Questions - AQA - A-Level Maths

AQA S3 2006 June Q3

11 marks Moderate -0.3

3 Each enquiry received by a business support unit is dealt with by Ewan, Fay or Gaby. The probabilities of them dealing with an enquiry are $0.2,0.3$ and 0.5 respectively. Of enquiries dealt with by Ewan, 60\% are answered immediately, 25\% are answered later the same day and the remainder are answered at a later date. Of enquiries dealt with by Fay, 75\% are answered immediately, 15\% are answered later the same day and the remainder are answered at a later date. Of enquiries dealt with by Gaby, 90\% are answered immediately and the remainder are answered at a later date.

Determine the probability that an enquiry:
1. is dealt with by Gaby and answered immediately;
2. is answered immediately;
3. is dealt with by Gaby, given that it is answered immediately.
Determine the probability that an enquiry is dealt with by Ewan, given that it is answered later the same day.

AQA S3 2006 June Q4

6 marks Moderate -0.3

4 The table below shows the probability distribution for the number of students, $R$, attending classes for a particular mathematics module.

$\boldsymbol { r }$	6	7	8
$\mathbf { P } ( \boldsymbol { R } = \boldsymbol { r } )$	0.1	0.6	0.3

Find values for $\mathrm { E } ( R )$ and $\operatorname { Var } ( R )$.
The number of students, $S$, attending classes for a different mathematics module is such that $$\mathrm { E } ( S ) = 10.9 , \quad \operatorname { Var } ( S ) = 1.69 \quad \text { and } \quad \rho _ { R S } = \frac { 2 } { 3 }$$ Find values for the mean and variance of:
1. $T = R + S$;
2. $\quad D = S - R$.

AQA S3 2006 June Q5

12 marks Standard +0.3

5 The number of letters per week received at home by Rosa may be modelled by a Poisson distribution with parameter 12.25.

Using a normal approximation, estimate the probability that, during a 4 -week period, Rosa receives at home at least 42 letters but at most 54 letters.
Rosa also receives letters at work. During a 16-week period, she receives at work a total of 248 letters.
1. Assuming that the number of letters received at work by Rosa may also be modelled by a Poisson distribution, calculate a $98 \%$ confidence interval for the average number of letters per week received at work by Rosa.
2. Hence comment on Rosa's belief that she receives, on average, fewer letters at home than at work.

AQA S3 2006 June Q6

8 marks Challenging +1.2

6 The random variable $X$ has a Poisson distribution with parameter $\lambda$.

Prove that $\mathrm { E } ( X ) = \lambda$.
By first proving that $\mathrm { E } ( X ( X - 1 ) ) = \lambda ^ { 2 }$, or otherwise, prove that $\operatorname { Var } ( X ) = \lambda$.

AQA S3 2006 June Q7

19 marks Challenging +1.2

7 A shop sells cooked chickens in two sizes: medium and large.
The weights, $X$ grams, of medium chickens may be assumed to be normally distributed with mean $\mu _ { X }$ and standard deviation 45. The weights, $Y$ grams, of large chickens may be assumed to be normally distributed with mean $\mu _ { Y }$ and standard deviation 65. A random sample of 20 medium chickens had a mean weight, $\bar { x }$ grams, of 936 .
A random sample of 10 large chickens had the following weights in grams: $$\begin{array} { l l l l l l l l l l } 1165 & 1202 & 1077 & 1144 & 1195 & 1275 & 1136 & 1215 & 1233 & 1288 \end{array}$$

Calculate the mean weight, $\bar { y }$ grams, of this sample of large chickens.
Hence investigate, at the $1 \%$ level of significance, the claim that the mean weight of large chickens exceeds that of medium chickens by more than 200 grams.
1. Deduce that, for your test in part (b), the critical value of $( \bar { y } - \bar { x } )$ is 253.24, correct to two decimal places.
2. Hence determine the power of your test in part (b), given that $\mu _ { Y } - \mu _ { X } = 275$.
3. Interpret, in the context of this question, the value that you obtained in part (c)(ii).
  (3 marks)

AQA S3 2007 June Q1

8 marks Moderate -0.3

1 As part of an investigation into the starting salaries of graduates in a European country, the following information was collected.

