| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2011 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Differential equations |
| Type | Geometric curve properties |
| Difficulty | Moderate -0.3 This is a straightforward separable differential equations question requiring standard techniques: setting up dy/dx = kxy, finding k from initial conditions, separating variables, and integrating. The sketch requires basic understanding of exponential behavior. While it involves multiple steps, each is routine for P3 level with no novel insight required, making it slightly easier than average. |
| Spec | 1.07t Construct differential equations: in context1.08k Separable differential equations: dy/dx = f(x)g(y) |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Show that the differential equation is \(\frac{dy}{dx} = 2xy\) | B1 | |
| Separate variables correctly and attempt integration of both sides | M1 | |
| Obtain term in \(y\), or equivalent | A1 | |
| Obtain term \(x^2\), or equivalent | A1 | |
| Evaluate a constant, or use limits \(x = 1, y = 2\), in a solution containing terms \(a\ln y\) and \(bx^2\) | M1 | |
| Obtain correct solution in any form | A1 | |
| Obtain the given answer correctly | A1 | [7] |
| (ii) State that the gradient at \((-1, 2)\) is \(-4\) | B1 | |
| Show the sketch of curve with correct concavity, positive \(y\)-intercept and axis of symmetry \(x = 0\) | B1 | [2] |
**(i)** Show that the differential equation is $\frac{dy}{dx} = 2xy$ | B1 |
Separate variables correctly and attempt integration of both sides | M1 |
Obtain term in $y$, or equivalent | A1 |
Obtain term $x^2$, or equivalent | A1 |
Evaluate a constant, or use limits $x = 1, y = 2$, in a solution containing terms $a\ln y$ and $bx^2$ | M1 |
Obtain correct solution in any form | A1 |
Obtain the given answer correctly | A1 | [7]
**(ii)** State that the gradient at $(-1, 2)$ is $-4$ | B1 |
Show the sketch of curve with correct concavity, positive $y$-intercept and axis of symmetry $x = 0$ | B1 | [2]
[SR: A solution with $k \neq 2$, or not evaluated, can earn B0M1A1A1M1A0 in part (i).]
[SR: If given answer is assumed valid, give B1 if $\frac{dy}{dx}$ is shown correctly to be equal to $2xy$, is stated to be proportional to $xy$, and shown to be equal to $4$ at $(1, 2)$.]6 A certain curve is such that its gradient at a point $( x , y )$ is proportional to $x y$. At the point $( 1,2 )$ the gradient is 4 .\\
(i) By setting up and solving a differential equation, show that the equation of the curve is $y = 2 \mathrm { e } ^ { x ^ { 2 } - 1 }$.\\
(ii) State the gradient of the curve at the point $( - 1,2 )$ and sketch the curve.
\hfill \mbox{\textit{CAIE P3 2011 Q6 [9]}}