Questions — Edexcel AS Paper 2 (73 questions)

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Edexcel AS Paper 2 2021 November Q5
  1. Two bags, \(\mathbf { A }\) and \(\mathbf { B }\), each contain balls which are either red or yellow or green.
Bag A contains 4 red, 3 yellow and \(n\) green balls.
Bag \(\mathbf { B }\) contains 5 red, 3 yellow and 1 green ball.
A ball is selected at random from bag \(\mathbf { A }\) and placed into bag \(\mathbf { B }\).
A ball is then selected at random from bag \(\mathbf { B }\) and placed into bag \(\mathbf { A }\).
The probability that bag \(\mathbf { A }\) now contains an equal number of red, yellow and green balls is \(p\). Given that \(p > 0\), find the possible values of \(n\) and \(p\).
Edexcel AS Paper 2 2018 June Q1
  1. A company is introducing a job evaluation scheme. Points ( \(x\) ) will be awarded to each job based on the qualifications and skills needed and the level of responsibility. Pay ( \(\pounds y\) ) will then be allocated to each job according to the number of points awarded.
Before the scheme is introduced, a random sample of 8 employees was taken and the linear regression equation of pay on points was \(y = 4.5 x - 47\)
  1. Describe the correlation between points and pay.
  2. Give an interpretation of the gradient of this regression line.
  3. Explain why this model might not be appropriate for all jobs in the company.
Edexcel AS Paper 2 2018 June Q2
  1. A factory buys \(10 \%\) of its components from supplier \(A , 30 \%\) from supplier \(B\) and the rest from supplier \(C\). It is known that \(6 \%\) of the components it buys are faulty.
Of the components bought from supplier \(A , 9 \%\) are faulty and of the components bought from supplier \(B , 3 \%\) are faulty.
  1. Find the percentage of components bought from supplier \(C\) that are faulty. A component is selected at random.
  2. Explain why the event "the component was bought from supplier \(B\) " is not statistically independent from the event "the component is faulty".
Edexcel AS Paper 2 2018 June Q3
  1. Naasir is playing a game with two friends. The game is designed to be a game of chance so that the probability of Naasir winning each game is \(\frac { 1 } { 3 }\)
    Naasir and his friends play the game 15 times.
    1. Find the probability that Naasir wins
      1. exactly 2 games,
      2. more than 5 games.
    Naasir claims he has a method to help him win more than \(\frac { 1 } { 3 }\) of the games. To test this claim, the three of them played the game again 32 times and Naasir won 16 of these games.
  2. Stating your hypotheses clearly, test Naasir's claim at the \(5 \%\) level of significance.
Edexcel AS Paper 2 2018 June Q4
  1. Helen is studying the daily mean wind speed for Camborne using the large data set from 1987. The data for one month are summarised in Table 1 below.
\begin{table}[h]
Windspeed\(\mathrm { n } / \mathrm { a }\)67891112131416
Frequency13232231212
\captionsetup{labelformat=empty} \caption{Table 1}
\end{table}
  1. Calculate the mean for these data.
  2. Calculate the standard deviation for these data and state the units. The means and standard deviations of the daily mean wind speed for the other months from the large data set for Camborne in 1987 are given in Table 2 below. The data are not in month order. \begin{table}[h]
    Month\(A\)\(B\)\(C\)\(D\)\(E\)
    Mean7.588.268.578.5711.57
    Standard Deviation2.933.893.463.874.64
    \captionsetup{labelformat=empty} \caption{Table 2}
    \end{table}
  3. Using your knowledge of the large data set, suggest, giving a reason, which month had a mean of 11.57 The data for these months are summarised in the box plots on the opposite page. They are not in month order or the same order as in Table 2.
    1. State the meaning of the * symbol on some of the box plots.
    2. Suggest, giving your reasons, which of the months in Table 2 is most likely to be summarised in the box plot marked \(Y\). \includegraphics[max width=\textwidth, alt={}, center]{2edcf965-9c93-4a9b-9395-2d3c023801af-11_1177_1216_324_427}
Edexcel AS Paper 2 2018 June Q5
5. A biased spinner can only land on one of the numbers \(1,2,3\) or 4 . The random variable \(X\) represents the number that the spinner lands on after a single spin and \(\mathrm { P } ( X = r ) = \mathrm { P } ( X = r + 2 )\) for \(r = 1,2\) Given that \(\mathrm { P } ( X = 2 ) = 0.35\)
  1. find the complete probability distribution of \(X\). Ambroh spins the spinner 60 times.
