| Exam Board | Edexcel |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2021 |
| Session | October |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Partial Fractions |
| Type | Find inverse function after simplification |
| Difficulty | Moderate -0.3 This is a straightforward multi-part question requiring routine algebraic manipulation. Part (a) involves factorizing and combining fractions over a common denominator—standard technique. Part (b) is a basic inverse function calculation. Part (c) uses the quotient rule and sign analysis. All steps are textbook exercises with no novel insight required, making it slightly easier than average. |
| Spec | 1.02k Simplify rational expressions: factorising, cancelling, algebraic division1.02u Functions: definition and vocabulary (domain, range, mapping)1.02v Inverse and composite functions: graphs and conditions for existence1.07i Differentiate x^n: for rational n and sums |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\frac{5x}{x^2+7x+12}+\frac{5x}{x+4}=\frac{5x+5x(x+3)}{(x+3)(x+4)}\) | M1 | Attempts to combine two fractions using common denominator; allow errors on numerator but at least one term must have been adapted; condone invisible brackets |
| \(=\frac{5x^2+20x}{(x+3)(x+4)}=\frac{5x(x+4)}{(x+3)(x+4)}\) | A1 | Correct unsimplified fraction with quadratic numerator and denominator |
| \(=\frac{5x}{x+3}\) | A1* | Correctly achieves given answer showing intermediate steps (cso); must see fractions combined then numerator and denominator factorised before cancelling |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(y=\frac{5x}{(x+3)} \Rightarrow xy+3y=5x \Rightarrow 5x-xy=3y\) | M1 | Attempts to change subject on \(y=\frac{5x}{x+3}\); look for cross multiplication with attempt to collect terms; do not follow through on part (a) |
| \(x=\frac{3y}{5-y}\), so \(f^{-1}(x)=\frac{3x}{5-x}\) | A1 | Must be in terms of \(x\); condone \(f^{-1}=\ldots\) or \(f^{-1}=y=\ldots\); do not allow just \(y=\ldots\) |
| Domain \(0 < x < 5\) | A1 | Correct domain |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(f(x)=\frac{5x}{(x+3)} \Rightarrow f'(x)=\frac{5(x+3)-5x}{(x+3)^2}=\frac{15}{(x+3)^2}\) | M1 A1 | M1: attempts quotient or product rule, look for form \(\frac{A(x+3)-5x}{(x+3)^2}\); A1: correct derivative \(\frac{15}{(x+3)^2}\) or \(15(x+3)^{-2}\); do not allow \(\frac{5}{x+3}-\frac{5x}{(x+3)^2}\) |
| States f is an increasing function since \((x+3)^2\) is positive | A1 | Must achieve \(f'(x)=\frac{15}{(x+3)^2}\) AND state increasing function AND reason that \((x+3)^2\) is positive |
# Question 1:
## Part (a)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{5x}{x^2+7x+12}+\frac{5x}{x+4}=\frac{5x+5x(x+3)}{(x+3)(x+4)}$ | M1 | Attempts to combine two fractions using common denominator; allow errors on numerator but at least one term must have been adapted; condone invisible brackets |
| $=\frac{5x^2+20x}{(x+3)(x+4)}=\frac{5x(x+4)}{(x+3)(x+4)}$ | A1 | Correct unsimplified fraction with quadratic numerator and denominator |
| $=\frac{5x}{x+3}$ | A1* | Correctly achieves given answer showing intermediate steps (cso); must see fractions combined then numerator and denominator factorised before cancelling |
## Part (b)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $y=\frac{5x}{(x+3)} \Rightarrow xy+3y=5x \Rightarrow 5x-xy=3y$ | M1 | Attempts to change subject on $y=\frac{5x}{x+3}$; look for cross multiplication with attempt to collect terms; do not follow through on part (a) |
| $x=\frac{3y}{5-y}$, so $f^{-1}(x)=\frac{3x}{5-x}$ | A1 | Must be in terms of $x$; condone $f^{-1}=\ldots$ or $f^{-1}=y=\ldots$; do not allow just $y=\ldots$ |
| Domain $0 < x < 5$ | A1 | Correct domain |
## Part (c)(i) and (ii)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $f(x)=\frac{5x}{(x+3)} \Rightarrow f'(x)=\frac{5(x+3)-5x}{(x+3)^2}=\frac{15}{(x+3)^2}$ | M1 A1 | M1: attempts quotient or product rule, look for form $\frac{A(x+3)-5x}{(x+3)^2}$; A1: correct derivative $\frac{15}{(x+3)^2}$ or $15(x+3)^{-2}$; do not allow $\frac{5}{x+3}-\frac{5x}{(x+3)^2}$ |
| States f is an increasing function since $(x+3)^2$ is positive | A1 | Must achieve $f'(x)=\frac{15}{(x+3)^2}$ AND state increasing function AND reason that $(x+3)^2$ is positive |
---\begin{enumerate}
\item The function f is defined by
\end{enumerate}
$$\mathrm { f } ( x ) = \frac { 5 x } { x ^ { 2 } + 7 x + 12 } + \frac { 5 x } { x + 4 } \quad x > 0$$
(a) Show that $\mathrm { f } ( x ) = \frac { 5 x } { x + 3 }$\\
(b) Find $\mathrm { f } ^ { - 1 }$\\
(c) (i) Find, in simplest form, $\mathrm { f } ^ { \prime } ( x )$.\\
(ii) Hence, state whether f is an increasing or a decreasing function, giving a reason for your answer.\\
(3)
\hfill \mbox{\textit{Edexcel P3 2021 Q1 [9]}}