Questions — Edexcel (10514 questions)

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Edexcel Paper 3 2020 October Q4
10 marks Standard +0.3
  1. The discrete random variable \(D\) has the following probability distribution
\(d\)1020304050
\(\mathrm { P } ( D = d )\)\(\frac { k } { 10 }\)\(\frac { k } { 20 }\)\(\frac { k } { 30 }\)\(\frac { k } { 40 }\)\(\frac { k } { 50 }\)
where \(k\) is a constant.
  1. Show that the value of \(k\) is \(\frac { 600 } { 137 }\) The random variables \(D _ { 1 }\) and \(D _ { 2 }\) are independent and each have the same distribution as \(D\).
  2. Find \(\mathrm { P } \left( D _ { 1 } + D _ { 2 } = 80 \right)\) Give your answer to 3 significant figures. A single observation of \(D\) is made.
    The value obtained, \(d\), is the common difference of an arithmetic sequence.
    The first 4 terms of this arithmetic sequence are the angles, measured in degrees, of quadrilateral \(Q\)
  3. Find the exact probability that the smallest angle of \(Q\) is more than \(50 ^ { \circ }\)
Edexcel Paper 3 2020 October Q5
15 marks Standard +0.3
A health centre claims that the time a doctor spends with a patient can be modelled by a normal distribution with a mean of 10 minutes and a standard deviation of 4 minutes.
  1. Using this model, find the probability that the time spent with a randomly selected patient is more than 15 minutes. Some patients complain that the mean time the doctor spends with a patient is more than 10 minutes. The receptionist takes a random sample of 20 patients and finds that the mean time the doctor spends with a patient is 11.5 minutes.
  2. Stating your hypotheses clearly and using a \(5 \%\) significance level, test whether or not there is evidence to support the patients' complaint. The health centre also claims that the time a dentist spends with a patient during a routine appointment, \(T\) minutes, can be modelled by the normal distribution where \(T \sim \mathrm {~N} \left( 5,3.5 ^ { 2 } \right)\)
  3. Using this model,
    1. find the probability that a routine appointment with the dentist takes less than 2 minutes
    2. find \(\mathrm { P } ( T < 2 \mid T > 0 )\)
    3. hence explain why this normal distribution may not be a good model for \(T\). The dentist believes that she cannot complete a routine appointment in less than 2 minutes.
      She suggests that the health centre should use a refined model only including values of \(T > 2\)
  4. Find the median time for a routine appointment using this new model, giving your answer correct to one decimal place.
Edexcel Paper 3 2021 October Q1
4 marks Moderate -0.8
  1. A particle \(P\) moves with constant acceleration \(( 2 \mathbf { i } - 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\)
At time \(t = 0 , P\) is moving with velocity \(4 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
  1. Find the velocity of \(P\) at time \(t = 2\) seconds. At time \(t = 0\), the position vector of \(P\) relative to a fixed origin \(O\) is \(( \mathbf { i } + \mathbf { j } ) \mathrm { m }\).
  2. Find the position vector of \(P\) relative to \(O\) at time \(t = 3\) seconds.
Edexcel Paper 3 2021 October Q2
12 marks Standard +0.3
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{63363c3e-13fc-49a1-8cef-951e6e97e801-04_396_993_246_536} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A small stone \(A\) of mass \(3 m\) is attached to one end of a string.
A small stone \(B\) of mass \(m\) is attached to the other end of the string.
Initially \(A\) is held at rest on a fixed rough plane.
The plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac { 3 } { 4 }\) The string passes over a pulley \(P\) that is fixed at the top of the plane.
The part of the string from \(A\) to \(P\) is parallel to a line of greatest slope of the plane.
Stone \(B\) hangs freely below \(P\), as shown in Figure 1.
The coefficient of friction between \(A\) and the plane is \(\frac { 1 } { 6 }\) Stone \(A\) is released from rest and begins to move down the plane.
The stones are modelled as particles.
The pulley is modelled as being small and smooth.
The string is modelled as being light and inextensible. Using the model for the motion of the system before \(B\) reaches the pulley,
  1. write down an equation of motion for \(A\)
  2. show that the acceleration of \(A\) is \(\frac { 1 } { 10 } \mathrm {~g}\)
  3. sketch a velocity-time graph for the motion of \(B\), from the instant when \(A\) is released from rest to the instant just before \(B\) reaches the pulley, explaining your answer. In reality, the string is not light.
