Questions — Edexcel Paper 3 (91 questions)

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Edexcel Paper 3 2019 June Q1
  1. \hspace{0pt} [In this question position vectors are given relative to a fixed origin \(O\) ]
At time \(t\) seconds, where \(t \geqslant 0\), a particle, \(P\), moves so that its velocity \(\mathbf { v ~ m ~ s } ^ { - 1 }\) is given by $$\mathbf { v } = 6 t \mathbf { i } - 5 t ^ { \frac { 3 } { 2 } } \mathbf { j }$$ When \(t = 0\), the position vector of \(P\) is \(( - 20 \mathbf { i } + 20 \mathbf { j } ) \mathrm { m }\).
  1. Find the acceleration of \(P\) when \(t = 4\)
  2. Find the position vector of \(P\) when \(t = 4\)
Edexcel Paper 3 2019 June Q2
  1. A particle, \(P\), moves with constant acceleration \(( 2 \mathbf { i } - 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\)
At time \(t = 0\), the particle is at the point \(A\) and is moving with velocity ( \(- \mathbf { i } + 4 \mathbf { j }\) ) \(\mathrm { m } \mathrm { s } ^ { - 1 }\)
At time \(t = T\) seconds, \(P\) is moving in the direction of vector ( \(3 \mathbf { i } - 4 \mathbf { j }\) )
  1. Find the value of \(T\). At time \(t = 4\) seconds, \(P\) is at the point \(B\).
  2. Find the distance \(A B\).
Edexcel Paper 3 2019 June Q3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8399dae8-1b9d-4564-a95b-7ab857368b86-06_339_812_242_628} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Two blocks, \(A\) and \(B\), of masses \(2 m\) and \(3 m\) respectively, are attached to the ends of a light string. Initially \(A\) is held at rest on a fixed rough plane.
The plane is inclined at angle \(\alpha\) to the horizontal ground, where \(\tan \alpha = \frac { 5 } { 12 }\)
The string passes over a small smooth pulley, \(P\), fixed at the top of the plane.
The part of the string from \(A\) to \(P\) is parallel to a line of greatest slope of the plane. Block \(B\) hangs freely below \(P\), as shown in Figure 1. The coefficient of friction between \(A\) and the plane is \(\frac { 2 } { 3 }\)
The blocks are released from rest with the string taut and \(A\) moves up the plane.
The tension in the string immediately after the blocks are released is \(T\).
The blocks are modelled as particles and the string is modelled as being inextensible.
  1. Show that \(T = \frac { 12 m g } { 5 }\) After \(B\) reaches the ground, \(A\) continues to move up the plane until it comes to rest before reaching \(P\).
  2. Determine whether \(A\) will remain at rest, carefully justifying your answer.
  3. Suggest two refinements to the model that would make it more realistic.
Edexcel Paper 3 2019 June Q4
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8399dae8-1b9d-4564-a95b-7ab857368b86-10_417_844_244_612} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A ramp, \(A B\), of length 8 m and mass 20 kg , rests in equilibrium with the end \(A\) on rough horizontal ground. The ramp rests on a smooth solid cylindrical drum which is partly under the ground. The drum is fixed with its axis at the same horizontal level as \(A\). The point of contact between the ramp and the drum is \(C\), where \(A C = 5 \mathrm {~m}\), as shown in Figure 2. The ramp is resting in a vertical plane which is perpendicular to the axis of the drum, at an angle \(\theta\) to the horizontal, where \(\tan \theta = \frac { 7 } { 24 }\) The ramp is modelled as a uniform rod.
  1. Explain why the reaction from the drum on the ramp at point \(C\) acts in a direction which is perpendicular to the ramp.
  2. Find the magnitude of the resultant force acting on the ramp at \(A\). The ramp is still in equilibrium in the position shown in Figure 2 but the ramp is not now modelled as being uniform. Given that the centre of mass of the ramp is assumed to be closer to \(A\) than to \(B\),
  3. state how this would affect the magnitude of the normal reaction between the ramp and the drum at \(C\).
Edexcel Paper 3 2019 June Q5
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8399dae8-1b9d-4564-a95b-7ab857368b86-14_223_855_239_605} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} The points \(A\) and \(B\) lie 50 m apart on horizontal ground.
At time \(t = 0\) two small balls, \(P\) and \(Q\), are projected in the vertical plane containing \(A B\).
