Question 6 - A-Level Maths

Edexcel FP2 2002 June — Question 6 11 marks

Exam Board	Edexcel
Module	FP2 (Further Pure Mathematics 2)
Year	2002
Session	June
Marks	11
Paper	Download PDF ↗
Mark scheme	Download PDF ↗
Topic	First order differential equations (integrating factor)
Type	Standard linear first order - variable coefficients
Difficulty	Standard +0.8 This is a standard integrating factor question from Further Maths FP2, requiring division by cos x, recognition of the integrating factor sec x, integration of sec³x (a non-trivial integral), and then geometric interpretation of the solution family. Part (b) requires finding where all solution curves intersect the x-axis, which demands understanding of the arbitrary constant's role. The sec³x integration and multi-part analysis elevate this above routine A-level questions but it follows a standard method.
Spec	4.10c Integrating factor: first order equations

6. (a) Find the general solution of the differential equation $$\cos x \frac { \mathrm {~d} y } { \mathrm {~d} x } + ( \sin x ) y = \cos ^ { 3 } x$$ (b) Show that, for $0 \leq x \leq 2 \pi$, there are two points on the $x$-axis through which all the solution curves for this differential equation pass.
(c) Sketch the graph, for $0 \leq x \leq 2 \pi$, of the particular solution for which $y = 0$ at $x = 0$.

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Question 6:

Part (a)

Answer	Marks	Guidance
Answer	Marks	Guidance
$\frac{dy}{dx}+y\left(\frac{\sin x}{\cos x}\right)=\cos^2 x$	M1
Int. factor $e^{\int\tan x\,dx}=e^{-\ln(\cos x)}=\sec x$	M1, A1
Integrate: $y\sec x=\int\cos x\,dx$	M1, A1
$y\sec x = \sin x + C$	A1
$(y=\sin x\cos x + C\cos x)$		(6)

Part (b)

Answer	Marks	Guidance
Answer	Marks	Guidance
When $y=0$: $\cos x(\sin x+C)=0$, $\cos x=0$	M1
2 solutions: $x=\frac{\pi}{2}, \frac{3\pi}{2}$	A1	(2)

Part (c)

Answer	Marks	Guidance
Answer	Marks	Guidance
$y=0$ at $x=0$: $C=0$, $y=\sin x\cos x$	M1
$\left(y=\frac{1}{2}\sin 2x\right)$	A1	Shape
A1	Scales (3)
(11 marks)

# Question 6:

## Part (a)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\frac{dy}{dx}+y\left(\frac{\sin x}{\cos x}\right)=\cos^2 x$ | M1 | |
| Int. factor $e^{\int\tan x\,dx}=e^{-\ln(\cos x)}=\sec x$ | M1, A1 | |
| Integrate: $y\sec x=\int\cos x\,dx$ | M1, A1 | |
| $y\sec x = \sin x + C$ | A1 | |
| $(y=\sin x\cos x + C\cos x)$ | | **(6)** |

## Part (b)
| Answer | Marks | Guidance |
|--------|-------|----------|
| When $y=0$: $\cos x(\sin x+C)=0$, $\cos x=0$ | M1 | |
| 2 solutions: $x=\frac{\pi}{2}, \frac{3\pi}{2}$ | A1 | **(2)** |

## Part (c)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $y=0$ at $x=0$: $C=0$, $y=\sin x\cos x$ | M1 | |
| $\left(y=\frac{1}{2}\sin 2x\right)$ | A1 | Shape |
| | A1 | Scales **(3)** |
| | | **(11 marks)** |

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6. (a) Find the general solution of the differential equation

$$\cos x \frac { \mathrm {~d} y } { \mathrm {~d} x } + ( \sin x ) y = \cos ^ { 3 } x$$

(b) Show that, for $0 \leq x \leq 2 \pi$, there are two points on the $x$-axis through which all the solution curves for this differential equation pass.\\
(c) Sketch the graph, for $0 \leq x \leq 2 \pi$, of the particular solution for which $y = 0$ at $x = 0$.\\

\hfill \mbox{\textit{Edexcel FP2 2002 Q6 [11]}}

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Q1 5 Q2 10 Q3 13 Q4 18 Q5 7 Q6 11 Q7 14 Q8 15 Q9 7 Q10 12