\multirow{2}{*}{}		Starting salary (€)
	Sample size	Sample mean	Sample standard deviation
Science graduates	175	19268	7321
Arts graduates	225	17896	8205

Stating a necessary assumption about the samples, construct a $98 \%$ confidence interval for the difference between the mean starting salary of science graduates and that of arts graduates.
What can be concluded from your confidence interval?

AQA S3 2007 June Q2

11 marks Moderate -0.8

2 A hill-top monument can be visited by one of three routes: road, funicular railway or cable car. The percentages of visitors using these routes are 25, 35 and 40 respectively. The age distribution, in percentages, of visitors using each route is shown in the table. For example, 15 per cent of visitors using the road were under 18 .

\multirow{2}{*}{}		Percentage of visitors using
		Road	Funicular railway	Cable car
\multirow{3}{*}{Age (years)}	Under 18	15	25	10
	18 to 64	80	60	55
	Over 64	5	15	35

Calculate the probability that a randomly selected visitor:

who used the road is aged 18 or over;
is aged between 18 and 64;
used the funicular railway and is aged over 64;
used the funicular railway, given that the visitor is aged over 64.

AQA S3 2007 June Q3

11 marks Standard +0.8

3 Kutz and Styler are two unisex hair salons. An analysis of a random sample of 150 customers at Kutz shows that 28 per cent are male. An analysis of an independent random sample of 250 customers at Styler shows that 34 per cent are male.

Test, at the $5 \%$ level of significance, the hypothesis that there is no difference between the proportion of male customers at Kutz and that at Styler.
State, with a reason, the probability of making a Type I error in the test in part (a) if, in fact, the actual difference between the two proportions is 0.05 .

AQA S3 2007 June Q4

6 marks Standard +0.8

4 A machine is used to fill 5-litre plastic containers with vinegar. The volume, in litres, of vinegar in a container filled by the machine may be assumed to be normally distributed with mean $\mu$ and standard deviation 0.08 . A quality control inspector requires a $99 \%$ confidence interval for $\mu$ to be constructed such that it has a width of at most 0.05 litres. Calculate, to the nearest 5, the sample size necessary in order to achieve the inspector's requirement.

AQA S3 2007 June Q5

7 marks Standard +0.3

5 The duration, $X$ minutes, of a timetabled 1-hour lesson may be assumed to be normally distributed with mean 54 and standard deviation 2. The duration, $Y$ minutes, of a timetabled $1 \frac { 1 } { 2 }$-hour lesson may be assumed to be normally distributed with mean 83 and standard deviation 3. Assuming the durations of lessons to be independent, determine the probability that the total duration of a random sample of three 1 -hour lessons is less than the total duration of a random sample of two $1 \frac { 1 } { 2 }$-hour lessons.
(7 marks)

AQA S3 2007 June Q6

20 marks Standard +0.3

6

The random variable $X$ has a binomial distribution with parameters $n$ and $p$.
1. Prove that $\mathrm { E } ( X ) = n p$.
2. Given that $\mathrm { E } \left( X ^ { 2 } \right) - \mathrm { E } ( X ) = n ( n - 1 ) p ^ { 2 }$, show that $\operatorname { Var } ( X ) = n p ( 1 - p )$.
3. Given that $X$ is found to have a mean of 3 and a variance of 2.97, find values for $n$ and $p$.
4. Hence use a distributional approximation to estimate $\mathrm { P } ( X > 2 )$.
Dressher is a nationwide chain of stores selling women's clothes. It claims that the probability that a customer who buys clothes from its stores uses a Dressher store card is 0.45 . Assuming this claim to be correct, use a distributional approximation to estimate the probability that, in a random sample of 500 customers who buy clothes from Dressher stores, at least half of them use a Dressher store card.

AQA S3 2007 June Q7

12 marks Standard +0.8

7 In a town, the total number, $R$, of houses sold during a week by estate agents may be modelled by a Poisson distribution with a mean of 13 . A new housing development is completed in the town. During the first week in which houses on this development are offered for sale by the developer, the estate agents sell a total of 10 houses.