Ball \(P\) is projected from \(A\) with speed \(20 \mathrm {~ms} ^ { - 1 }\) at \(30 ^ { \circ }\) to \(A B\).
Ball \(Q\) is projected from \(B\) with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at angle \(\theta\) to \(B A\), as shown in Figure 3.
At time \(t = 2\) seconds, \(P\) and \(Q\) collide.
Until they collide, the balls are modelled as particles moving freely under gravity.
  1. Find the velocity of \(P\) at the instant before it collides with \(Q\).
  2. Find
    1. the size of angle \(\theta\),
    2. the value of \(u\).
  3. State one limitation of the model, other than air resistance, that could affect the accuracy of your answers.
Edexcel Paper 3 2019 June Q1
  1. Three bags, \(A , B\) and \(C\), each contain 1 red marble and some green marbles.
Bag \(A\) contains 1 red marble and 9 green marbles only
Bag \(B\) contains 1 red marble and 4 green marbles only
Bag \(C\) contains 1 red marble and 2 green marbles only
Sasha selects at random one marble from bag \(A\).
If he selects a red marble, he stops selecting.
If the marble is green, he continues by selecting at random one marble from bag \(B\).
If he selects a red marble, he stops selecting.
If the marble is green, he continues by selecting at random one marble from bag \(C\).
  1. Draw a tree diagram to represent this information.
  2. Find the probability that Sasha selects 3 green marbles.
  3. Find the probability that Sasha selects at least 1 marble of each colour.
  4. Given that Sasha selects a red marble, find the probability that he selects it from bag \(B\).
Edexcel Paper 3 2019 June Q2
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{d1eaaae7-c1dc-4aee-ab54-59f35519a7a4-06_321_1822_294_127} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The partially completed box plot in Figure 1 shows the distribution of daily mean air temperatures using the data from the large data set for Beijing in 2015 An outlier is defined as a value
more than \(1.5 \times\) IQR below \(Q _ { 1 }\) or
more than \(1.5 \times\) IQR above \(Q _ { 3 }\)
The three lowest air temperatures in the data set are \(7.6 ^ { \circ } \mathrm { C } , 8.1 ^ { \circ } \mathrm { C }\) and \(9.1 ^ { \circ } \mathrm { C }\)
The highest air temperature in the data set is \(32.5 ^ { \circ } \mathrm { C }\)
  1. Complete the box plot in Figure 1 showing clearly any outliers.
  2. Using your knowledge of the large data set, suggest from which month the two outliers are likely to have come. Using the data from the large data set, Simon produced the following summary statistics for the daily mean air temperature, \(x ^ { \circ } \mathrm { C }\), for Beijing in 2015 $$n = 184 \quad \sum x = 4153.6 \quad \mathrm {~S} _ { x x } = 4952.906$$
  3. Show that, to 3 significant figures, the standard deviation is \(5.19 ^ { \circ } \mathrm { C }\) Simon decides to model the air temperatures with the random variable $$T \sim \mathrm {~N} \left( 22.6,5.19 ^ { 2 } \right)$$
  4. Using Simon's model, calculate the 10th to 90th interpercentile range. Simon wants to model another variable from the large data set for Beijing using a normal distribution.
  5. State two variables from the large data set for Beijing that are not suitable to be modelled by a normal distribution. Give a reason for each answer.
    \includegraphics[max width=\textwidth, alt={}, center]{d1eaaae7-c1dc-4aee-ab54-59f35519a7a4-09_473_1813_2161_127}
    (Total for Question 2 is 11 marks)
Edexcel Paper 3 2019 June Q3
3. Barbara is investigating the relationship between average income (GDP per capita), \(x\) US dollars, and average annual carbon dioxide ( \(\mathrm { CO } _ { 2 }\) ) emissions, \(y\) tonnes, for different countries. She takes a random sample of 24 countries and finds the product moment correlation coefficient between average annual \(\mathrm { CO } _ { 2 }\) emissions and average income to be 0.446
  1. Stating your hypotheses clearly, test, at the \(5 \%\) level of significance, whether or not the product moment correlation coefficient for all countries is greater than zero. Barbara believes that a non-linear model would be a better fit to the data.
