Questions AS Paper 2 (308 questions)

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Edexcel AS Paper 2 2023 June Q1
1. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d9615c4f-d8fa-4e44-978a-cf34b2b1c0b5-02_720_1490_283_299} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Two children, Pat \(( P )\) and Sam \(( S )\), run a race along a straight horizontal track.
Both children start from rest at the same time and cross the finish line at the same time.
In a model of the motion:
Pat accelerates at a constant rate from rest for 5 s until reaching a speed of \(4 \mathrm {~ms} ^ { - 1 }\) and then maintains a constant speed of \(4 \mathrm {~ms} ^ { - 1 }\) until crossing the finish line. Sam accelerates at a constant rate of \(1 \mathrm {~ms} ^ { - 2 }\) from rest until reaching a speed of \(X \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and then maintains a constant speed of \(X \mathrm {~m} \mathrm {~s} ^ { - 1 }\) until crossing the finish line. Both children take 27.5 s to complete the race.
The velocity-time graphs shown in Figure 1 describe the model of the motion of each child from the instant they start to the instant they cross the finish line together. Using the model,
  1. explain why the areas under the two graphs are equal,
  2. find the acceleration of Pat during the first 5 seconds,
  3. find, in metres, the length of the race,
  4. find the value of \(X\), giving your answer to 3 significant figures.
Edexcel AS Paper 2 2023 June Q2
  1. A small stone is projected vertically upwards with speed \(39.2 \mathrm {~ms} ^ { - 1 }\) from a point \(O\).
The stone is modelled as a particle moving freely under gravity from when it is projected until it hits the ground 10s later. Using the model, find
  1. the height of \(O\) above the ground,
  2. the total length of time for which the speed of the stone is less than or equal to \(24.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
  3. State one refinement that could be made to the model that would make your answer to part (a) more accurate.
Edexcel AS Paper 2 2023 June Q3
  1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
A fixed point \(O\) lies on a straight line.
A particle \(P\) moves along the straight line such that at time \(t\) seconds, \(t \geqslant 0\), after passing through \(O\), the velocity of \(P , v \mathrm {~ms} ^ { - 1 }\), is modelled as $$v = 15 - t ^ { 2 } - 2 t$$
  1. Verify that \(P\) comes to instantaneous rest when \(t = 3\)
  2. Find the magnitude of the acceleration of \(P\) when \(t = 3\)
  3. Find the total distance travelled by \(P\) in the interval \(0 \leqslant t \leqslant 4\)
Edexcel AS Paper 2 2023 June Q4
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d9615c4f-d8fa-4e44-978a-cf34b2b1c0b5-10_211_1527_294_269} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A car of mass 1200 kg is towing a trailer of mass 400 kg along a straight horizontal road using a tow rope, as shown in Figure 2.
The rope is horizontal and parallel to the direction of motion of the car.
  • The resistance to motion of the car is modelled as a constant force of magnitude \(2 R\) newtons
  • The resistance to motion of the trailer is modelled as a constant force of magnitude \(R\) newtons
  • The rope is modelled as being light and inextensible
  • The acceleration of the car is modelled as \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\)
The driving force of the engine of the car is 7400 N and the tension in the tow rope is 2400 N . Using the model,
  1. find the value of \(a\) In a refined model, the rope is modelled as having mass and the acceleration of the car is found to be \(a _ { 1 } \mathrm {~ms} ^ { - 2 }\)
  2. State how the value of \(a _ { 1 }\) compares with the value of \(a\)
  3. State one limitation of the model used for the resistance to motion of the car.
Edexcel AS Paper 2 2023 June Q1
  1. The histogram and its frequency polygon below give information about the weights, in grams, of 50 plums.
    \includegraphics[max width=\textwidth, alt={}, center]{854568d2-b32d-44de-8a9c-26372e509c20-02_908_1307_328_386}
    1. Show that an estimate for the mean weight of the 50 plums is 63.72 grams.
    2. Calculate an estimate for the standard deviation of the 50 plums.
    Later it was discovered that the scales used to weigh the plums were broken.
    Each plum actually weighs 5 grams less than originally thought.
