Questions AS Paper 2 (308 questions)

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Edexcel AS Paper 2 2019 June Q1
  1. At time \(t = 0\), a parachutist falls vertically from rest from a helicopter which is hovering at a height of 550 m above horizontal ground.
The parachutist, who is modelled as a particle, falls for 3 seconds before her parachute opens.
While she is falling, and before her parachute opens, she is modelled as falling freely under gravity. The acceleration due to gravity is modelled as being \(10 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  1. Using this model, find the speed of the parachutist at the instant her parachute opens. When her parachute is open, the parachutist continues to fall vertically.
    Immediately after her parachute opens, she decelerates at \(12 \mathrm {~ms} ^ { - 2 }\) for 2 seconds before reaching a constant speed and she reaches the ground with this speed. The total time taken by the parachutist to fall the 550 m from the helicopter to the ground is \(T\) seconds.
  2. Sketch a speed-time graph for the motion of the parachutist for \(0 \leqslant t \leqslant T\).
  3. Find, to the nearest whole number, the value of \(T\). In a refinement of the model of the motion of the parachutist, the effect of air resistance is included before her parachute opens and this refined model is now used to find a new value of \(T\).
  4. How would this new value of \(T\) compare with the value found, using the initial model, in part (c)?
  5. Suggest one further refinement to the model, apart from air resistance, to make the model more realistic.
Edexcel AS Paper 2 2019 June Q2
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{ad0eca04-7b0b-4163-b0da-3a6dc85fec22-06_711_1264_251_402} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A small ball, \(P\), of mass 0.8 kg , is held at rest on a smooth horizontal table and is attached to one end of a thin rope. The rope passes over a pulley that is fixed at the edge of the table.
The other end of the rope is attached to another small ball, \(Q\), of mass 0.6 kg , that hangs freely below the pulley. Ball \(P\) is released from rest, with the rope taut, with \(P\) at a distance of 1.5 m from the pulley and with \(Q\) at a height of 0.4 m above the horizontal floor, as shown in Figure 1. Ball \(Q\) descends, hits the floor and does not rebound.
The balls are modelled as particles, the rope as a light and inextensible string and the pulley as small and smooth. Using this model,
  1. show that the acceleration of \(Q\), as it falls, is \(4.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\)
  2. find the time taken by \(P\) to hit the pulley from the instant when \(P\) is released.
  3. State one limitation of the model that will affect the accuracy of your answer to part (a).
Edexcel AS Paper 2 2019 June Q3
  1. A particle, \(P\), moves along a straight line such that at time \(t\) seconds, \(t \geqslant 0\), the velocity of \(P\), \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), is modelled as
$$v = 12 + 4 t - t ^ { 2 }$$ Find
  1. the magnitude of the acceleration of \(P\) when \(P\) is at instantaneous rest,
  2. the distance travelled by \(P\) in the interval \(0 \leqslant t \leqslant 3\)
Edexcel AS Paper 2 2019 June Q1
  1. A sixth form college has 84 students in Year 12 and 56 students in Year 13
The head teacher selects a stratified sample of 40 students, stratified by year group.
  1. Describe how this sample could be taken. The head teacher is investigating the relationship between the amount of sleep, s hours, that each student had the night before they took an aptitude test and their performance in the test, \(p\) marks.
    For the sample of 40 students, he finds the equation of the regression line of \(p\) on \(s\) to be $$p = 26.1 + 5.60 s$$
  2. With reference to this equation, describe the effect that an extra 0.5 hours of sleep may have, on average, on a student's performance in the aptitude test.
  3. Describe one limitation of this regression model.
Edexcel AS Paper 2 2019 June Q2
  1. The Venn diagram shows three events, \(A\), \(B\) and \(C\), and their associated probabilities.
    \includegraphics[max width=\textwidth, alt={}, center]{e73193ee-339e-48ab-811c-9ab6817f786d-04_680_780_296_644}
Events \(B\) and \(C\) are mutually exclusive.
Events \(A\) and \(C\) are independent.
Showing your working, find the value of \(x\), the value of \(y\) and the value of \(z\).
Edexcel AS Paper 2 2019 June Q3
  1. A fair 5 -sided spinner has sides numbered \(1,2,3,4\) and 5
The spinner is spun once and the score of the side it lands on is recorded.
  1. Write down the name of the distribution that can be used to model the score of the side it lands on. The spinner is spun 28 times.