  4. State how this would affect the working in part (b).
Edexcel Paper 3 2021 October Q3
10 marks Standard +0.3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{63363c3e-13fc-49a1-8cef-951e6e97e801-08_796_750_242_660} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A beam \(A B\) has mass \(m\) and length \(2 a\).
The beam rests in equilibrium with \(A\) on rough horizontal ground and with \(B\) against a smooth vertical wall. The beam is inclined to the horizontal at an angle \(\theta\), as shown in Figure 2.
The coefficient of friction between the beam and the ground is \(\mu\) The beam is modelled as a uniform rod resting in a vertical plane that is perpendicular to the wall. Using the model,
  1. show that \(\mu \geqslant \frac { 1 } { 2 } \cot \theta\) A horizontal force of magnitude \(k m g\), where \(k\) is a constant, is now applied to the beam at \(A\). This force acts in a direction that is perpendicular to the wall and towards the wall.
    Given that \(\tan \theta = \frac { 5 } { 4 } , \mu = \frac { 1 } { 2 }\) and the beam is now in limiting equilibrium,
  2. use the model to find the value of \(k\).
Edexcel Paper 3 2021 October Q4
10 marks Standard +0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{63363c3e-13fc-49a1-8cef-951e6e97e801-12_453_990_244_539} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A small stone is projected with speed \(65 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(O\) at the top of a vertical cliff.
Point \(O\) is 70 m vertically above the point \(N\).
Point \(N\) is on horizontal ground.
The stone is projected at an angle \(\alpha\) above the horizontal, where \(\tan \alpha = \frac { 5 } { 12 }\) The stone hits the ground at the point \(A\), as shown in Figure 3.
The stone is modelled as a particle moving freely under gravity.
The acceleration due to gravity is modelled as having magnitude \(\mathbf { 1 0 m ~ s } \mathbf { m ~ } ^ { \mathbf { - 2 } }\) Using the model,
  1. find the time taken for the stone to travel from \(O\) to \(A\),
  2. find the speed of the stone at the instant just before it hits the ground at \(A\). One limitation of the model is that it ignores air resistance.
  3. State one other limitation of the model that could affect the reliability of your answers.
Edexcel Paper 3 2021 October Q5
14 marks Standard +0.3
  1. At time \(t\) seconds, a particle \(P\) has velocity \(\mathbf { v } \mathrm { ms } ^ { - 1 }\), where
$$\mathbf { v } = 3 t ^ { \frac { 1 } { 2 } } \mathbf { i } - 2 t \mathbf { j } \quad t > 0$$
  1. Find the acceleration of \(P\) at time \(t\) seconds, where \(t > 0\)
  2. Find the value of \(t\) at the instant when \(P\) is moving in the direction of \(\mathbf { i } - \mathbf { j }\) At time \(t\) seconds, where \(t > 0\), the position vector of \(P\), relative to a fixed origin \(O\), is \(\mathbf { r }\) metres. When \(t = 1 , \mathbf { r } = - \mathbf { j }\)
  3. Find an expression for \(\mathbf { r }\) in terms of \(t\).
  4. Find the exact distance of \(P\) from \(O\) at the instant when \(P\) is moving with speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
Edexcel Paper 3 2021 October Q1
7 marks Moderate -0.8
  1. State one disadvantage of using quota sampling compared with simple random sampling. In a university 8\% of students are members of the university dance club.
    A random sample of 36 students is taken from the university.
    The random variable \(X\) represents the number of these students who are members of the dance club.
  2. Using a suitable model for \(X\), find
    1. \(\mathrm { P } ( X = 4 )\)
    2. \(\mathrm { P } ( X \geqslant 7 )\) Only 40\% of the university dance club members can dance the tango.
  3. Find the probability that a student is a member of the university dance club and can dance the tango. A random sample of 50 students is taken from the university.
  4. Find the probability that fewer than 3 of these students are members of the university dance club and can dance the tango. \section*{Question 1 continued.}
Edexcel Paper 3 2021 October Q2
6 marks Standard +0.3
  1. Marc took a random sample of 16 students from a school and for each student recorded
  • the number of letters, \(x\), in their last name
  • the number of letters, \(y\), in their first name
His results are shown in the scatter diagram on the next page.
  1. Describe the correlation between \(x\) and \(y\). Marc suggests that parents with long last names tend to give their children shorter first names.
  2. Using the scatter diagram comment on Marc's suggestion, giving a reason for your answer. The results from Marc's random sample of 16 observations are given in the table below.
