Questions — Edexcel (9685 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 Mechanics 1 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pure 1 S1 S2 S3 S4 Stats 1 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 Mechanics 1 PURE Pure 1 S1 S2 S3 S4 Stats 1 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
Sort by: Default | Easiest first | Hardest first
Edexcel S4 2017 June Q6
19 marks Challenging +1.2
6. The independent random variables \(X _ { 1 }\) and \(X _ { 2 }\) are each distributed \(\mathrm { B } ( n , p )\), where \(n > 1\) An unbiased estimator for \(p\) is given by $$\hat { p } = \frac { a X _ { 1 } + b X _ { 2 } } { n }$$ where \(a\) and \(b\) are constants.
[0pt] [You may assume that if \(X _ { 1 }\) and \(X _ { 2 }\) are independent then \(\mathrm { E } \left( X _ { 1 } X _ { 2 } \right) = \mathrm { E } \left( X _ { 1 } \right) \mathrm { E } \left( X _ { 2 } \right)\) ]
  1. Show that \(a + b = 1\)
  2. Show that \(\operatorname { Var } ( \hat { p } ) = \frac { \left( 2 a ^ { 2 } - 2 a + 1 \right) p ( 1 - p ) } { n }\)
  3. Hence, justifying your answer, determine the value of \(a\) and the value of \(b\) for which \(\hat { p }\) has minimum variance.
    1. Show that \(\hat { p } ^ { 2 }\) is a biased estimator for \(p ^ { 2 }\)
    2. Show that the bias \(\rightarrow 0\) as \(n \rightarrow \infty\)
  5. By considering \(\mathrm { E } \left[ X _ { 1 } \left( X _ { 1 } - 1 \right) \right]\) find an unbiased estimator for \(p ^ { 2 }\)
Edexcel S4 2018 June Q1
5 marks Moderate -0.3
  1. A machine fills packets with almonds. The weight, in grams, of almonds in a packet is modelled by \(\mathrm { N } \left( \mu , \sigma ^ { 2 } \right)\). To check that the machine is working properly, a random sample of 10 packets is selected and unbiased estimates for \(\mu\) and \(\sigma ^ { 2 }\) are
$$\bar { x } = 202 \quad \text { and } \quad s ^ { 2 } = 3.6$$ Stating your hypotheses clearly, test, at the \(1 \%\) level of significance, whether or not the mean weight of almonds in a packet is more than 200 g .
Edexcel S4 2018 June Q2
13 marks Standard +0.3
  1. Jeremiah currently uses a Fruity model of juicer. He agrees to trial a new model of juicer, Zesty. The amounts of juice extracted, \(x \mathrm { ml }\), from each of 9 randomly selected oranges, using the Zesty are summarised as
$$\sum x = 468 \quad \sum x ^ { 2 } = 24560$$ Given that the amounts of juice extracted follow a normal distribution,
  1. calculate a 95\% confidence interval for
    1. the mean amount of juice extracted from an orange using the Zesty,
    2. the standard deviation of the amount of juice extracted from an orange using the Zesty. Jeremiah knows that, for his Fruity, the mean amount of juice extracted from an orange is 38 ml and the standard deviation of juice extracted from an orange is 5 ml . He decides that he will replace his Fruity with a Zesty if both
      • the mean for the Zesty is more than \(20 \%\) higher than the mean for his Fruity and
  2. the standard deviation for the Zesty is less than 5.5 ml .
  3. Using your answers to part (a), explain whether or not Jeremiah should replace his Fruity with the Zesty.
Edexcel S4 2018 June Q3
10 marks Challenging +1.2
  1. A random sample of 8 students is selected from a school database.
Each student's reaction time is measured at the start of the school day and again at the end of the school day. The reaction times, in milliseconds, are recorded below.