    The random variable \(X\) represents the number of times the spinner lands on 2
    1. Find the probability that the spinner lands on 2 at least 7 times.
    2. Find \(\mathrm { P } ( 4 \leqslant X < 8 )\)
Edexcel AS Paper 2 2019 June Q4
  1. Joshua is investigating the daily total rainfall in Hurn for May to October 2015
Using the information from the large data set, Joshua wishes to calculate the mean of the daily total rainfall in Hurn for May to October 2015
  1. Using your knowledge of the large data set, explain why Joshua needs to clean the data before calculating the mean. Using the information from the large data set, he produces the grouped frequency table below.
    Daily total rainfall ( \(r \mathrm {~mm}\) )FrequencyMidpoint ( \(\boldsymbol { x } \mathbf { m m }\) )
    \(0 \leqslant r < 0.5\)1210.25
    \(0.5 \leqslant r < 1.0\)100.75
    \(1.0 \leqslant r < 5.0\)243.0
    \(5.0 \leqslant r < 10.0\)127.5
    \(10.0 \leqslant r < 30.0\)1720.0
    $$\text { You may use } \sum \mathrm { f } x = 539.75 \text { and } \sum \mathrm { f } x ^ { 2 } = 7704.1875$$
  2. Use linear interpolation to calculate an estimate for the upper quartile of the daily total rainfall.
  3. Calculate an estimate for the standard deviation of the daily total rainfall in Hurn for May to October 2015
    1. State the assumption involved with using class midpoints to calculate an estimate of a mean from a grouped frequency table.
    2. Using your knowledge of the large data set, explain why this assumption does not hold in this case.
    3. State, giving a reason, whether you would expect the actual mean daily total rainfall in Hurn for May to October 2015 to be larger than, smaller than or the same as an estimate based on the grouped frequency table.
Edexcel AS Paper 2 2019 June Q5
  1. Past records show that \(15 \%\) of customers at a shop buy chocolate. The shopkeeper believes that moving the chocolate closer to the till will increase the proportion of customers buying chocolate.
After moving the chocolate closer to the till, a random sample of 30 customers is taken and 8 of them are found to have bought chocolate. Julie carries out a hypothesis test, at the 5\% level of significance, to test the shopkeeper's belief.
Julie's hypothesis test is shown below.
\(\mathrm { H } _ { 0 } : p = 0.15\)
\(\mathrm { H } _ { 1 } : p \geqslant 0.15\)
Let \(X =\) the number of customers who buy chocolate.
\(X \sim \mathrm {~B} ( 30,0.15 )\)
\(\mathrm { P } ( X = 8 ) = 0.0420\)
\(0.0420 < 0.05\) so reject \(\mathrm { H } _ { 0 }\)
There is sufficient evidence to suggest that the proportion of customers buying chocolate has increased.
  1. Identify the first two errors that Julie has made in her hypothesis test.
  2. Explain whether or not these errors will affect the conclusion of her hypothesis test. Give a reason for your answer.
  3. Find, using a 5\% level of significance, the critical region for a one-tailed test of the shopkeeper's belief. The probability in the tail should be less than 0.05
  4. Find the actual level of significance of this test.
Edexcel AS Paper 2 2020 June Q1
  1. At time \(t = 0\), a small ball is projected vertically upwards with speed \(U \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(A\) that is 16.8 m above horizontal ground.
The speed of the ball at the instant immediately before it hits the ground for the first time is \(19 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) The ball hits the ground for the first time at time \(t = T\) seconds.
The motion of the ball, from the instant it is projected until the instant just before it hits the ground for the first time, is modelled as that of a particle moving freely under gravity. The acceleration due to gravity is modelled as having magnitude \(10 \mathrm {~m} \mathrm {~s} ^ { - 2 }\)
Using the model,
  1. show that \(U = 5\)
  2. find the value of \(T\),
  3. find the time from the instant the ball is projected until the instant when the ball is 1.2 m below \(A\).
  4. Sketch a velocity-time graph for the motion of the ball for \(0 \leqslant t \leqslant T\), stating the coordinates of the start point and the end point of your graph. In a refinement of the model of the motion of the ball, the effect of air resistance on the ball is included and this refined model is now used to find the value of \(U\).
  5. State, with a reason, how this new value of \(U\) would compare with the value found in part (a), using the initial unrefined model.
  6. Suggest one further refinement that could be made to the model, apart from including air resistance, that would make the model more realistic.
Edexcel AS Paper 2 2020 June Q2
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0fd98465-9db5-4125-b53f-7a9a3467ac41-06_526_415_244_826} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} One end of a string is attached to a small ball \(P\) of mass \(4 m\).