  2. State the effect this will have on the estimate of the standard deviation in part (b). Give a reason for your answer.
Edexcel AS Paper 2 2023 June Q2
  1. Fred and Nadine are investigating whether there is a linear relationship between Daily Mean Pressure, \(p \mathrm { hPa }\), and Daily Mean Air Temperature, \(t ^ { \circ } \mathrm { C }\), in Beijing using the 2015 data from the large data set.
Fred randomly selects one month from the data set and draws the scatter diagram in Figure 1 using the data from that month. The scale has been left off the horizontal axis. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{854568d2-b32d-44de-8a9c-26372e509c20-04_794_1539_589_264} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure}
  1. Describe the correlation shown in Figure 1. Nadine chooses to use all of the data for Beijing from 2015 and draws the scatter diagram in Figure 2. She uses the same scales as Fred. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{854568d2-b32d-44de-8a9c-26372e509c20-04_777_1509_1841_278} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure}
  2. Explain, in context, what Nadine can infer about the relationship between \(p\) and \(t\) using the information shown in Figure 2.
  3. Using your knowledge of the large data set, state a value of \(p\) for which interpolation can be used with Figure 2 to predict a value of \(t\).
  4. Using your knowledge of the large data set, explain why it is not meaningful to look for a linear relationship between Daily Mean Wind Speed (Beaufort Conversion) and Daily Mean Air Temperature in Beijing in 2015.
  5. Explain, in context, what Nadine can infer about the relationship between \(p\) and \(t\) using the information shown in Figure 2.
Edexcel AS Paper 2 2023 June Q3
3. In an after-school club, students can choose to take part in Art, Music, both or neither. There are 45 students that attend the after-school club. Of these
  • 25 students take part in Art
  • 12 students take part in both Art and Music
  • the number of students that take part in Music is \(x\)
    1. Find the range of possible values of \(x\)
One of the 45 students is selected at random.
Event \(A\) is the event that the student selected takes part in Art.
Event \(M\) is the event that the student selected takes part in Music.
  • Determine whether or not it is possible for the events \(A\) and \(M\) to be independent.
    •
    Edexcel AS Paper 2 2023 June Q4
    1. Past information shows that \(25 \%\) of adults in a large population have a particular allergy.
    Rylan believes that the proportion that has the allergy differs from 25\%
    He takes a random sample of 50 adults from the population.
    Rylan carries out a test of the null hypothesis \(\mathrm { H } _ { 0 } : p = 0.25\) using a \(5 \%\) level of significance.
    1. Write down the alternative hypothesis for Rylan's test.
    2. Find the critical region for this test. You should state the probability associated with each tail, which should be as close to \(2.5 \%\) as possible.
    3. State the actual probability of incorrectly rejecting \(\mathrm { H } _ { 0 }\) for this test. Rylan finds that 10 of the adults in his sample have the allergy.
    4. State the conclusion of Rylan's hypothesis test.
    Edexcel AS Paper 2 2023 June Q5
    1. Julia selects 3 letters at random, one at a time without replacement, from the word
    \section*{VARIANCE} The discrete random variable \(X\) represents the number of times she selects a letter A.
    1. Find the complete probability distribution of \(X\). Yuki selects 10 letters at random, one at a time with replacement, from the word \section*{DEVIATION}
    2. Find the probability that he selects the letter E at least 4 times.
    Edexcel AS Paper 2 2024 June Q1
    1. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{34fc8023-cf31-420a-bb92-a31735fe5bdb-02_630_1537_296_264} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} Figure 1 shows the speed-time graph for the journey of a car moving in a long queue of traffic on a straight horizontal road. At time \(\mathrm { t } = 0\), the car is at rest at the point A .
    The car then accelerates uniformly for 5 seconds until it reaches a speed of \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
    For the next 15 seconds the car travels at a constant speed of \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
    The car then decelerates uniformly until it comes to rest at the point B.
    The total journey time is 30 seconds.
    1. Find the distance AB .
    2. Sketch a distance-time graph for the journey of the car from A to B .
    Edexcel AS Paper 2 2024 June Q2
    1. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.
    A particle is moving along a straight line.
