AQA AS Paper 1 2019 June — Question 13 9 marks

Exam BoardAQA
ModuleAS Paper 1 (AS Paper 1)
Year2019
SessionJune
Marks9
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicApplied differentiation
TypeKinematics: displacement-velocity-acceleration
DifficultyModerate -0.3 This is a straightforward AS-level mechanics question testing standard calculus techniques: integration for distance, differentiation for maximum speed, and interpretation of acceleration. All parts follow routine procedures with no conceptual challenges, making it slightly easier than average but requiring multiple correct applications of calculus.
Spec1.07k Differentiate trig: sin(kx), cos(kx), tan(kx)1.08d Evaluate definite integrals: between limits3.02f Non-uniform acceleration: using differentiation and integration
A car, starting from rest, is driven along a horizontal track. The velocity of the car, \(v \text{m s}^{-1}\), at time \(t\) seconds, is modelled by the equation $$v = 0.48t^2 - 0.024t^3 \text{ for } 0 \leq t \leq 15$$
  1. Find the distance the car travels during the first 10 seconds of its journey. [3 marks]
  2. Find the maximum speed of the car. Give your answer to three significant figures. [4 marks]
  3. Deduce the range of values of \(t\) for which the car is modelled as decelerating. [2 marks]
Question 13:
AnswerMarks Guidance
13(a)Integrates given velocity equation,
with at least one term correct PI3.4 M1
2 3
๐‘ ๐‘  = ๏ฟฝ 0.48๐‘ก๐‘ก โˆ’0.024๐‘ก๐‘ก ๐‘‘๐‘‘๐‘ก๐‘ก
0
3 4 10
๐‘ ๐‘  = [0.16๐‘ก๐‘ก โˆ’0.006๐‘ก๐‘ก ]0
m
๐‘ ๐‘  = 100
AnswerMarks Guidance
Obtains fully correct integral PI1.1b A1
Substitutes correct limits to obtain
correct answer. CAO. Condone
AnswerMarks Guidance
omission of units.1.1b A1
Differentiates v (at least one term
AnswerMarks Guidance
correct) PI3.4 M1
= 0.96๐‘ก๐‘กโˆ’0.072๐‘ก๐‘ก
๐‘‘๐‘‘๐‘ก๐‘ก
2
0.96๐‘ก๐‘กโˆ’0.072๐‘ก๐‘ก = 0
40
๐‘ก๐‘ก = 0 ๐‘œ๐‘œ๐‘œ๐‘œ ๐‘ก๐‘ก =
3
When then ms-1
40
Equates their equation to zero
๐‘‘๐‘‘๐‘‘๐‘‘
PI
AnswerMarks Guidance
๐‘‘๐‘‘๐‘‘๐‘‘1.1a M1
Finds correct non-zero value for t1.1b A1
Finds correct maximum speed 28.4
Condone exact answer .
4
Condone omission of units.
AnswerMarks Guidance
2891.1b A1
AnswerMarks
13(c)Deduces lower critical value of t
with correct associated inequality
Follow through their value of t from
AnswerMarks Guidance
part (b) provided 0 < t < 152.2a R1F
40
< ๐‘ก๐‘ก โ‰ค 15
3
AnswerMarks Guidance
States2.5 R1
Total ๐‘ก๐‘ก โ‰ค 159 
Question 13:
--- 13(a) ---
13(a) | Integrates given velocity equation,
with at least one term correct PI | 3.4 | M1 | 10
2 3
๐‘ ๐‘  = ๏ฟฝ 0.48๐‘ก๐‘ก โˆ’0.024๐‘ก๐‘ก ๐‘‘๐‘‘๐‘ก๐‘ก
0
3 4 10
๐‘ ๐‘  = [0.16๐‘ก๐‘ก โˆ’0.006๐‘ก๐‘ก ]0
m
๐‘ ๐‘  = 100
Obtains fully correct integral PI | 1.1b | A1
Substitutes correct limits to obtain
correct answer. CAO. Condone
omission of units. | 1.1b | A1
Differentiates v (at least one term
correct) PI | 3.4 | M1 | ๐‘‘๐‘‘๐‘‘๐‘‘ 2
= 0.96๐‘ก๐‘กโˆ’0.072๐‘ก๐‘ก
๐‘‘๐‘‘๐‘ก๐‘ก
2
0.96๐‘ก๐‘กโˆ’0.072๐‘ก๐‘ก = 0
40
๐‘ก๐‘ก = 0 ๐‘œ๐‘œ๐‘œ๐‘œ ๐‘ก๐‘ก =
3
When then ms-1
40
Equates their equation to zero
๐‘‘๐‘‘๐‘‘๐‘‘
PI
๐‘‘๐‘‘๐‘‘๐‘‘ | 1.1a | M1
Finds correct non-zero value for t | 1.1b | A1
Finds correct maximum speed 28.4
Condone exact answer .
4
Condone omission of units.
289 | 1.1b | A1
--- 13(c) ---
13(c) | Deduces lower critical value of t
with correct associated inequality
Follow through their value of t from
part (b) provided 0 < t < 15 | 2.2a | R1F | ๐‘ก๐‘ก = 3 ๐‘‘๐‘‘ = 28.4
40
< ๐‘ก๐‘ก โ‰ค 15
3
States | 2.5 | R1
Total ๐‘ก๐‘ก โ‰ค 15 | 9
A car, starting from rest, is driven along a horizontal track.

The velocity of the car, $v \text{m s}^{-1}$, at time $t$ seconds, is modelled by the equation
$$v = 0.48t^2 - 0.024t^3 \text{ for } 0 \leq t \leq 15$$

\begin{enumerate}[label=(\alph*)]
\item Find the distance the car travels during the first 10 seconds of its journey.
[3 marks]

\item Find the maximum speed of the car.

Give your answer to three significant figures.
[4 marks]

\item Deduce the range of values of $t$ for which the car is modelled as decelerating.
[2 marks]
\end{enumerate}

\hfill \mbox{\textit{AQA AS Paper 1 2019 Q13 [9]}}
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