| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2010 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Differential equations |
| Type | First-order integration |
| Difficulty | Moderate -0.8 This is a straightforward first-order differential equation requiring direct integration of a simple power function, followed by using a boundary condition to find the constant. Part (ii) involves basic stationary point analysis (setting derivative to zero and using second derivative test). Both parts are routine AS-level techniques with no problem-solving insight required, making this easier than average. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.07n Stationary points: find maxima, minima using derivatives1.08b Integrate x^n: where n != -1 and sums |
| Answer | Marks | Guidance |
|---|---|---|
| \(y = \frac{3x^{\frac{3}{2}}}{\frac{3}{2}} - 6x\ (+c)\) | B2,1 | Loses 1 for each error – ignore \(+c\) |
| \((9,\ 2):\ 2 = 54 - 54 + c \rightarrow c = 2\) | M1 A1 [4] | Uses \((9, 2)\) with integration to find \(c\). co |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{dy}{dx} = 0 \rightarrow x = 4\) | B1 | Ignore any \(y\)-value |
| \(\frac{d^2y}{dx^2} = \frac{3x^{-\frac{1}{2}}}{2}\), \(\rightarrow\) +ve (or \(\frac{3}{4}\)) Minimum | M1 A1 [3] | Any valid method. co |
## Question 6:
$\frac{dy}{dx} = 3\sqrt{x} - 6\quad (9, 2)$
**Part (i):**
$y = \frac{3x^{\frac{3}{2}}}{\frac{3}{2}} - 6x\ (+c)$ | B2,1 | Loses 1 for each error – ignore $+c$
$(9,\ 2):\ 2 = 54 - 54 + c \rightarrow c = 2$ | M1 A1 [4] | Uses $(9, 2)$ with integration to find $c$. co
**Part (ii):**
$\frac{dy}{dx} = 0 \rightarrow x = 4$ | B1 | Ignore any $y$-value
$\frac{d^2y}{dx^2} = \frac{3x^{-\frac{1}{2}}}{2}$, $\rightarrow$ +ve (or $\frac{3}{4}$) Minimum | M1 A1 [3] | Any valid method. co
---6 A curve is such that $\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { \frac { 1 } { 2 } } - 6$ and the point $( 9,2 )$ lies on the curve.\\
(i) Find the equation of the curve.\\
(ii) Find the $x$-coordinate of the stationary point on the curve and determine the nature of the stationary point.
\hfill \mbox{\textit{CAIE P1 2010 Q6 [7]}}