Question 3 - A-Level Maths

AQA FP3 2007 June — Question 3 8 marks

Exam Board	AQA
Module	FP3 (Further Pure Mathematics 3)
Year	2007
Session	June
Marks	8
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 a straightforward setup. While it requires knowing the integrating factor method and integrating sec x (which gives ln\|sec x + tan x\|), the structure is textbook-standard with no conceptual surprises. The initial condition application is routine. Slightly above average difficulty due to being Further Maths content and requiring fluency with trig integrals, but still a direct application of a learned technique.
Spec	4.10c Integrating factor: first order equations

3 By using an integrating factor, find the solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + ( \tan x ) y = \sec x$$ given that $y = 3$ when $x = 0$.

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Question 3:

Answer	Marks	Guidance
Answer/Working	Marks	Guidance
IF is $e^{\int \tan x\, dx}$	M1
$= e^{-\ln\cos x} = e^{\ln\sec x}$	A1	Accept either
$= \sec x$	A1ft	ft on earlier sign error
$\frac{d}{dx}(y\sec x) = \sec^2 x$	M1A1
$y\sec x = \int \sec^2 x\, dx$
$y\sec x = \tan x + c$	A1	Condone missing $c$
$y = 3$ when $x = 0 \Rightarrow 3\sec 0 = 0 + c$	m1
$c = 3 \Rightarrow y\sec x = \tan x + 3$	A1	8 marks

## Question 3:

| Answer/Working | Marks | Guidance |
|---|---|---|
| IF is $e^{\int \tan x\, dx}$ | M1 | |
| $= e^{-\ln\cos x} = e^{\ln\sec x}$ | A1 | Accept either |
| $= \sec x$ | A1ft | ft on earlier sign error |
| $\frac{d}{dx}(y\sec x) = \sec^2 x$ | M1A1 | |
| $y\sec x = \int \sec^2 x\, dx$ | | |
| $y\sec x = \tan x + c$ | A1 | Condone missing $c$ |
| $y = 3$ when $x = 0 \Rightarrow 3\sec 0 = 0 + c$ | m1 | |
| $c = 3 \Rightarrow y\sec x = \tan x + 3$ | A1 | **8 marks** | OE; condone solution finishing at $c = 3$ provided no errors |

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

$$\frac { \mathrm { d } y } { \mathrm {~d} x } + ( \tan x ) y = \sec x$$

given that $y = 3$ when $x = 0$.

\hfill \mbox{\textit{AQA FP3 2007 Q3 [8]}}

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Q1 10 Q2 9 Q3 8 Q4 14 Q5 12 Q6 15 Q7 7