StudentA\(B\)CD\(E\)\(F\)G\(H\)
Reaction time at the start of the school day10.87.28.76.89.410.911.17.6
Reaction time at the end of the school day106.18.85.78.78.19.86.8
  1. State one assumption that needs to be made in order to carry out a paired \(t\)-test.
    (1) The random variable \(R\) is the reaction time at the start of the school day minus the reaction time at the end of the school day. The mean of \(R\) is \(\mu\). John uses a paired \(t\)-test to test the hypotheses $$\mathrm { H } _ { 0 } : \mu = m \quad \mathrm { H } _ { 1 } : \mu \neq m$$ Given that \(\mathrm { H } _ { 0 }\) is rejected at the 5\% level of significance but accepted at the 1\% level of significance,
  2. find the ranges of possible values for \(m\).
Edexcel S4 2018 June Q4
17 marks Challenging +1.2
  1. A glue supplier claims that Goglue is stronger than Tackfast. A company is presently using Tackfast but agrees to change to Goglue if, at the 5\% significance level,
  • the standard deviation of the force required for Goglue to fail is not greater than the standard deviation of the force required for Tackfast to fail and
  • the mean force required for Goglue to fail is more than 4 newtons greater than the mean force for Tackfast to fail.
A series of trials is carried out, using Goglue and Tackfast, and the glues are tested to destruction. The force, \(x\) newtons, at which each glue fails is recorded.
Sample size \(( n )\)Sample mean \(( \bar { x } )\)Standard deviation \(( s )\)
Tackfast \(( T )\)65.270.31
Goglue \(( G )\)510.120.66
It can be assumed that the force at which each glue fails is normally distributed.
  1. Test, at the \(5 \%\) level of significance, whether or not there is evidence that the standard deviation of the force required for Goglue to fail is greater than the standard deviation of the force required for the Tackfast to fail. State your hypotheses clearly. The supplier claims that the mean force required for its Goglue to fail is more than 4 newtons greater than the mean force required for Tackfast to fail.
  2. Stating your hypotheses clearly and using a \(5 \%\) level of significance, test the supplier's claim.
  3. Show that, at the \(5 \%\) level of significance, the supplier's claim will be accepted if \(\bar { X } _ { G } - \bar { X } _ { T } > 4.55\), where \(\bar { X } _ { G }\) and \(\bar { X } _ { T }\) are the mean forces required for Goglue to fail and Tackfast to fail respectively. Later, it was found that an error had been made when recording the results for Goglue. This resulted in all the forces recorded for Goglue being 0.5 newtons more than they should have been. The results for Tackfast were correct.
  4. Explain whether or not this information affects the decision about which glue the supplier decides to use.
Edexcel S4 2018 June Q5
11 marks Challenging +1.2
  1. A machine makes posts. The length of a post is normally distributed with unknown mean \(\mu\) and standard deviation 4 cm .
A random sample of size \(n\) is taken to test, at the \(5 \%\) significance level, the hypotheses $$\mathrm { H } _ { 0 } : \mu = 150 \quad \mathrm { H } _ { 1 } : \mu > 150$$
  1. State the probability of a Type I error for this test. The manufacturer requires the probability of a Type II error to be less than 0.1 when the actual value of \(\mu\) is 152
  2. Calculate the minimum value of \(n\).
Edexcel S4 2018 June Q6
19 marks Challenging +1.2
  1. The continuous random variable \(X\) has probability density function \(\mathrm { f } ( x )\)
$$f ( x ) = \left\{ \begin{array} { c c } \frac { x } { 2 \theta ^ { 2 } } & 0 \leqslant x \leqslant 2 \theta \\ 0 & \text { otherwise } \end{array} \right.$$ where \(\theta\) is a constant.
