Question 10 - A-Level Maths

WJEC Further Unit 4 2019 June — Question 10 8 marks

Exam Board	WJEC
Module	Further Unit 4 (Further Unit 4)
Year	2019
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	Challenging +1.8 This is a first-order linear ODE requiring identification of integrating factor (cos x), careful integration involving substitution (u = sec x leads to non-standard integrals), and application of initial conditions. The 8-mark allocation and non-routine integrating factor make this substantially harder than typical A-level ODE questions, though the overall structure follows standard methods.
Spec	4.10c Integrating factor: first order equations

Given the differential equation $$\sec x \frac{\mathrm{d}y}{\mathrm{d}x} + y\cos \sec x = 2$$ and $x = \frac{\pi}{2}$ when $y = 3$, find the value of $y$ when $x = \frac{\pi}{4}$. [8]

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Answer	Marks	Guidance
Dividing both sides by $\sec x$:	M1
$\frac{dy}{dx} + \frac{y\cos x}{\sin x} = 2\cos x$
Integrating factor: $e^{\int\frac{\cos x}{\sin x}dx}$	M1
$= e^{\ln\sin x} = \sin x$	A1
Multiplying both sides:	M1
$\sin x \frac{dy}{dx} + y\cos x = 2\sin x\cos x$ (= $\sin 2x$)	m1
Integrating:	A1	Or equivalent
$y\sin x = -\frac{\cos 2x}{2} + c$
Substituting: $3 = \frac{1}{2} + c \Rightarrow c = \frac{5}{2}$	B1
Solution: $y\sin x = \frac{-\cos 2x}{2} + \frac{5}{2}$
When $x = \frac{\pi}{4}$, $y\sin\frac{\pi}{4} = \frac{5}{\sqrt{2}} \Rightarrow y = \frac{5\sqrt{2}}{2}$ (3.536)	B1

Dividing both sides by $\sec x$: | M1 |
$\frac{dy}{dx} + \frac{y\cos x}{\sin x} = 2\cos x$ | |
Integrating factor: $e^{\int\frac{\cos x}{\sin x}dx}$ | M1 |
$= e^{\ln\sin x} = \sin x$ | A1 |
Multiplying both sides: | M1 |
$\sin x \frac{dy}{dx} + y\cos x = 2\sin x\cos x$ (= $\sin 2x$) | m1 |
Integrating: | A1 | Or equivalent
$y\sin x = -\frac{\cos 2x}{2} + c$ | |
Substituting: $3 = \frac{1}{2} + c \Rightarrow c = \frac{5}{2}$ | B1 |
Solution: $y\sin x = \frac{-\cos 2x}{2} + \frac{5}{2}$ | |
When $x = \frac{\pi}{4}$, $y\sin\frac{\pi}{4} = \frac{5}{\sqrt{2}} \Rightarrow y = \frac{5\sqrt{2}}{2}$ (3.536) | B1 |

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Given the differential equation
$$\sec x \frac{\mathrm{d}y}{\mathrm{d}x} + y\cos \sec x = 2$$
and $x = \frac{\pi}{2}$ when $y = 3$, find the value of $y$ when $x = \frac{\pi}{4}$. [8]

\hfill \mbox{\textit{WJEC Further Unit 4 2019 Q10 [8]}}

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Q1 8 Q2 9 Q3 8 Q4 16 Q5 8 Q6 10 Q7 6 Q8 10 Q9 14 Q10 8 Q11 9 Q12 14

Answer	Marks	Guidance
Dividing both sides by \(\sec x\):	M1
\(\frac{dy}{dx} + \frac{y\cos x}{\sin x} = 2\cos x\)
Integrating factor: \(e^{\int\frac{\cos x}{\sin x}dx}\)	M1
\(= e^{\ln\sin x} = \sin x\)	A1
Multiplying both sides:	M1
\(\sin x \frac{dy}{dx} + y\cos x = 2\sin x\cos x\) (= \(\sin 2x\))	m1
Integrating:	A1	Or equivalent
\(y\sin x = -\frac{\cos 2x}{2} + c\)
Substituting: \(3 = \frac{1}{2} + c \Rightarrow c = \frac{5}{2}\)	B1
Solution: \(y\sin x = \frac{-\cos 2x}{2} + \frac{5}{2}\)
When \(x = \frac{\pi}{4}\), \(y\sin\frac{\pi}{4} = \frac{5}{\sqrt{2}} \Rightarrow y = \frac{5\sqrt{2}}{2}\) (3.536)	B1