Question 8 - A-Level Maths

AQA C4 2008 January — Question 8 5 marks

Exam Board	AQA
Module	C4 (Core Mathematics 4)
Year	2008
Session	January
Marks	5
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	Differential equations
Type	Separable variables - standard (polynomial/exponential x-side)
Difficulty	Moderate -0.8 This is a straightforward separable variables question requiring only standard integration techniques (cos 3x and a simple constant). The method is routine: separate, integrate both sides, apply initial condition, and rearrange to the requested form. No problem-solving insight needed beyond recognizing the standard separation technique.
Spec	1.08k Separable differential equations: dy/dx = f(x)g(y)

8 Solve the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 3 \cos 3 x } { y }$$ given that $y = 2$ when $x = \frac { \pi } { 2 }$. Give your answer in the form $y ^ { 2 } = \mathrm { f } ( x )$.

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Answer	Marks	Guidance
$\int y \, dy = \int 3\cos 3x \, dx$	M1	attempt to separate and integrate $py^2 = q\sin 3x$ seen $\Rightarrow$ implies separation
$\frac{1}{2}y^2 = \sin 3x \, (+C)$	A1 A1	integrals – accept $\frac{1}{3} \times 3\sin 3x$
$\left(\frac{\pi}{2}, 2\right)$: $\frac{1}{2} \times 4 = \sin\frac{3\pi}{2} + C$	M1	use $\left(\frac{\pi}{2}, 2\right)$ to find constant
$C = 3$
$y^2 = 2\sin 3x + 6$	A1	CSO (in any correct form)

Total: 5 marks

| $\int y \, dy = \int 3\cos 3x \, dx$ | M1 | attempt to separate and integrate $py^2 = q\sin 3x$ seen $\Rightarrow$ implies separation |
| $\frac{1}{2}y^2 = \sin 3x \, (+C)$ | A1 A1 | integrals – accept $\frac{1}{3} \times 3\sin 3x$ |
| $\left(\frac{\pi}{2}, 2\right)$: $\frac{1}{2} \times 4 = \sin\frac{3\pi}{2} + C$ | M1 | use $\left(\frac{\pi}{2}, 2\right)$ to find constant |
| $C = 3$ | | |
| $y^2 = 2\sin 3x + 6$ | A1 | CSO (in any correct form) |

**Total: 5 marks**

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8 Solve the differential equation

$$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 3 \cos 3 x } { y }$$

given that $y = 2$ when $x = \frac { \pi } { 2 }$. Give your answer in the form $y ^ { 2 } = \mathrm { f } ( x )$.

\hfill \mbox{\textit{AQA C4 2008 Q8 [5]}}

This paper (9 questions)

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Q1 5 Q2 10 Q3 6 Q4 9 Q5 10 Q6 5 Q7 14 Q8 5 Q9 11

Answer	Marks	Guidance
\(\int y \, dy = \int 3\cos 3x \, dx\)	M1	attempt to separate and integrate \(py^2 = q\sin 3x\) seen \(\Rightarrow\) implies separation
\(\frac{1}{2}y^2 = \sin 3x \, (+C)\)	A1 A1	integrals – accept \(\frac{1}{3} \times 3\sin 3x\)
\(\left(\frac{\pi}{2}, 2\right)\): \(\frac{1}{2} \times 4 = \sin\frac{3\pi}{2} + C\)	M1	use \(\left(\frac{\pi}{2}, 2\right)\) to find constant
\(C = 3\)
\(y^2 = 2\sin 3x + 6\)	A1	CSO (in any correct form)