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Edexcel AS Paper 2 Specimen Q9
10 marks Moderate -0.8
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
9 marks Moderate -0.8
  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
8 marks Standard +0.3
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
6 marks Standard +0.3
  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
7 marks Moderate -0.3
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.
Edexcel Paper 3 2018 June Q1
5 marks Moderate -0.8
  1. Helen believes that the random variable \(C\), representing cloud cover from the large data set, can be modelled by a discrete uniform distribution.
    1. Write down the probability distribution for \(C\).
    2. Using this model, find the probability that cloud cover is less than 50\%
    Helen used all the data from the large data set for Hurn in 2015 and found that the proportion of days with cloud cover of less than \(50 \%\) was 0.315
  2. Comment on the suitability of Helen's model in the light of this information.
  3. Suggest an appropriate refinement to Helen’s model.
Edexcel Paper 3 2018 June Q2
7 marks Standard +0.3
  1. Tessa owns a small clothes shop in a seaside town. She records the weekly sales figures, \(\pounds w\), and the average weekly temperature, \(t ^ { \circ } \mathrm { C }\), for 8 weeks during the summer.
    The product moment correlation coefficient for these data is - 0.915
    1. Stating your hypotheses clearly and using a \(5 \%\) level of significance, test whether or not the correlation between sales figures and average weekly temperature is negative.
    2. Suggest a possible reason for this correlation.
    Tessa suggests that a linear regression model could be used to model these data.
  2. State, giving a reason, whether or not the correlation coefficient is consistent with Tessa’s suggestion.
  3. State, giving a reason, which variable would be the explanatory variable. Tessa calculated the linear regression equation as \(w = 10755 - 171 t\)
  4. Give an interpretation of the gradient of this regression equation.
Edexcel Paper 3 2018 June Q3
11 marks Moderate -0.3
  1. In an experiment a group of children each repeatedly throw a dart at a target. For each child, the random variable \(H\) represents the number of times the dart hits the target in the first 10 throws.
Peta models \(H\) as \(\mathrm { B } ( 10,0.1 )\)
  1. State two assumptions Peta needs to make to use her model.
  2. Using Peta's model, find \(\mathrm { P } ( H \geqslant 4 )\) For each child the random variable \(F\) represents the number of the throw on which the dart first hits the target. Using Peta's assumptions about this experiment,
  3. find \(\mathrm { P } ( F = 5 )\) Thomas assumes that in this experiment no child will need more than 10 throws for the dart to hit the target for the first time. He models \(\mathrm { P } ( F = n )\) as $$\mathrm { P } ( F = n ) = 0.01 + ( n - 1 ) \times \alpha$$ where \(\alpha\) is a constant.
  4. Find the value of \(\alpha\)
  5. Using Thomas' model, find \(\mathrm { P } ( F = 5 )\)
  6. Explain how Peta's and Thomas' models differ in describing the probability that a dart hits the target in this experiment.
Edexcel Paper 3 2018 June Q4
13 marks Easy -1.3
  1. Charlie is studying the time it takes members of his company to travel to the office. He stands by the door to the office from 0840 to 0850 one morning and asks workers, as they arrive, how long their journey was.
    1. State the sampling method Charlie used.
    2. State and briefly describe an alternative method of non-random sampling Charlie could have used to obtain a sample of 40 workers.
    Taruni decided to ask every member of the company the time, \(x\) minutes, it takes them to travel to the office.
  2. State the data selection process Taruni used. Taruni's results are summarised by the box plot and summary statistics below. \includegraphics[max width=\textwidth, alt={}, center]{65e4b254-fb7b-45c2-9702-32f034018193-10_378_1349_1050_367} $$n = 95 \quad \sum x = 4133 \quad \sum x ^ { 2 } = 202294$$
  3. Write down the interquartile range for these data.
  4. Calculate the mean and the standard deviation for these data.
  5. State, giving a reason, whether you would recommend using the mean and standard deviation or the median and interquartile range to describe these data. Rana and David both work for the company and have both moved house since Taruni collected her data. Rana's journey to work has changed from 75 minutes to 35 minutes and David's journey to work has changed from 60 minutes to 33 minutes. Taruni drew her box plot again and only had to change two values.