The other end of the string is attached to another small ball \(Q\) of mass \(3 m\).
The string passes over a fixed pulley.
Ball \(P\) is held at rest with the string taut and the hanging parts of the string vertical, as shown in Figure 1. Ball \(P\) is released.
The string is modelled as being light and inextensible, the balls are modelled as particles, the pulley is modelled as being smooth and air resistance is ignored.
  1. Using the model, find, in terms of \(m\) and \(g\), the magnitude of the force exerted on the pulley by the string while \(P\) is falling and before \(Q\) hits the pulley.
  2. State one limitation of the model, apart from ignoring air resistance, that will affect the accuracy of your answer to part (a).
Edexcel AS Paper 2 2020 June Q3
  1. A particle \(P\) moves along a straight line such that at time \(t\) seconds, \(t \geqslant 0\), after leaving the point \(O\) on the line, the velocity, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), of \(P\) is modelled as
$$v = ( 7 - 2 t ) ( t + 2 )$$
  1. Find the value of \(t\) at the instant when \(P\) stops accelerating.
  2. Find the distance of \(P\) from \(O\) at the instant when \(P\) changes its direction of motion. In this question, solutions relying on calculator technology are not acceptable.
Edexcel AS Paper 2 2020 June Q1
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d62e5a00-cd23-417f-b244-8b3e24da4aa2-02_849_1271_246_303} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The histogram in Figure 1 shows the times taken to complete a crossword by a random sample of students. The number of students who completed the crossword in more than 15 minutes is 78
Estimate the percentage of students who took less than 11 minutes to complete the crossword.
Edexcel AS Paper 2 2020 June Q2
  1. Jerry is studying visibility for Camborne using the large data set June 1987.
The table below contains two extracts from the large data set.
It shows the daily maximum relative humidity and the daily mean visibility.
Date
Daily Maximum
Relative Humidity
Daily Mean Visibility
Units\(\%\)
\(10 / 06 / 1987\)905300
\(28 / 06 / 1987\)1000
(The units for Daily Mean Visibility are deliberately omitted.)
Given that daily mean visibility is given to the nearest 100,
  1. write down the range of distances in metres that corresponds to the recorded value 0 for the daily mean visibility. Jerry drew the following scatter diagram, Figure 2, and calculated some statistics using the June 1987 data for Camborne from the large data set. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{d62e5a00-cd23-417f-b244-8b3e24da4aa2-04_823_1764_1281_137} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} Jerry defines an outlier as a value that is more than 1.5 times the interquartile range above \(Q _ { 3 }\) or more than 1.5 times the interquartile range below \(Q _ { 1 }\).
  2. Show that the point circled on the scatter diagram is an outlier for visibility.
  3. Interpret the correlation between the daily mean visibility and the daily maximum relative humidity. Jerry drew the following scatter diagram, Figure 3, using the June 1987 data for Camborne from the large data set, but forgot to label the \(x\)-axis.
    \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{d62e5a00-cd23-417f-b244-8b3e24da4aa2-05_730_1056_342_386} \captionsetup{labelformat=empty} \caption{Figure 3}
    \end{figure}
  4. Using your knowledge of the large data set, suggest which variable the \(x\)-axis on this scatter diagram represents.
Edexcel AS Paper 2 2020 June Q3
  1. In a game, a player can score \(0,1,2,3\) or 4 points each time the game is played.
The random variable \(S\), representing the player's score, has the following probability distribution where \(a , b\) and \(c\) are constants.
\(s\)01234
\(\mathrm { P } ( S = s )\)\(a\)\(b\)\(c\)0.10.15
The probability of scoring less than 2 points is twice the probability of scoring at least 2 points. Each game played is independent of previous games played.
John plays the game twice and adds the two scores together to get a total.
Calculate the probability that the total is 6 points.
Edexcel AS Paper 2 2020 June Q4
  1. A lake contains three different types of carp.
There are an estimated 450 mirror carp, 300 leather carp and 850 common carp.
Tim wishes to investigate the health of the fish in the lake.
He decides to take a sample of 160 fish.
  1. Give a reason why stratified random sampling cannot be used.
  2. Explain how a sample of size 160 could be taken to ensure that the estimated populations of each type of carp are fairly represented. You should state the name of the sampling method used. As part of the health check, Tim weighed the fish.
    His results are given in the table below.