  2. Find the probability that more than half of the spins land on the number 4 Give your answer to 3 significant figures. The random variable \(Y = \frac { 12 } { X }\)
  3. Find \(\mathrm { P } ( Y - X \leqslant 4 )\)
Edexcel AS Paper 2 2018 June Q6
  1. A man throws a tennis ball into the air so that, at the instant when the ball leaves his hand, the ball is 2 m above the ground and is moving vertically upwards with speed \(9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
The motion of the ball is modelled as that of a particle moving freely under gravity and the acceleration due to gravity is modelled as being of constant magnitude \(10 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) The ball hits the ground \(T\) seconds after leaving the man's hand.
Using the model, find the value of \(T\).
Edexcel AS Paper 2 2018 June Q7
  1. A train travels along a straight horizontal track between two stations, \(A\) and \(B\).
In a model of the motion, the train starts from rest at \(A\) and moves with constant acceleration \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for 80 s .
The train then moves at constant velocity before it moves with a constant deceleration of \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), coming to rest at \(B\).
  1. For this model of the motion of the train between \(A\) and \(B\),
    1. state the value of the constant velocity of the train,
    2. state the time for which the train is decelerating,
    3. sketch a velocity-time graph. The total distance between the two stations is 4800 m .
  2. Using the model, find the total time taken by the train to travel from \(A\) to \(B\).
  3. Suggest one improvement that could be made to the model of the motion of the train from \(A\) to \(B\) in order to make the model more realistic.
Edexcel AS Paper 2 2018 June Q8
  1. A particle, \(P\), moves along the \(x\)-axis. At time \(t\) seconds, \(t \geqslant 0\), the displacement, \(x\) metres, of \(P\) from the origin \(O\), is given by \(x = \frac { 1 } { 2 } t ^ { 2 } \left( t ^ { 2 } - 2 t + 1 \right)\)
    1. Find the times when \(P\) is instantaneously at rest.
    2. Find the total distance travelled by \(P\) in the time interval \(0 \leqslant t \leqslant 2\)
    3. Show that \(P\) will never move along the negative \(x\)-axis.
Edexcel AS Paper 2 2018 June Q9
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2edcf965-9c93-4a9b-9395-2d3c023801af-26_551_276_210_890} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Two small balls, \(P\) and \(Q\), have masses \(2 m\) and \(k m\) respectively, where \(k < 2\).
The balls are attached to the ends of a string that passes over a fixed pulley.
The system is held at rest with the string taut and the hanging parts of the string vertical, as shown in Figure 1. The system is released from rest and, in the subsequent motion, \(P\) moves downwards with an acceleration of magnitude \(\frac { 5 g } { 7 }\) The balls are modelled as particles moving freely.
The string is modelled as being light and inextensible.
The pulley is modelled as being small and smooth.
Using the model,
  1. find, in terms of \(m\) and \(g\), the tension in the string,
  2. explain why the acceleration of \(Q\) also has magnitude \(\frac { 5 g } { 7 }\)
  3. find the value of \(k\).
  4. Identify one limitation of the model that will affect the accuracy of your answer to part (c).
Edexcel AS Paper 2 Specimen Q1
  1. Sara is investigating the variation in daily maximum gust, \(t \mathrm { kn }\), for Camborne in June and July 1987.
She used the large data set to select a sample of size 20 from the June and July data for 1987. Sara selected the first value using a random number from 1 to 4 and then selected every third value after that.
  1. State the sampling technique Sara used.
  2. From your knowledge of the large data set explain why this process may not generate a sample of size 20 . The data Sara collected are summarised as follows $$n = 20 \quad \sum t = 374 \quad \sum t ^ { 2 } = 7600$$
  3. Calculate the standard deviation.
Edexcel AS Paper 2 Specimen Q2
  1. The partially completed histogram and the partially completed table show the time, to the nearest minute, that a random sample of motorists was delayed by roadworks on a stretch of motorway.
    \includegraphics[max width=\textwidth, alt={}, center]{8f3dbcb4-3260-4493-a230-12577b4ed691-04_1227_1465_354_301}
Delay (minutes)Number of motorists
4-66
7-8
917
10-1245
13-159
16-20
Estimate the percentage of these motorists who were delayed by the roadworks for between 8.5 and 13.5 minutes.
