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AQA Further AS Paper 2 Statistics 2021 June Q7

11 marks Standard +0.3

7 Two employees, $A$ and $B$, both produce the same toy for a company. The company records the total number of errors made per day by each employee during a 40-day period. The results are summarised in the following table. Employee

Number of errors made per day
	0	1	2	3 or more	Total
$A$	8	10	20	2	40
B	18	4	15	3	40
Total	26	14	35	5	80

The company claims that there is an association between employee and number of errors made per day. 7

Test the company's claim, using the $5 \%$ level of significance.
7
By considering observed and expected frequencies, interpret in context the association between employee and number of errors made per day. \includegraphics[max width=\textwidth, alt={}, center]{9be40ed6-6df8-426a-8afd-fefc17287de6-12_2492_1723_217_150}
\includegraphics[max width=\textwidth, alt={}]{9be40ed6-6df8-426a-8afd-fefc17287de6-16_2496_1721_214_148}

AQA Further AS Paper 2 Statistics Specimen Q1

1 marks Easy -1.2

1 The random variable $T$ has probability density defined by $$\mathrm { f } ( t ) = \left\{ \begin{array} { c c } \frac { t } { 8 } & 0 \leq t \leq k \\ 0 & \text { otherwise } \end{array} \right.$$ Find the value of $k$ [0pt] [1 mark] $$\begin{array} { l l l l } \frac { 1 } { 16 } & \frac { 1 } { 4 } & 4 & 16 \end{array}$$

AQA Further AS Paper 2 Statistics Specimen Q2

1 marks Easy -2.0

2 The discrete random variable $X$ has probability distribution defined by $$\mathrm { P } ( X = x ) = \begin{cases} 0.1 & x = 0,1,2,3,4,5,6,7,8,9 \\ 0 & \text { otherwise } \end{cases}$$ Find the value of $\mathrm { P } ( 4 \leq X \leq 7 )$ Circle your answer.

0.2

0.3

0.4

0.5

AQA Further AS Paper 2 Statistics Specimen Q3

4 marks Standard +0.3

3 The discrete random variable $R$ has the following probability distribution.

$\boldsymbol { r }$	- 2	0	$a$	4
$\mathbf { P } ( \boldsymbol { R } = \boldsymbol { r } )$	0.3	$b$	$c$	0.1

It is known that $\mathrm { E } ( R ) = 0.2$ and $\operatorname { Var } ( R ) = 3.56$ Find the values of $a , b$ and $c$.
[0pt] [4 marks]

AQA Further AS Paper 2 Statistics Specimen Q4

3 marks Moderate -0.3

4 The number of printers, $V$, bought during one day from the Verigood store can be modelled by a Poisson distribution with mean 4.5 The number of printers, $W$, bought during one day from the Winnerprint store can be modelled by a Poisson distribution with mean 5.5 4

Find the probability that the total number of printers bought during one day from Verigood and Winnerprint stores is greater than 10.
[0pt] [2 marks] 4
State the circumstance under which the distributional model you used in part (a) would not be valid.
[0pt] [1 mark]

AQA Further AS Paper 2 Statistics Specimen Q5

5 marks Standard +0.8

5 Participants in a school jumping competition gain a total score for each jump based on the length, $L$ metres, jumped beyond a fixed point and a mark, $S$, for style. $L$ may be regarded as a continuous random variable with probability density function $$\mathrm { f } ( l ) = \left\{ \begin{array} { c c } w l & 0 \leq l \leq 15 \\ 0 & \text { otherwise } \end{array} \right.$$ where $w$ is a constant. $S$ may be regarded as a discrete random variable with probability function $$\mathrm { P } ( S = s ) = \left\{ \begin{array} { c l } \frac { 1 } { 15 } s & s = 1,2,3,4,5 \\ 0 & \text { otherwise } \end{array} \right.$$ Assume that $L$ and $S$ are independent. The total score for a participant in this competition, $T$, is given by $T = L ^ { 2 } + \frac { 1 } { 2 } S$ Show that the expected total score for a participant is $114 \frac { 1 } { 3 }$

AQA Further AS Paper 2 Statistics Specimen Q6

8 marks Moderate -0.3

6 The continuous random variable $T$ has probability density function defined by $$\mathrm { f } ( t ) = \left\{ \begin{array} { c c } \frac { 1 } { 3 } & 0 \leq t \leq \frac { 3 } { 2 } \\ \frac { 9 - 2 t } { 18 } & \frac { 3 } { 2 } \leq t \leq \frac { 9 } { 2 } \\ 0 & \text { otherwise } \end{array} \right.$$ 6