Using the $10 \%$ level of significance, investigate whether the offer for sale of houses by the developer has resulted in a reduction in the mean value of $R$.
Determine, for your test in part (a), the critical region for $R$.
Assuming that the offer for sale of houses on the new housing development has reduced the mean value of $R$ to 6.5, determine, for a test at the 10\% level of significance, the probability of a Type II error.
(4 marks)

AQA M1 Q1

Moderate -0.8

1 A particle $A$ moves across a smooth horizontal surface in a straight line. The particle $A$ has mass 2 kg and speed $6 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. A particle $B$, which has mass 3 kg , is at rest on the surface. The particle $A$ collides with the particle $B$. \includegraphics[max width=\textwidth, alt={}, center]{6151e6ab-30af-4d1c-ab4a-e7dbad170cbf-003_147_506_644_733}

If, after the collision, $A$ is at rest and $B$ moves away from $A$, find the speed of $B$.
If, after the collision, $A$ and $B$ move away from each other with speeds $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and $4 v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ respectively, as shown in the diagram below, find the value of $v$. \includegraphics[max width=\textwidth, alt={}, center]{6151e6ab-30af-4d1c-ab4a-e7dbad170cbf-003_138_506_1144_730}

AQA M1 Q4

Standard +0.3

4 Water flows in a constant direction at a constant speed of $u \mathrm {~m} \mathrm {~s} ^ { - 1 }$. A boat travels in the water at a speed of $12 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ relative to the water.

The direction in which the boat travels relative to the water is perpendicular to the direction of motion of the water. The resultant velocity of the boat is $V \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at an angle of $74 ^ { \circ }$ to the direction of motion of the water, as shown in the diagram. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6151e6ab-30af-4d1c-ab4a-e7dbad170cbf-004_120_164_662_488} \captionsetup{labelformat=empty} \caption{Velocity of the water}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6151e6ab-30af-4d1c-ab4a-e7dbad170cbf-004_126_186_667_890} \captionsetup{labelformat=empty} \caption{Velocity of the boat relative to the water}
\end{figure}
1. Find $V$.
2. Show that $u = 3.44$, correct to three significant figures.
The boat changes course so that it travels relative to the water at an angle of $45 ^ { \circ }$ to the direction of motion of the water. The resultant velocity of the boat is now of magnitude $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The velocity of the water is unchanged, as shown in the diagram below. $$\xrightarrow { 3.44 \mathrm {~m} \mathrm {~s} ^ { - 1 } }$$
\includegraphics[max width=\textwidth, alt={}]{6151e6ab-30af-4d1c-ab4a-e7dbad170cbf-004_132_273_1493_895}
Velocity of the boat relative to the water \includegraphics[max width=\textwidth, alt={}, center]{6151e6ab-30af-4d1c-ab4a-e7dbad170cbf-004_232_355_1498_1384} Find the value of $v$.
(4 marks)

AQA M1 Q5

Moderate -0.8

5 A golf ball is projected from a point $O$ with initial velocity $V$ at an angle $\alpha$ to the horizontal. The ball first hits the ground at a point $A$ which is at the same horizontal level as $O$, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{6151e6ab-30af-4d1c-ab4a-e7dbad170cbf-005_227_602_484_735} It is given that $V \cos \alpha = 6 u$ and $V \sin \alpha = 2.5 u$.

Show that the time taken for the ball to travel from $O$ to $A$ is $\frac { 5 u } { g }$.
Find, in terms of $g$ and $u$, the distance $O A$.
Find $V$, in terms of $u$.
State, in terms of $u$, the least speed of the ball during its flight from $O$ to $A$.

AQA M1 Q6

Moderate -0.8

6 A van moves from rest on a straight horizontal road.

In a simple model, the first 30 seconds of the motion are represented by three separate stages, each lasting 10 seconds and each with a constant acceleration. During the first stage, the van accelerates from rest to a velocity of $4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
During the second stage, the van accelerates from $4 \mathrm {~ms} ^ { - 1 }$ to $12 \mathrm {~ms} ^ { - 1 }$.
During the third stage, the van accelerates from $12 \mathrm {~ms} ^ { - 1 }$ to $16 \mathrm {~ms} ^ { - 1 }$.
1. Sketch a velocity-time graph to represent the motion of the van during the first 30 seconds of its motion.
2. Find the total distance that the van travels during the 30 seconds.
3. Find the average speed of the van during the 30 seconds.
4. Find the greatest acceleration of the van during the 30 seconds.
In another model of the 30 seconds of the motion, the acceleration of the van is assumed to vary during the first and third stages of the motion, but to be constant during the second stage, as shown in the velocity-time graph below. \includegraphics[max width=\textwidth, alt={}, center]{6151e6ab-30af-4d1c-ab4a-e7dbad170cbf-006_554_1138_1432_539} The velocity of the van takes the same values at the beginning and the end of each stage of the motion as in part (a).
1. State, with a reason, whether the distance travelled by the van during the first 10 seconds of the motion in this model is greater or less than the distance travelled during the same time interval in the model in part (a).
2. Give one reason why this model represents the motion of the van more realistically than the model in part (a).

