Questions - Edexcel Paper 3

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 $\mu \geqslant \frac { 5 } { 12 }$

Edexcel Paper 3 2024 June Q4

In this question you must show all stages of your working.

\section*{Solutions relying entirely on calculator technology are not acceptable.} [In this question, $\mathbf { i }$ is a unit vector due east and $\mathbf { j }$ is a unit vector due north.
Position vectors are given relative to a fixed origin $O$.] At time $t$ seconds, $t \geqslant 1$, the position vector of a particle $P$ is $\mathbf { r }$ metres, where $$\mathbf { r } = c t ^ { \frac { 1 } { 2 } } \mathbf { i } - \frac { 3 } { 8 } t ^ { 2 } \mathbf { j }$$ and $c$ is a constant.
When $t = 4$, the bearing of $P$ from $O$ is $135 ^ { \circ }$

Show that $c = 3$
Find the speed of $P$ when $t = 4$ When $t = T , P$ is accelerating in the direction of ( $\mathbf { - i } - \mathbf { 2 7 j }$ ).
Find the value of $T$.

Edexcel Paper 3 2024 June Q5

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{184043b7-1222-44fb-bc9f-3f484f72147b-12_270_1109_244_470} \captionsetup{labelformat=empty} \caption{Figure 4}

\end{figure} At time $t = 0$, a small stone is projected with velocity $35 \mathrm {~ms} ^ { - 1 }$ from a point $O$ on horizontal ground. The stone is projected at an angle $\alpha$ to the horizontal, where $\tan \alpha = \frac { 3 } { 4 }$
In an initial model

the stone is modelled as a particle $P$ moving freely under gravity
the stone hits the ground at the point $A$

Figure 4 shows the path of $P$ from $O$ to $A$.
For the motion of $P$ from $O$ to $A$

at time $t$ seconds, the horizontal distance of $P$ from $O$ is $x$ metres
at time $t$ seconds, the vertical distance of $P$ above the ground is $y$ metres
1. Using the model, show that

$$y = \frac { 3 } { 4 } x - \frac { 1 } { 160 } x ^ { 2 }$$

Use the answer to (a), or otherwise, to find the length $O A$. Using the model, the greatest height of the stone above the ground is found to be $H$ metres.

Use the answer to (a), or otherwise, to find the value of $H$.

The model is refined to include air resistance.

Using this new model, the greatest height of the stone above the ground is found to be $K$ metres.

State which is greater, $H$ or $K$, justifying your answer.

State one limitation of this refined model.

Edexcel Paper 3 2024 June Q6

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{184043b7-1222-44fb-bc9f-3f484f72147b-16_458_798_258_630} \captionsetup{labelformat=empty} \caption{Figure 5}

\end{figure} Figure 5 shows a uniform rod $A B$ of mass $M$ and length $2 a$.

the rod has its end $A$ on rough horizontal ground
the rod rests in equilibrium against a small smooth fixed horizontal peg $P$
the point $C$ on the rod, where $A C = 1.5 a$, is the point of contact between the rod and the peg
the rod is at an angle $\theta$ to the ground, where $\tan \theta = \frac { 4 } { 3 }$

The rod lies in a vertical plane perpendicular to the peg.
The magnitude of the normal reaction of the peg on the rod at $C$ is $S$.

Show that $S = \frac { 2 } { 5 } M g$ The coefficient of friction between the rod and the ground is $\mu$.
Given that the rod is in limiting equilibrium,
find the value of $\mu$.

Edexcel Paper 3 2024 June Q1

Xian rolls a fair die 10 times.

The random variable $X$ represents the number of times the die lands on a six.

Using a suitable distribution for $X$, find
1. $\mathrm { P } ( X = 3 )$
2. $\mathrm { P } ( X < 3 )$ Xian repeats this experiment each day for 60 days and records the number of days when $X = 3$
Find the probability that there were at least 12 days when $X = 3$
Find an estimate for the total number of sixes that Xian will roll during these 60 days.
Use a normal approximation to estimate the probability that Xian rolls a total of more than 95 sixes during these 60 days.

Edexcel Paper 3 2024 June Q2

Amar is studying the flight of a bird from its nest.