  6. Explain which two values Taruni must have changed and whether each of these values has increased or decreased.
Edexcel Paper 3 2018 June Q5
14 marks Challenging +1.2
  1. The lifetime, \(L\) hours, of a battery has a normal distribution with mean 18 hours and standard deviation 4 hours.
Alice's calculator requires 4 batteries and will stop working when any one battery reaches the end of its lifetime.
  1. Find the probability that a randomly selected battery will last for longer than 16 hours. At the start of her exams Alice put 4 new batteries in her calculator. She has used her calculator for 16 hours, but has another 4 hours of exams to sit.
  2. Find the probability that her calculator will not stop working for Alice's remaining exams. Alice only has 2 new batteries so, after the first 16 hours of her exams, although her calculator is still working, she randomly selects 2 of the batteries from her calculator and replaces these with the 2 new batteries.
  3. Show that the probability that her calculator will not stop working for the remainder of her exams is 0.199 to 3 significant figures. After her exams, Alice believed that the lifetime of the batteries was more than 18 hours. She took a random sample of 20 of these batteries and found that their mean lifetime was 19.2 hours.
  4. Stating your hypotheses clearly and using a \(5 \%\) level of significance, test Alice's belief.
Edexcel Paper 3 2018 June Q6
6 marks Standard +0.3
6. At time \(t\) seconds, where \(t \geqslant 0\), a particle \(P\) moves in the \(x - y\) plane in such a way that its velocity \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\) is given by $$\mathbf { v } = t ^ { - \frac { 1 } { 2 } } \mathbf { i } - 4 \mathbf { j }$$ When \(t = 1 , P\) is at the point \(A\) and when \(t = 4 , P\) is at the point \(B\).
Find the exact distance \(A B\).
Edexcel Paper 3 2018 June Q7
8 marks Standard +0.3
7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{65e4b254-fb7b-45c2-9702-32f034018193-20_264_698_246_685} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A wooden crate of mass 20 kg is pulled in a straight line along a rough horizontal floor using a handle attached to the crate.
The handle is inclined at an angle \(\alpha\) to the floor, as shown in Figure 1, where \(\tan \alpha = \frac { 3 } { 4 }\) The tension in the handle is 40 N .
The coefficient of friction between the crate and the floor is 0.14
The crate is modelled as a particle and the handle is modelled as a light rod.
Using the model,
  1. find the acceleration of the crate. The crate is now pushed along the same floor using the handle. The handle is again inclined at the same angle \(\alpha\) to the floor, and the thrust in the handle is 40 N as shown in Figure 2 below. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{65e4b254-fb7b-45c2-9702-32f034018193-20_220_923_1457_571} \captionsetup{labelformat=empty} \caption{Figure 2}
    \end{figure}
  2. Explain briefly why the acceleration of the crate would now be less than the acceleration of the crate found in part (a).
Edexcel Paper 3 2018 June Q8
8 marks Moderate -0.3
  1. \hspace{0pt} [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal unit vectors due east and due north respectively and position vectors are given relative to the fixed point \(O\).]
A particle \(P\) moves with constant acceleration.
At time \(t = 0\), the particle is at \(O\) and is moving with velocity ( \(2 \mathbf { i } - 3 \mathbf { j }\) ) \(\mathrm { ms } ^ { - 1 }\) At time \(t = 2\) seconds, \(P\) is at the point \(A\) with position vector ( \(7 \mathbf { i } - 10 \mathbf { j }\) ) m.
  1. Show that the magnitude of the acceleration of \(P\) is \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) At the instant when \(P\) leaves the point \(A\), the acceleration of \(P\) changes so that \(P\) now moves with constant acceleration ( \(4 \mathbf { i } + 8.8 \mathbf { j }\) ) \(\mathrm { m } \mathrm { s } ^ { - 2 }\) At the instant when \(P\) reaches the point \(B\), the direction of motion of \(P\) is north east.
