AQA C4 2010 January — Question 7 6 marks

Exam BoardAQA
ModuleC4 (Core Mathematics 4)
Year2010
SessionJanuary
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 technique: separate variables, integrate both sides (using simple substitution for cos(x/3)), apply initial condition, and rearrange. The integration is routine and the algebra is minimal, making it slightly easier than an average A-level question.
Spec1.08k Separable differential equations: dy/dx = f(x)g(y)
7 Solve the differential equation \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { y } \cos \left( \frac { x } { 3 } \right)\), given that \(y = 1\) when \(x = \frac { \pi } { 2 }\).
Write your answer in the form \(y ^ { 2 } = \mathrm { f } ( x )\).
Question 7:
AnswerMarks Guidance
Answer/WorkingMark Guidance
\(\int y \, dy = \int \cos\left(\frac{x}{3}\right)dx\)B1 Separate variables; condone missing integral signs
\(\frac{1}{2}y^2 = 3\sin\left(\frac{x}{3}\right) + C\)B1 B1 Accept \(\frac{\sin\left(\frac{x}{3}\right)}{\frac{1}{3}}\)
\(\left(\frac{\pi}{2}, 1\right)\): \(\frac{1}{2} = 3\sin\frac{\pi}{6} + C\)M1 Use \(\left(\frac{\pi}{2}, 1\right)\) to find \(C\); must be in form \(py^2 = q\sin\left(\frac{x}{3}\right) + C\)
\(C = -1\)A1F
\(y^2 = 6\sin\left(\frac{x}{3}\right) - 2\)A1 CSO 
## Question 7:

| Answer/Working | Mark | Guidance |
|---|---|---|
| $\int y \, dy = \int \cos\left(\frac{x}{3}\right)dx$ | B1 | Separate variables; condone missing integral signs |
| $\frac{1}{2}y^2 = 3\sin\left(\frac{x}{3}\right) + C$ | B1 B1 | Accept $\frac{\sin\left(\frac{x}{3}\right)}{\frac{1}{3}}$ |
| $\left(\frac{\pi}{2}, 1\right)$: $\frac{1}{2} = 3\sin\frac{\pi}{6} + C$ | M1 | Use $\left(\frac{\pi}{2}, 1\right)$ to find $C$; must be in form $py^2 = q\sin\left(\frac{x}{3}\right) + C$ |
| $C = -1$ | A1F | |
| $y^2 = 6\sin\left(\frac{x}{3}\right) - 2$ | A1 | CSO |

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7 Solve the differential equation $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { y } \cos \left( \frac { x } { 3 } \right)$, given that $y = 1$ when $x = \frac { \pi } { 2 }$.\\
Write your answer in the form $y ^ { 2 } = \mathrm { f } ( x )$.

\hfill \mbox{\textit{AQA C4 2010 Q7 [6]}}
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