Questions Paper 3 - A-Level Maths

1. find the magnitude of the frictional force acting on $B$,
2. state the direction of the frictional force acting on $B$. The horizontal force of magnitude $X$ newtons is now removed and $B$ moves down the plane. Given that the coefficient of friction between $B$ and the plane is 0.5
find the acceleration of $B$ down the plane.

Edexcel Paper 3 2022 June Q3

\hspace{0pt} [In this question, $\mathbf { i }$ and $\mathbf { j }$ are horizontal unit vectors.]

A particle $P$ of mass 4 kg is at rest at the point $A$ on a smooth horizontal plane.
At time $t = 0$, two forces, $\mathbf { F } _ { 1 } = ( 4 \mathbf { i } - \mathbf { j } ) \mathrm { N }$ and $\mathbf { F } _ { 2 } = ( \lambda \mathbf { i } + \mu \mathbf { j } ) \mathrm { N }$, where $\lambda$ and $\mu$ are constants, are applied to $P$ Given that $P$ moves in the direction of the vector ( $3 \mathbf { i } + \mathbf { j }$ )

show that $$\lambda - 3 \mu + 7 = 0$$ At time $t = 4$ seconds, $P$ passes through the point $B$.
Given that $\lambda = 2$
find the length of $A B$.

Edexcel Paper 3 2022 June Q4

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{414946db-64d7-44b8-801d-2c7805ee9cc6-12_716_1191_246_438} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} A uniform rod $A B$ has mass $M$ and length $2 a$
A particle of mass $2 M$ is attached to the rod at the point $C$, where $A C = 1.5 a$
The rod rests with its end $A$ on rough horizontal ground.
The rod is held in equilibrium at an angle $\theta$ to the ground by a light string that is attached to the end $B$ of the rod. The string is perpendicular to the rod, as shown in Figure 2.

Explain why the frictional force acting on the rod at $A$ acts horizontally to the right on the diagram. The tension in the string is $T$
Show that $T = 2 M g \cos \theta$ Given that $\cos \theta = \frac { 3 } { 5 }$
show that the magnitude of the vertical force exerted by the ground on the rod at $A$ is $\frac { 57 M g } { 25 }$ The coefficient of friction between the rod and the ground is $\mu$
Given that the rod is in limiting equilibrium,
show that $\mu = \frac { 8 } { 19 }$

Edexcel Paper 3 2022 June Q5

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{414946db-64d7-44b8-801d-2c7805ee9cc6-16_303_1266_237_404} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} A golf ball is at rest at the point $A$ on horizontal ground.
The ball is hit and initially moves at an angle $\alpha$ to the ground.
The ball first hits the ground at the point $B$, where $A B = 120 \mathrm {~m}$, as shown in Figure 3.
The motion of the ball is modelled as that of a particle, moving freely under gravity, whose initial speed is $U \mathrm {~m} \mathrm {~s} ^ { - 1 }$ Using this model,

show that $U ^ { 2 } \sin \alpha \cos \alpha = 588$ The ball reaches a maximum height of 10 m above the ground.
Show that $U ^ { 2 } = 1960$ In a refinement to the model, the effect of air resistance is included.
The motion of the ball, from $A$ to $B$, is now modelled as that of a particle whose initial speed is $V \mathrm {~m} \mathrm {~s} ^ { - 1 }$ This refined model is used to calculate a value for $V$
State which is greater, $U$ or $V$, giving a reason for your answer.
State one further refinement to the model that would make the model more realistic.

Edexcel Paper 3 2022 June Q1

George throws a ball at a target 15 times.

Each time George throws the ball, the probability of the ball hitting the target is 0.48
The random variable $X$ represents the number of times George hits the target in 15 throws.

Find
1. $\mathrm { P } ( X = 3 )$
2. $\mathrm { P } ( X \geqslant 5 )$ George now throws the ball at the target 250 times.
Use a normal approximation to calculate the probability that he will hit the target more than 110 times.

Edexcel Paper 3 2022 June Q2

A manufacturer uses a machine to make metal rods.

