| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2005 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Differentiating Transcendental Functions |
| Type | Find stationary points - logarithmic functions |
| Difficulty | Moderate -0.3 Part (a) is a straightforward application of the product rule to find dy/dx = ln x + 1, then solving ln x + 1 = 0 for the stationary point. Part (b) requires quotient rule differentiation and showing the derivative is never zero, which is routine algebraic manipulation. Both parts are standard textbook exercises requiring only direct application of differentiation rules with minimal problem-solving. |
| Spec | 1.07n Stationary points: find maxima, minima using derivatives1.07q Product and quotient rules: differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| (a) Attempt use of product rule | *M1 | |
| Obtain \(\ln x + 1\) | A1 | [or unsimplified equiv] |
| Equate attempt at first derivative to zero and obtain value involving \(e\) | M1 | [dependent on *M] |
| Obtain \(e^{-1}\) | A1 | Total: 4 marks [or exact equiv] |
| (b) Attempt use of quotient rule | M1 | [or equiv using product rule or …] |
| Obtain \(\frac{(4x - c)4 - 4(4x + c)}{(4x - c)^2}\) | A1 | [or equiv] |
| Show that first derivative cannot be zero | A1 | Total: 3 marks [AG; derivative must be correct] |
**(a)** Attempt use of product rule | *M1 |
Obtain $\ln x + 1$ | A1 | [or unsimplified equiv]
Equate attempt at first derivative to zero and obtain value involving $e$ | M1 | [dependent on *M]
Obtain $e^{-1}$ | A1 | Total: 4 marks [or exact equiv]
**(b)** Attempt use of quotient rule | M1 | [or equiv using product rule or …]
Obtain $\frac{(4x - c)4 - 4(4x + c)}{(4x - c)^2}$ | A1 | [or equiv]
Show that first derivative cannot be zero | A1 | Total: 3 marks [AG; derivative must be correct]6
\begin{enumerate}[label=(\alph*)]
\item Find the exact value of the $x$-coordinate of the stationary point of the curve $y = x \ln x$.
\item The equation of a curve is $y = \frac { 4 x + c } { 4 x - c }$, where $c$ is a non-zero constant. Show by differentiation that this curve has no stationary points.
\end{enumerate}
\hfill \mbox{\textit{OCR C3 2005 Q6 [7]}}