AQA M1 Q7

Moderate -0.8

7 A builder ties two identical buckets, $P$ and $Q$, to the ends of a light inextensible rope. He hangs the rope over a smooth beam so that the buckets hang in equilibrium, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{6151e6ab-30af-4d1c-ab4a-e7dbad170cbf-007_360_296_502_904} The buckets are each of mass 0.6 kg .

1. State the magnitude of the tension in the rope.
2. State the magnitude and direction of the force exerted on the beam by the rope.
The bucket $Q$ is held at rest while a stone, of mass 0.2 kg , is placed inside it. The system is then released from rest and, in the subsequent motion, bucket $Q$ moves vertically downwards with the stone inside.
1. By forming an equation of motion for each bucket, show that the magnitude of the tension in the rope during the motion is 6.72 newtons, correct to three significant figures.
2. State the magnitude of the force exerted on the beam by the rope while the motion takes place.

AQA M1 Q8

Standard +0.3

8 A rough slope is inclined at an angle of $25 ^ { \circ }$ to the horizontal. A box of weight 80 newtons is on the slope. A rope is attached to the box and is parallel to the slope. The tension in the rope is of magnitude $T$ newtons. The diagram shows the slope, the box and the rope. \includegraphics[max width=\textwidth, alt={}, center]{6151e6ab-30af-4d1c-ab4a-e7dbad170cbf-008_307_469_500_840}

The box is held in equilibrium by the rope.
1. Show that the normal reaction force between the box and the slope is 72.5 newtons, correct to three significant figures.
2. The coefficient of friction between the box and the slope is 0.32 . Find the magnitude of the maximum value of the frictional force which can act on the box.
3. Find the least possible tension in the rope to prevent the box from moving down the slope.
4. Find the greatest possible tension in the rope.
5. Show that the mass of the box is approximately 8.16 kg .
The rope is now released and the box slides down the slope. Find the acceleration of the box. General Certificate of Education
June 2006
Advanced Subsidiary Examination ASSESSMENT and
REALIFIEATIONS
ALLIANCE Tuesday 6 June 20061.30 pm to 2.45 pm \section*{For this paper you must have:}
- an 8-page answer book
- the blue AQA booklet of formulae and statistical tables
You may use a graphics calculator. Time allowed: 1 hour 15 minutes \section*{Instructions}
- Use blue or black ink or ball-point pen. Pencil should only be used for drawing.
- Write the information required on the front of your answer book. The Examining Body for this paper is AQA. The Paper Reference is MM1A/W.
- Answer all questions.
- Show all necessary working; otherwise marks for method may be lost.
- The final answer to questions requiring the use of calculators should be given to three significant figures, unless stated otherwise.
- Take $g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }$, unless stated otherwise.
\section*{Information}
- The maximum mark for this paper is 60 .
- The marks for questions are shown in brackets.
- Unit Mechanics 1A has a written paper and coursework.
\section*{Advice}
- Unless stated otherwise, you may quote formulae, without proof, from the booklet.
Answer all questions. 1 A small stone is dropped from a high bridge and falls vertically.
Find the distance that the stone falls during the first 4 seconds of its motion. (3 marks)
Find the speed of the stone when it has been falling for 4 seconds. 2 A car travels along a straight horizontal road. The motion of the car can be modelled as three separate stages. During the first stage, the car accelerates uniformly from rest to a velocity of $10 \mathrm {~ms} ^ { - 1 }$ in 6 seconds. During the second stage, the car travels with a constant velocity of $10 \mathrm {~ms} ^ { - 1 }$ for a further 4 seconds. During the third stage of the motion, the car travels with a uniform retardation of magnitude $0.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ until it comes to rest.
Show that the time taken for the third stage of the motion is 12.5 seconds.
Sketch a velocity-time graph for the car during the three stages of the motion.
Find the total distance travelled by the car during the motion. 3 A stone rests in equilibrium on a rough plane inclined at an angle of $16 ^ { \circ }$ to the horizontal, as shown in the diagram. The mass of the stone is 0.5 kg . \includegraphics[max width=\textwidth, alt={}, center]{6151e6ab-30af-4d1c-ab4a-e7dbad170cbf-010_222_528_1978_731}
Draw a diagram to show the forces acting on the stone.
Show that the magnitude of the frictional force acting on the stone is 1.35 newtons, correct to three significant figures.
Find the magnitude of the normal reaction force between the stone and the plane.
Hence find an inequality for the value of $\mu$, the coefficient of friction between the stone and the plane. 4 A block $P$ is attached to a can $Q$ by a light inextensible string. The string hangs over a smooth peg so that $P$ and $Q$ hang freely, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{6151e6ab-30af-4d1c-ab4a-e7dbad170cbf-011_246_259_470_886} The block $P$ and the can $Q$ each has mass 0.2 kg . The can $Q$ contains a small stone of mass 0.1 kg . The system is released from rest and the can $Q$ and the stone move vertically downwards.
By forming two equations of motion, show that the magnitude of the acceleration of $P$ and $Q$ is $1.96 \mathrm {~m} \mathrm {~s} ^ { - 2 }$.
Find the magnitude of the reaction force between the can and the stone. 5 The points $A$ and $B$ have position vectors $( 3 \mathbf { i } + 2 \mathbf { j } )$ metres and $( 6 \mathbf { i } - 4 \mathbf { j } )$ metres respectively. The vectors $\mathbf { i }$ and $\mathbf { j }$ are in a horizontal plane.
A particle moves from $A$ to $B$ with constant velocity $( \mathbf { i } - 2 \mathbf { j } ) \mathrm { ms } ^ { - 1 }$. Calculate the time that the particle takes to move from $A$ to $B$.
The particle then moves from $B$ to a point $C$ with a constant acceleration of $2 \mathbf { j } \mathrm {~m} \mathrm {~s} ^ { - 2 }$. It takes 4 seconds to move from $B$ to $C$.
1. Find the position vector of $C$.
2. Find the distance $A C$.