He measures the bird's height above the ground, $h$ metres, at time $t$ seconds for 10 values of $t$
Amar finds the equation of the regression line for the data to be $h = 38.6 - 1.28 t$

Interpret the gradient of this line. The product moment correlation coefficient between $h$ and $t$ is - 0.510
Test whether or not there is evidence of a negative correlation between the height above the ground and the time during the flight.
You should
- state your hypotheses clearly
- use a $5 \%$ level of significance
- state the critical value used
Jane draws the following scatter diagram for Amar’s data.
\includegraphics[max width=\textwidth, alt={}, center]{ab7f7951-e6fe-4853-bb69-8016cf3e796c-06_1024_1033_1135_516}
With reference to the scatter diagram, state, giving a reason, whether or not the regression line $h = 38.6 - 1.28 t$ is an appropriate model for these data. Jane suggests an improved model using the variable $u = ( t - k ) ^ { 2 }$ where $k$ is a constant.
She obtains the equation $h = 38.1 - 0.78 u$
Choose a suitable value for $k$ to write Jane's improved model for $h$ in terms of $t$ only.

Edexcel Paper 3 2024 June Q3

Ming is studying the large data set for Perth in 2015

He intended to use all the data available to find summary statistics for the Daily Mean Air Temperature, $x { } ^ { \circ } \mathrm { C }$.
Unfortunately, Ming selected an incorrect variable on the spreadsheet.
This incorrect variable gave a mean of 5.3 and a standard deviation of 12.4

Using your knowledge of the large data set, suggest which variable Ming selected. The correct values for the Daily Mean Air Temperature are summarised as $$n = 184 \quad \sum x = 2801.2 \quad \sum x ^ { 2 } = 44695.4$$
Calculate the mean and standard deviation for these data. One of the months from the large data set for Perth in 2015 has
- mean $\bar { X } = 19.4$
- standard deviation $\sigma _ { x } = 2.83$
  for Daily Mean Air Temperature.
- Suggest, giving a reason, a month these data may have come from.

Edexcel Paper 3 2024 June Q4

The proportion of left-handed adults in a country is $10 \%$

Freya believes that the proportion of left-handed adults under the age of 25 in this country is different from 10\% She takes a random sample of 40 adults under the age of 25 from this country to investigate her belief.

Find the critical region for a suitable test to assess Freya's belief. You should
- state your hypotheses clearly
- use a $5 \%$ level of significance
- state the probability of rejection in each tail
- Write down the actual significance level of your test in part (a)
In Freya's sample 7 adults were left-handed.
With reference to your answer in part (a) comment on Freya's belief.

Edexcel Paper 3 2024 June Q5

The records for a school athletics club show that the height, $H$ metres, achieved by students in the high jump is normally distributed with mean 1.4 metres and standard deviation 0.15 metres.
1. Find the proportion of these students achieving a height of more than 1.6 metres.
The records also show that the time, $T$ seconds, to run 1500 metres is normally distributed with mean 330 seconds and standard deviation 26 seconds. The school's Head would like to use these distributions to estimate the proportion of students from the school athletics club who can jump higher than 1.6 metres and can run 1500 metres in less than 5 minutes.
State a necessary assumption about $H$ and $T$ for the Head to calculate an estimate of this proportion.
Find the Head's estimate of this proportion. Students in the school athletics club also throw the discus.
The random variable $D \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)$ represents the distance, in metres, that a student can throw the discus. Given that $\mathrm { P } ( D < 16.3 ) = 0.30$ and $\mathrm { P } ( D > 29.0 ) = 0.10$
calculate the value of $\mu$ and the value of $\sigma$

Edexcel Paper 3 2024 June Q6

The Venn diagram, where $p , q$ and $r$ are probabilities, shows the events $A , B , C$ and $D$ and associated probabilities.

\begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{ab7f7951-e6fe-4853-bb69-8016cf3e796c-18_527_1074_358_494} \captionsetup{labelformat=empty} \caption{$r$}

\end{figure}

State any pair of mutually exclusive events from $A$, $B$, $C$ and $D$ The events $B$ and $C$ are independent.
Find the value of $p$
Find the greatest possible value of $\mathrm { P } \left( A \mid B ^ { \prime } \right)$ Given that $\mathrm { P } \left( B \mid A ^ { \prime } \right) = 0.5$
find the value of $q$ and the value of $r$
Find $\mathrm { P } \left( [ A \cup B ] ^ { \prime } \cap C \right)$
Use set notation to write an expression for the event with probability $p$

Edexcel Paper 3 2020 October Q1

A rough plane is inclined to the horizontal at an angle $\alpha$, where $\tan \alpha = \frac { 3 } { 4 }$

A brick $P$ of mass $m$ is placed on the plane.
The coefficient of friction between $P$ and the plane is $\mu$
Brick $P$ is in equilibrium and on the point of sliding down the plane.
Brick $P$ is modelled as a particle.
Using the model,

find, in terms of $m$ and $g$, the magnitude of the normal reaction of the plane on brick $P$
show that $\mu = \frac { 3 } { 4 }$ For parts (c) and (d), you are not required to do any further calculations.
Brick $P$ is now removed from the plane and a much heavier brick $Q$ is placed on the plane. The coefficient of friction between $Q$ and the plane is also $\frac { 3 } { 4 }$
Explain briefly why brick $Q$ will remain at rest on the plane. Brick $Q$ is now projected with speed $0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ down a line of greatest slope of the plane.
Brick $Q$ is modelled as a particle.
Using the model,
describe the motion of brick $Q$, giving a reason for your answer.