  1. Use integration to show that \(\mathrm { E } \left( X ^ { N } \right) = \frac { 2 ^ { N + 1 } } { N + 2 } \theta ^ { N }\)
  2. Hence
    1. write down an expression for \(\mathrm { E } ( X )\) in terms of \(\theta\)
    2. find \(\operatorname { Var } ( X )\) in terms of \(\theta\) A random sample \(X _ { 1 } , X _ { 2 } , \ldots , X _ { n }\) where \(n \geqslant 2\) is taken to estimate the value of \(\theta\) The random variable \(S _ { 1 } = q \bar { X }\) is an unbiased estimator of \(\theta\)
  3. Write down the value of \(q\) and show that \(S _ { 1 }\) is a consistent estimator of \(\theta\) The continuous random variable \(Y\) is independent of \(X\) and is uniformly distributed over the interval \(\left[ 0 , \frac { 2 \theta } { 3 } \right]\), where \(\theta\) is the same unknown constant as in \(\mathrm { f } ( x )\). The random variable \(S _ { 2 } = a X + b Y\) is an unbiased estimator of \(\theta\) and is based on one observation of \(X\) and one observation of \(Y\).
  4. Find the value of \(a\) and the value of \(b\) for which \(S _ { 2 }\) has minimum variance.
  5. Show that the minimum variance of \(S _ { 2 }\) is \(\frac { \theta ^ { 2 } } { 11 }\)
  6. Explain which of \(S _ { 1 }\) or \(S _ { 2 }\) is the better estimator for \(\theta\)
Edexcel S4 Q1
8 marks Standard +0.3
  1. The weights of the contents of jars of jam are normally distributed with a stated mean of 100 g . A random sample of 7 jars was taken and the contents of each jar, \(x\) grams, was weighed. The results are summarised by the following statistics.
$$\sum x = 710.9 , \sum x ^ { 2 } = 72219.45 .$$ Test at the \(5 \%\) level of significance whether or not there is evidence that the mean weight of the contents of the jars is greater than 100 g . State your hypotheses clearly.
(8 marks)
Edexcel S4 Q2
8 marks Standard +0.3
2. An engineer decided to investigate whether or not the strength of rope was affected by water. A random sample of 9 pieces of rope was taken and each piece was cut in half. One half of each piece was soaked in water over night, and then each piece of rope was tested to find its strength. The results, in coded units, are given in the table below
Rope no.123456789
Dry rope9.78.56.38.37.25.46.88.15.9
Wet rope9.19.58.29.78.54.98.48.77.7
Assuming that the strength of rope follows a normal distribution, test whether or not there is any difference between the mean strengths of dry and wet rope. State your hypotheses clearly and use a \(1 \%\) level of significance.
(8 marks)
Edexcel S4 Q3
13 marks Standard +0.8
3. A certain vaccine is known to be only \(70 \%\) effective against a particular virus; thus \(30 \%\) of those vaccinated will actually catch the virus. In order to test whether or not a new and more expensive vaccine provides better protection against the same virus, a random sample of 30 people were chosen and given the new vaccine. If fewer than 6 people contracted the virus the new vaccine would be considered more effective than the current one.
  1. Write down suitable hypotheses for this test.
  2. Find the probability of making a Type I error.
  3. Find the power of this test if the new vaccine is
    1. \(80 \%\) effective,
    2. \(90 \%\) effective. An independent research organisation decided to test the new vaccine on a random sample of 50 people to see if it could be considered more than \(70 \%\) effective. They required the probability of a Type I error to be as close as possible to 0.05 .
  4. Find the critical region for this test.
  5. State the size of this critical region.
  6. Find the power of this test if the new vaccine is
    1. \(80 \%\) effective,
    2. \(90 \%\) effective.
  7. Give one advantage and one disadvantage of the second test.
Edexcel S4 Q4
14 marks Standard +0.8
4. Gill, a member of the accounts department in a large company, is studying the expenses claims of company employees. She assumes that the claims, in \(\pounds\), follow a normal distribution with mean \(\mu\) and variance \(\sigma ^ { 2 }\). As a first stage in her investigation she took the following random sample of 10 claims. $$30.85,99.75,142.73,223.16,75.43,28.57,53.90,81.43,68.62,43.45 .$$
  1. Find a 95\% confidence interval for \(\mu\). The chief accountant would like a \(95 \%\) confidence interval where the difference between the upper confidence limit and the lower confidence limit is less than 20 .
