right]$ **or** $\frac{-\frac{3}{x-2}+p}{4}$ | M1 | OE Make $x$ (or $y$) the subject |
| $\left[f^{-1}(x) =\right] \frac{p}{4} - \frac{3}{4x-8}$ | A1 | OE in correct form (must be in terms of $x$) |
### Part 8(c):
| $[p =]\ 8$ | B1 | |
|---|---|---|
---8 The function f is defined by $\mathrm { f } ( x ) = 2 - \frac { 3 } { 4 x - p }$ for $x > \frac { p } { 4 }$, where $p$ is a constant.
\begin{enumerate}[label=(\alph*)]
\item Find $\mathrm { f } ^ { \prime } ( x )$ and hence determine whether f is an increasing function, a decreasing function or neither.
\item Express $\mathrm { f } ^ { - 1 } ( x )$ in the form $\frac { p } { a } - \frac { b } { c x - d }$, where $a , b , c$ and $d$ are integers.
\item Hence state the value of $p$ for which $\mathrm { f } ^ { - 1 } ( x ) \equiv \mathrm { f } ( x )$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2022 Q8 [8]}}