    Weight (wkg)Frequency (f)Midpoint (m kg)
    \(2 \leqslant w < 3.5\)82.75
    \(3.5 \leqslant w < 4\)323.75
    \(4 \leqslant w < 4.5\)644.25
    \(4.5 \leqslant w < 5\)404.75
    \(5 \leqslant w < 6\)165.5
    $$\left( \text { You may use } \sum \mathrm { f } m = 692 \quad \text { and } \quad \sum \mathrm { f } m ^ { 2 } = 3053 \right)$$
  3. Calculate an estimate for the standard deviation of the weight of the carp. Tim realised that he had transposed the figures for 2 of the weights of the fish.
    He had recorded in the table 2.3 instead of 3.2 and 4.6 instead of 6.4
  4. Without calculating a new estimate for the standard deviation, state what effect
    1. using the correct figure of 3.2 instead of 2.3
    2. using the correct figure of 6.4 instead of 4.6
      would have on your estimated standard deviation.
      Give a reason for each of your answers.
Edexcel AS Paper 2 2020 June Q5
  1. Afrika works in a call centre.
She assumes that calls are independent and knows, from past experience, that on each sales call that she makes there is a probability of \(\frac { 1 } { 6 }\) that it is successful. Afrika makes 9 sales calls.
  1. Calculate the probability that at least 3 of these sales calls will be successful. The probability of Afrika making a successful sales call is the same each day.
    Afrika makes 9 sales calls on each of 5 different days.
  2. Calculate the probability that at least 3 of the sales calls will be successful on exactly 1 of these days. Rowan works in the same call centre as Afrika and believes he is a more successful salesperson. To check Rowan’s belief, Afrika monitors the next 35 sales calls Rowan makes and finds that 11 of the sales calls are successful.
  3. Stating your hypotheses clearly test, at the \(5 \%\) level of significance, whether or not there is evidence to support Rowan’s belief.
Edexcel AS Paper 2 2022 June Q1
  1. The point \(A\) is 1.8 m vertically above horizontal ground.
At time \(t = 0\), a small stone is projected vertically upwards with speed \(U \mathrm {~ms} ^ { - 1 }\) from the point \(A\). At time \(t = T\) seconds, the stone hits the ground.
The speed of the stone as it hits the ground is \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
In an initial model of the motion of the stone as it moves from \(A\) to where it hits the ground
  • the stone is modelled as a particle moving freely under gravity
  • the acceleration due to gravity is modelled as having magnitude \(\mathbf { 1 0 } \mathbf { m ~ s } ^ { \mathbf { - 2 } }\)
Using the model,
  1. find the value of \(U\),
  2. find the value of \(T\).
  3. Suggest one refinement, apart from including air resistance, that would make the model more realistic. In reality the stone will not move freely under gravity and will be subject to air resistance.
  4. Explain how this would affect your answer to part (a).
Edexcel AS Paper 2 2022 June Q2
  1. A train travels along a straight horizontal track from station \(P\) to station \(Q\).
In a model of the motion of the train, at time \(t = 0\) the train starts from rest at \(P\), and moves with constant acceleration until it reaches its maximum speed of \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) The train then travels at this constant speed of \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) before finally moving with constant deceleration until it comes to rest at \(Q\). The time spent decelerating is four times the time spent accelerating.
The journey from \(P\) to \(Q\) takes 700 s .
Using the model,
  1. sketch a speed-time graph for the motion of the train between the two stations \(P\) and \(Q\). The distance between the two stations is 15 km .
    Using the model,
  2. show that the time spent accelerating by the train is 40 s ,
  3. find the acceleration, in \(\mathrm { m } \mathrm { s } ^ { - 2 }\), of the train,
  4. find the speed of the train 572s after leaving \(P\).
  5. State one limitation of the model which could affect your answers to parts (b) and (c).
Edexcel AS Paper 2 2022 June Q3
  1. A fixed point \(O\) lies on a straight line.
A particle \(P\) moves along the straight line.
At time \(t\) seconds, \(t \geqslant 0\), the distance, \(s\) metres, of \(P\) from \(O\) is given by $$s = \frac { 1 } { 3 } t ^ { 3 } - \frac { 5 } { 2 } t ^ { 2 } + 6 t$$
  1. Find the acceleration of \(P\) at each of the times when \(P\) is at instantaneous rest.
  2. Find the total distance travelled by \(P\) in the interval \(0 \leqslant t \leqslant 4\)
Edexcel AS Paper 2 2022 June Q4
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{278a61e9-e27f-4fd5-895a-db01886aadfe-14_545_314_248_877} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A vertical rope \(P Q\) has its end \(Q\) attached to the top of a small lift cage.
The lift cage has mass 40 kg and carries a block of mass 10 kg , as shown in Figure 1.
