| Exam Board | AQA |
|---|---|
| Module | Further Paper 1 (Further Paper 1) |
| Session | Specimen |
| Marks | 7 |
| 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 first-order linear differential equation requiring recognition of the standard form, finding an integrating factor (1/cos x), and performing non-trivial integration involving trigonometric functions. While the method is standard for Further Maths students, the algebraic manipulation and integration steps require careful execution, placing it moderately above average difficulty. |
| Spec | 4.10c Integrating factor: first order equations |
| Answer | Marks |
|---|---|
| 6(a) | Selects appropriate method for |
| Answer | Marks | Guidance |
|---|---|---|
| equation by dividing by tan x | AO3.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| Finds correct integrating factor | AO1.1b | B1 |
| ALT | ALT | Alt. finds an integrating factor by |
| Answer | Marks | Guidance |
|---|---|---|
| (PI) | Alt. finds an integrating factor by | AO3.1a |
| Answer | Marks |
|---|---|
| (PI) | dy |
| Answer | Marks | Guidance |
|---|---|---|
| finds integrating factor = cos x | AO1.1b | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| y×sinx PI | AO1.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| method for RHS of ‘their’ equation | AO1.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| correct solution | AO1.1b | A1 |
| (b) | Uses boundary condition after |
| Answer | Marks | Guidance |
|---|---|---|
| ysinx =... or y=... form OE | AO1.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| solution | AO1.1b | A1 |
| Total | 7 | |
| Q | Marking Instructions | AO |
Question 6:
--- 6(a) ---
6(a) | Selects appropriate method for
example by changing to reduced
equation by dividing by tan x | AO3.1a | M1 | dy
+(cotx)y=sinx
dx
∫(cotx)dx
Integrating factor =e
=eln(sinx) =sinx
Finds correct integrating factor | AO1.1b | B1
ALT | ALT | Alt. finds an integrating factor by
inspection, using original equation.
(PI) | Alt. finds an integrating factor by | AO3.1a | AO3.1a | M1 | M1 | dy
cosxtanx + ycosx=sinxtanxcosx
dx
inspection, using original equation.
(PI) | dy
sinx +( cosx ) y =sin2x
dx
finds integrating factor = cos x | AO1.1b | B1
Multiples reduced or original
equation by ‘their’ integrating factor
and identifies LHS as differential of
y×sinx PI | AO1.1a | M1 | dy
sinx +(cosx)y =sin2x
dx
d
[ysinx]=sin2x
dx
⌠1
ysinx= (1−cos2x)dx
⌡2
1 1
ysinx = x− sin2x+C
2 4
Uses appropriate integration
method for RHS of ‘their’ equation | AO1.1a | M1
Integrates correctly to obtain
correct solution | AO1.1b | A1
(b) | Uses boundary condition after
integration completed in either
ysinx =... or y=... form OE | AO1.1a | M1 | π 1 1 π 1 π
sin . = . − sin +C
4 2 2 2 4 4 2
1 π
C = −
2 8
1 1 1 π
ysinx = x− sin2x+ −
2 4 2 8
States fully correct particular
solution | AO1.1b | A1
Total | 7
Q | Marking Instructions | AO | Marks | Typical Solution\begin{enumerate}[label=(\alph*)]
\item Obtain the general solution of the differential equation $$\tan x \frac{dy}{dx} + y = \sin x \tan x$$
where $0 < x < \frac{\pi}{2}$
[5 marks]
\item Hence find the particular solution of this differential equation, given that $y = \frac{1}{2\sqrt{2}}$ when $x = \frac{\pi}{4}$
[2 marks]
\end{enumerate}
\hfill \mbox{\textit{AQA Further Paper 1 Q6 [7]}}