Edexcel AS Paper 2 Specimen Q3
  1. The Venn diagram shows the probabilities for students at a college taking part in various sports.
    \(A\) represents the event that a student takes part in Athletics.
    \(T\) represents the event that a student takes part in Tennis.
    \(C\) represents the event that a student takes part in Cricket.
    \(p\) and \(q\) are probabilities.
    \includegraphics[max width=\textwidth, alt={}, center]{8f3dbcb4-3260-4493-a230-12577b4ed691-06_668_935_596_566}
The probability that a student selected at random takes part in Athletics or Tennis is 0.75
  1. Find the value of \(p\).
  2. State, giving a reason, whether or not the events \(A\) and \(T\) are statistically independent. Show your working clearly.
  3. Find the probability that a student selected at random does not take part in Athletics or Cricket.
Edexcel AS Paper 2 Specimen Q4
  1. Sara was studying the relationship between rainfall, \(r \mathrm {~mm}\), and humidity, \(h \%\), in the UK. She takes a random sample of 11 days from May 1987 for Leuchars from the large data set.
She obtained the following results.
\(h\)9386959786949797879786
\(r\)1.10.33.720.6002.41.10.10.90.1
Sara examined the rainfall figures and found $$Q _ { 1 } = 0.1 \quad Q _ { 2 } = 0.9 \quad Q _ { 3 } = 2.4$$ A value that is more than 1.5 times the interquartile range (IQR) above \(Q _ { 3 }\) is called an outlier.
  1. Show that \(r = 20.6\) is an outlier.
  2. Give a reason why Sara might:
    1. include
    2. exclude
      this day's reading. Sara decided to exclude this day's reading and drew the following scatter diagram for the remaining 10 days' values of \(r\) and \(h\).
      \includegraphics[max width=\textwidth, alt={}, center]{8f3dbcb4-3260-4493-a230-12577b4ed691-08_988_1081_1555_420}
  3. Give an interpretation of the correlation between rainfall and humidity. The equation of the regression line of \(r\) on \(h\) for these 10 days is \(r = - 12.8 + 0.15 h\)
  4. Give an interpretation of the gradient of this regression line.
    1. Comment on the suitability of Sara's sampling method for this study.
    2. Suggest how Sara could make better use of the large data set for her study.
Edexcel AS Paper 2 Specimen Q5
5. (a) The discrete random variable \(X \sim \mathrm {~B} ( 40,0.27 )\) $$\text { Find } \quad \mathrm { P } ( X \geqslant 16 )$$ Past records suggest that \(30 \%\) of customers who buy baked beans from a large supermarket buy them in single tins. A new manager suspects that there has been a change in the proportion of customers who buy baked beans in single tins. A random sample of 20 customers who had bought baked beans was taken.
(b) Write down the hypotheses that should be used to test the manager's suspicion.
(c) Using a \(10 \%\) level of significance, find the critical region for a two-tailed test to answer the manager's suspicion. You should state the probability of rejection in each tail, which should be less than 0.05
(d) Find the actual significance level of a test based on your critical region from part (c). One afternoon the manager observes that 12 of the 20 customers who bought baked beans, bought their beans in single tins.
(e) Comment on the manager's suspicion in the light of this observation. Later it was discovered that the local scout group visited the supermarket that afternoon to buy food for their camping trip.
(f) Comment on the validity of the model used to obtain the answer to part (e), giving a reason for your answer.
Edexcel AS Paper 2 Specimen Q6
6. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8f3dbcb4-3260-4493-a230-12577b4ed691-12_520_1072_616_388} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A car moves along a straight horizontal road. At time \(t = 0\), the velocity of the car is \(U \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The car then accelerates with constant acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for \(T\) seconds. The car travels a distance \(D\) metres during these \(T\) seconds. Figure 1 shows the velocity-time graph for the motion of the car for \(0 \leqslant t \leqslant T\).
Using the graph, show that \(D = U T + 1 / 2 a T ^ { 2 }\).
(No credit will be given for answers which use any of the kinematics (suvat) formulae listed under Mechanics in the AS Mathematics section of the formulae booklet.)