  2. Find the time it takes for \(P\) to travel from \(A\) to \(B\).
Edexcel Paper 3 2018 June Q9
13 marks Standard +0.3
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{65e4b254-fb7b-45c2-9702-32f034018193-28_684_908_246_580} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A plank, \(A B\), of mass \(M\) and length \(2 a\), rests with its end \(A\) against a rough vertical wall. The plank is held in a horizontal position by a rope. One end of the rope is attached to the plank at \(B\) and the other end is attached to the wall at the point \(C\), which is vertically above \(A\). A small block of mass \(3 M\) is placed on the plank at the point \(P\), where \(A P = x\). The plank is in equilibrium in a vertical plane which is perpendicular to the wall. The angle between the rope and the plank is \(\alpha\), where \(\tan \alpha = \frac { 3 } { 4 }\), as shown in Figure 3 .
The plank is modelled as a uniform rod, the block is modelled as a particle and the rope is modelled as a light inextensible string.
  1. Using the model, show that the tension in the rope is \(\frac { 5 M g ( 3 x + a ) } { 6 a }\) The magnitude of the horizontal component of the force exerted on the plank at \(A\) by the wall is \(2 M g\).
  2. Find \(x\) in terms of \(a\). The force exerted on the plank at \(A\) by the wall acts in a direction which makes an angle \(\beta\) with the horizontal.
  3. Find the value of \(\tan \beta\) The rope will break if the tension in it exceeds \(5 M g\).
  4. Explain how this will restrict the possible positions of \(P\). You must justify your answer carefully.
Edexcel Paper 3 2018 June Q10
15 marks Standard +0.3
10. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{65e4b254-fb7b-45c2-9702-32f034018193-32_435_1257_244_402} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} A boy throws a ball at a target. At the instant when the ball leaves the boy's hand at the point \(A\), the ball is 2 m above horizontal ground and is moving with speed \(U\) at an angle \(\alpha\) above the horizontal. In the subsequent motion, the highest point reached by the ball is 3 m above the ground. The target is modelled as being the point \(T\), as shown in Figure 4.
The ball is modelled as a particle moving freely under gravity.
Using the model,
  1. show that \(U ^ { 2 } = \frac { 2 g } { \sin ^ { 2 } \alpha }\). The point \(T\) is at a horizontal distance of 20 m from \(A\) and is at a height of 0.75 m above the ground. The ball reaches \(T\) without hitting the ground.
  2. Find the size of the angle \(\alpha\)
  3. State one limitation of the model that could affect your answer to part (b).
  4. Find the time taken for the ball to travel from \(A\) to \(T\).
Edexcel Paper 3 Specimen Q1
13 marks Easy -1.3
  1. The number of hours of sunshine each day, \(y\), for the month of July at Heathrow are summarised in the table below.
Hours\(0 \leqslant y < 5\)\(5 \leqslant y < 8\)\(8 \leqslant y < 11\)\(11 \leqslant y < 12\)\(12 \leqslant y < 14\)
Frequency126832
A histogram was drawn to represent these data. The \(8 \leqslant y < 11\) group was represented by a bar of width 1.5 cm and height 8 cm .
  1. Find the width and the height of the \(0 \leqslant y < 5\) group.
  2. Use your calculator to estimate the mean and the standard deviation of the number of hours of sunshine each day, for the month of July at Heathrow.
    Give your answers to 3 significant figures. The mean and standard deviation for the number of hours of daily sunshine for the same month in Hurn are 5.98 hours and 4.12 hours respectably.
    Thomas believes that the further south you are the more consistent should be the number of hours of daily sunshine.
  3. State, giving a reason, whether or not the calculations in part (b) support Thomas' belief.
  4. Estimate the number of days in July at Heathrow where the number of hours of sunshine is more than 1 standard deviation above the mean. Helen models the number of hours of sunshine each day, for the month of July at Heathrow by \(\mathrm { N } \left( 6.6,3.7 ^ { 2 } \right)\).
  5. Use Helen's model to predict the number of days in July at Heathrow when the number of hours of sunshine is more than 1 standard deviation above the mean.