    She codes the data using the coding \(m = \log _ { 10 } x\) and \(c = \log _ { 10 } y\) and obtains the model \(c = - 1.82 + 0.89 m\) The product moment correlation coefficient between \(c\) and \(m\) is found to be 0.882
  2. Explain how this value supports Barbara's belief.
  3. Show that the relationship between \(y\) and \(x\) can be written in the form \(y = a x ^ { n }\) where \(a\) and \(n\) are constants to be found.
Edexcel Paper 3 2019 June Q4
  1. Magali is studying the mean total cloud cover, in oktas, for Leuchars in 1987 using data from the large data set. The daily mean total cloud cover for all 184 days from the large data set is summarised in the table below.
Daily mean total cloud cover (oktas)012345678
Frequency (number of days)01471030525228
One of the 184 days is selected at random.
  1. Find the probability that it has a daily mean total cloud cover of 6 or greater. Magali is investigating whether the daily mean total cloud cover can be modelled using a binomial distribution. She uses the random variable \(X\) to denote the daily mean total cloud cover and believes that \(X \sim \mathrm {~B} ( 8,0.76 )\) Using Magali's model,
    1. find \(\mathrm { P } ( X \geqslant 6 )\)
    2. find, to 1 decimal place, the expected number of days in a sample of 184 days with a daily mean total cloud cover of 7
  3. Explain whether or not your answers to part (b) support the use of Magali's model. There were 28 days that had a daily mean total cloud cover of 8 For these 28 days the daily mean total cloud cover for the following day is shown in the table below.
    Daily mean total cloud cover (oktas)012345678
    Frequency (number of days)001121599
  4. Find the proportion of these days when the daily mean total cloud cover was 6 or greater.
  5. Comment on Magali's model in light of your answer to part (d).
Edexcel Paper 3 2019 June Q5
  1. A machine puts liquid into bottles of perfume. The amount of liquid put into each bottle, \(D \mathrm { ml }\), follows a normal distribution with mean 25 ml
Given that 15\% of bottles contain less than 24.63 ml
  1. find, to 2 decimal places, the value of \(k\) such that \(\mathrm { P } ( 24.63 < D < k ) = 0.45\) A random sample of 200 bottles is taken.
  2. Using a normal approximation, find the probability that fewer than half of these bottles contain between 24.63 ml and \(k \mathrm { ml }\) The machine is adjusted so that the standard deviation of the liquid put in the bottles is now 0.16 ml Following the adjustments, Hannah believes that the mean amount of liquid put in each bottle is less than 25 ml She takes a random sample of 20 bottles and finds the mean amount of liquid to be 24.94 ml
  3. Test Hannah's belief at the \(5 \%\) level of significance. You should state your hypotheses clearly.
Edexcel Paper 3 2022 June Q1
  1. \hspace{0pt} [In this question, position vectors are given relative to a fixed origin.] At time \(t\) seconds, where \(t > 0\), a particle \(P\) has velocity \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\) where
$$\mathbf { v } = 3 t ^ { 2 } \mathbf { i } - 6 t ^ { \frac { 1 } { 2 } } \mathbf { j }$$
  1. Find the speed of \(P\) at time \(t = 2\) seconds.
  2. Find an expression, in terms of \(t , \mathbf { i }\) and \(\mathbf { j }\), for the acceleration of \(P\) at time \(t\) seconds, where \(t > 0\) At time \(t = 4\) seconds, the position vector of \(P\) is ( \(\mathbf { i } - 4 \mathbf { j }\) ) m.
  3. Find the position vector of \(P\) at time \(t = 1\) second.
Edexcel Paper 3 2022 June Q2
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{414946db-64d7-44b8-801d-2c7805ee9cc6-04_282_627_246_721} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A rough plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac { 3 } { 4 }\)
A small block \(B\) of mass 5 kg is held in equilibrium on the plane by a horizontal force of magnitude \(X\) newtons, as shown in Figure 1. The force acts in a vertical plane which contains a line of greatest slope of the inclined plane. The block \(B\) is modelled as a particle.
The magnitude of the normal reaction of the plane on \(B\) is 68.6 N .
Using the model,
    1. find the magnitude of the frictional force acting on \(B\),
    2. state the direction of the frictional force acting on \(B\). The horizontal force of magnitude \(X\) newtons is now removed and \(B\) moves down the plane. Given that the coefficient of friction between \(B\) and the plane is 0.5
  2. find the acceleration of \(B\) down the plane.
