OCR MEI C3 — Question 8 7 marks

Exam BoardOCR MEI
ModuleC3 (Core Mathematics 3)
Marks7
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicProduct & Quotient Rules
TypeFind stationary points coordinates
DifficultyStandard +0.8 This question requires applying the quotient rule to a function involving logarithms, then solving the resulting equation dy/dx = 0 to find stationary points. While the quotient rule application is standard C3 content, the algebraic manipulation to solve for the stationary point (particularly dealing with the ln x term) requires careful work and is more demanding than routine differentiation exercises. The 7-mark allocation reflects this multi-step nature, placing it moderately above average difficulty.
Spec1.07l Derivative of ln(x): and related functions1.07n Stationary points: find maxima, minima using derivatives1.07q Product and quotient rules: differentiation
A curve has equation \(y = \frac{x}{2 + 3\ln x}\). Find \(\frac{dy}{dx}\). Hence find the exact coordinates of the stationary point of the curve. [7]
AnswerMarks Guidance
\(y = \frac{x}{2 + 3\ln x}\)M1 Quotient rule consistent with their derivatives or product rule + chain rule on \((2+3x)^{-1}\)
\(\frac{dy}{dx} = \frac{(2+3\ln x) \cdot 1 - x \cdot \frac{3}{x}}{(2+3\ln x)^2}\)B1 \(\frac{d}{dx}(\ln x) = \frac{1}{x}\) soi correct expression
\(= \frac{2 + 3\ln x - 3}{(2+3\ln x)^2}\)A1
\(= \frac{3\ln x - 1}{(2+3\ln x)^2}\)M1 their numerator = 0 (or equivalent step from product rule formulation) M0 if denominator = 0 is pursued
When \(\frac{dy}{dx} = 0\), \(3\ln x - 1 = 0\)A1cao \(x = e^{1/3}\)
\(\Rightarrow \ln x = 1/3\)M1 A1cao substituting for their \(x\) (correctly) Must be exact: \(-0.46\ldots\) is M1A0
\(\Rightarrow x = e^{1/3}\)[7]
\(\Rightarrow y = \frac{e^{1/3}}{2+1} = \frac{1}{3}e^{1/3}\) 
$y = \frac{x}{2 + 3\ln x}$ | M1 | Quotient rule consistent with their derivatives or product rule + chain rule on $(2+3x)^{-1}$
$\frac{dy}{dx} = \frac{(2+3\ln x) \cdot 1 - x \cdot \frac{3}{x}}{(2+3\ln x)^2}$ | B1 | $\frac{d}{dx}(\ln x) = \frac{1}{x}$ soi correct expression
$= \frac{2 + 3\ln x - 3}{(2+3\ln x)^2}$ | A1 |
$= \frac{3\ln x - 1}{(2+3\ln x)^2}$ | M1 | their numerator = 0 (or equivalent step from product rule formulation) M0 if denominator = 0 is pursued
When $\frac{dy}{dx} = 0$, $3\ln x - 1 = 0$ | A1cao | $x = e^{1/3}$
$\Rightarrow \ln x = 1/3$ | M1 A1cao | substituting for their $x$ (correctly) Must be exact: $-0.46\ldots$ is M1A0
$\Rightarrow x = e^{1/3}$ | [7] |
$\Rightarrow y = \frac{e^{1/3}}{2+1} = \frac{1}{3}e^{1/3}$ |
A curve has equation $y = \frac{x}{2 + 3\ln x}$. Find $\frac{dy}{dx}$. Hence find the exact coordinates of the stationary point of the curve. [7]

\hfill \mbox{\textit{OCR MEI C3  Q8 [7]}}
This paper (8 questions)
View full paper
Q1 18 Q2 4 Q3 6 Q4 8 Q5 4 Q6 18 Q7 7 Q8 7