  6. Use your answers to part (d) and part (e) to comment on the suitability of Helen's model.
Edexcel Paper 3 Specimen Q2
6 marks Standard +0.3
  1. A meteorologist believes that there is a relationship between the daily mean windspeed, \(w \mathrm { kn }\), and the daily mean temperature, \(t ^ { \circ } \mathrm { C }\). A random sample of 9 consecutive days is taken from past records from a town in the UK in July and the relevant data is given in the table below.
\(\boldsymbol { t }\)13.316.215.716.616.316.419.317.113.2
\(\boldsymbol { w }\)711811138151011
The meteorologist calculated the product moment correlation coefficient for the 9 days and obtained \(r = 0.609\)
  1. Explain why a linear regression model based on these data is unreliable on a day when the mean temperature is \(24 ^ { \circ } \mathrm { C }\)
  2. State what is measured by the product moment correlation coefficient.
  3. Stating your hypotheses clearly test, at the \(5 \%\) significance level, whether or not the product moment correlation coefficient for the population is greater than zero. Using the same 9 days a location from the large data set gave \(\bar { t } = 27.2\) and \(\bar { w } = 3.5\)
  4. Using your knowledge of the large data set, suggest, giving your reason, the location that gave rise to these statistics.
Edexcel Paper 3 Specimen Q3
12 marks Standard +0.3
  1. A machine cuts strips of metal to length \(L \mathrm {~cm}\), where \(L\) is normally distributed with standard deviation 0.5 cm .
Strips with length either less than 49 cm or greater than 50.75 cm cannot be used.
Given that 2.5\% of the cut lengths exceed 50.98 cm ,
  1. find the probability that a randomly chosen strip of metal can be used. Ten strips of metal are selected at random.
  2. Find the probability fewer than 4 of these strips cannot be used. A second machine cuts strips of metal of length \(X \mathrm {~cm}\), where \(X\) is normally distributed with standard deviation 0.6 cm A random sample of 15 strips cut by this second machine was found to have a mean length of 50.4 cm
  3. Stating your hypotheses clearly and using a \(1 \%\) level of significance, test whether or not the mean length of all the strips, cut by the second machine, is greater than 50.1 cm
Edexcel Paper 3 Specimen Q4
10 marks Standard +0.3
  1. Given that
$$\mathrm { P } ( A ) = 0.35 \quad \mathrm { P } ( B ) = 0.45 \quad \text { and } \quad \mathrm { P } ( A \cap B ) = 0.13$$ find
  1. \(\mathrm { P } \left( A ^ { \prime } \mid B ^ { \prime } \right)\)
  2. Explain why the events \(A\) and \(B\) are not independent. The event \(C\) has \(\mathrm { P } ( C ) = 0.20\) The events \(A\) and \(C\) are mutually exclusive and the events \(B\) and \(C\) are statistically independent.
  3. Draw a Venn diagram to illustrate the events \(A , B\) and \(C\), giving the probabilities for each region.
  4. Find \(\mathrm { P } \left( [ B \cup C ] ^ { \prime } \right)\)
Edexcel Paper 3 Specimen Q5
9 marks Moderate -0.3
  1. A company sells seeds and claims that \(55 \%\) of its pea seeds germinate.
    1. Write down a reason why the company should not justify their claim by testing all the pea seeds they produce.
    A random selection of the pea seeds is planted in 10 trays with 24 seeds in each tray.
  2. Assuming that the company's claim is correct, calculate the probability that in at least half of the trays 15 or more of the seeds germinate.
  3. Write down two conditions under which the normal distribution may be used as an approximation to the binomial distribution. A random sample of 240 pea seeds was planted and 150 of these seeds germinated.
  4. Assuming that the company's claim is correct, use a normal approximation to find the probability that at least 150 pea seeds germinate.