Edexcel Paper 3 2022 June Q3
  1. \hspace{0pt} [In this question, \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal unit vectors.]
A particle \(P\) of mass 4 kg is at rest at the point \(A\) on a smooth horizontal plane.
At time \(t = 0\), two forces, \(\mathbf { F } _ { 1 } = ( 4 \mathbf { i } - \mathbf { j } ) \mathrm { N }\) and \(\mathbf { F } _ { 2 } = ( \lambda \mathbf { i } + \mu \mathbf { j } ) \mathrm { N }\), where \(\lambda\) and \(\mu\) are constants, are applied to \(P\) Given that \(P\) moves in the direction of the vector ( \(3 \mathbf { i } + \mathbf { j }\) )
  1. show that $$\lambda - 3 \mu + 7 = 0$$ At time \(t = 4\) seconds, \(P\) passes through the point \(B\).
    Given that \(\lambda = 2\)
  2. find the length of \(A B\).
Edexcel Paper 3 2022 June Q4
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{414946db-64d7-44b8-801d-2c7805ee9cc6-12_716_1191_246_438} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A uniform rod \(A B\) has mass \(M\) and length \(2 a\)
A particle of mass \(2 M\) is attached to the rod at the point \(C\), where \(A C = 1.5 a\)
The rod rests with its end \(A\) on rough horizontal ground.
The rod is held in equilibrium at an angle \(\theta\) to the ground by a light string that is attached to the end \(B\) of the rod. The string is perpendicular to the rod, as shown in Figure 2.
  1. Explain why the frictional force acting on the rod at \(A\) acts horizontally to the right on the diagram. The tension in the string is \(T\)
  2. Show that \(T = 2 M g \cos \theta\) Given that \(\cos \theta = \frac { 3 } { 5 }\)
  3. show that the magnitude of the vertical force exerted by the ground on the rod at \(A\) is \(\frac { 57 M g } { 25 }\) The coefficient of friction between the rod and the ground is \(\mu\)
    Given that the rod is in limiting equilibrium,
  4. show that \(\mu = \frac { 8 } { 19 }\)
Edexcel Paper 3 2022 June Q5
5. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{414946db-64d7-44b8-801d-2c7805ee9cc6-16_303_1266_237_404} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A golf ball is at rest at the point \(A\) on horizontal ground.
The ball is hit and initially moves at an angle \(\alpha\) to the ground.
The ball first hits the ground at the point \(B\), where \(A B = 120 \mathrm {~m}\), as shown in Figure 3.
The motion of the ball is modelled as that of a particle, moving freely under gravity, whose initial speed is \(U \mathrm {~m} \mathrm {~s} ^ { - 1 }\) Using this model,
  1. show that \(U ^ { 2 } \sin \alpha \cos \alpha = 588\) The ball reaches a maximum height of 10 m above the ground.
  2. Show that \(U ^ { 2 } = 1960\) In a refinement to the model, the effect of air resistance is included.
    The motion of the ball, from \(A\) to \(B\), is now modelled as that of a particle whose initial speed is \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) This refined model is used to calculate a value for \(V\)
  3. State which is greater, \(U\) or \(V\), giving a reason for your answer.
  4. State one further refinement to the model that would make the model more realistic.
Edexcel Paper 3 2022 June Q1
  1. George throws a ball at a target 15 times.
Each time George throws the ball, the probability of the ball hitting the target is 0.48
The random variable \(X\) represents the number of times George hits the target in 15 throws.
  1. Find
    1. \(\mathrm { P } ( X = 3 )\)
    2. \(\mathrm { P } ( X \geqslant 5 )\) George now throws the ball at the target 250 times.
  2. Use a normal approximation to calculate the probability that he will hit the target more than 110 times.
Edexcel Paper 3 2022 June Q2
  1. A manufacturer uses a machine to make metal rods.
The length of a metal rod, \(L \mathrm {~cm}\), is normally distributed with
  • a mean of 8 cm
  • a standard deviation of \(x \mathrm {~cm}\)
Given that the proportion of metal rods less than 7.902 cm in length is \(2.5 \%\)
  1. show that \(x = 0.05\) to 2 decimal places.
