AQA Paper 1 Specimen — Question 15 8 marks

Exam BoardAQA
ModulePaper 1 (Paper 1)
SessionSpecimen
Marks8
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Mark schemeDownload PDF ↗
TopicDifferential equations
TypeSeparable variables - standard (polynomial/exponential x-side)
DifficultyStandard +0.3 This is a separable differential equation requiring standard integration techniques (separation of variables, integrating sin(2t) and √x, applying initial conditions). Part (b) requires finding the maximum by setting the derivative to zero. While it involves multiple steps and careful algebraic manipulation, it follows a well-established procedure taught in A-level Further Maths or single maths differential equations topics, making it slightly easier than average overall.
Spec1.05k Further identities: sec^2=1+tan^2 and cosec^2=1+cot^21.08k Separable differential equations: dy/dx = f(x)g(y)
The height \(x\) metres, of a column of water in a fountain display satisfies the differential equation \(\frac{dx}{dt} = \frac{8\sin 2t}{3\sqrt{x}}\), where \(t\) is the time in seconds after the display begins.
  1. Solve the differential equation, given that initially the column of water has zero height. Express your answer in the form \(x = f(t)\) [7 marks]
  2. Find the maximum height of the column of water, giving your answer to the nearest cm. [1 mark]
Question 15:
AnswerMarks Guidance
15(a)Separates variables, at least one
side correct.AO3.1a M1
3 x 8sin2t
dt
3 x dx  8sin2t dt
1
3x2 dx8sin2t dt
3
2x2 4cos2t(c)
3
2(0)2  4cos(20)c
c = 4
3
x2 =22cos2t
2
x 22cos2t 3
AnswerMarks Guidance
Obtains correct separation PIAO1.1b A1
integrates ‘their’ expressions, at
AnswerMarks Guidance
least one of ‘their’ sides correctAO1.1a M1
Obtains correct integral (condone
AnswerMarks Guidance
missing + c) CAOAO1.1b A1
Substitutes initial conditions, to find
AnswerMarks Guidance
+ c.AO3.1b M1
Obtains a correct solution
AnswerMarks Guidance
ACFAO1.1b A1
Obtains correct solution of the form
AnswerMarks Guidance
xf(t)AO2.5 A1
(b)Obtains correct max height, in cm
Award FT from correct substitution
into incorrect equation x f(t)but
only if all three M1 marks have been
AnswerMarks Guidance
awarded, must have correct units.AO3.4 A1F
Max height  43 252 cm
AnswerMarks Guidance
Total8
QMarking Instructions AO 
Question 15:
--- 15(a) ---
15(a) | Separates variables, at least one
side correct. | AO3.1a | M1 | dx
3 x 8sin2t
dt
3 x dx  8sin2t dt
1
3x2 dx8sin2t dt
3
2x2 4cos2t(c)
3
2(0)2  4cos(20)c
c = 4
3
x2 =22cos2t
2
x 22cos2t 3
Obtains correct separation PI | AO1.1b | A1
integrates ‘their’ expressions, at
least one of ‘their’ sides correct | AO1.1a | M1
Obtains correct integral (condone
missing + c) CAO | AO1.1b | A1
Substitutes initial conditions, to find
+ c. | AO3.1b | M1
Obtains a correct solution
ACF | AO1.1b | A1
Obtains correct solution of the form
xf(t) | AO2.5 | A1
(b) | Obtains correct max height, in cm
Award FT from correct substitution
into incorrect equation x f(t)but
only if all three M1 marks have been
awarded, must have correct units. | AO3.4 | A1F | 2
Max height  43 252 cm
Total | 8
Q | Marking Instructions | AO | Marks | Typical Solution
The height $x$ metres, of a column of water in a fountain display satisfies the differential equation $\frac{dx}{dt} = \frac{8\sin 2t}{3\sqrt{x}}$, where $t$ is the time in seconds after the display begins.

\begin{enumerate}[label=(\alph*)]
\item Solve the differential equation, given that initially the column of water has zero height.

Express your answer in the form $x = f(t)$
[7 marks]
\item Find the maximum height of the column of water, giving your answer to the nearest cm.
[1 mark]
\end{enumerate}

\hfill \mbox{\textit{AQA Paper 1  Q15 [8]}}
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Q1 1 Q2 1 Q3 3 Q4 6 Q5 8 Q6 4 Q7 4 Q8 7 Q9 8 Q10 10 Q11 8 Q12 8 Q13 3 Q14 10 Q15 8 Q16 5 Q17 6