The length of a metal rod, $L \mathrm {~cm}$, is normally distributed with

a mean of 8 cm
a standard deviation of $x \mathrm {~cm}$

Given that the proportion of metal rods less than 7.902 cm in length is $2.5 \%$

show that $x = 0.05$ to 2 decimal places.
Calculate the proportion of metal rods that are between 7.94 cm and 8.09 cm in length. The cost of producing a single metal rod is 20p
A metal rod
- where $L < 7.94$ is sold for scrap for 5 p
- where $7.94 \leqslant L \leqslant 8.09$ is sold for 50 p
- where $L > 8.09$ is shortened for an extra cost of 10 p and then sold for 50 p
- Calculate the expected profit per 500 of the metal rods.
Give your answer to the nearest pound. The same manufacturer makes metal hinges in large batches.
The hinges each have a probability of 0.015 of having a fault.
A random sample of 200 hinges is taken from each batch and the batch is accepted if fewer than 6 hinges are faulty. The manufacturer's aim is for 95\% of batches to be accepted.
Explain whether the manufacturer is likely to achieve its aim.

Edexcel Paper 3 2022 June Q3

Dian uses the large data set to investigate the Daily Total Rainfall, $r \mathrm {~mm}$, for Camborne.
1. Write down how a value of $0 < r \leqslant 0.05$ is recorded in the large data set.
Dian uses the data for the 31 days of August 2015 for Camborne and calculates the following statistics $$n = 31 \quad \sum r = 174.9 \quad \sum r ^ { 2 } = 3523.283$$
Use these statistics to calculate
1. the mean of the Daily Total Rainfall in Camborne for August 2015,
2. the standard deviation of the Daily Total Rainfall in Camborne for August 2015. Dian believes that the mean Daily Total Rainfall in August is less in the South of the UK than in the North of the UK.
  The mean Daily Total Rainfall in Leuchars for August 2015 is 1.72 mm to 2 decimal places.
State, giving a reason, whether this provides evidence to support Dian's belief. Dian uses the large data set to estimate the proportion of days with no rain in Camborne for 1987 to be 0.27 to 2 decimal places.
Explain why the distribution $\mathrm { B } ( 14,0.27 )$ might not be a reasonable model for the number of days without rain for a 14-day summer event.

Edexcel Paper 3 2022 June Q4

A dentist knows from past records that $10 \%$ of customers arrive late for their appointment.

A new manager believes that there has been a change in the proportion of customers who arrive late for their appointment. A random sample of 50 of the dentist's customers is taken.

Write down
- a null hypothesis corresponding to no change in the proportion of customers who arrive late
- an alternative hypothesis corresponding to the manager's belief
- Using a $5 \%$ level of significance, find the critical region for a two-tailed test of the null hypothesis in (a) You should state the probability of rejection in each tail, which should be less than 0.025
- Find the actual level of significance of the test based on your critical region from part (b)
The manager observes that 15 of the 50 customers arrived late for their appointment.
With reference to part (b), comment on the manager's belief.

Edexcel Paper 3 2022 June Q5

A company has 1825 employees.

The employees are classified as professional, skilled or elementary.
The following table shows

the number of employees in each classification
the two areas, $A$ or $B$, where the employees live

\cline { 2 - 3 } \multicolumn{1}{c\|}{}	$\boldsymbol { A }$	$\boldsymbol { B }$
Professional	740	380
Skilled	275	90
Elementary	260	80

An employee is chosen at random.
Find the probability that this employee

is skilled,
lives in area $B$ and is not a professional. Some classifications of employees are more likely to work from home.
- $65 \%$ of professional employees in both area $A$ and area $B$ work from home
- $40 \%$ of skilled employees in both area $A$ and area $B$ work from home
- $5 \%$ of elementary employees in both area $A$ and area $B$ work from home
- Event $F$ is that the employee is a professional
- Event $H$ is that the employee works from home
- Event $R$ is that the employee is from area $A$
- Using this information, complete the Venn diagram on the opposite page.
- Find $\mathrm { P } \left( R ^ { \prime } \cap F \right)$
- Find $\mathrm { P } \left( [ H \cup R ] ^ { \prime } \right)$
- Find $\mathrm { P } ( F \mid H )$
\includegraphics[max width=\textwidth, alt={}]{3a09f809-fa28-4b3d-bb69-ea074433bd8f-13_872_1020_294_525}
Turn over for a spare diagram if you need to redraw your Venn diagram. Only use this diagram if you need to redraw your Venn diagram.
\includegraphics[max width=\textwidth, alt={}, center]{3a09f809-fa28-4b3d-bb69-ea074433bd8f-15_872_1017_392_525}