    \(x\)368753113454971066
    \(y\)7744685584745563
  3. Use your calculator to find the product moment correlation coefficient between \(x\) and \(y\) for these data.
  4. Test whether or not there is evidence of a negative correlation between the number of letters in the last name and the number of letters in the first name. You should
    \section*{Question 2 continued.}
    \includegraphics[max width=\textwidth, alt={}]{10736735-3050-43eb-9e76-011ca6fa48b8-05_1125_1337_294_372}
    \section*{Question 2 continued.} \section*{Question 2 continued.}
Edexcel Paper 3 2021 October Q3
8 marks Easy -1.2
  1. Stav is studying the large data set for September 2015
He codes the variable Daily Mean Pressure, \(x\), using the formula \(y = x - 1010\) The data for all 30 days from Hurn are summarised by $$\sum y = 214 \quad \sum y ^ { 2 } = 5912$$
  1. State the units of the variable \(x\)
  2. Find the mean Daily Mean Pressure for these 30 days.
  3. Find the standard deviation of Daily Mean Pressure for these 30 days. Stav knows that, in the UK, winds circulate
    The table gives the Daily Mean Pressure for 3 locations from the large data set on 26/09/2015
    LocationHeathrowHurnLeuchars
    Daily Mean Pressure102910281028
    Cardinal Wind Direction
    The Cardinal Wind Directions for these 3 locations on 26/09/2015 were, in random order, $$\begin{array} { l l l } W & N E & E \end{array}$$ You may assume that these 3 locations were under a single region of pressure.
  4. Using your knowledge of the large data set, place each of these Cardinal Wind Directions in the correct location in the table.
    Give a reason for your answer. \section*{Question 3 continued.}
Edexcel Paper 3 2021 October Q4
11 marks Standard +0.3
  1. A large college produces three magazines.
One magazine is about green issues, one is about equality and one is about sports.
A student at the college is selected at random and the events \(G , E\) and \(S\) are defined as follows \(G\) is the event that the student reads the magazine about green issues \(E\) is the event that the student reads the magazine about equality \(S\) is the event that the student reads the magazine about sports
The Venn diagram, where \(p , q , r\) and \(t\) are probabilities, gives the probability for each subset. \includegraphics[max width=\textwidth, alt={}, center]{10736735-3050-43eb-9e76-011ca6fa48b8-10_508_862_756_603}
  1. Find the proportion of students in the college who read exactly one of these magazines. No students read all three magazines and \(\mathrm { P } ( G ) = 0.25\)
  2. Find
    1. the value of \(p\)
    2. the value of \(q\) Given that \(\mathrm { P } ( S \mid E ) = \frac { 5 } { 12 }\)
  3. find
    1. the value of \(r\)
    2. the value of \(t\)
  4. Determine whether or not the events ( \(S \cap E ^ { \prime }\) ) and \(G\) are independent. Show your working clearly. \section*{Question 4 continued.} \section*{Question 4 continued.} \section*{Question 4 continued.}
Edexcel Paper 3 2021 October Q5
11 marks Standard +0.3
  1. The heights of females from a country are normally distributed with
  • a mean of 166.5 cm
  • a standard deviation of 6.1 cm
Given that \(1 \%\) of females from this country are shorter than \(k \mathrm {~cm}\),
  1. find the value of \(k\)
  2. Find the proportion of females from this country with heights between 150 cm and 175 cm A female, from this country, is chosen at random from those with heights between 150 cm and 175 cm
  3. Find the probability that her height is more than 160 cm The heights of females from a different country are normally distributed with a standard deviation of 7.4 cm Mia believes that the mean height of females from this country is less than 166.5 cm
    Mia takes a random sample of 50 females from this country and finds the mean of her sample is 164.6 cm
  4. Carry out a suitable test to assess Mia's belief. You should
    \section*{Question 5 continued.} \section*{Question 5 continued.} \section*{Question 5 continued.}
Edexcel Paper 3 2021 October Q6
7 marks Challenging +1.2
  1. The discrete random variable \(X\) has the following probability distribution
\(x\)\(a\)\(b\)\(c\)
\(\mathrm { P } ( X = x )\)\(\log _ { 36 } a\)\(\log _ { 36 } b\)\(\log _ { 36 } c\)
where
  • \(\quad a , b\) and \(c\) are distinct integers \(( a < b < c )\)
  • all the probabilities are greater than zero
    1. Find
      1. the value of a
      2. the value of \(b\)
      3. the value of \(c\)
Show your working clearly. The independent random variables \(X _ { 1 }\) and \(X _ { 2 }\) each have the same distribution as \(X\)
  • Find \(\mathrm { P } \left( X _ { 1 } = X _ { 2 } \right)\) \section*{Question 6 continued.} \section*{Question 6 continued.}
    •
    Edexcel S1 2016 June Q1
    12 marks Moderate -0.8
    1. The percentage oil content, \(p\), and the weight, \(w\) milligrams, of each of 10 randomly selected sunflower seeds were recorded. These data are summarised below.