    At time t seconds, \(\mathrm { t } > 0\), the velocity of the particle is \(\mathrm { Vms } ^ { - 1 }\), where $$v = 2 t - 7 \sqrt { t } + 6$$
    1. Find the acceleration of the particle when \(t = 4\) When \(\mathrm { t } = 1\) the particle is at the point X .
      When \(\mathrm { t } = 2\) the particle is at the point Y . Given that the particle does not come to instantaneous rest in the interval \(1 < \mathrm { t } < 2\)
    2. show that \(X Y = \frac { 1 } { 3 } ( 41 - 28 \sqrt { 2 } )\) metres.
    Edexcel AS Paper 2 2024 June Q3
    1. \hspace{0pt} [In this question, \(\mathbf { i }\) and \(\mathbf { j }\) are perpendi cular unit vectors in a horizontal plane]
    A particle P is moving on a smooth horizontal surface under the action of two forces.
    Given that
    • the mass of P is 2 kg
    • the two forces are \(( 2 \mathbf { i } + 4 \mathbf { j } ) \mathrm { N }\) and \(( \mathbf { i } - 2 \mathbf { j } ) \mathrm { N }\), where C is a constant
    • the magnitude of the acceleration of P is \(\sqrt { 5 } \mathrm {~m} \mathrm {~s} ^ { - 2 }\)
      find the two possible values of C .
    Edexcel AS Paper 2 2024 June Q4
    4. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{34fc8023-cf31-420a-bb92-a31735fe5bdb-08_225_1239_280_413} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure} Figure 2 shows a car towing a trailer along a straight horizontal road.
    The mass of the car is 800 kg and the mass of the trailer is 600 kg .
    The trailer is attached to the car by a towbar which is parallel to the road and parallel to the direction of motion of the car and the trailer. The towbar is modelled as a light rod.
    The resistance to the motion of the car is modelled as a constant force of magnitude 400 N .
    The resistance to the motion of the trailer is modelled as a constant force of magnitude R newtons. The engine of the car is producing a constant driving force that is horizontal and of magnitude 1740 N. The acceleration of the car is \(0.6 \mathrm {~ms} ^ { - 2 }\) and the tension in the towbar is T newtons.
    Using the model,
    1. show that \(\mathrm { R } = 500\)
    2. find the value of T . At the instant when the speed of the car and the trailer is \(12.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the towbar breaks.
      The trailer moves a further distance d metres before coming to rest.
      The resistance to the motion of the trailer is modelled as a constant force of magnitude 500 N. Using the model,
    3. show that, after the towbar breaks, the deceleration of the trailer is \(\frac { 5 } { 6 } \mathrm {~ms} ^ { - 2 }\)
    4. find the value of d. In reality, the distance d metres is likely to be different from the answer found in part (d).
    5. Give two different reasons why this is the case.
    Edexcel AS Paper 2 2024 June Q1
    1. A coach recorded the heights of some adult rugby players and found the following summary statistics.
    $$\begin{array} { r } \text { Median } = 1.85 \mathrm {~m}
    \text { Range } = 0.28 \mathrm {~m}
    \text { Interquartile range } = 0.11 \mathrm {~m} \end{array}$$ The coach also noticed that
    • the height of the shortest player is 1.72 m
    • \(25 \%\) of the players' heights are below the height of a player whose height is 1.81 m
    Draw a box and whisker plot to represent this information on the grid below.
    \includegraphics[max width=\textwidth, alt={}, center]{6a0b46f8-7a6a-4ed8-8c7a-9772787f155a-02_342_1096_1027_488}
    Edexcel AS Paper 2 2024 June Q2
    1. Keith is studying the variable Daily Mean Wind Direction, in degrees, from the large data set.
    Keith summarised the data for Camborne from 1987 into 4 directions \(A , B , C\) and \(D\) representing North, South, East and West in some order.
    Direction\(A\)\(B\)\(C\)\(D\)
    Frequency22485658
    1. Using your knowledge of the large data set state, giving a reason, which direction \(A\) represents. The entry for Hurn on 27th September 1987 was 999
    2. State, giving a reason, what Keith should do with this value.
    Edexcel AS Paper 2 2024 June Q3
    1. Customers in a shop have to queue to pay.
    The partially completed table below and partially completed histogram opposite, give information about the time, \(x\) minutes, spent in the queue by each of 112 customers one day.