Edexcel Paper 3 2022 June Q6

6. Anna is investigating the relationship between exercise and resting heart rate. She takes a random sample of 19 people in her year at school and records for each person

their resting heart rate, $h$ beats per minute
the number of minutes, $m$, spent exercising each week

Her results are shown on the scatter diagram.
\includegraphics[max width=\textwidth, alt={}, center]{3a09f809-fa28-4b3d-bb69-ea074433bd8f-16_531_551_653_740}

Interpret the nature of the relationship between $h$ and $m$ Anna codes the data using the formulae $$\begin{aligned} & x = \log _ { 10 } m
& y = \log _ { 10 } h \end{aligned}$$ The product moment correlation coefficient between $x$ and $y$ is - 0.897
Test whether or not there is significant evidence of a negative correlation between $x$ and $y$
You should
- state your hypotheses clearly
- use a $5 \%$ level of significance
- state the critical value used
The equation of the line of best fit of $y$ on $x$ is $$y = - 0.05 x + 1.92$$
Use the equation of the line of best fit of $y$ on $x$ to find a model for $h$ on $m$ in the form $$h = a m ^ { k }$$ where $a$ and $k$ are constants to be found.

Edexcel Paper 3 2023 June Q1

A car is initially at rest on a straight horizontal road.

The car then accelerates along the road with a constant acceleration of $3.2 \mathrm {~ms} ^ { - 2 }$
Find

the speed of the car after 5 s ,
the distance travelled by the car in the first 5 s .

Edexcel Paper 3 2023 June Q2

2. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{f9dc8158-8ed8-4138-9c75-050cf52e6f7e-04_83_659_267_703} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} A particle $P$ has mass 5 kg .
The particle is pulled along a rough horizontal plane by a horizontal force of magnitude 28 N . The only resistance to motion is a frictional force of magnitude $F$ newtons, as shown in Figure 1.

Find the magnitude of the normal reaction of the plane on $P$ The particle is accelerating along the plane at $1.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }$
Find the value of $F$ The coefficient of friction between $P$ and the plane is $\mu$
Find the value of $\mu$, giving your answer to 2 significant figures.

Edexcel Paper 3 2023 June Q3

At time $t$ seconds, where $t \geqslant 0$, a particle $P$ has velocity $\mathbf { v } \mathrm { ms } ^ { - 1 }$ where

$$\mathbf { v } = \left( t ^ { 2 } - 3 t + 7 \right) \mathbf { i } + \left( 2 t ^ { 2 } - 3 \right) \mathbf { j }$$ Find

the speed of $P$ at time $t = 0$
the value of $t$ when $P$ is moving parallel to $( \mathbf { i } + \mathbf { j } )$
the acceleration of $P$ at time $t$ seconds
the value of $t$ when the direction of the acceleration of $P$ is perpendicular to $\mathbf { i }$

Edexcel Paper 3 2023 June Q4

\hspace{0pt} [In this question, $\mathbf { i }$ and $\mathbf { j }$ are horizontal unit vectors and position vectors are given relative to a fixed origin $O$ ]

A particle $P$ is moving on a smooth horizontal plane.
The particle has constant acceleration $( 2.4 \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }$
At time $t = 0 , P$ passes through the point $A$.
At time $t = 5 \mathrm {~s} , P$ passes through the point $B$.
The velocity of $P$ as it passes through $A$ is $( - 16 \mathbf { i } - 3 \mathbf { j } ) \mathrm { ms } ^ { - 1 }$

Find the speed of $P$ as it passes through $B$. The position vector of $A$ is $( 44 \mathbf { i } - 10 \mathbf { j } ) \mathrm { m }$.
At time $t = T$ seconds, where $T > 5 , P$ passes through the point $C$.
The position vector of $C$ is $( 4 \mathbf { i } + c \mathbf { j } ) \mathrm { m }$.
Find the value of $T$.
Find the value of $c$.