    $$\sum w ^ { 2 } = 41252 \quad \sum w p = 27557.8 \quad \sum w = 640 \quad \sum p = 431 \quad \mathrm {~S} _ { p p } = 2.72$$
    1. Find the value of \(\mathrm { S } _ { w w }\) and the value of \(\mathrm { S } _ { w p }\)
    2. Calculate the product moment correlation coefficient between \(p\) and \(w\)
    3. Give an interpretation of your product moment correlation coefficient. The equation of the regression line of \(p\) on \(w\) is given in the form \(p = a + b w\)
    4. Find the equation of the regression line of \(p\) on \(w\)
    5. Hence estimate the percentage oil content of a sunflower seed which weighs 60 milligrams.
    Edexcel S1 2016 June Q2
    3 marks Easy -1.2
    2. The time taken to complete a puzzle, in minutes, is recorded for each person in a club. The times are summarised in a grouped frequency distribution and represented by a histogram. One of the class intervals has a frequency of 20 and is shown by a bar of width 1.5 cm and height 12 cm on the histogram. The total area under the histogram is \(94.5 \mathrm {~cm} ^ { 2 }\) Find the number of people in the club.
    (3)
    VILM SIMI NI JIIIM I ON OC
    VILV SIHI NI JAHM ION OC
    VJ4V SIHI NI JIIYM ION OC
    Edexcel S1 2016 June Q3
    14 marks Easy -1.2
    3. The discrete random variable \(X\) has probability distribution $$\mathrm { P } ( X = x ) = \frac { 1 } { 5 } \quad x = 1,2,3,4,5$$
    1. Write down the name given to this distribution. Find
    2. \(\mathrm { P } ( X = 4 )\)
    3. \(\mathrm { F } ( 3 )\)
    4. \(\mathrm { P } ( 3 X - 3 > X + 4 )\)
    5. Write down \(\mathrm { E } ( X )\)
    6. Find \(\mathrm { E } \left( X ^ { 2 } \right)\)
    7. Hence find \(\operatorname { Var } ( X )\) Given that \(\mathrm { E } ( a X - 3 ) = 11.4\)
    8. find \(\operatorname { Var } ( a X - 3 )\)
    Edexcel S1 2016 June Q4
    12 marks Moderate -0.8
    4. A researcher recorded the time, \(t\) minutes, spent using a mobile phone during a particular afternoon, for each child in a club. The researcher coded the data using \(v = \frac { t - 5 } { 10 }\) and the results are summarised in the table below.
    Coded Time (v)Frequency ( \(\boldsymbol { f }\) )Coded Time Midpoint (m)
    \(0 \leqslant v < 5\)202.5
    \(5 \leqslant v < 10\)24\(a\)
    \(10 \leqslant v < 15\)1612.5
    \(15 \leqslant v < 20\)1417.5
    \(20 \leqslant v < 30\)6\(b\)
    $$\text { (You may use } \sum f m = 825 \text { and } \sum f m ^ { 2 } = 12012.5 \text { ) }$$
    1. Write down the value of \(a\) and the value of \(b\).
    2. Calculate an estimate of the mean of \(v\).
    3. Calculate an estimate of the standard deviation of \(v\).
    4. Use linear interpolation to estimate the median of \(v\).
    5. Hence describe the skewness of the distribution. Give a reason for your answer.
    6. Calculate estimates of the mean and the standard deviation of the time spent using a mobile phone during the afternoon by the children in this club.
