| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2020 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Differential equations |
| Type | Geometric curve properties |
| Difficulty | Standard +0.3 This is a straightforward separable differential equation requiring standard integration techniques. Setting up dy/dx = k·y/(x√x), separating variables, integrating (using ln y and x^(1/2)), then applying two boundary conditions to find constants k and C is methodical but routine. Part (b) requires simple limit analysis of the exponential form. Slightly above average due to the two-condition setup and fractional power integration, but still a standard textbook exercise. |
| Spec | 1.07t Construct differential equations: in context1.08k Separable differential equations: dy/dx = f(x)g(y) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State \(\dfrac{dy}{dx} = k\dfrac{y}{x\sqrt{x}}\), or equivalent | B1 | |
| Separate variables correctly and attempt integration of at least one side | M1 | |
| Obtain term \(\ln y\), or equivalent | A1 | |
| Obtain term \(-2k\dfrac{1}{\sqrt{x}}\), or equivalent | A1 | |
| Use given coordinates to find \(k\) or constant of integration \(c\) in a solution containing terms of the form \(a\ln y\) and \(\dfrac{b}{\sqrt{x}}\), where \(ab\neq 0\) | M1 | |
| Obtain \(k=1\) and \(c=2\) | A1+A1 | |
| Obtain final answer \(y = \exp\!\left(-\dfrac{2}{\sqrt{x}}+2\right)\), or equivalent | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State that \(y\) approaches \(e^2\) | B1FT | FT their \(c\) in part (a) of the correct form |
## Question 8(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| State $\dfrac{dy}{dx} = k\dfrac{y}{x\sqrt{x}}$, or equivalent | B1 | |
| Separate variables correctly and attempt integration of at least one side | M1 | |
| Obtain term $\ln y$, or equivalent | A1 | |
| Obtain term $-2k\dfrac{1}{\sqrt{x}}$, or equivalent | A1 | |
| Use given coordinates to find $k$ or constant of integration $c$ in a solution containing terms of the form $a\ln y$ and $\dfrac{b}{\sqrt{x}}$, where $ab\neq 0$ | M1 | |
| Obtain $k=1$ and $c=2$ | A1+A1 | |
| Obtain final answer $y = \exp\!\left(-\dfrac{2}{\sqrt{x}}+2\right)$, or equivalent | A1 | |
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## Question 8(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| State that $y$ approaches $e^2$ | B1FT | FT their $c$ in part (a) of the correct form |
---8 A certain curve is such that its gradient at a point $( x , y )$ is proportional to $\frac { y } { x \sqrt { x } }$. The curve passes through the points with coordinates $( 1,1 )$ and $( 4 , \mathrm { e } )$.
\begin{enumerate}[label=(\alph*)]
\item By setting up and solving a differential equation, find the equation of the curve, expressing $y$ in terms of $x$.
\item Describe what happens to $y$ as $x$ tends to infinity.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2020 Q8 [9]}}