AQA M1 2006 January Q1

6 marks Moderate -0.8

1 A particle $A$ moves across a smooth horizontal surface in a straight line. The particle $A$ has mass 2 kg and speed $6 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. A particle $B$, which has mass 3 kg , is at rest on the surface. The particle $A$ collides with the particle $B$. \includegraphics[max width=\textwidth, alt={}, center]{c220e6c4-2676-4022-8301-7d720dc082b2-2_147_506_644_733}

If, after the collision, $A$ is at rest and $B$ moves away from $A$, find the speed of $B$.
If, after the collision, $A$ and $B$ move away from each other with speeds $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and $4 v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ respectively, as shown in the diagram below, find the value of $v$. \includegraphics[max width=\textwidth, alt={}, center]{c220e6c4-2676-4022-8301-7d720dc082b2-2_138_506_1144_730}

AQA M1 2006 January Q2

5 marks Moderate -0.8

2 A particle $P$ moves with acceleration $( - 3 \mathbf { i } + 12 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }$. Initially the velocity of $P$ is $4 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }$.

Find the velocity of $P$ at time $t$ seconds.
Find the speed of $P$ when $t = 0.5$.

AQA M1 2006 January Q3

6 marks Easy -1.2

3

A small stone is dropped from a height of 25 metres above the ground.
1. Find the time taken for the stone to reach the ground.
2. Find the speed of the stone as it reaches the ground.
A large package is dropped from the same height as the stone. Explain briefly why the time taken for the package to reach the ground is likely to be different from that for the stone.
(2 marks)

AQA M1 2006 January Q4

7 marks Moderate -0.3

4 Water flows in a constant direction at a constant speed of $u \mathrm {~m} \mathrm {~s} ^ { - 1 }$. A boat travels in the water at a speed of $12 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ relative to the water.