  2. Show that \(\frac { \sigma ^ { 2 } } { n } < 26.03\) (to 2 decimal places), where \(n\) is the size of the sample required to achieve this. Gill decides to use her original sample of 10 to obtain a value for \(\sigma ^ { 2 }\) so that the chance of her value being an underestimate is 0.01 .
  3. Find such a value for \(\sigma ^ { 2 }\).
  4. Use this value for \(\sigma ^ { 2 }\) to estimate the size of sample the chief accountant requires.
Edexcel S4 Q5
16 marks Standard +0.8
5. An educational researcher is testing the effectiveness of a new method of teaching a topic in mathematics. A random sample of 10 children were taught by the new method and a second random sample of 9 children, of similar age and ability, were taught by the conventional method. At the end of the teaching, the same test was given to both groups of children. The marks obtained by the two groups are summarised in the table below.
New methodConventional method
Mean \(( \bar { x } )\)82.378.2
Standard deviation \(( s )\)3.55.7
Number of students \(( n )\)109
  1. Stating your hypotheses clearly and using a \(5 \%\) level of significance, investigate whether or not
    1. the variance of the marks of children taught by the conventional method is greater than that of children taught by the new method,
    2. the mean score of children taught by the conventional method is lower than the mean score of those taught by the new method.
      [0pt] [In each case you should give full details of the calculation of the test statistics.]
  2. State any assumptions you made in order to carry out these tests.
  3. Find a 95\% confidence interval for the common variance of the marks of the two groups.
Edexcel S4 Q6
18 marks Standard +0.3
6. A statistics student is trying to estimate the probability, \(p\), of rolling a 6 with a particular die. The die is rolled 10 times and the random variable \(X _ { 1 }\) represents the number of sixes obtained. The random variable \(R _ { 1 } = \frac { X _ { 1 } } { 10 }\) is proposed as an estimator of \(p\).
  1. Show that \(R _ { 1 }\) is an unbiased estimator of \(p\). The student decided to roll the die again \(n\) times ( \(n > 10\) ) and the random variable \(X _ { 2 }\) represents the number of sixes in these \(n\) rolls. The random variable \(R _ { 2 } = \frac { X _ { 2 } } { n }\) and the random variable \(Y = \frac { 1 } { 2 } \left( R _ { 1 } + R _ { 2 } \right)\).
  2. Show that both \(R _ { 2 }\) and \(Y\) are unbiased estimators of \(p\).
  3. Find \(\operatorname { Var } \left( R _ { 2 } \right)\) and \(\operatorname { Var } ( Y )\).
  4. State giving a reason which of the 3 estimators \(R _ { 1 } , R _ { 2 }\) and \(Y\) are consistent estimators of \(p\).
  5. For the case \(n = 20\) state, giving a reason, which of the 3 estimators \(R _ { 1 } , R _ { 2 }\) and \(Y\) you would recommend. The student's teacher pointed out that a better estimator could be found based on the random variable \(X _ { 1 } + X _ { 2 }\).
  6. Find a suitable estimator and explain why it is better than \(R _ { 1 } , R _ { 2 }\) and \(Y\). END
Edexcel M1 Q1
6 marks Moderate -0.8
  1. A bee flies in a straight line from \(A\) to \(B\), where \(\overrightarrow { A B } = \left( 3 \frac { 1 } { 2 } \mathbf { i } - 12 \mathbf { j } \right) \mathrm { m }\), in 5 seconds at a constant speed. Find
    1. the straight-line distance \(A B\),
    2. the speed of the bee,
    3. the velocity vector of the bee.
    4. A small ball \(B\), of mass 0.8 kg , is suspended from a horizontal ceiling by two light inextensible strings. \(B\) is in equilibrium under gravity with both strings inclined at \(30 ^ { \circ }\) to the horizontal, as shown. \includegraphics[max width=\textwidth, alt={}, center]{3e495748-ccb7-4c99-8387-160c4f0f9d4f-1_163_438_758_1480}
    5. Find the tension, in N , in either string.
    6. Calculate the magnitude of the least horizontal force that must be applied to \(B\) in this position to cause one string to become slack.