1. Sketch this probability density function below. \includegraphics[max width=\textwidth, alt={}, center]{6ccf7d1d-5a7b-47d1-b38e-c7e762204746-07_1009_1041_1073_520} 6
  1. (ii) State the median of $T$. 6
2. 1. Find $\mathrm { E } ( T )$ [0pt] [2 marks]
    6
3. (ii) Given that $\mathrm { E } \left( T ^ { 2 } \right) = \frac { 15 } { 4 }$, find $\operatorname { Var } ( 4 T - 5 )$ [3 marks]

AQA Further AS Paper 2 Statistics Specimen Q7

9 marks Standard +0.3

7 A dairy industry researcher, Robyn, decided to investigate the milk yield, classified as low, medium or high, obtained from four different breeds of cow, $\mathrm { A } , \mathrm { B } , \mathrm { C }$ and D . The milk yield of a sample of 105 cows was monitored and the results are summarised in contingency Table 1.

\multirow{2}{*}{Table 1}		Yield
		Low	Medium	High	Total
\multirow{4}{*}{Breed}	A	4	5	12	21
	B	10	6	4	20
	C	8	17	7	32
	D	5	20	7	32
	Total	27	48	30	105

The sample of cows may be regarded as random.
Robyn decides to carry out a $\chi ^ { 2 }$-test for association between milk yield and breed using the information given in Table 1. 7

Contingency Table 2 gives some of the expected frequencies for this test.
Complete Table 2 with the missing expected values.
\multirow[t]{2}{*}{Table 2} Yield
Low Medium High
\multirow{4}{*}{Breed} A 6
B 5.14 9.14 5.71
C
D 8.23 14.63 9.14
7
(i) For Robyn's test, the test statistic $\sum \frac { ( O - E ) ^ { 2 } } { E } = 19.4$ correct to three significant figures.
Use this information to carry out Robyn's test, using the $1 \%$ level of significance.
7 (b) (ii) By considering the observed frequencies given in Table 1 with the expected frequencies in Table 2, interpret, in context, the association, if any, between milk yield and breed.

AQA Further AS Paper 2 Statistics Specimen Q8

9 marks Standard +0.3

8 In a small town, the number of properties sold during a week in spring by a local estate agent, Keith, can be regarded as occurring independently and with constant mean $\mu$. Data from several years have shown the value of $\mu$ to be 3.5 . A new housing development was built on the outskirts of the town and the properties on this development were offered for sale by the builder of the development, not by the local estate agents. During the first four weeks in spring, when properties on the new development were offered for sale by the builder, Keith sold a total of 8 properties. Keith claims that the sale of new properties by the builder reduced his mean number of properties sold during a week in spring. 8

Investigate Keith's claim, using the $5 \%$ level of significance.
[0pt] [6 marks]
8
For your test carried out in part (a) state, in context, the meaning of a Type II error.
[0pt] [1 mark]
8
State one advantage and one disadvantage of using a 1\% significance level rather than a 5\% level of significance in a hypothesis test.
[0pt] [2 marks]

AQA Further AS Paper 2 Mechanics 2020 June Q1

1 marks Moderate -0.8

1 In this question use $g = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ A particle of mass 2 kg is attached to one end of a light elastic string of natural length 0.5 metres and modulus of elasticity 100 N . The other end of the string is attached to the point $O$. Find the extension of the elastic string when the particle hangs in equilibrium vertically below $O$. Circle your answer.
0.01 m
0.1 m
0.2 m
0.4 m

AQA Further AS Paper 2 Mechanics 2020 June Q2

1 marks Moderate -0.8

2 An object moves under the action of a single force $F$ newtons.
It is given that $F = 6 x ^ { 2 }$, where $x$ represents the displacement in metres from the initial position of the object. Find the work done by $F$ in moving the object from $x = 1$ to $x = 2$ Circle your answer.
[0pt] [1 mark]
12 J
14 J
18J
42 J

AQA Further AS Paper 2 Mechanics 2020 June Q3

3 marks Easy -1.2

3 The time taken for the moon to make one complete orbit around Earth is approximately 27.3 days. Model this orbit as circular, with a radius of $3.84 \times 10 ^ { 8 }$ metres.
Find the approximate speed of the moon relative to Earth, in metres per second.