Edexcel Paper 3 2023 June Q5

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{f9dc8158-8ed8-4138-9c75-050cf52e6f7e-12_965_1226_244_422} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} A small ball is projected with speed $28 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ from a point $O$ on horizontal ground. After moving for $T$ seconds, the ball passes through the point $A$. The point $A$ is 40 m horizontally and 20 m vertically from the point $O$, as shown in Figure 2. The motion of the ball from $O$ to $A$ is modelled as that of a particle moving freely under gravity. Given that the ball is projected at an angle $\alpha$ to the ground, use the model to

show that $T = \frac { 10 } { 7 \cos \alpha }$
show that $\tan ^ { 2 } \alpha - 4 \tan \alpha + 3 = 0$
find the greatest possible height, in metres, of the ball above the ground as the ball moves from $O$ to $A$. The model does not include air resistance.
State one other limitation of the model.

Edexcel Paper 3 2023 June Q6

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{f9dc8158-8ed8-4138-9c75-050cf52e6f7e-16_408_967_246_539} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} A $\operatorname { rod } A B$ has mass $M$ and length $2 a$.
The rod has its end $A$ on rough horizontal ground and its end $B$ against a smooth vertical wall. The rod makes an angle $\theta$ with the ground, as shown in Figure 3.
The rod is at rest in limiting equilibrium.

State the direction (left or right on Figure 3 above) of the frictional force acting on the $\operatorname { rod }$ at $A$. Give a reason for your answer. The magnitude of the normal reaction of the wall on the rod at $B$ is $S$.
In an initial model, the rod is modelled as being uniform.
Use this initial model to answer parts (b), (c) and (d).
By taking moments about $A$, show that $$S = \frac { 1 } { 2 } M g \cot \theta$$ The coefficient of friction between the rod and the ground is $\mu$
Given that $\tan \theta = \frac { 3 } { 4 }$
find the value of $\mu$
find, in terms of $M$ and $g$, the magnitude of the resultant force acting on the rod at $A$. In a new model, the rod is modelled as being non-uniform, with its centre of mass closer to $B$ than it is to $A$. A new value for $S$ is calculated using this new model, with $\tan \theta = \frac { 3 } { 4 }$
State whether this new value for $S$ is larger, smaller or equal to the value that $S$ would take using the initial model. Give a reason for your answer.

Edexcel Paper 3 2023 June Q1

The Venn diagram, where $p$ and $q$ are probabilities, shows the three events $A , B$ and $C$ and their associated probabilities.
\includegraphics[max width=\textwidth, alt={}, center]{a067577e-e2a6-440b-9d22-d558fade15f0-02_745_935_347_566}
1. Find $\mathrm { P } ( A )$
The events $B$ and $C$ are independent.
Find the value of $p$ and the value of $q$
Find $\mathrm { P } \left( A \mid B ^ { \prime } \right)$

Edexcel Paper 3 2023 June Q2

A machine fills packets with sweets and $\frac { 1 } { 7 }$ of the packets also contain a prize.

The packets of sweets are placed in boxes before being delivered to shops. There are 40 packets of sweets in each box. The random variable $T$ represents the number of packets of sweets that contain a prize in each box.

State a condition needed for $T$ to be modelled by $\mathrm { B } \left( 40 , \frac { 1 } { 7 } \right)$ A box is selected at random.
Using $T \sim \mathrm {~B} \left( 40 , \frac { 1 } { 7 } \right)$ find
1. the probability that the box has exactly 6 packets containing a prize,
2. the probability that the box has fewer than 3 packets containing a prize. Kamil's sweet shop buys 5 boxes of these sweets.
Find the probability that exactly 2 of these 5 boxes have fewer than 3 packets containing a prize. Kamil claims that the proportion of packets containing a prize is less than $\frac { 1 } { 7 }$
A random sample of 110 packets is taken and 9 packets contain a prize.
Use a suitable test to assess Kamil's claim. You should
- state your hypotheses clearly
- use a $5 \%$ level of significance

Edexcel Paper 3 2023 June Q3

Ben is studying the Daily Total Rainfall, $x \mathrm {~mm}$, in Leeming for 1987

He used all the data from the large data set and summarised the information in the following table.