    Edexcel S1 2016 June Q5
    8 marks Moderate -0.3
    5. A biased tetrahedral die has faces numbered \(0,1,2\) and 3 . The die is rolled and the number face down on the die, \(X\), is recorded. The probability distribution of \(X\) is
    \(x\)0123
    \(\mathrm { P } ( X = x )\)\(\frac { 1 } { 6 }\)\(\frac { 1 } { 6 }\)\(\frac { 1 } { 6 }\)\(\frac { 1 } { 2 }\)
    If \(X = 3\) then the final score is 3
    If \(X \neq 3\) then the die is rolled again and the final score is the sum of the two numbers. The random variable \(T\) is the final score.
    1. Find \(\mathrm { P } ( T = 2 )\)
    2. Find \(\mathrm { P } ( T = 3 )\)
    3. Given that the die is rolled twice, find the probability that the final score is 3
    Edexcel S1 2016 June Q6
    11 marks Moderate -0.3
    6. Three events \(A , B\) and \(C\) are such that $$\mathrm { P } ( A ) = \frac { 2 } { 5 } \quad \mathrm { P } ( C ) = \frac { 1 } { 2 } \quad \mathrm { P } ( A \cup B ) = \frac { 5 } { 8 }$$ Given that \(A\) and \(C\) are mutually exclusive find
    1. \(\mathrm { P } ( A \cup C )\) Given that \(A\) and \(B\) are independent
    2. show that \(\mathrm { P } ( B ) = \frac { 3 } { 8 }\)
    3. Find \(\mathrm { P } ( A \mid B )\) Given that \(\mathrm { P } \left( C ^ { \prime } \cap B ^ { \prime } \right) = 0.3\)
    4. draw a Venn diagram to represent the events \(A , B\) and \(C\)
    Edexcel S1 2016 June Q7
    15 marks Standard +0.3
    7. A machine fills bottles with water. The volume of water delivered by the machine to a bottle is \(X \mathrm { ml }\) where \(X \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)\) One of these bottles of water is selected at random. Given that \(\mu = 503\) and \(\sigma = 1.6\)
    1. find
      1. \(\mathrm { P } ( X > 505 )\)
      2. \(\mathrm { P } ( 501 < X < 505 )\)
    2. Find \(w\) such that \(\mathrm { P } ( 1006 - w < X < w ) = 0.9426\) Following adjustments to the machine, the volume of water delivered by the machine to a bottle is such that \(\mu = 503\) and \(\sigma = q\) Given that \(\mathrm { P } ( X < r ) = 0.01\) and \(\mathrm { P } ( X > r + 6 ) = 0.05\)
    3. find the value of \(r\) and the value of \(q\)
    Edexcel S1 2018 June Q1
    13 marks Moderate -0.3
    1. A random sample of 10 cars of different makes and sizes is taken and the published miles per gallon, \(p\), and the actual miles per gallon, \(m\), are recorded. The data are coded using variables \(x = \frac { p } { 10 }\) and \(y = m - 25\)
    The results for the coded data are summarised below.
    \(\boldsymbol { x }\)6.893.675.925.044.873.924.715.143.655.23
    \(\boldsymbol { y }\)30322151381513.5319
    (You may use \(\sum y ^ { 2 } = 2628.25 \quad \sum x y = 768.58 \quad \mathrm {~S} _ { x x } = 9.25924 \quad \mathrm {~S} _ { x y } = 74.664\) )
    1. Show that \(\mathrm { S } _ { y y } = 626.025\)
    2. Find the product moment correlation coefficient between \(x\) and \(y\).
    3. Give a reason to support fitting a regression model of the form \(y = a + b x\) to these data.
    4. Find the equation of the regression line of \(y\) on \(x\), giving your answer in the form \(y = a + b x\).
      Give the value of \(a\) and the value of \(b\) to 3 significant figures. A car's published miles per gallon is 44
    5. Estimate the actual miles per gallon for this particular car.
    6. Comment on the reliability of your estimate in part (e). Give a reason for your answer.
    Edexcel S1 2018 June Q2
    11 marks Easy -1.3
    2. Two youth clubs, Eastyou and Westyou, decided to raise money for charity by running a 5 km race. All the members of the youth clubs took part and the time, in minutes, taken for each member to run the 5 km was recorded. The times for the Westyou members are summarised in Figure 1. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{b115bffa-1190-4a2b-b6f2-b006580e8dbd-06_349_1378_497_274} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure}
    1. Write down the time that is exceeded by \(75 \%\) of Westyou members. The times for the Eastyou members are summarised by the stem and leaf diagram below.