AQA Further AS Paper 2 Mechanics 2020 June Q4

4 marks Moderate -0.3

4 A particle $P$, of mass $m \mathrm {~kg}$, collides with a particle $Q$, of mass 2 kg Immediately before the collision the velocity of $P$ is $\left[ \begin{array} { c } 4 \\ - 2 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }$ and the velocity of $Q$ is $\left[ \begin{array} { c } - 3 \\ 5 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }$ As a result of the collision the particles coalesce into a single particle which moves with velocity $\left[ \begin{array} { l } k \\ 0 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }$, where $k$ is a constant. Find the value of $k$.

AQA Further AS Paper 2 Mechanics 2020 June Q5

5 marks Moderate -0.3

5 A train consisting of an engine and eight carriages moves on a straight horizontal track. A constant resistive force of 2400 N acts on the engine.
A constant resistive force of 300 N acts on each of the eight carriages.
The maximum speed of the train on the track is $120 \mathrm {~km} \mathrm {~h} ^ { - 1 }$ Find the maximum power output of the engine.
Fully justify your answer.

AQA Further AS Paper 2 Mechanics 2020 June Q6

5 marks Standard +0.3

6 The magnitude of the gravitational force $F$ between two planets of masses $m _ { 1 }$ and $m _ { 2 }$ with centres at a distance $d$ apart is given by $$F = \frac { G m _ { 1 } m _ { 2 } } { d ^ { 2 } }$$ where $G$ is a constant.
6

Show that $G$ must have dimensions $L ^ { 3 } M ^ { - 1 } T ^ { - 2 }$, where $L$ represents length, $M$ represents mass and $T$ represents time.
6
The lifetime $t$ of a planet is thought to depend on its mass $m$, its radius $r$, the constant $G$ and a dimensionless constant $k$ such that $$t = k m ^ { a } r ^ { b } G ^ { c }$$ where $a , b$ and $c$ are constants.
Determine the values of $a , b$ and $c$.

AQA Further AS Paper 2 Mechanics 2020 June Q7

9 marks Standard +0.3

7 In this question use $g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ As part of a competition, Jo-Jo makes a small pop-up rocket.
It is operated by pressing the rocket vertically downwards to compress a light spring, which is positioned underneath the rocket. The rocket is released from rest and moves vertically upwards.
The mass of the rocket is 18 grams and the stiffness constant of the spring is $60 \mathrm { Nm } ^ { - 1 }$ Initially the spring is compressed by 3 cm
7

Find the speed of the rocket when the spring first reaches its natural length.
7
By considering energy find the distance that the rocket rises. 7
In order to win a prize in the competition, the rocket must reach a point which is 15 cm vertically above its starting position. With reference to the assumptions you have made, determine if Jo-Jo wins a prize or not. Fully justify your answer.

AQA Further AS Paper 2 Mechanics 2020 June Q14

4 marks Moderate -0.5

14 J
18J
42 J 3 The time taken for the moon to make one complete orbit around Earth is approximately 27.3 days. Model this orbit as circular, with a radius of $3.84 \times 10 ^ { 8 }$ metres.
Find the approximate speed of the moon relative to Earth, in metres per second.
4 A particle $P$, of mass $m \mathrm {~kg}$, collides with a particle $Q$, of mass 2 kg Immediately before the collision the velocity of $P$ is $\left[ \begin{array} { c } 4 \\ - 2 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }$ and the velocity of $Q$ is $\left[ \begin{array} { c } - 3 \\ 5 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }$ As a result of the collision the particles coalesce into a single particle which moves with velocity $\left[ \begin{array} { l } k \\ 0 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }$, where $k$ is a constant. Find the value of $k$.
5 A train consisting of an engine and eight carriages moves on a straight horizontal track. A constant resistive force of 2400 N acts on the engine.
A constant resistive force of 300 N acts on each of the eight carriages.
The maximum speed of the train on the track is $120 \mathrm {~km} \mathrm {~h} ^ { - 1 }$ Find the maximum power output of the engine.
Fully justify your answer.
6 The magnitude of the gravitational force $F$ between two planets of masses $m _ { 1 }$ and $m _ { 2 }$ with centres at a distance $d$ apart is given by $$F = \frac { G m _ { 1 } m _ { 2 } } { d ^ { 2 } }$$ where $G$ is a constant.
6

Show that $G$ must have dimensions $L ^ { 3 } M ^ { - 1 } T ^ { - 2 }$, where $L$ represents length, $M$ represents mass and $T$ represents time.
6