The lift cage is raised vertically by moving the end \(P\) of the rope vertically upwards with constant acceleration \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) The rope is modelled as being light and inextensible and air resistance is ignored.
Using the model,
  1. find the tension in the rope \(P Q\)
  2. find the magnitude of the force exerted on the block by the lift cage.
Edexcel AS Paper 2 2022 June Q1
  1. The relationship between two variables \(p\) and \(t\) is modelled by the regression line with equation
$$p = 22 - 1.1 t$$ The model is based on observations of the independent variable, \(t\), between 1 and 10
  1. Describe the correlation between \(p\) and \(t\) implied by this model. Given that \(p\) is measured in centimetres and \(t\) is measured in days,
  2. state the units of the gradient of the regression line. Using the model,
  3. calculate the change in \(p\) over a 3-day period. Tisam uses this model to estimate the value of \(p\) when \(t = 19\)
  4. Comment, giving a reason, on the reliability of this estimate.
Edexcel AS Paper 2 2022 June Q2
  1. A manufacturer of sweets knows that \(8 \%\) of the bags of sugar delivered from supplier \(A\) will be damp.
    A random sample of 35 bags of sugar is taken from supplier \(A\).
    1. Using a suitable model, find the probability that the number of bags of sugar that are damp is
      1. exactly 2
      2. more than 3
    Supplier \(B\) claims that when it supplies bags of sugar, the proportion of bags that are damp is less than \(8 \%\) The manufacturer takes a random sample of 70 bags of sugar from supplier \(B\) and finds that only 2 of the bags are damp.
  2. Carry out a suitable test to assess supplier B's claim. You should state your hypotheses clearly and use a \(10 \%\) level of significance.
Edexcel AS Paper 2 2022 June Q3
  1. The histogram summarises the heights of 256 seedlings two weeks after they were planted.
    \includegraphics[max width=\textwidth, alt={}, center]{08e3b0b0-2155-4b37-83e3-343c317ca10c-06_1242_1810_287_132}
    1. Use linear interpolation to estimate the median height of the seedlings.
      (4)
    Chris decides to model the frequency density for these 256 seedlings by a curve with equation $$y = k x ( 8 - x ) \quad 0 \leqslant x \leqslant 8$$ where \(k\) is a constant.
  2. Find the value of \(k\) Using this model,
  3. write down the median height of the seedlings.
Edexcel AS Paper 2 2022 June Q4
  1. Jiang is studying the variable Daily Mean Pressure from the large data set.
He drew the following box and whisker plot for these data for one of the months for one location using a linear scale but
  • he failed to label all the values on the scale
  • he gave an incorrect value for the median
    \includegraphics[max width=\textwidth, alt={}, center]{08e3b0b0-2155-4b37-83e3-343c317ca10c-09_248_1264_573_402}
Daily Mean Pressure (hPa)
Using your knowledge of the large data set, suggest a suitable value for
  1. the median,
  2. the range.
    (You are not expected to have memorised values from the large data set. The question is simply looking for sensible answers.)
    1. Jiang is studying the variable Daily Mean Pressure from the large data set.
    He drew the following box and whisker plot for these data for one of the months for one nong asing at
    • he gave an incorrect value for the median "
    Using your knowledge of the large data set, suggest a suitable value for
  3. the median,
    "
    \includegraphics[max width=\textwidth, alt={}, center]{08e3b0b0-2155-4b37-83e3-343c317ca10c-09_42_31_1213_1304}
    "
Edexcel AS Paper 2 2022 June Q5
5. Manon has two biased spinners, one red and one green. The random variable \(R\) represents the score when the red spinner is spun.
The random variable \(G\) represents the score when the green spinner is spun.
The probability distributions for \(R\) and \(G\) are given below.
\(r\)23
\(\mathrm { P } ( R = r )\)\(\frac { 1 } { 4 }\)\(\frac { 3 } { 4 }\)
\(g\)14
\(\mathrm { P } ( G = g )\)\(\frac { 2 } { 3 }\)\(\frac { 1 } { 3 }\)
Manon spins each spinner once and adds the two scores.
  1. Find the probability that
    1. the sum of the two scores is 7
    2. the sum of the two scores is less than 4 The random variable \(X = m R + n G\) where \(m\) and \(n\) are integers. $$\mathrm { P } ( X = 20 ) = \frac { 1 } { 6 } \quad \text { and } \quad \mathrm { P } ( X = 50 ) = \frac { 1 } { 4 }$$
  2. Find the value of \(m\) and the value of \(n\)