| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2021 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Parametric differentiation |
| Type | Find gradient at given parameter |
| Difficulty | Standard +0.3 This is a straightforward parametric differentiation question requiring standard techniques: solving a logarithm equation using log laws in part (a), then applying the chain rule dy/dx = (dy/dt)/(dx/dt) with product rule for dy/dt in part (b). All steps are routine for P2 level with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.03g Parametric equations: of curves and conversion to cartesian1.07s Parametric and implicit differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Equate \(x\) to \(\ln4\) and use relevant logarithm property | M1 | |
| Obtain equation with no logarithm present, \(\dfrac{2t+6}{t} = 4\) | A1 | OE |
| Obtain \(t = 3\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Obtain \(\dfrac{dx}{dt} = \dfrac{2}{2t+6} - \dfrac{1}{t}\) | B1 | |
| Use product rule to find \(\dfrac{dy}{dt}\) | M1 | |
| Obtain \(\ln t + t \times \dfrac{1}{t}\) | A1 | |
| Divide to obtain \(\dfrac{dy}{dx}\) using *their* \(\dfrac{dy}{dt}\) and \(\dfrac{dx}{dt}\) correctly | DM1 | Must have at least B1 or M1; Do not condone incorrect inverting of terms unless a correct statement is seen initially |
| Obtain \(-6(\ln3 + 1)\) | A1 | or exact equivalents |
## Question 4(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Equate $x$ to $\ln4$ and use relevant logarithm property | M1 | |
| Obtain equation with no logarithm present, $\dfrac{2t+6}{t} = 4$ | A1 | OE |
| Obtain $t = 3$ | A1 | |
---
## Question 4(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Obtain $\dfrac{dx}{dt} = \dfrac{2}{2t+6} - \dfrac{1}{t}$ | B1 | |
| Use product rule to find $\dfrac{dy}{dt}$ | M1 | |
| Obtain $\ln t + t \times \dfrac{1}{t}$ | A1 | |
| Divide to obtain $\dfrac{dy}{dx}$ using *their* $\dfrac{dy}{dt}$ and $\dfrac{dx}{dt}$ correctly | DM1 | Must have at least B1 or M1; Do not condone incorrect inverting of terms unless a correct statement is seen initially |
| Obtain $-6(\ln3 + 1)$ | A1 | or exact equivalents |
---4 A curve has parametric equations
$$x = \ln ( 2 t + 6 ) - \ln t , \quad y = t \ln t$$
\begin{enumerate}[label=(\alph*)]
\item Find the value of $t$ at the point $P$ on the curve for which $x = \ln 4$.
\item Find the exact gradient of the curve at $P$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P2 2021 Q4 [8]}}