Edexcel AS Paper 2 Specimen Q7
  1. A car is moving along a straight horizontal road with constant acceleration. There are three points \(A , B\) and \(C\), in that order, on the road, where \(A B = 22 \mathrm {~m}\) and \(B C = 104 \mathrm {~m}\). The car takes 2 s to travel from \(A\) to \(B\) and 4 s to travel from \(B\) to \(C\).
Find
  1. the acceleration of the car,
  2. the speed of the car at the instant it passes \(A\).
Edexcel AS Paper 2 Specimen Q8
  1. A bird leaves its nest at time \(t = 0\) for a short flight along a straight line.
The bird then returns to its nest.
The bird is modelled as a particle moving in a straight horizontal line.
The distance, \(s\) metres, of the bird from its nest at time \(t\) seconds is given by $$s = \frac { 1 } { 10 } \left( t ^ { 4 } - 20 t ^ { 3 } + 100 t ^ { 2 } \right) , \quad \text { where } 0 \leqslant t \leqslant 10$$
  1. Explain the restriction, \(0 \leqslant t \leqslant 10\)
  2. Find the distance of the bird from the nest when the bird first comes to instantaneous rest.
Edexcel AS Paper 2 Specimen Q9
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8f3dbcb4-3260-4493-a230-12577b4ed691-18_694_1262_223_406} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A small ball \(A\) of mass 2.5 kg is held at rest on a rough horizontal table.
The ball is attached to one end of a string.
The string passes over a pulley \(P\) which is fixed at the edge of the table. The other end of the string is attached to a small ball \(B\) of mass 1.5 kg hanging freely, vertically below \(P\) and with \(B\) at a height of 1 m above the horizontal floor. The system is release from rest, with the string taut, as shown in Figure 2.
The resistance to the motion of \(A\) from the rough table is modelled as having constant magnitude 12.7 N . Ball \(B\) reaches the floor before ball \(A\) reaches the pulley. The balls are modelled as particles, the string is modelled as being light and inextensible, the pulley is modelled as being small and smooth and the acceleration due to gravity, \(g\), is modelled as being \(9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
    1. Write down an equation of motion for \(A\).
    2. Write down an equation of motion for \(B\).
  2. Hence find the acceleration of \(B\).
  3. Using the model, find the time it takes, from release, for \(B\) to reach the floor.
  4. Suggest two improvements that could be made in the model.
Edexcel AS Paper 2 Specimen Q1
  1. A company manager is investigating the time taken, \(t\) minutes, to complete an aptitude test. The human resources manager produced the table below of coded times, \(x\) minutes, for a random sample of 30 applicants.
Coded time ( \(x\) minutes)Frequency (f)Coded time midpoint (y minutes)
\(0 \leq x < 5\)32.5
\(5 \leq x < 10\)157.5
\(10 \leq x < 15\)212.5
\(15 \leq x < 25\)920
\(25 \leq x < 35\)130
(You may use \(\sum f y = 355\) and \(\sum f y ^ { 2 } = 5675\) )
  1. Use linear interpolation to estimate the median of the coded times.
  2. Estimate the standard deviation of the coded times. The company manager is told by the human resources manager that he subtracted 15 from each of the times and then divided by 2 , to calculate the coded times.
  3. Calculate an estimate for the median and the standard deviation of \(t\).
    (3) The following year, the company has 25 positions available. The company manager decides not to offer a position to any applicant who takes 35 minutes or more to complete the aptitude test. The company has 60 applicants.
  4. Comment on whether or not the company manager's decision will result in the company being able to fill the 25 positions available from these 60 applicants. Give a reason for your answer.
Edexcel AS Paper 2 Specimen Q2
2. The discrete random variable \(X \sim \mathrm {~B} ( 30,0.28 )\)
  1. Find \(\mathrm { P } ( 5 \leq X < 12 )\). Past records from a large supermarket show that \(25 \%\) of people who buy eggs, buy organic eggs. On one particular day a random sample of 40 people is taken from those that had bought eggs and 16 people are found to have bought organic eggs.
  2. Test, at the \(1 \%\) significance level, whether or not the proportion \(p\) of people who bought organic eggs that day had increased. State your hypotheses clearly.
