CAIE P3 2012 November — Question 4 6 marks

Exam BoardCAIE
ModuleP3 (Pure Mathematics 3)
Year2012
SessionNovember
Marks6
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicDifferential equations
TypeSeparable variables - standard (polynomial/exponential x-side)
DifficultyModerate -0.3 This is a straightforward separable variables question requiring standard techniques: separate variables, integrate both sides (using substitution for the left side and log rule for the right), apply initial condition to find the constant, and rearrange. The integration is routine and the algebra is clean, making this slightly easier than average for A-level.
Spec1.08k Separable differential equations: dy/dx = f(x)g(y)
4 The variables \(x\) and \(y\) are related by the differential equation $$\left( x ^ { 2 } + 4 \right) \frac { d y } { d x } = 6 x y$$ It is given that \(y = 32\) when \(x = 0\). Find an expression for \(y\) in terms of \(x\).
Question 4:
AnswerMarks Guidance
Answer/WorkingMark Guidance
Separate variables correctly and integrate one sideM1
Obtain \(\ln y = \ldots\) or equivalentA1
Obtain \(= 3\ln(x^2+4)\) or equivalentA1
Evaluate a constant or use \(x=0\), \(y=32\) as limits in a solution containing terms \(a\ln y\) and \(b\ln(x^2+4)\)M1
Obtain \(\ln y = 3\ln(x^2+4) + \ln 32 - 3\ln 4\) or equivalentA1
Obtain \(y = \frac{1}{2}(x^2+4)^3\) or equivalentA1 [6] 
## Question 4:

| Answer/Working | Mark | Guidance |
|---|---|---|
| Separate variables correctly and integrate one side | M1 | |
| Obtain $\ln y = \ldots$ or equivalent | A1 | |
| Obtain $= 3\ln(x^2+4)$ or equivalent | A1 | |
| Evaluate a constant or use $x=0$, $y=32$ as limits in a solution containing terms $a\ln y$ and $b\ln(x^2+4)$ | M1 | |
| Obtain $\ln y = 3\ln(x^2+4) + \ln 32 - 3\ln 4$ or equivalent | A1 | |
| Obtain $y = \frac{1}{2}(x^2+4)^3$ or equivalent | A1 | [6] |

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4 The variables $x$ and $y$ are related by the differential equation

$$\left( x ^ { 2 } + 4 \right) \frac { d y } { d x } = 6 x y$$

It is given that $y = 32$ when $x = 0$. Find an expression for $y$ in terms of $x$.

\hfill \mbox{\textit{CAIE P3 2012 Q4 [6]}}
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