| Exam Board | Edexcel |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2022 |
| Session | October |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Composite & Inverse Functions |
| Type | Make x the subject |
| Difficulty | Moderate -0.8 Part (a) is a straightforward rearrangement using basic logarithm laws (converting log form to exponential form), requiring only 2 marks. Part (b) applies the standard inverse function derivative formula dx/dy = 1/(dy/dx), which is a direct application of a known result. Both parts are routine procedural questions with no problem-solving or novel insight required, making this easier than average. |
| Spec | 1.06c Logarithm definition: log_a(x) as inverse of a^x1.06d Natural logarithm: ln(x) function and properties1.07l Derivative of ln(x): and related functions |
| \(\begin{array} { c } \text { Leave } | |
| \text { blank } \end{array}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(y=\log_{10}(2x+1)\Rightarrow 10^y=2x+1 \Rightarrow x=...\) | M1 | Correctly undoes logarithm and rearranges to make \(x\) subject |
| \(x=\frac{10^y-1}{2}\) | A1 | Correct expression for \(x\); isw after correct answer |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\frac{dx}{dy}=\frac{1}{2}10^y\ln10\) | M1 | Differentiates \(10^y\); accept \(\alpha10^y\to\beta10^y\ln10\) |
| \(\frac{dy}{dx}=1\Big/\frac{dx}{dy}=\frac{1}{\frac{1}{2}10^y\ln10}\) | M1 | Applies reciprocal \(\frac{dy}{dx}=1\big/\frac{dx}{dy}\); variables must be consistent |
| \(\frac{dy}{dx}=\frac{2}{(2x+1)\ln10}\) | A1 | Correct answer; accept equivalents in terms of \(x\) e.g. with \(10^{\log(2x+1)}\) isw |
# Question 4:
## Part (a)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $y=\log_{10}(2x+1)\Rightarrow 10^y=2x+1 \Rightarrow x=...$ | M1 | Correctly undoes logarithm and rearranges to make $x$ subject |
| $x=\frac{10^y-1}{2}$ | A1 | Correct expression for $x$; isw after correct answer |
## Part (b)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{dx}{dy}=\frac{1}{2}10^y\ln10$ | M1 | Differentiates $10^y$; accept $\alpha10^y\to\beta10^y\ln10$ |
| $\frac{dy}{dx}=1\Big/\frac{dx}{dy}=\frac{1}{\frac{1}{2}10^y\ln10}$ | M1 | Applies reciprocal $\frac{dy}{dx}=1\big/\frac{dx}{dy}$; variables must be consistent |
| $\frac{dy}{dx}=\frac{2}{(2x+1)\ln10}$ | A1 | Correct answer; accept equivalents in terms of $x$ e.g. with $10^{\log(2x+1)}$ isw |
---4.
$$y = \log _ { 10 } ( 2 x + 1 )$$
\begin{enumerate}[label=(\alph*)]
\item Express $x$ in terms of $y$.
\item Hence, giving your answer in terms of $x$, find $\frac { \mathrm { d } y } { \mathrm {~d} x }$\\
4.\\
(a) Express $x$ in terms of $y$.
\begin{center}
\begin{tabular}{ l | c }
& $\begin{array} { c } \text { Leave } \\ \text { blank } \end{array}$ \\
\end{tabular}
\end{center} (2)
(b) Hence, giving your answer in terms of $x$, find $\frac { \mathrm { d } y } { \mathrm {~d} x }$\\
\includegraphics[max width=\textwidth, alt={}, center]{83e12fa4-1abb-4bea-bff4-8d36757bd9c3-10_2662_111_107_1950}
\end{enumerate}
\hfill \mbox{\textit{Edexcel P3 2022 Q4 [5]}}