| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2017 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Differentiating Transcendental Functions |
| Type | Find gradient at a point - at special curve features |
| Difficulty | Standard +0.3 This is a straightforward differentiation question using the product rule and chain rule on a logarithmic function. Part (i) requires finding where y=0, then evaluating dy/dx at that point. Part (ii) requires setting dy/dx=0 and solving. Both are standard techniques with no novel insight required, making it slightly easier than average. |
| Spec | 1.07l Derivative of ln(x): and related functions1.07n Stationary points: find maxima, minima using derivatives1.07q Product and quotient rules: differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Apply product rule to find first derivative | \*M1 | |
| Obtain \(6x\ln\left(\frac{1}{6}x\right) + 3x\) or equivalent | A1 | Allow unsimplified for A1 |
| Identify \(x = 6\) at \(P\) | B1 | |
| Substitute their value of \(x\) at \(P\) into attempt at first derivative | DM1 | dep \*M |
| Obtain \(18\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Equate their first derivative to zero and attempt solution of equation of form \(k\ln\left(\frac{1}{6}x\right) + m = 0\) | \*M1 | |
| Obtain \(x\)-coordinate of form \(a_1e^{a_2}\) | DM1 | dep \*M |
| Obtain \(x = 6e^{-\frac{1}{2}}\) or exact equivalent | A1 | |
| Substitute exact \(x\)-value in the form \(a_1e^{a_2}\) and attempt simplification to remove ln | M1 | |
| Obtain \(-54e^{-1}\) or exact equivalent | A1 |
## Question 8(i):
| Answer | Mark | Guidance |
|--------|------|----------|
| Apply product rule to find first derivative | \*M1 | |
| Obtain $6x\ln\left(\frac{1}{6}x\right) + 3x$ or equivalent | A1 | Allow unsimplified for A1 |
| Identify $x = 6$ at $P$ | B1 | |
| Substitute their value of $x$ at $P$ into attempt at first derivative | DM1 | dep \*M |
| Obtain $18$ | A1 | |
**Total: 5**
---
## Question 8(ii):
| Answer | Mark | Guidance |
|--------|------|----------|
| Equate their first derivative to zero and attempt solution of equation of form $k\ln\left(\frac{1}{6}x\right) + m = 0$ | \*M1 | |
| Obtain $x$-coordinate of form $a_1e^{a_2}$ | DM1 | dep \*M |
| Obtain $x = 6e^{-\frac{1}{2}}$ or exact equivalent | A1 | |
| Substitute exact $x$-value in the form $a_1e^{a_2}$ and attempt simplification to remove ln | M1 | |
| Obtain $-54e^{-1}$ or exact equivalent | A1 | |
**Total: 5**8\\
\includegraphics[max width=\textwidth, alt={}, center]{6295873e-7db4-4e7e-8dcd-912ad9c41675-10_643_414_260_863}
The diagram shows the curve with equation
$$y = 3 x ^ { 2 } \ln \left( \frac { 1 } { 6 } x \right) .$$
The curve crosses the $x$-axis at the point $P$ and has a minimum point $M$.\\
(i) Find the gradient of the curve at the point $P$.\\
(ii) Find the exact coordinates of the point $M$.\\
\hfill \mbox{\textit{CAIE P2 2017 Q8 [10]}}