    7. A particle \(P\) moves in a straight line through a fixed point \(O\) with constant acceleration \(a \mathrm {~ms} ^ { - 2 }\). 3 seconds after passing through \(O , P\) is 6 m from \(O\).
      After a further 6 seconds, \(P\) has travelled a further 33 m in the same direction. Calculate
    8. the value of \(a\),
    9. the speed with which \(P\) passed through \(O\).
    10. A force of magnitude \(F \mathrm {~N}\) is applied to a block of mass \(M \mathrm {~kg}\) which is initially at rest on a horizontal plane. The block starts to move with acceleration \(3 \mathrm {~ms} ^ { - 2 }\). Modelling the block as a particle, \includegraphics[max width=\textwidth, alt={}, center]{3e495748-ccb7-4c99-8387-160c4f0f9d4f-1_134_405_1672_1540}
    11. if the plane is smooth, find an expression for \(F\) in terms of \(M\).
    If the plane is rough, and the coefficient of friction between the block and the plane is \(\mu\),
  2. express \(F\) in terms of \(M , \mu\) and \(g\).
  3. Calculate the value of \(\mu\) if \(F = \frac { 1 } { 2 } M g\).
Edexcel M1 Q5
12 marks Standard +0.3
5. Two metal weights \(A\) and \(B\), of masses 2.4 kg and 1.8 kg respectively, are attached to the ends of a light inextensible string which passes over a smooth fixed pulley so that the string hangs vertically on each side. The system is released from rest with the string taut.
  1. Calculate the acceleration of each weight and the tension in the string. \(A\) is now replaced by a different weight of mass \(m \mathrm {~kg}\), where \(m < 1 \cdot 8\), and the system is again released from rest. The magnitude of the acceleration has half of its previous value.
  2. Calculate the value of \(m\).
    (6 marks) \section*{MECHANICS 1 (A)TEST PAPER 1 Page 2}
Edexcel M1 Q6
12 marks Standard +0.3
  1. The diagram shows the speed-time graph for a particle during a period of \(9 T\) seconds. \includegraphics[max width=\textwidth, alt={}, center]{3e495748-ccb7-4c99-8387-160c4f0f9d4f-2_406_1162_319_351}
    1. If \(T = 5\), find
      1. the acceleration for each section of the motion,
      2. the total distance travelled by the particle.
    2. Sketch, for this motion,
      1. an acceleration-time graph,
      2. a displacement-time graph.
    3. Calculate the value of \(T\) for which the distance travelled over the \(9 T\) seconds is 3.708 km .
    4. Two smooth spheres \(A\) and \(B\), of masses 60 grams and 90 grams respectively, are at rest on a smooth horizontal table. \(A\) is projected towards \(B\) with speed \(4 \mathrm {~ms} ^ { - 1 }\) and the particles collide. After the collision, \(A\) and \(B\) move in the same direction as each other, with speeds \(u \mathrm {~ms} ^ { - 1 }\) and \(6 u \mathrm {~ms} ^ { - 1 }\) respectively. Calculate
    5. the value of \(u\),
    6. the magnitude of the impulse exerted by \(A\) on \(B\), stating the units of your answer.
      (3 marks) \(A\) and \(B\) are now replaced in their original positions and projected towards each other with speeds \(2 \mathrm {~ms} ^ { - 1 }\) and \(8 \mathrm {~ms} ^ { - 1 }\) respectively. They collide again, after which \(A\) moves with speed \(7 \mathrm {~ms} ^ { - 1 }\), its direction of motion being reversed.
