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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 .

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)

OCR MEI S3 Q2

20 marks Standard +0.3

2 Geoffrey is a university lecturer. He has to prepare five questions for an examination. He knows by experience that it takes about 3 hours to prepare a question, and he models the time (in minutes) taken to prepare one by the Normally distributed random variable $X$ with mean 180 and standard deviation 12, independently for all questions.

One morning, Geoffrey has a gap of 2 hours 50 minutes ( 170 minutes) between other activities. Find the probability that he can prepare a question in this time.
One weekend, Geoffrey can devote 14 hours to preparing the complete examination paper. Find the probability that he can prepare all five questions in this time. A colleague, Helen, has to check the questions.
She models the time (in minutes) to check a question by the Normally distributed random variable $Y$ with mean 50 and standard deviation 6, independently for all questions and independently of $X$. Find the probability that the total time for Geoffrey to prepare a question and Helen to check it exceeds 4 hours.
When working under pressure of deadlines, Helen models the time to check a question in a different way. She uses the Normally distributed random variable $\frac { 1 } { 4 } X$, where $X$ is as above. Find the length of time, as given by this model, which Helen needs to ensure that, with probability 0.9 , she has time to check a question. Ian, an educational researcher, suggests that a better model for the time taken to prepare a question would be a constant $k$ representing "thinking time" plus a random variable $T$ representing the time required to write the question itself, independently for all questions.
Taking $k$ as 45 and $T$ as Normally distributed with mean 120 and standard deviation 10 (all units are minutes), find the probability according to Ian's model that a question can be prepared in less than 2 hours 30 minutes. Juliet, an administrator, proposes that the examination should be reduced in time and shorter questions should be used.
Juliet suggests that Ian's model should be used for the time taken to prepare such shorter questions but with $k = 30$ and $T$ replaced by $\frac { 3 } { 5 } T$. Find the probability as given by this model that a question can be prepared in less than $1 \frac { 3 } { 4 }$ hours.

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 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

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

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.

AQA M1 2006 January Q8

16 marks 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]{c220e6c4-2676-4022-8301-7d720dc082b2-7_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.

AQA M1 2010 January Q1

3 marks Easy -1.2

1 Two particles, $A$ and $B$, are travelling in the same direction along a straight line on a smooth horizontal surface. Particle $A$ has mass 3 kg and particle $B$ has mass 7 kg . Particle $A$ has a speed of $20 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and particle $B$ has a speed of $10 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{fe8c1ea4-cf4d-4741-8af5-03e8c2c88559-2_186_835_653_593} Particle $A$ and particle $B$ collide and coalesce to form a single particle. Find the speed of this single particle after the collision.

AQA M1 2010 January Q2

10 marks Easy -1.2

2 A sprinter accelerates from rest at a constant rate for the first 10 metres of a 100 -metre race. He takes 2.5 seconds to run the first 10 metres.

Find the acceleration of the sprinter during the first 2.5 seconds of the race.
Show that the speed of the sprinter at the end of the first 2.5 seconds of the race is $8 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
The sprinter completes the 100 -metre race, travelling the remaining 90 metres at a constant speed of $8 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. Find the total time taken for the sprinter to travel the 100 metres.
Calculate the average speed of the sprinter during the 100 -metre race.

AQA M1 2010 January Q3

5 marks Easy -1.2

3 A particle of mass 3 kg is on a smooth slope inclined at $60 ^ { \circ }$ to the horizontal. The particle is held at rest by a force of $T$ newtons parallel to the slope, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{fe8c1ea4-cf4d-4741-8af5-03e8c2c88559-2_337_284_2023_879}

Draw a diagram to show all the forces acting on the particle.
Show that the magnitude of the normal reaction acting on the particle is 14.7 newtons.
Find $T$.

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