$x$	0	$0.1 - 0.5$	$0.6 - 1.0$	$1.1 - 1.9$	$2.0 - 4.0$	$4.1 - 6.9$	$7.0 - 12.0$	$12.1 - 20.9$	$21.0 - 32.0$	$\operatorname { tr }$
Frequency	55	18	18	21	17	9	9	6	2	29

Explain how the data will need to be cleaned before Ben can start to calculate statistics such as the mean and standard deviation. Using all 184 of these values, Ben estimates $\sum x = 390$ and $\sum x ^ { 2 } = 4336$
Calculate estimates for
1. the mean Daily Total Rainfall,
2. the standard deviation of the Daily Total Rainfall. Ben suggests using the statistic calculated in part (b)(i) to estimate the annual mean Daily Total Rainfall in Leeming for 1987
Using your knowledge of the large data set,
1. give a reason why these data would not be suitable,
2. state, giving a reason, how you would expect the estimate in part (b)(i) to differ from the actual annual mean Daily Total Rainfall in Leeming for 1987

Edexcel Paper 3 2023 June Q4

A study was made of adult men from region $A$ of a country. It was found that their heights were normally distributed with a mean of 175.4 cm and standard deviation 6.8 cm .
1. Find the proportion of these men that are taller than 180 cm .
A student claimed that the mean height of adult men from region $B$ of this country was different from the mean height of adult men from region $A$. A random sample of 52 adult men from region $B$ had a mean height of 177.2 cm
The student assumed that the standard deviation of heights of adult men was 6.8 cm both for region $A$ and region $B$.
Use a suitable test to assess the student's claim. You should
- state your hypotheses clearly
- use a $5 \%$ level of significance
- Find the $p$-value for the test in part (b)

Edexcel Paper 3 2023 June Q5

Tisam is playing a game.

She uses a ball, a cup and a spinner.
The random variable $X$ represents the number the spinner lands on when it is spun. The probability distribution of $X$ is given in the following table

$x$	20	50	80	100
$\mathrm { P } ( X = x )$	$a$	$b$	$c$	$d$

where $a , b , c$ and $d$ are probabilities.
To play the game

the spinner is spun to obtain a value of $x$
Tisam then stands $x \mathrm {~cm}$ from the cup and tries to throw the ball into the cup

The event $S$ represents the event that Tisam successfully throws the ball into the cup.
To model this game Tisam assumes that

$\mathrm { P } ( S \mid \{ X = x \} ) = \frac { k } { x }$ where $k$ is a constant
$\mathrm { P } ( S \cap \{ X = x \} )$ should be the same whatever value of $x$ is obtained from the spinner

Using Tisam's model,

show that $c = \frac { 8 } { 5 } b$
find the probability distribution of $X$ Nav tries, a large number of times, to throw the ball into the cup from a distance of 100 cm .
He successfully gets the ball in the cup $30 \%$ of the time.
State, giving a reason, why Tisam's model of this game is not suitable to describe Nav playing the game for all values of $X$

Edexcel Paper 3 2023 June Q6

A medical researcher is studying the number of hours, $T$, a patient stays in hospital following a particular operation.

The histogram on the page opposite summarises the results for a random sample of 90 patients.