Edexcel Paper 3 2020 October Q2

A particle $P$ moves with acceleration $( 4 \mathbf { i } - 5 \mathbf { j } ) \mathrm { ms } ^ { - 2 }$

At time $t = 0 , P$ is moving with velocity $( - 2 \mathbf { i } + 2 \mathbf { j } ) \mathrm { ms } ^ { - 1 }$

Find the velocity of $P$ at time $t = 2$ seconds. At time $t = 0 , P$ passes through the origin $O$.
At time $t = T$ seconds, where $T > 0$, the particle $P$ passes through the point $A$.
The position vector of $A$ is ( $\lambda \mathbf { i } - 4.5 \mathbf { j }$ )m relative to $O$, where $\lambda$ is a constant.
Find the value of $T$.
Hence find the value of $\lambda$

Edexcel Paper 3 2020 October Q3

1. At time $t$ seconds, where $t \geqslant 0$, a particle $P$ moves so that its acceleration a $\mathrm { ms } ^ { - 2 }$ is given by

$$\mathbf { a } = ( 1 - 4 t ) \mathbf { i } + \left( 3 - t ^ { 2 } \right) \mathbf { j }$$ At the instant when $t = 0$, the velocity of $P$ is $36 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }$

Find the velocity of $P$ when $t = 4$
Find the value of $t$ at the instant when $P$ is moving in a direction perpendicular to i
(ii) At time $t$ seconds, where $t \geqslant 0$, a particle $Q$ moves so that its position vector $\mathbf { r }$ metres, relative to a fixed origin $O$, is given by $$\mathbf { r } = \left( t ^ { 2 } - t \right) \mathbf { i } + 3 t \mathbf { j }$$ Find the value of $t$ at the instant when the speed of $Q$ is $5 \mathrm {~m} \mathrm {~s} ^ { - 1 }$

Edexcel Paper 3 2020 October Q4

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{d1989e18-1a4a-47e9-9f12-3beb8985ed87-12_803_767_239_647} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} A ladder $A B$ has mass $M$ and length $6 a$.
The end $A$ of the ladder is on rough horizontal ground.
The ladder rests against a fixed smooth horizontal rail at the point $C$.
The point $C$ is at a vertical height $4 a$ above the ground.
The vertical plane containing $A B$ is perpendicular to the rail.
The ladder is inclined to the horizontal at an angle $\alpha$, where $\sin \alpha = \frac { 4 } { 5 }$, as shown in Figure 1.
The coefficient of friction between the ladder and the ground is $\mu$.
The ladder rests in limiting equilibrium.
The ladder is modelled as a uniform rod.
Using the model,

show that the magnitude of the force exerted on the ladder by the rail at $C$ is $\frac { 9 M g } { 25 }$
Hence, or otherwise, find the value of $\mu$.

Edexcel Paper 3 2020 October Q5

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{d1989e18-1a4a-47e9-9f12-3beb8985ed87-16_532_1002_237_533} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} A small ball is projected with speed $U \mathrm {~ms} ^ { - 1 }$ from a point $O$ at the top of a vertical cliff. The point $O$ is 25 m vertically above the point $N$ which is on horizontal ground. The ball is projected at an angle of $45 ^ { \circ }$ above the horizontal.
The ball hits the ground at a point $A$, where $A N = 100 \mathrm {~m}$, as shown in Figure 2 .
The motion of the ball is modelled as that of a particle moving freely under gravity.
Using this initial model,

show that $U = 28$
find the greatest height of the ball above the horizontal ground $N A$. In a refinement to the model of the motion of the ball from $O$ to $A$, the effect of air resistance is included. This refined model is used to find a new value of $U$.
How would this new value of $U$ compare with 28, the value given in part (a)?
State one further refinement to the model that would make the model more realistic. \section*{" " $_ { \text {" } } ^ { \text {" } }$ " "}

\(x\)	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
\(\mathrm { P } ( X = x )\)	\(a\)	\(b\)	\(c\)	\(d\)

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