    7. Find the speed of \(B\) after this collision and state whether its direction of motion has been reversed.
    8. In a theatre, three lights \(A , B\) and \(C\) are suspended from a horizontal beam \(X Y\) of length \(4.5 \mathrm {~m} . A\) and \(C\) are each of mass 8 kg and \(B\) is of mass 6 kg . The beam \(X Y\) is held in place by vertical ropes \(P X\) and \(Q Y\), as shown. \includegraphics[max width=\textwidth, alt={}, center]{3e495748-ccb7-4c99-8387-160c4f0f9d4f-2_282_643_2104_1316}
    In a simple mathematical model of this situation, \(X Y\) is modelled as a light rod.
  2. Calculate the tension in each of \(P X\) and \(Q Y\). In a refined model, \(X Y\) is modelled as a uniform rod of mass \(m \mathrm {~kg}\).
  3. If the tension in \(P X\) is 1.5 times that in \(Q Y\), calculate the value of \(m\).
Edexcel M1 Q1
5 marks Moderate -0.8
  1. A plank of wood \(A B\), of mass 8 kg and length 6 m , rests on a support at \(P\), where \(A P = 4 \mathrm {~m}\). When particles of mass 1 kg and \(k \mathrm {~kg}\) are suspended from \(A\) and \(B\) respectively, the plank rests horizontally in equilibrium.
    Modelling the plank as a uniform rod, find
    1. the value of \(k\),
    2. the magnitude of the force exerted by the support on the plank at \(P\).
    3. Forces of magnitude 4 N and 6 N act in directions which make an angle of \(40 ^ { \circ }\) with each other, as shown. Calculate \includegraphics[max width=\textwidth, alt={}, center]{9c9b6087-d5a1-4fb0-b771-5ccc13a04bc4-1_209_478_840_1348}
    4. the magnitude of the resultant of the two forces,
    5. the angle, in degrees, between the resultant and the 4 N force.
    6. A stone is dropped from rest at a height of 7 m above horizontal ground. It falls vertically, hits the ground and rebounds vertically upwards with half the speed with which it hit the ground. Calculate
    7. the time taken for the stone to fall to the ground,
    8. the speed with which the stone hits the ground,
    9. the height to which the stone rises before it comes to instantaneous rest.
    State two modelling assumptions that you have made.
Edexcel M1 Q4
12 marks Moderate -0.3
4. A boy starts at the corner \(O\) of a rectangular playing field and runs across the field with constant velocity vector \(( \mathbf { i } + 2 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\), where \(\mathbf { i }\) and \(\mathbf { j }\) are unit vectors in the directions of two perpendicular sides of the field. After 40 seconds, at the point \(P\) in the field, he changes speed and direction so that his new velocity vector is \(( 2 \cdot 4 \mathbf { i } - 1 \cdot 8 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) and maintains this velocity until he reaches the point \(Q\), where \(P Q = 75 \mathrm {~m}\).
Calculate (a) the distance \(O P\),
(b) the time taken to travel from \(P\) to \(Q\),
(c) the position vector of \(Q\) relative to \(O\). Another boy travels directly from \(O\) to \(Q\) with constant velocity \(( a \mathbf { i } + b \mathbf { j } ) \mathrm { ms } ^ { - 1 }\), leaving \(O\) and reaching \(Q\) at the same times as the first boy.
(d) Find the values of the constants \(a\) and \(b\). \section*{MECHANICS 1 (A)TEST PAPER 2 Page 2}
Edexcel M1 Q5
12 marks Standard +0.3
  1. Two railway trucks \(A\) and \(B\), of masses 10000 kg and 7000 kg respectively, are moving towards each other along a horizontal straight track. The trucks collide, and in the collision \(A\) exerts an impulse on \(B\) of magnitude 84000 Ns. Immediately after the collision, the trucks move together with speed \(10 \mathrm {~ms} ^ { - 1 }\). Modelling the trucks as particles,
    1. find the speed of each truck immediately before the collision.
    When the trucks are moving together along the track, the coefficient of friction between them and the track is 0.15 . Assuming that no other resisting forces act on the trucks, calculate
  2. the magnitude of the resisting force on the trucks,
  3. the time taken after the collision for the trucks to come to rest.