    Time in queue ( \(\boldsymbol { x }\) minutes)Frequency
    \(1 - 2\)64
    \(2 - 3\)
    \(3 - 4\)13
    \(4 - 6\)
    \(6 - 8\)3
    No customer spent less than 1 minute or longer than 8 minutes in the queue.
    1. Complete the table.
    2. Complete the histogram. Ting decides to model the frequency density for these 112 customers by a curve with equation $$y = \frac { k } { x ^ { 2 } } \quad 1 \leqslant x \leqslant 8$$ where \(k\) is a constant.
    3. Find the value of \(k\)
      \includegraphics[max width=\textwidth, alt={}]{6a0b46f8-7a6a-4ed8-8c7a-9772787f155a-07_1584_1189_285_443}
      Only use this grid if you need to redraw your histogram.
      \includegraphics[max width=\textwidth, alt={}, center]{6a0b46f8-7a6a-4ed8-8c7a-9772787f155a-09_1582_1192_367_440}
      \includegraphics[max width=\textwidth, alt={}, center]{6a0b46f8-7a6a-4ed8-8c7a-9772787f155a-09_2267_51_307_36}
    Edexcel AS Paper 2 2024 June Q4
    4. The random variable \(X \sim \mathrm {~B} ( 27,0.35 )\)
    1. Find
      1. \(\mathrm { P } ( X = 10 )\)
      2. \(\mathrm { P } ( 12 \leqslant X < 15 )\) Historical records show that the proportion of defective items produced by a machine is 0.12 Following a maintenance service of the machine, a random sample of 60 items is taken and 3 defective items are found.
    2. Carry out a suitable test to determine whether the proportion of defective items produced by the machine has decreased following the maintenance service. You should state your hypotheses clearly and use a \(5 \%\) level of significance.
    3. Write down the \(p\)-value for your test in part (b)
    Edexcel AS Paper 2 2024 June Q5
    1. A biased 4 -sided spinner has the numbers \(6,7,8\) and 10 on it.
    The discrete random variable \(X\) represents the score when the spinner is spun once and has the following probability distribution,
    \(x\)67810
    \(\mathrm { P } ( X = x )\)0.50.2\(q\)\(q\)
    where \(q\) is a probability.
    1. Find the value of \(q\) Karen spins the spinner repeatedly until she either gets a 7 or she has taken 4 spins.
    2. Show that the probability that Karen stops after taking her 3rd spin is 0.128 The random variable \(S\) represents the number of spins Karen takes.
    3. Find the probability distribution for \(S\) The random variable \(N\) represents the number of times Karen gets a 7
    4. Find \(\mathrm { P } ( S > N )\)
    Edexcel AS Paper 2 2021 November Q1
    1. At time \(t = 0\), a small stone is thrown vertically upwards with speed \(14.7 \mathrm {~ms} ^ { - 1 }\) from a point \(A\).
    At time \(t = T\) seconds, the stone passes through \(A\), moving downwards.
    The stone is modelled as a particle moving freely under gravity throughout its motion.
    Using the model,
    1. find the value of \(T\),
    2. find the total distance travelled by the stone in the first 4 seconds of its motion.
    3. State one refinement that could be made to the model, apart from air resistance, that would make the model more realistic.
    Edexcel AS Paper 2 2021 November Q2
    1. A particle \(P\) moves along a straight line.
    At time \(t\) seconds, the velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) of \(P\) is modelled as $$v = 10 t - t ^ { 2 } - k \quad t \geqslant 0$$ where \(k\) is a constant.
    1. Find the acceleration of \(P\) at time \(t\) seconds. The particle \(P\) is instantaneously at rest when \(t = 6\)
    2. Find the other value of \(t\) when \(P\) is instantaneously at rest.
    3. Find the total distance travelled by \(P\) in the interval \(0 \leqslant t \leqslant 6\)
    Edexcel AS Paper 2 2021 November Q3
    3. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{4a022ec0-7640-4664-87a6-1963309cad6a-08_761_595_210_735} \captionsetup{labelformat=empty} \caption{Figure 1}
    \end{figure} A ball \(P\) of mass \(2 m\) is attached to one end of a string.