  2. Calculate the proportion of metal rods that are between 7.94 cm and 8.09 cm in length. The cost of producing a single metal rod is 20p
    A metal rod
    • where \(L < 7.94\) is sold for scrap for 5 p
    • where \(7.94 \leqslant L \leqslant 8.09\) is sold for 50 p
    • where \(L > 8.09\) is shortened for an extra cost of 10 p and then sold for 50 p
    • Calculate the expected profit per 500 of the metal rods.
    Give your answer to the nearest pound. The same manufacturer makes metal hinges in large batches.
    The hinges each have a probability of 0.015 of having a fault.
    A random sample of 200 hinges is taken from each batch and the batch is accepted if fewer than 6 hinges are faulty. The manufacturer's aim is for 95\% of batches to be accepted.
  3. Explain whether the manufacturer is likely to achieve its aim.
Edexcel Paper 3 2022 June Q3
  1. Dian uses the large data set to investigate the Daily Total Rainfall, \(r \mathrm {~mm}\), for Camborne.
    1. Write down how a value of \(0 < r \leqslant 0.05\) is recorded in the large data set.
    Dian uses the data for the 31 days of August 2015 for Camborne and calculates the following statistics $$n = 31 \quad \sum r = 174.9 \quad \sum r ^ { 2 } = 3523.283$$
  2. Use these statistics to calculate
    1. the mean of the Daily Total Rainfall in Camborne for August 2015,
    2. the standard deviation of the Daily Total Rainfall in Camborne for August 2015. Dian believes that the mean Daily Total Rainfall in August is less in the South of the UK than in the North of the UK.
      The mean Daily Total Rainfall in Leuchars for August 2015 is 1.72 mm to 2 decimal places.
  3. State, giving a reason, whether this provides evidence to support Dian's belief. Dian uses the large data set to estimate the proportion of days with no rain in Camborne for 1987 to be 0.27 to 2 decimal places.
  4. Explain why the distribution \(\mathrm { B } ( 14,0.27 )\) might not be a reasonable model for the number of days without rain for a 14-day summer event.
Edexcel Paper 3 2022 June Q4
  1. A dentist knows from past records that \(10 \%\) of customers arrive late for their appointment.
A new manager believes that there has been a change in the proportion of customers who arrive late for their appointment. A random sample of 50 of the dentist's customers is taken.
  1. Write down
    • a null hypothesis corresponding to no change in the proportion of customers who arrive late
    • an alternative hypothesis corresponding to the manager's belief
    • Using a \(5 \%\) level of significance, find the critical region for a two-tailed test of the null hypothesis in (a) You should state the probability of rejection in each tail, which should be less than 0.025
    • Find the actual level of significance of the test based on your critical region from part (b)
    The manager observes that 15 of the 50 customers arrived late for their appointment.
  2. With reference to part (b), comment on the manager's belief.
Edexcel Paper 3 2022 June Q5
  1. A company has 1825 employees.
The employees are classified as professional, skilled or elementary.
The following table shows
  • the number of employees in each classification
  • the two areas, \(A\) or \(B\), where the employees live
\cline { 2 - 3 } \multicolumn{1}{c|}{}\(\boldsymbol { A }\)\(\boldsymbol { B }\)
Professional740380
Skilled27590
Elementary26080
An employee is chosen at random.
Find the probability that this employee
  1. is skilled,
  2. lives in area \(B\) and is not a professional. Some classifications of employees are more likely to work from home.
    • \(65 \%\) of professional employees in both area \(A\) and area \(B\) work from home
    • \(40 \%\) of skilled employees in both area \(A\) and area \(B\) work from home
    • \(5 \%\) of elementary employees in both area \(A\) and area \(B\) work from home
    • Event \(F\) is that the employee is a professional
    • Event \(H\) is that the employee works from home
    • Event \(R\) is that the employee is from area \(A\)
    • Using this information, complete the Venn diagram on the opposite page.
    • Find \(\mathrm { P } \left( R ^ { \prime } \cap F \right)\)
    • Find \(\mathrm { P } \left( [ H \cup R ] ^ { \prime } \right)\)
    • Find \(\mathrm { P } ( F \mid H )\)
    \includegraphics[max width=\textwidth, alt={}]{3a09f809-fa28-4b3d-bb69-ea074433bd8f-13_872_1020_294_525}
    Turn over for a spare diagram if you need to redraw your Venn diagram. Only use this diagram if you need to redraw your Venn diagram.
