| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2014 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Connected Rates of Change |
| Type | Curve motion: find x-coordinate |
| Difficulty | Standard +0.3 This is a straightforward connected rates of change question requiring the chain rule (dy/dt = dy/dx × dx/dt). Part (i) is routine differentiation using the chain rule or quotient rule. Part (ii) involves substituting given rates and solving a quadratic equation. While it requires multiple steps, the techniques are standard P1 material with no novel insight needed, making it slightly easier than average. |
| Spec | 1.07b Gradient as rate of change: dy/dx notation1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Differential \(= -12(3-2x)^{-2} \times -2\) | B1, B1 [2] | co co (even if 1st B mark lost) |
| Answer | Marks | Guidance |
|---|---|---|
| \(\rightarrow x = 0\) or \(3\) | M1, M1, A1, A1 [4] | Chain rule used correctly (AEF); Equates their \(\frac{dy}{dx}\) with their \(\frac{8}{3}\) or \(\frac{3}{8}\); co co |
$y = \frac{12}{3-2x}$
**(i)** Differential $= -12(3-2x)^{-2} \times -2$ | B1, B1 [2] | co co (even if 1st B mark lost)
**(ii)** $\frac{dy}{dx} = \frac{dy}{dr} \cdot \frac{dx}{dt} = 0.4 \div 0.15$
$\rightarrow \frac{24}{(3-2x)^2} = \frac{8}{3}$
$\rightarrow x = 0$ or $3$ | M1, M1, A1, A1 [4] | Chain rule used correctly (AEF); Equates their $\frac{dy}{dx}$ with their $\frac{8}{3}$ or $\frac{3}{8}$; co coA curve has equation $y = \frac{12}{5 - 2x}$.
\begin{enumerate}[label=(\roman*)]
\item Find $\frac{dy}{dx}$. [2]
\end{enumerate}
A point moves along this curve. As the point passes through $A$, the $x$-coordinate is increasing at a rate of 0.15 units per second and the $y$-coordinate is increasing at a rate of 0.4 units per second.
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{1}
\item Find the possible $x$-coordinates of $A$. [4]
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2014 Q4 [6]}}