Edexcel M1 Q6
15 marks Standard +0.3
6. A small package \(P\), of mass 1 kg , is initially at rest on the rough horizontal top surface of a wooden packing case which is 1.5 m long and 1 m high and stands on a horizontal floor. The coefficient of friction between \(P\) and the case is 0.2 . \includegraphics[max width=\textwidth, alt={}, center]{9c9b6087-d5a1-4fb0-b771-5ccc13a04bc4-2_287_517_941_1428} \(P\) is attached by a light inextensible string, which passes over a smooth fixed pulley, to a weight \(Q\) of mass \(M \mathrm {~kg}\) which rests against the smooth vertical side of the case.
The system is released from rest with \(P 0.75 \mathrm {~m}\) from the pulley and \(Q 0.5 \mathrm {~m}\) from the pulley. \(P\) and \(Q\) start to move with acceleration \(0.4 \mathrm {~ms} ^ { - 2 }\). Calculate
  1. the tension in the string, in N ,
  2. the value of \(M\),
  3. the time taken for \(Q\) to hit the floor. Given that \(Q\) does not rebound from the floor,
  4. calculate the distance of \(P\) from the pulley when it comes to rest.
Edexcel M1 Q7
16 marks Standard +0.3
7. A car starts from rest at time \(t = 0\) and moves along a straight road with constant acceleration 4 \(\mathrm { ms } ^ { - 2 }\) for 10 seconds. It then travels at a constant speed for 50 seconds before decelerating to rest over a further distance of 240 m .
  1. Sketch a graph of velocity against time for the total period of the car's motion.
  2. Find the car's average speed for the whole journey. In reality the car's acceleration \(a \mathrm {~ms} ^ { - 2 }\) in the first 10 seconds is not constant, but increases from 0 to \(4 \mathrm {~ms} ^ { - 2 }\) in the first 5 seconds and then decreases to 0 again. A refined model designed to take account of this uses the formula \(a = k \left( m t - t ^ { 2 } \right)\) for \(0 \leq t \leq 10\).
  3. Calculate the values of the constants \(k\) and \(m\).
  4. Find the acceleration of the car when \(t = 2\) according to this model.
Edexcel M1 Q1
7 marks Standard +0.3
  1. A particle \(P\), of mass 2.5 kg , initially at rest at the point \(O\), moves on a smooth horizontal surface with constant acceleration \(( \mathbf { i } + 2 \mathbf { j } ) \mathrm { ms } ^ { - 2 }\), where \(\mathbf { i }\) and \(\mathbf { j }\) are unit vectors in the directions due East and due North respectively. Find
    1. the velocity vector of \(P\) at time \(t\) seconds after it leaves \(O\),
    2. the magnitude and direction of the velocity of \(P\) when \(t = 7\),
    3. the magnitude, in N , of the force acting on \(P\).
    4. An iron bar \(A B\), of length 4 m , is kept in a horizontal position by a support at \(A\) and a wire attached to the point \(P\) on the bar, where \(P B = 0.85 \mathrm {~m}\). The bar is modelled as a non-uniform rod whose centre of mass is at \(G\), where \(A G = 1.4 \mathrm {~m}\), and the wire is modelled as a light inextensible string. Given that the tension in the wire is 12 N , calculate
    5. the weight of the bar,
    6. the magnitude of the reaction on the bar at \(A\).
    7. State briefly how you have used the given modelling assumption about the bar.
    \includegraphics[max width=\textwidth, alt={}]{f8386a80-e428-43a7-acc8-f7ab11b2a53a-1_201_453_1399_378}
    A small packet, of mass 1.2 kg , is at rest on a rough plane inclined at an angle \(\alpha\) to the horizontal. The coefficient of friction between the packet and the plane is \(\frac { 1 } { 8 }\).
    When a force of magnitude 8.4 N , acting parallel to the plane, is applied to the packet as shown, the packet is just on the point of moving up the plane. Modelling the packet as a particle,
  2. show that \(7 ( \cos \alpha + 8 \sin \alpha ) = 40\). Given that the solution of this equation is \(\alpha = 38 ^ { \circ }\),
  3. find the acceleration with which the packet moves down the plane when it is released from rest with no external force applied.