The lifetime $t$ of a planet is thought to depend on its mass $m$, its radius $r$, the constant $G$ and a dimensionless constant $k$ such that $$t = k m ^ { a } r ^ { b } G ^ { c }$$ where $a , b$ and $c$ are constants.
Determine the values of $a , b$ and $c$.
7 In this question use $g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ As part of a competition, Jo-Jo makes a small pop-up rocket.
It is operated by pressing the rocket vertically downwards to compress a light spring, which is positioned underneath the rocket. The rocket is released from rest and moves vertically upwards.
The mass of the rocket is 18 grams and the stiffness constant of the spring is $60 \mathrm { Nm } ^ { - 1 }$ Initially the spring is compressed by 3 cm
7

Find the speed of the rocket when the spring first reaches its natural length.
7
By considering energy find the distance that the rocket rises. 7
In order to win a prize in the competition, the rocket must reach a point which is 15 cm vertically above its starting position. With reference to the assumptions you have made, determine if Jo-Jo wins a prize or not. Fully justify your answer.
8 Two smooth spheres $A$ and $B$ have the same radius and are free to move on a smooth horizontal surface. The masses of $A$ and $B$ are $2 m$ and $m$ respectively.
Both $A$ and $B$ are initially at rest.
The sphere $A$ is set in motion directly towards $B$ with speed $3 u$ and at the same time $B$ is set in motion directly towards $A$ with speed $2 u$. Subsequently $A$ and $B$ collide directly. $A$ The coefficient of restitution between the spheres is $e$.
8
Show that the speed of $B$ after the collision is given by $$\frac { 2 u ( 2 + 5 e ) } { 3 }$$ \section*{Question 8 continues on the next page} 8
Given that the direction of the velocity of $A$ is reversed during the collision, find the range of possible values of $e$. Fully justify your answer.
[0pt] [4 marks]
8

Given that the magnitude of the impulse that $A$ exerts on $B$ is $\frac { 19 m u } { 3 }$, find the value of $e$.

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AQA Further AS Paper 2 Mechanics Specimen Q1

1 marks Easy -1.2

1 A child, of mass 40 kg , moves at constant speed of $5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ on a fairground ride.
The path of the child is a circle of radius 4 metres.
Find the magnitude of the resultant force acting on the child.
Circle your answer.
[0pt] [1 mark]
6.3 N
50 N
130 N
250 N

AQA Further AS Paper 2 Mechanics Specimen Q2

1 marks Easy -1.2

2 The graph shows how a force, $F$, varies with time during a period of 0.8 seconds. \includegraphics[max width=\textwidth, alt={}, center]{18522f4c-4aa2-4ef5-898f-5ad2b06e287c-03_440_960_568_516} Find the magnitude of the impulse of $F$ during the 0.8 seconds.
Circle your answer.
[0pt] [1 mark]
1.0 Ns
1.6 Ns
2.2 Ns
3.2 Ns Turn over for the next question

AQA Further AS Paper 2 Mechanics Specimen Q3

4 marks Moderate -0.3

3 A tank full of liquid has a hole made in its base.
Two students, Sarah and David, propose two different models for the speed, $v$, at which liquid exits the tank. David thinks that $v$ will depend on the height of the liquid in the tank, $h$, the acceleration due to gravity, $g$, and the density of the liquid, $\rho$, such that $v \propto g ^ { a } h ^ { b } \rho ^ { c }$ where $a$, $b$ and $c$ are constants. Sarah thinks that $v$ will not depend on the density of the liquid and suggests the model $v \propto g ^ { a } h ^ { b }$ 3

By considering dimensions, explain which student's model should be rejected.
[0pt] [2 marks]
3
Find the values of the constants in order for the model that you did not reject in part (a) to be dimensionally consistent.
[0pt] [2 marks]

AQA Further AS Paper 2 Mechanics Specimen Q4

5 marks Moderate -0.3

4 A cricket ball of mass 156 grams is thrown from a point which is 1.5 metres above the ground, with a speed of $12 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ A tennis ball of mass 58 grams is thrown from the same point, with the same speed.
Prove that both balls hit the ground with the same speed.
Clearly state any assumptions you have made and how you have used them.
[0pt] [5 marks]

AQA Further AS Paper 2 Mechanics Specimen Q5

4 marks Standard +0.3

5 Two small smooth discs, $C$ and $D$, have equal radii and masses of 2 kg and 3 kg respectively. The discs are sliding on a smooth horizontal surface towards each other and collide directly. Disc $C$ has speed $4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and disc $D$ has speed $2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ as they collide. The coefficient of restitution between $C$ and $D$ is 0.6 The diagram shows the discs, viewed from above, before the collision. \includegraphics[max width=\textwidth, alt={}, center]{18522f4c-4aa2-4ef5-898f-5ad2b06e287c-06_343_712_868_753} 5