    The other end of the string is attached to a ball \(Q\) of mass \(5 m\).
    The string passes over a fixed pulley.
    The system is held at rest with the balls hanging freely and the string taut.
    The hanging parts of the string are vertical with \(P\) at a height \(2 h\) above horizontal ground and with \(Q\) at a height \(h\) above the ground, as shown in Figure 1. The system is released from rest.
    In the subsequent motion, \(Q\) does not rebound when it hits the ground and \(P\) does not hit 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.
    Air resistance is modelled as being negligible.
    Using this model,
      1. write down an equation of motion for \(P\),
      2. write down an equation of motion for \(Q\),
    2. find, in terms of \(h\) only, the height above the ground at which \(P\) first comes to instantaneous rest.
    3. State one limitation of modelling the balls as particles that could affect your answer to part (b). In reality, the string will not be inextensible.
    4. State how this would affect the accelerations of the particles.
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    Edexcel AS Paper 2 2021 November Q1
    1.
    \includegraphics[max width=\textwidth, alt={}, center]{6dfefd72-338f-40be-ac37-aef56bfaccaa-02_399_743_248_662} The Venn diagram, where \(p\) is a probability, shows the 3 events \(A , B\) and \(C\) with their associated probabilities.
    1. Find the value of \(p\).
    2. Write down a pair of mutually exclusive events from \(A , B\) and \(C\).
    Edexcel AS Paper 2 2021 November Q2
    1. The partially completed table and partially completed histogram give information about the ages of passengers on an airline.
    There were no passengers aged 90 or over.
    Age ( \(x\) years)\(0 \leqslant x < 5\)\(5 \leqslant x < 20\)\(20 \leqslant x < 40\)\(40 \leqslant x < 65\)\(65 \leqslant x < 80\)\(80 \leqslant x < 90\)
    Frequency545901
    \includegraphics[max width=\textwidth, alt={}, center]{6dfefd72-338f-40be-ac37-aef56bfaccaa-04_1173_1792_721_139}
    1. Complete the histogram.
    2. Use linear interpolation to estimate the median age. An outlier is defined as a value greater than \(Q _ { 3 } + 1.5 \times\) interquartile range.
      Given that \(Q _ { 1 } = 27.3\) and \(Q _ { 3 } = 58.9\)
    3. determine, giving a reason, whether or not the oldest passenger could be considered as an outlier.
      (2)
    Edexcel AS Paper 2 2021 November Q3
    1. Helen is studying one of the qualitative variables from the large data set for Heathrow from 2015.
    She started with the data from 3rd May and then took every 10th reading.
    There were only 3 different outcomes with the following frequencies
    Outcome\(A\)\(B\)\(C\)
    Frequency1621
    1. State the sampling technique Helen used.
    2. From your knowledge of the large data set
      1. suggest which variable was being studied,
      2. state the name of outcome \(A\). George is also studying the same variable from the large data set for Heathrow from 2015. He started with the data from 5th May and then took every 10th reading and obtained the following
        Outcome\(A\)\(B\)\(C\)
        Frequency1611
        Helen and George decided they should examine all of the data for this variable for Heathrow from 2015 and obtained the following
        Outcome\(A\)\(B\)\(C\)
        Frequency155263
    3. State what inference Helen and George could reliably make from their original samples about the outcomes of this variable at Heathrow, for the period covered by the large data set in 2015.
    Edexcel AS Paper 2 2021 November Q4
    1. A nursery has a sack containing a large number of coloured beads of which \(14 \%\) are coloured red.
    Aliya takes a random sample of 18 beads from the sack to make a bracelet.
    1. State a suitable binomial distribution to model the number of red beads in Aliya’s bracelet.
    2. Use this binomial distribution to find the probability that
      1. Aliya has just 1 red bead in her bracelet,
      2. there are at least 4 red beads in Aliya’s bracelet.
    3. Comment on the suitability of a binomial distribution to model this situation. After several children have used beads from the sack, the nursery teacher decides to test whether or not the proportion of red beads in the sack has changed. She takes a random sample of 75 beads and finds 4 red beads.
    4. Stating your hypotheses clearly, use a 5\% significance level to carry out a suitable test for the teacher.
    5. Find the \(p\)-value in this case.