Edexcel M1 Q4
11 marks Moderate -0.3
4. A car moves in a straight line from \(P\) to \(Q\), a distance of 420 m , with constant acceleration. At \(P\) the speed of the car is \(8 \mathrm {~ms} ^ { - 1 }\). At \(Q\) the speed of the car is \(20 \mathrm {~ms} ^ { - 1 }\). Find
  1. the time taken to travel from \(P\) to \(Q\),
  2. the acceleration of the car,
  3. the time taken for the car to travel 240 m from \(P\). Given that the mass of the car is 1200 kg and the tractive force of the car is 900 N ,
  4. find the magnitude of the resistance to the car's motion. \section*{MECHANICS 1 (A) TEST PAPER 3 Page 2}
Edexcel M1 Q5
11 marks Standard +0.3
  1. Two smooth spheres \(X\) and \(Y\), of masses \(x \mathrm {~kg}\) and \(y \mathrm {~kg}\) respectively, are free to move in a smooth straight groove in a horizontal table. \(X\) is projected with speed \(6 \mathrm {~ms} ^ { - 1 }\) towards \(Y\), which is stationary. After the collision \(X\) moves with speed \(2 \mathrm {~ms} ^ { - 1 }\) and \(Y\) moves with speed \(3 \mathrm {~ms} ^ { - 1 }\).
    1. Calculate the two possible values of the ratio \(x : y\).
    2. State a modelling assumption that you have made concerning \(X\) and \(Y\). \(Y\) now strikes a vertical barrier and rebounds along the groove with speed \(k \mathrm {~ms} ^ { - 1 }\), colliding again with \(X\) which is still moving at \(2 \mathrm {~ms} ^ { - 1 }\). Given that in this impact \(Y\) is brought to rest and the direction of motion of \(X\) is reversed,
    3. show that \(k > 1 \cdot 5\).
    4. Two particles \(P\) and \(Q\), of masses 3 kg and 2 kg respectively, rest on the smooth faces of a wedge whose cross-section is a triangle with angles \(30 ^ { \circ } , 60 ^ { \circ }\) and \(90 ^ { \circ }\), as shown. \(P\) and \(Q\) are connected by a light \includegraphics[max width=\textwidth, alt={}, center]{f8386a80-e428-43a7-acc8-f7ab11b2a53a-2_255_607_1078_1311}
      string, parallel to the lines of greatest slope of the two planes, which passes over a fixed pulley at the highest point of the wedge.
      The system is released from rest with \(P 0.8 \mathrm {~m}\) from the pulley and \(Q 1 \mathrm {~m}\) from the bottom of the wedge, and \(Q\) starts to move down. Calculate
    5. the acceleration of either particle,
    6. the tension in the string,
    7. the speed with which \(P\) reaches the pulley.
    Two modelling assumptions have been made about the string and the pulley.
  2. State these two assumptions and briefly describe how you have used each one in your solution.
Edexcel M1 Q7
15 marks Standard +0.8
7. Two stones are projected simultaneously from a point \(O\) on horizontal ground. Stone \(A\) is thrown vertically upwards with speed \(98 \mathrm {~ms} ^ { - 1 }\). Stone \(B\) is projected along the smooth ground in a straight line at \(24 \cdot 5 \mathrm {~ms} ^ { - 1 }\).
  1. Find the distances of the two stones from \(O\) after \(t\) seconds, where \(0 \leq t \leq 20\).
  2. Show that the distance \(d \mathrm {~m}\) between the two stones after \(t\) seconds is given by $$d ^ { 2 } = 24 \cdot 01 \left( t ^ { 4 } - 40 t ^ { 3 } + 425 t ^ { 2 } \right) .$$
  3. Hence find the range of values of \(t\) for which the distance between the stones is decreasing.