| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2012 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Tangents, normals and gradients |
| Type | Related rates problems |
| Difficulty | Standard +0.3 This question involves standard integration twice to find the curve equation using given conditions, followed by a straightforward related rates problem using the chain rule. Part (i) requires integrating -4x twice and applying the maximum point conditions (gradient = 0 at x=2, point lies on curve), which is routine calculus. Part (ii) is a basic application of dy/dt = (dy/dx)(dx/dt). While it requires multiple techniques, all are standard A-level procedures with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates1.08b Integrate x^n: where n != -1 and sums |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(\frac{dy}{dx} = -2x^2 + c\); \(\frac{dy}{dx} = 0\) when \(x = 2, \to c = 8\); \(y = -\frac{2x^3}{3} + 8x\) (+C); Subs (2, 12) \(\to C = \frac{4}{3}\) | B1, B1, B1, B1, M1, A1 | For \(-2x^2\); \(c = 8\); For each term – ✓ on "c" – ignore (+C); Uses (2, 12) to find C |
| (iii) \(\frac{dy}{dt} = \frac{dy}{dx} \times \frac{dx}{dt} = -10 \times 0.05 \to\) decreasing at 0.5 units per second | M1, A1 | Must use; Enough to see product of gradient and rate; bod over notation |
$\frac{d^2y}{dx^2} = -4x$
**(i)** $\frac{dy}{dx} = -2x^2 + c$; $\frac{dy}{dx} = 0$ when $x = 2, \to c = 8$; $y = -\frac{2x^3}{3} + 8x$ (+C); Subs (2, 12) $\to C = \frac{4}{3}$ | B1, B1, B1, B1, M1, A1 | For $-2x^2$; $c = 8$; For each term – ✓ on "c" – ignore (+C); Uses (2, 12) to find C
**(iii)** $\frac{dy}{dt} = \frac{dy}{dx} \times \frac{dx}{dt} = -10 \times 0.05 \to$ decreasing at 0.5 units per second | M1, A1 | Must use; Enough to see product of gradient and rate; bod over notationA curve is such that $\frac{d^2y}{dx^2} = -4x$. The curve has a maximum point at $(2, 12)$.
\begin{enumerate}[label=(\roman*)]
\item Find the equation of the curve. [6]
\end{enumerate}
A point $P$ moves along the curve in such a way that the $x$-coordinate is increasing at 0.05 units per second.
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{1}
\item Find the rate at which the $y$-coordinate is changing when $x = 3$, stating whether the $y$-coordinate is increasing or decreasing. [2]
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2012 Q9 [8]}}