  5. Using your answer to part (d), comment on whether or not the proportion of the company's pea seeds that germinate is different from the company's claim of \(55 \%\)
Edexcel Paper 3 Specimen Q6
6 marks Moderate -0.3
6. At time \(t\) seconds, where \(t \geqslant 0\), a particle \(P\) moves so that its acceleration \(\mathbf { a } \mathrm { m } \mathrm { s } ^ { - 2 }\) is given by $$\mathbf { a } = 5 t \mathbf { i } - 15 t ^ { \frac { 1 } { 2 } } \mathbf { j }$$ When \(t = 0\), the velocity of \(P\) is \(20 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) Find the speed of \(P\) when \(t = 4\)
Edexcel Paper 3 Specimen Q7
8 marks Standard +0.3
  1. A rough plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac { 3 } { 4 }\).
A particle of mass \(m\) is placed on the plane and then projected up a line of greatest slope of the plane. The coefficient of friction between the particle and the plane is \(\mu\).
The particle moves up the plane with a constant deceleration of \(\frac { 4 } { 5 } \mathrm {~g}\).
  1. Find the value of \(\mu\). The particle comes to rest at the point \(A\) on the plane.
  2. Determine whether the particle will remain at \(A\), carefully justifying your answer.
Edexcel Paper 3 Specimen Q8
10 marks Moderate -0.3
  1. \hspace{0pt} [In this question \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal unit vectors due east and due north respectively]
A radio controlled model boat is placed on the surface of a large pond.
The boat is modelled as a particle.
At time \(t = 0\), the boat is at the fixed point \(O\) and is moving due north with speed \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
Relative to \(O\), the position vector of the boat at time \(t\) seconds is \(\mathbf { r }\) metres.
At time \(t = 15\), the velocity of the boat is \(( 10.5 \mathbf { i } - 0.9 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).
The acceleration of the boat is constant.
  1. Show that the acceleration of the boat is \(( 0.7 \mathbf { i } - 0.1 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\).
  2. Find \(\mathbf { r }\) in terms of \(t\).
  3. Find the value of \(t\) when the boat is north-east of \(O\).
  4. Find the value of \(t\) when the boat is moving in a north-east direction.
Edexcel Paper 3 Specimen Q9
13 marks Challenging +1.2
9. Figure 1 A uniform ladder \(A B\), of length \(2 a\) and weight \(W\), has its end \(A\) on rough horizontal ground. The coefficient of friction between the ladder and the ground is \(\frac { 1 } { 4 }\).
The end \(B\) of the ladder is resting against a smooth vertical wall, as shown in Figure 1.
A builder of weight \(7 W\) stands at the top of the ladder.
To stop the ladder from slipping, the builder's assistant applies a horizontal force of magnitude \(P\) to the ladder at \(A\), towards the wall.
The force acts in a direction which is perpendicular to the wall.
The ladder rests in equilibrium in a vertical plane perpendicular to the wall and makes an angle \(\alpha\) with the horizontal ground, where \(\tan \alpha = \frac { 5 } { 2 }\).
The builder is modelled as a particle and the ladder is modelled as a uniform rod.
  1. Show that the reaction of the wall on the ladder at \(B\) has magnitude \(3 W\).
  2. Find, in terms of \(W\), the range of possible values of \(P\) for which the ladder remains in equilibrium. Often in practice, the builder's assistant will simply stand on the bottom of the ladder.
  3. Explain briefly how this helps to stop the ladder from slipping.
Edexcel Paper 3 Specimen Q10
13 marks Standard +0.3
10. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e678bf51-6dca-4ad7-808b-dfa31b04dc63-22_719_1333_246_365} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A boy throws a stone with speed \(U \mathrm {~ms} ^ { - 1 }\) from a point \(O\) at the top of a vertical cliff. The point \(O\) is 18 m above sea level.
The stone is thrown at an angle \(\alpha\) above the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\).
The stone hits the sea at the point \(S\) which is at a horizontal distance of 36 m from the foot of the cliff, as shown in Figure 2.
The stone is modelled as a particle moving freely under gravity with \(g = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) Find
  1. the value of \(U\),
  2. the speed of the stone when it is 10.8 m above sea level, giving your answer to 2 significant figures.
  3. Suggest two improvements that could be made to the model.