| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2012 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Connected Rates of Change |
| Type | Curve motion: find dy/dt |
| Difficulty | Moderate -0.8 This is a straightforward connected rates of change question requiring basic differentiation of power functions and direct application of the chain rule formula dy/dt = (dy/dx)(dx/dt). Part (i) is routine differentiation, and part (ii) involves simple substitution with no conceptual challenges—easier than average A-level questions. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates |
| Answer | Marks | Guidance |
|---|---|---|
| \(y = 4\sqrt{x} + \frac{2}{\sqrt{x}}\) | ||
| (i) \(\frac{dy}{dx} = 4 \cdot \frac{1}{2}x^{-\frac{1}{2}} + 2(-\frac{1}{2})x^{-1.5}\) | M1 | Reducing "their" power by 1 once. Allow unsimplified |
| or \(y = \frac{2}{\sqrt{x}} - \frac{1}{x^{1.5}}\) | A1 A1 | [3] |
| (ii) \(\frac{dv}{dt} = \frac{dy}{dx} \times \frac{dx}{dt}\) used | M1 | Must be used correctly |
| \(\to \frac{7}{8} \times 0.12 = 0.105\) | A1 | co – fraction or decimal. [2] |
$y = 4\sqrt{x} + \frac{2}{\sqrt{x}}$ | |
(i) $\frac{dy}{dx} = 4 \cdot \frac{1}{2}x^{-\frac{1}{2}} + 2(-\frac{1}{2})x^{-1.5}$ | M1 | Reducing "their" power by 1 once. Allow unsimplified
or $y = \frac{2}{\sqrt{x}} - \frac{1}{x^{1.5}}$ | A1 A1 | [3]
(ii) $\frac{dv}{dt} = \frac{dy}{dx} \times \frac{dx}{dt}$ used | M1 | Must be used correctly
$\to \frac{7}{8} \times 0.12 = 0.105$ | A1 | co – fraction or decimal. [2]The equation of a curve is $y = 4\sqrt{x} + \frac{2}{\sqrt{x}}$.
\begin{enumerate}[label=(\roman*)]
\item Obtain an expression for $\frac{dy}{dx}$. [3]
\item A point is moving along the curve in such a way that the $x$-coordinate is increasing at a constant rate of $0.12$ units per second. Find the rate of change of the $y$-coordinate when $x = 4$. [2]
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2012 Q2 [5]}}