Use the histogram to estimate $\mathrm { P } ( 10 < T < 30 )$ For these 90 patients the time spent in hospital following the operation had
- a mean of 14.9 hours
- a standard deviation of 9.3 hours
Tomas suggests that $T$ can be modelled by $\mathrm { N } \left( 14.9,9.3 ^ { 2 } \right)$
With reference to the histogram, state, giving a reason, whether or not Tomas' model could be suitable. Xiang suggests that the frequency polygon based on this histogram could be modelled by a curve with equation $$y = k x \mathrm { e } ^ { - x } \quad 0 \leqslant x \leqslant 4$$ where
- $x$ is measured in tens of hours
- $k$ is a constant
- Use algebraic integration to show that
$$\int _ { 0 } ^ { n } x \mathrm { e } ^ { - x } \mathrm {~d} x = 1 - ( n + 1 ) \mathrm { e } ^ { - n }$$
Show that, for Xiang's model, $k = 99$ to the nearest integer.
Estimate $\mathrm { P } ( 10 < T < 30 )$ using
1. Tomas' model of $T \sim \mathrm {~N} \left( 14.9,9.3 ^ { 2 } \right)$
2. Xiang's curve with equation $y = 99 x \mathrm { e } ^ { - x }$ and the answer to part (c) The researcher decides to use Xiang's curve to model $\mathrm { P } ( a < T < b )$
State one limitation of Xiang's model. \begin{figure}[h]
\captionsetup{labelformat=empty} \caption{Question 6 continued} \includegraphics[alt={},max width=\textwidth]{a067577e-e2a6-440b-9d22-d558fade15f0-17_1164_1778_294_146}
\end{figure} Time in hours

Edexcel Paper 3 2024 June Q1

1. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{184043b7-1222-44fb-bc9f-3f484f72147b-02_108_997_242_534} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a particle $P$ of mass 0.5 kg at rest on a rough horizontal plane.

Find the magnitude of the normal reaction of the plane on $P$. The coefficient of friction between $P$ and the plane is $\frac { 2 } { 7 }$
A horizontal force of magnitude $X$ newtons is applied to $P$.
Given that $P$ is now in limiting equilibrium,
find the value of $X$.

Edexcel Paper 3 2024 June Q2

2. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{184043b7-1222-44fb-bc9f-3f484f72147b-04_675_1499_242_258} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 shows a speed-time graph for a model of the motion of an athlete running a $\mathbf { 2 0 0 m }$ race in 24 s . The athlete

starts from rest at time $t = 0$ and accelerates at a constant rate, reaching a speed of $10 \mathrm {~ms} ^ { - 1 }$ at $t = 4$
then moves at a constant speed of $10 \mathrm {~ms} ^ { - 1 }$ from $t = 4$ to $t = 18$
then decelerates at a constant rate from $t = 18$ to $t = 24$, crossing the finishing line with speed $U \mathrm {~m} \mathrm {~s} ^ { - 1 }$

Using the model,

find the acceleration of the athlete during the first 4 s of the race, stating the units of your answer,
find the distance covered by the athlete during the first 18s of the race,
find the value of $U$.

Edexcel Paper 3 2024 June Q3

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{184043b7-1222-44fb-bc9f-3f484f72147b-08_408_606_246_731} \captionsetup{labelformat=empty} \caption{Figure 3}

\end{figure} A particle $P$ of mass $m$ is held at rest at a point on a rough inclined plane, as shown in Figure 3. It is given that

the plane is inclined to the horizontal at an angle $\alpha$, where $\tan \alpha = \frac { 5 } { 12 }$
the coefficient of friction between $P$ and the plane is $\mu$, where $\mu < \frac { 5 } { 12 }$

The particle $P$ is released from rest and slides down the plane.
Air resistance is modelled as being negligible.
Using the model,

find, in terms of $m$ and $g$, the magnitude of the normal reaction of the plane on $P$,
show that, as $P$ slides down the plane, the acceleration of $P$ down the plane is $$\frac { 1 } { 13 } g ( 5 - 12 \mu )$$
State what would happen to $P$ if it is released from rest but $\mu \geqslant \frac { 5 } { 12 }$

\(x\)	0	\(0.1 - 0.5\)	\(0.6 - 1.0\)	\(1.1 - 1.9\)	\(2.0 - 4.0\)	\(4.1 - 6.9\)	\(7.0 - 12.0\)	\(12.1 - 20.9\)	\(21.0 - 32.0\)	\(\operatorname { tr }\)
Frequency	55	18	18	21	17	9	9	6	2	29

\(x\)	20	50	80	100
\(\mathrm { P } ( X = x )\)	\(a\)	\(b\)	\(c\)	\(d\)

\cline { 2 - 3 } \multicolumn{1}{c\|}{}	\(\boldsymbol { A }\)	\(\boldsymbol { B }\)
Professional	740	380
Skilled	275	90
Elementary	260	80

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