  3. State the conclusion you would have reached if a \(5 \%\) significance level had been used for this test. \section*{(Total for Question 2 is 8 marks)}
Edexcel AS Paper 2 Specimen Q3
  1. Pete is investigating the relationship between daily rainfall, \(w \mathrm {~mm}\), and daily mean pressure, \(p\) hPa , in Perth during 2015. He used the large data set to take a sample of size 12.
He obtained the following results.
\(p\)100710121013100910191010101010101013101110141022
\(w\)102.063.063.038.438.035.034.232.030.428.028.015
Pete drew the following scatter diagram for the values of \(w\) and \(p\) and calculated the quartiles.
Q 1Q 2Q 3
\(p\)10101011.51013.5
\(w\)29.234.650.7
\includegraphics[max width=\textwidth, alt={}]{b29b0411-8401-420b-9227-befe25c245d8-04_818_1081_989_477}
An outlier is a value which is more than 1.5 times the interquartile range above Q3 or more than 1.5 times the interquartile range below Q1.
  1. Show that the 3 points circled on the scatter diagram above are outliers.
    (2)
  2. Describe the effect of removing the 3 outliers on the correlation between daily rainfall and daily mean pressure in this sample.
    (1) John has also been studying the large data set and believes that the sample Pete has taken is not random.
  3. From your knowledge of the large data set, explain why Pete's sample is unlikely to be a random sample. John finds that the equation of the regression line of \(w\) on \(p\), using all the data in the large data set, is $$w = 1023 - 0.223 p$$
  4. Give an interpretation of the figure - 0.223 in this regression line. John decided to use the regression line to estimate the daily rainfall for a day in December when the daily mean pressure is 1011 hPa .
  5. Using your knowledge of the large data set, comment on the reliability of John's estimate.
    (Total for Question 3 is 6 marks)
Edexcel AS Paper 2 Specimen Q4
4. Alyona, Dawn and Sergei are sometimes late for school. The events \(A , D\) and \(S\) are as follows:
A Alyona is late for school
D Dawn is late for school
S Sergei is late for school The Venn diagram below shows the three events \(A , D\) and \(S\) and the probabilities associated with each region of \(D\). The constants \(p , q\) and \(r\) each represent probabilities associated with the three separate regions outside \(D\).
\includegraphics[max width=\textwidth, alt={}, center]{b29b0411-8401-420b-9227-befe25c245d8-06_624_1068_845_479}
  1. Write down 2 of the events \(A , D\) and \(S\) that are mutually exclusive. Give a reason for your answer. The probability that Sergei is late for school is 0.2 . The events \(A\) and \(D\) are independent.
  2. Find the value of \(r\).
    (4) Dawn and Sergei's teacher believes that when Sergei is late for school, Dawn tends to be late for school.
  3. State whether or not \(D\) and \(S\) are independent, giving a reason for your answer.
    (1)
  4. Comment on the teacher's belief in the light of your answer to part (c).
    (1)
    (Total for Question 4 is 7 marks) \section*{Pearson Edexcel Level 3} \section*{GCE Mathematics} \section*{Paper 2: Mechanics}
    Specimen paper
    Time: \(\mathbf { 3 5 }\) minutes
    		Paper Reference(s)
    \(\mathbf { 8 M A 0 } / \mathbf { 0 2 }\)
    You must have:
    Mathematical Formulae and Statistical Tables, calculator
    Candidates may use any calculator permitted by Pearson regulations. Calculators must not have the facility for algebraic manipulation, differentiation and integration, or have retrievable mathematical formulae stored in them. \section*{Instructions}
    • Use black ink or ball-point pen.
    • If pencil is used for diagrams/sketches/graphs it must be dark (HB or B).
    • Fill in the boxes at the top of this page with your name, centre number and candidate number.
    • Answer all the questions in Section B.
    • Answer the questions in the spaces provided - there may be more space than you need.
    • You should show sufficient working to make your methods clear. Answers without working may not gain full credit.
    • Inexact answers should be given to three significant figures unless otherwise stated.
    \section*{Information}
    • A booklet 'Mathematical Formulae and Statistical Tables' is provided.
    • There are 4 questions in this section. The total mark for Part B of this paper is 30.
    • The marks for each question are shown in brackets - use this as a guide as to how much time to spend on each question.
    \section*{Advice}
    • Read each question carefully before you start to answer it.
    • Try to answer every question.
    • Check your answers if you have time at the end.
    • If you change your mind about an answer, cross it out and put your new answer and any working underneath.