      StemLeaf
      20234\(( 4 )\)
      25688899
      300000111222234\(( 14 )\)
      355579\(( 5 )\)
      Key: 2|0 means 20 minutes
    2. Find the value of the median and interquartile range for the Eastyou members. An outlier is a value that falls either
    3. On the grid on page 7, draw a box plot to represent the times of the Eastyou members.
    4. State the skewness of each distribution. Give reasons for your answers. $$\begin{aligned} & \text { more than } 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right) \text { above } Q _ { 3 } \\ & \text { or more than } 1.5 \times \left( Q _ { 3 } - Q _ { 1 } \right) \text { below } Q _ { 1 } \end{aligned}$$
      \includegraphics[max width=\textwidth, alt={}]{b115bffa-1190-4a2b-b6f2-b006580e8dbd-06_2255_50_314_1976}
      \includegraphics[max width=\textwidth, alt={}, center]{b115bffa-1190-4a2b-b6f2-b006580e8dbd-07_406_1390_2224_262} Turn over for a spare grid if you need to redraw your box plot. \begin{figure}[h]
      \captionsetup{labelformat=empty} \caption{Only use this grid if you need to redraw your box plot.} \includegraphics[alt={},max width=\textwidth]{b115bffa-1190-4a2b-b6f2-b006580e8dbd-09_401_1399_2261_258}
      \end{figure}
    Edexcel S1 2018 June Q3
    7 marks Moderate -0.3
    1. A manufacturer of electric generators buys engines for its generators from three companies, \(R , S\) and \(T\).
    Company \(R\) supplies 40\% of the engines. Company \(S\) supplies \(25 \%\) of the engines. The rest of the engines are supplied by company \(T\). It is known that \(2 \%\) of the engines supplied by company \(R\) are faulty, \(1 \%\) of the engines supplied by company \(S\) are faulty and \(2 \%\) of the engines supplied by company \(T\) are faulty. An engine is chosen at random.
    1. Draw a tree diagram to show all the possible outcomes and the associated probabilities.
    2. Calculate the probability that the engine is from company \(R\) and is not faulty.
    3. Calculate the probability that the engine is faulty. Given that the engine is faulty,
    4. find the probability that the engine did not come from company \(S\).
    Edexcel S1 2018 June Q4
    10 marks Moderate -0.3
    4. A discrete random variable \(X\) has probability function $$\mathrm { P } ( X = x ) = \left\{ \begin{array} { c l } k ( 2 - x ) & x = 0,1 \\ k ( 3 - x ) & x = 2,3 \\ k ( x + 1 ) & x = 4 \\ 0 & \text { otherwise } \end{array} \right.$$ where \(k\) is a constant.
    1. Show that \(k = \frac { 1 } { 9 }\) Find the exact value of
    2. \(\mathrm { P } ( 1 \leqslant X < 4 )\)
    3. \(\mathrm { E } ( X )\)
    4. \(\mathrm { E } \left( X ^ { 2 } \right)\)
    5. \(\operatorname { Var } ( 3 X + 1 )\)
    Edexcel S1 2018 June Q5
    13 marks Moderate -0.8
    5. The weights, in grams, of a random sample of 48 broad beans are summarised in the table.
    Weight in grams ( \(\boldsymbol { x }\) )Frequency (f)Class midpoint (y)
    \(0.9 < x \leqslant 1.1\)91.0
    \(1.1 < x \leqslant 1.3\)121.2
    \(1.3 < x \leqslant 1.5\)111.4
    \(1.5 < x \leqslant 1.7\)81.6
    \(1.7 < x \leqslant 1.9\)31.8
    \(1.9 < x \leqslant 2.1\)32.0
    \(2.1 < x \leqslant 2.7\)22.4
    (You may assume \(\sum \mathrm { fy } { } ^ { 2 } = 101.56\) ) A histogram was drawn to represent these data. The \(2.1 < x \leqslant 2.7\) class was represented by a bar of width 1.5 cm and height 1 cm .
    1. Find the width and height of the \(0.9 < x \leqslant 1.1\) class.
    2. Give a reason to justify the use of a histogram to represent these data.
    3. Estimate the mean and the standard deviation of the weights of these broad beans.
    4. Use linear interpolation to estimate the median of the weights of these broad beans. One of these broad beans is selected at random.
    5. Estimate the probability that its weight lies between 1.1 grams and 1.6 grams. One of these broad beans having a recorded weight of 0.95 grams was incorrectly weighed. The correct weight is 1.4 grams.
    6. State, giving a reason, the effect this would have on your answers to part (c). Do not carry out any further calculations.