Question 2 - A-Level Maths

AQA FP3 2009 June — Question 2 9 marks

Exam Board	AQA
Module	FP3 (Further Pure Mathematics 3)
Year	2009
Session	June
Marks	9
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	First order differential equations (integrating factor)
Type	Standard linear first order - variable coefficients
Difficulty	Standard +0.3 This is a standard integrating factor question with straightforward identification of P(x) = -tan x, leading to integrating factor cos x. The integration steps are routine (standard integrals of sin x cos x), and applying the initial condition is direct. While it's a Further Maths topic, it follows the textbook method exactly with no tricks or novel insights required, making it slightly easier than average overall.
Spec	4.10c Integrating factor: first order equations

2 By using an integrating factor, find the solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } - y \tan x = 2 \sin x$$ given that $y = 2$ when $x = 0$.
(9 marks)

Show mark scheme Show mark scheme source

Question 2:

Answer	Marks	Guidance
Answer/Working	Marks	Guidance
IF is $e^{\int -\tan x \, dx}$	M1	Award even if negative sign missing
$= e^{\ln(\cos x) (+c)}$	A1	OE Condone missing $c$
$= (k)\cos x$	A1F	ft earlier sign error
$\cos x \frac{dy}{dx} - y\tan x \cos x = 2\sin x \cos x$
$\frac{d}{dx}(y\cos x) = 2\sin x \cos x$	M1	LHS as $\frac{d}{dx}(y \times \text{IF})$, PI
$y\cos x = \int 2\sin x \cos x \, dx$	A1F	ft on $c$'s IF provided no exp or logs
$y\cos x = \int \sin 2x \, dx$	m1	Double angle or substitution OE for integrating $2\sin x \cos x$
$y\cos x = -\frac{1}{2}\cos 2x (+c)$	A1	ACF
$2 = -\frac{1}{2} + c$, so $c = \frac{5}{2}$	m1	Boundary condition used to find $c$
$y\cos x = -\frac{1}{2}\cos 2x + \frac{5}{2}$	A1	Total: 9 marks. ACF e.g. $y\cos x - 2 + \sin^2 x$. Apply ISW after ACF

# Question 2:

| Answer/Working | Marks | Guidance |
|---|---|---|
| IF is $e^{\int -\tan x \, dx}$ | M1 | Award even if negative sign missing |
| $= e^{\ln(\cos x) (+c)}$ | A1 | OE Condone missing $c$ |
| $= (k)\cos x$ | A1F | ft earlier sign error |
| $\cos x \frac{dy}{dx} - y\tan x \cos x = 2\sin x \cos x$ | | |
| $\frac{d}{dx}(y\cos x) = 2\sin x \cos x$ | M1 | LHS as $\frac{d}{dx}(y \times \text{IF})$, PI |
| $y\cos x = \int 2\sin x \cos x \, dx$ | A1F | ft on $c$'s IF provided no exp or logs |
| $y\cos x = \int \sin 2x \, dx$ | m1 | Double angle or substitution OE for integrating $2\sin x \cos x$ |
| $y\cos x = -\frac{1}{2}\cos 2x (+c)$ | A1 | ACF |
| $2 = -\frac{1}{2} + c$, so $c = \frac{5}{2}$ | m1 | Boundary condition used to find $c$ |
| $y\cos x = -\frac{1}{2}\cos 2x + \frac{5}{2}$ | A1 | Total: 9 marks. ACF e.g. $y\cos x - 2 + \sin^2 x$. Apply ISW after ACF |

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2 By using an integrating factor, find the solution of the differential equation

$$\frac { \mathrm { d } y } { \mathrm {~d} x } - y \tan x = 2 \sin x$$

given that $y = 2$ when $x = 0$.\\
(9 marks)

\hfill \mbox{\textit{AQA FP3 2009 Q2 [9]}}

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Q1 6 Q2 9 Q3 8 Q4 5 Q5 11 Q6 10 Q7 14 Q8 12