| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2016 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Differential equations |
| Type | Geometric curve properties |
| Difficulty | Standard +0.3 This is a straightforward differential equations question requiring students to translate a geometric condition into a DE (dy/dx = ½xy), then solve a separable equation with an initial condition. The setup is slightly above routine due to the geometric interpretation, but the mathematical techniques (separation of variables, integration, applying boundary conditions) are standard P3 fare with no novel insights required. |
| Spec | 1.02n Sketch curves: simple equations including polynomials1.07t Construct differential equations: in context1.08k Separable differential equations: dy/dx = f(x)g(y) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| State equation \(\frac{dy}{dx} = \frac{1}{2}xy\) | B1 | [1] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Separate variables correctly and attempt to integrate one side | M1 | |
| Obtain terms of the form \(a\ln y\) and \(bx^2\) | A1 | |
| Use \(x=0\) and \(y=2\) to evaluate constant, or as limits, in expression containing \(a\ln y\) or \(bx^2\) | M1 | |
| Obtain correct solution in any form, e.g. \(\ln y = \frac{1}{4}x^2 + \ln 2\) | A1 | |
| Obtain correct expression for \(y\), e.g. \(y = 2e^{\frac{1}{4}x^2}\) | A1 | [5] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Show correct sketch for \(x \geq 0\). Needs through \((0,2)\) and rapidly increasing positive gradient | B1 | [1] |
## Question 5:
### Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| State equation $\frac{dy}{dx} = \frac{1}{2}xy$ | B1 | [1] |
### Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Separate variables correctly and attempt to integrate one side | M1 | |
| Obtain terms of the form $a\ln y$ and $bx^2$ | A1 | |
| Use $x=0$ and $y=2$ to evaluate constant, or as limits, in expression containing $a\ln y$ or $bx^2$ | M1 | |
| Obtain correct solution in any form, e.g. $\ln y = \frac{1}{4}x^2 + \ln 2$ | A1 | |
| Obtain correct expression for $y$, e.g. $y = 2e^{\frac{1}{4}x^2}$ | A1 | [5] |
### Part (iii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Show correct sketch for $x \geq 0$. Needs through $(0,2)$ and rapidly increasing positive gradient | B1 | [1] |
---5\\
\includegraphics[max width=\textwidth, alt={}, center]{ccadf73b-16f5-463a-8f69-1394839d5325-2_346_437_1155_854}
The diagram shows a variable point $P$ with coordinates $( x , y )$ and the point $N$ which is the foot of the perpendicular from $P$ to the $x$-axis. $P$ moves on a curve such that, for all $x \geqslant 0$, the gradient of the curve is equal in value to the area of the triangle $O P N$, where $O$ is the origin.\\
(i) State a differential equation satisfied by $x$ and $y$.
The point with coordinates $( 0,2 )$ lies on the curve.\\
(ii) Solve the differential equation to obtain the equation of the curve, expressing $y$ in terms of $x$.\\
(iii) Sketch the curve.
\hfill \mbox{\textit{CAIE P3 2016 Q5 [7]}}