    \includegraphics[max width=\textwidth, alt={}, center]{3a09f809-fa28-4b3d-bb69-ea074433bd8f-15_872_1017_392_525}
Edexcel Paper 3 2022 June Q6
6. Anna is investigating the relationship between exercise and resting heart rate. She takes a random sample of 19 people in her year at school and records for each person
  • their resting heart rate, \(h\) beats per minute
  • the number of minutes, \(m\), spent exercising each week
Her results are shown on the scatter diagram.
\includegraphics[max width=\textwidth, alt={}, center]{3a09f809-fa28-4b3d-bb69-ea074433bd8f-16_531_551_653_740}
  1. Interpret the nature of the relationship between \(h\) and \(m\) Anna codes the data using the formulae $$\begin{aligned} & x = \log _ { 10 } m
    & y = \log _ { 10 } h \end{aligned}$$ The product moment correlation coefficient between \(x\) and \(y\) is - 0.897
  2. Test whether or not there is significant evidence of a negative correlation between \(x\) and \(y\)
    You should
    • state your hypotheses clearly
    • use a \(5 \%\) level of significance
    • state the critical value used
    The equation of the line of best fit of \(y\) on \(x\) is $$y = - 0.05 x + 1.92$$
  3. Use the equation of the line of best fit of \(y\) on \(x\) to find a model for \(h\) on \(m\) in the form $$h = a m ^ { k }$$ where \(a\) and \(k\) are constants to be found.
Edexcel Paper 3 2023 June Q1
  1. A car is initially at rest on a straight horizontal road.
The car then accelerates along the road with a constant acceleration of \(3.2 \mathrm {~ms} ^ { - 2 }\)
Find
  1. the speed of the car after 5 s ,
  2. the distance travelled by the car in the first 5 s .
Edexcel Paper 3 2023 June Q2
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f9dc8158-8ed8-4138-9c75-050cf52e6f7e-04_83_659_267_703} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A particle \(P\) has mass 5 kg .
The particle is pulled along a rough horizontal plane by a horizontal force of magnitude 28 N . The only resistance to motion is a frictional force of magnitude \(F\) newtons, as shown in Figure 1.
  1. Find the magnitude of the normal reaction of the plane on \(P\) The particle is accelerating along the plane at \(1.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\)
  2. Find the value of \(F\) The coefficient of friction between \(P\) and the plane is \(\mu\)
  3. Find the value of \(\mu\), giving your answer to 2 significant figures.
Edexcel Paper 3 2023 June Q3
  1. At time \(t\) seconds, where \(t \geqslant 0\), a particle \(P\) has velocity \(\mathbf { v } \mathrm { ms } ^ { - 1 }\) where
$$\mathbf { v } = \left( t ^ { 2 } - 3 t + 7 \right) \mathbf { i } + \left( 2 t ^ { 2 } - 3 \right) \mathbf { j }$$ Find
  1. the speed of \(P\) at time \(t = 0\)
  2. the value of \(t\) when \(P\) is moving parallel to \(( \mathbf { i } + \mathbf { j } )\)
  3. the acceleration of \(P\) at time \(t\) seconds
  4. the value of \(t\) when the direction of the acceleration of \(P\) is perpendicular to \(\mathbf { i }\)
Edexcel Paper 3 2023 June Q4
  1. \hspace{0pt} [In this question, \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal unit vectors and position vectors are given relative to a fixed origin \(O\) ]
A particle \(P\) is moving on a smooth horizontal plane.
The particle has constant acceleration \(( 2.4 \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\)
At time \(t = 0 , P\) passes through the point \(A\).
At time \(t = 5 \mathrm {~s} , P\) passes through the point \(B\).
The velocity of \(P\) as it passes through \(A\) is \(( - 16 \mathbf { i } - 3 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\)
  1. Find the speed of \(P\) as it passes through \(B\). The position vector of \(A\) is \(( 44 \mathbf { i } - 10 \mathbf { j } ) \mathrm { m }\).
    At time \(t = T\) seconds, where \(T > 5 , P\) passes through the point \(C\).
    The position vector of \(C\) is \(( 4 \mathbf { i } + c \mathbf { j } ) \mathrm { m }\).
  2. Find the value of \(T\).
  3. Find the value of \(c\).