| Exam Board | Edexcel |
|---|---|
| Module | C34 (Core Mathematics 3 & 4) |
| Year | 2018 |
| Session | January |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Implicit equations and differentiation |
| Type | Find dy/dx at a point |
| Difficulty | Standard +0.3 This is a straightforward implicit differentiation question requiring application of standard rules (exponential, product rule) and algebraic manipulation to solve for dy/dx, then substitution of given coordinates. While it involves multiple techniques, it follows a predictable pattern with no novel insight required, making it slightly easier than average. |
| Spec | 1.07s Parametric and implicit differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance Notes |
| Differentiates wrt \(x\): \(3^x \ln 3 + x\frac{dy}{dx} + y = 1 + 2y\frac{dy}{dx}\) | B1, B1, M1, A1 | B1: \(3^x \rightarrow 3^x \ln 3\) or \(e^{x\ln 3} \rightarrow e^{x\ln 3}\ln 3\); B1: Correct product rule on \(xy\) finding \(x\frac{dy}{dx} + y\); M1: Implicit differentiation \(y^2 \rightarrow ky\frac{dy}{dx}\); A1: All terms correct other than \(3^x\) term |
| Substitutes \((4, 11)\) AND rearranges to get \(\frac{dy}{dx} = \ldots\) Nb \(\frac{dy}{dx} = \frac{3^x \ln 3 + y - 1}{2y - x}\) | M1 | Must substitute both \(x=4\), \(y=11\) into expression with two \(\frac{dy}{dx}\) terms and find numerical value |
| \(\Rightarrow 81\ln 3 + 4\frac{dy}{dx} + 11 = 1 + 22\frac{dy}{dx} \Rightarrow \frac{dy}{dx} = \frac{81\ln 3 + 10}{18} = \frac{5}{9} + \frac{9}{2}\ln 3\) | A1 | Exact answer only; accept equivalents e.g. \(\frac{10}{18} + \frac{81}{18}\ln 3\), \(\frac{5}{9} + 4.5\ln 3\) |
# Question 1:
| Answer/Working | Marks | Guidance Notes |
|---|---|---|
| Differentiates wrt $x$: $3^x \ln 3 + x\frac{dy}{dx} + y = 1 + 2y\frac{dy}{dx}$ | B1, B1, M1, A1 | B1: $3^x \rightarrow 3^x \ln 3$ or $e^{x\ln 3} \rightarrow e^{x\ln 3}\ln 3$; B1: Correct product rule on $xy$ finding $x\frac{dy}{dx} + y$; M1: Implicit differentiation $y^2 \rightarrow ky\frac{dy}{dx}$; A1: All terms correct other than $3^x$ term |
| Substitutes $(4, 11)$ AND rearranges to get $\frac{dy}{dx} = \ldots$ Nb $\frac{dy}{dx} = \frac{3^x \ln 3 + y - 1}{2y - x}$ | M1 | Must substitute both $x=4$, $y=11$ into expression with two $\frac{dy}{dx}$ terms and find numerical value |
| $\Rightarrow 81\ln 3 + 4\frac{dy}{dx} + 11 = 1 + 22\frac{dy}{dx} \Rightarrow \frac{dy}{dx} = \frac{81\ln 3 + 10}{18} = \frac{5}{9} + \frac{9}{2}\ln 3$ | A1 | Exact answer only; accept equivalents e.g. $\frac{10}{18} + \frac{81}{18}\ln 3$, $\frac{5}{9} + 4.5\ln 3$ |
---\begin{enumerate}
\item A curve $C$ has equation
\end{enumerate}
$$3 ^ { x } + x y = x + y ^ { 2 } , \quad y > 1$$
The point $P$ with coordinates $( 4,11 )$ lies on $C$.\\
Find the exact value of $\frac { \mathrm { d } y } { \mathrm {~d} x }$ at the point $P$.
Give your answer in the form $a + b \ln 3$, where $a$ and $b$ are rational numbers.\\
\hfill \mbox{\textit{Edexcel C34 2018 Q1 [6]}}