Show that the speed of $D$ immediately after the collision is $1.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }$, correct to 2 significant figures.
5
Find the speed of $C$ immediately after the collision.
[0pt] [2 marks]
5
In fact the horizontal surface on which the discs are sliding is not smooth.
Explain how the introduction of friction will affect your answer to part (b).
[0pt] [2 marks]
Turn over for the next question

AQA Further AS Paper 2 Mechanics Specimen Q6

4 marks Standard +0.3

6 A car, of mass 1200 kg , moves on a straight horizontal road where it has a maximum speed of $40 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ When the car travels at a speed of $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ it experiences a resistance force which can be modelled as being of magnitude 30 v newtons. 6

Show that the power output of the car is 48000 W , when it is travelling at its maximum speed. 6
Find the maximum acceleration of the car when it is travelling at a speed of $25 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ [0pt] [4 marks]

AQA Further AS Paper 2 Mechanics Specimen Q7

3 marks Standard +0.3

7 A disc, of mass 0.15 kg , slides across a smooth horizontal table and collides with a vertical wall which is perpendicular to the path of the disc. The disc is in contact with the wall for 0.02 seconds and then rebounds.
A possible model for the force, $F$ newtons, exerted on the disc by the wall, whilst in contact, is given by $$F = k t ^ { 2 } ( t - b ) ^ { 2 } \quad \text { for } \quad 0 \leq t \leq 0.020$$ where $k$ and $b$ are constants.
The force is initially zero and becomes zero again as the disc loses contact with the wall. 7

State the value of $b$.
7
Find the magnitude of the impulse on the disc, giving your answer in terms of $k$.
7
The disc is travelling at $4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ when it hits the wall.
The disc rebounds with a speed of $2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ Find $k$.
[0pt] [3 marks]

AQA Further AS Paper 2 Mechanics Specimen Q8

6 marks Challenging +1.2

8 In this question use $g = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }$.
A particle, of mass 2 kg , is attached to one end of a light elastic string of natural length 0.2 metres. The other end of the string is attached to a fixed point $O$.
The particle is pulled down and released from rest at a point 0.6 metres directly below $O$.
The particle then moves vertically and next comes to rest when it is 0.1 metres below $O$.
Assume that no air resistance acts on the particle.
8

Find the maximum speed of the particle.
[0pt] [6 marks]
8
Describe one way in which the model you have used could be refined.

\(\boldsymbol { r }\)	- 2	0	\(a\)	4
\(\mathbf { P } ( \boldsymbol { R } = \boldsymbol { r } )\)	0.3	\(b\)	\(c\)	0.1

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AQA Further AS Paper 2 Statistics 2021 June Q7

AQA Further AS Paper 2 Statistics Specimen Q1

AQA Further AS Paper 2 Statistics Specimen Q2

AQA Further AS Paper 2 Statistics Specimen Q3

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AQA Further AS Paper 2 Statistics Specimen Q5

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AQA Further AS Paper 2 Statistics Specimen Q7

AQA Further AS Paper 2 Statistics Specimen Q8

AQA Further AS Paper 2 Mechanics 2020 June Q1

AQA Further AS Paper 2 Mechanics 2020 June Q2

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AQA Further AS Paper 2 Mechanics 2020 June Q4

AQA Further AS Paper 2 Mechanics 2020 June Q5

AQA Further AS Paper 2 Mechanics 2020 June Q6

AQA Further AS Paper 2 Mechanics 2020 June Q7

AQA Further AS Paper 2 Mechanics 2020 June Q14

AQA Further AS Paper 2 Mechanics Specimen Q1

AQA Further AS Paper 2 Mechanics Specimen Q2

AQA Further AS Paper 2 Mechanics Specimen Q3

AQA Further AS Paper 2 Mechanics Specimen Q4

AQA Further AS Paper 2 Mechanics Specimen Q5

AQA Further AS Paper 2 Mechanics Specimen Q6

AQA Further AS Paper 2 Mechanics Specimen Q7

AQA Further AS Paper 2 Mechanics Specimen Q8

\multirow[t]{2}{*}{Table 2}		Yield
		Low	Medium	High
\multirow{4}{*}{Breed}	A			6
	B	5.14	9.14	5.71
	C
	D	8.23	14.63	9.14

Question number	Additional page, if required. Write the question numbers in the left-hand margin.
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