The direction in which the boat travels relative to the water is perpendicular to the direction of motion of the water. The resultant velocity of the boat is $V \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at an angle of $74 ^ { \circ }$ to the direction of motion of the water, as shown in the diagram. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c220e6c4-2676-4022-8301-7d720dc082b2-3_120_164_662_488} \captionsetup{labelformat=empty} \caption{Velocity of the water}
\end{figure} \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c220e6c4-2676-4022-8301-7d720dc082b2-3_126_186_667_890} \captionsetup{labelformat=empty} \caption{Velocity of the boat relative to the water}
\end{figure}
1. Find $V$.
2. Show that $u = 3.44$, correct to three significant figures.
The boat changes course so that it travels relative to the water at an angle of $45 ^ { \circ }$ to the direction of motion of the water. The resultant velocity of the boat is now of magnitude $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$. The velocity of the water is unchanged, as shown in the diagram below. $$\xrightarrow { 3.44 \mathrm {~m} \mathrm {~s} ^ { - 1 } }$$
\includegraphics[max width=\textwidth, alt={}]{c220e6c4-2676-4022-8301-7d720dc082b2-3_132_273_1493_895}
Velocity of the boat relative to the water \includegraphics[max width=\textwidth, alt={}, center]{c220e6c4-2676-4022-8301-7d720dc082b2-3_232_355_1498_1384} Find the value of $v$.
(4 marks)

AQA M1 2006 January Q5

9 marks Moderate -0.8

5 A golf ball is projected from a point $O$ with initial velocity $V$ at an angle $\alpha$ to the horizontal. The ball first hits the ground at a point $A$ which is at the same horizontal level as $O$, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{c220e6c4-2676-4022-8301-7d720dc082b2-4_227_602_484_735} It is given that $V \cos \alpha = 6 u$ and $V \sin \alpha = 2.5 u$.

Show that the time taken for the ball to travel from $O$ to $A$ is $\frac { 5 u } { g }$.
Find, in terms of $g$ and $u$, the distance $O A$.
Find $V$, in terms of $u$.
State, in terms of $u$, the least speed of the ball during its flight from $O$ to $A$.

AQA M1 2006 January Q6

16 marks Moderate -0.8

6 A van moves from rest on a straight horizontal road.

In a simple model, the first 30 seconds of the motion are represented by three separate stages, each lasting 10 seconds and each with a constant acceleration. During the first stage, the van accelerates from rest to a velocity of $4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
During the second stage, the van accelerates from $4 \mathrm {~ms} ^ { - 1 }$ to $12 \mathrm {~ms} ^ { - 1 }$.
During the third stage, the van accelerates from $12 \mathrm {~ms} ^ { - 1 }$ to $16 \mathrm {~ms} ^ { - 1 }$.
1. Sketch a velocity-time graph to represent the motion of the van during the first 30 seconds of its motion.
2. Find the total distance that the van travels during the 30 seconds.
3. Find the average speed of the van during the 30 seconds.
4. Find the greatest acceleration of the van during the 30 seconds.
In another model of the 30 seconds of the motion, the acceleration of the van is assumed to vary during the first and third stages of the motion, but to be constant during the second stage, as shown in the velocity-time graph below. \includegraphics[max width=\textwidth, alt={}, center]{c220e6c4-2676-4022-8301-7d720dc082b2-5_554_1138_1432_539} The velocity of the van takes the same values at the beginning and the end of each stage of the motion as in part (a).
1. State, with a reason, whether the distance travelled by the van during the first 10 seconds of the motion in this model is greater or less than the distance travelled during the same time interval in the model in part (a).
2. Give one reason why this model represents the motion of the van more realistically than the model in part (a).

AQA M1 2006 January Q7

5 marks Moderate -0.8

7 A builder ties two identical buckets, $P$ and $Q$, to the ends of a light inextensible rope. He hangs the rope over a smooth beam so that the buckets hang in equilibrium, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{c220e6c4-2676-4022-8301-7d720dc082b2-6_360_296_502_904} The buckets are each of mass 0.6 kg .

1. State the magnitude of the tension in the rope.
2. State the magnitude and direction of the force exerted on the beam by the rope.
The bucket $Q$ is held at rest while a stone, of mass 0.2 kg , is placed inside it. The system is then released from rest and, in the subsequent motion, bucket $Q$ moves vertically downwards with the stone inside.
1. By forming an equation of motion for each bucket, show that the magnitude of the tension in the rope during the motion is 6.72 newtons, correct to three significant figures.
2. State the magnitude of the force exerted on the beam by the rope while the motion takes place.

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AQA M1 2006 January Q6

AQA M1 2006 January Q7

\(\boldsymbol { r }\)	6	7	8
\(\mathbf { P } ( \boldsymbol { R } = \boldsymbol { r } )\)	0.1	0.6	0.3