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Edexcel Paper 3 2020 October Q1
8 marks Moderate -0.8
  1. The Venn diagram shows the probabilities associated with four events, \(A , B , C\) and \(D\) \includegraphics[max width=\textwidth, alt={}, center]{2b63aa7f-bc50-4422-8dc0-e661b521c221-02_505_861_296_602}
    1. Write down any pair of mutually exclusive events from \(A , B , C\) and \(D\)
    Given that \(\mathrm { P } ( B ) = 0.4\)
  2. find the value of \(p\) Given also that \(A\) and \(B\) are independent
  3. find the value of \(q\) Given further that \(\mathrm { P } \left( B ^ { \prime } \mid C \right) = 0.64\)
  4. find
    1. the value of \(r\)
    2. the value of \(s\)
Edexcel Paper 3 2020 October Q2
7 marks Moderate -0.8
  1. A random sample of 15 days is taken from the large data set for Perth in June and July 1987. The scatter diagram in Figure 1 displays the values of two of the variables for these 15 days.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{2b63aa7f-bc50-4422-8dc0-e661b521c221-04_722_709_376_677} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure}
  1. Describe the correlation. The variable on the \(x\)-axis is Daily Mean Temperature measured in \({ } ^ { \circ } \mathrm { C }\).
  2. Using your knowledge of the large data set,
    1. suggest which variable is on the \(y\)-axis,
    2. state the units that are used in the large data set for this variable. Stav believes that there is a correlation between Daily Total Sunshine and Daily Maximum Relative Humidity at Heathrow. He calculates the product moment correlation coefficient between these two variables for a random sample of 30 days and obtains \(r = - 0.377\)
  3. Carry out a suitable test to investigate Stav's belief at a \(5 \%\) level of significance. State clearly
    • your hypotheses
    • your critical value
    On a random day at Heathrow the Daily Maximum Relative Humidity was 97\%
  4. Comment on the number of hours of sunshine you would expect on that day, giving a reason for your answer.
Edexcel Paper 3 2020 October Q3
10 marks Moderate -0.3
  1. Each member of a group of 27 people was timed when completing a puzzle.
The time taken, \(x\) minutes, for each member of the group was recorded.
These times are summarised in the following box and whisker plot. \includegraphics[max width=\textwidth, alt={}, center]{2b63aa7f-bc50-4422-8dc0-e661b521c221-08_353_1436_458_319}
  1. Find the range of the times.
  2. Find the interquartile range of the times. For these 27 people \(\sum x = 607.5\) and \(\sum x ^ { 2 } = 17623.25\)
  3. calculate the mean time taken to complete the puzzle,
  4. calculate the standard deviation of the times taken to complete the puzzle. Taruni defines an outlier as a value more than 3 standard deviations above the mean.
  5. State how many outliers Taruni would say there are in these data, giving a reason for your answer. Adam and Beth also completed the puzzle in \(a\) minutes and \(b\) minutes respectively, where \(a > b\).
    When their times are included with the data of the other 27 people
    • the median time increases
    • the mean time does not change
    • Suggest a possible value for \(a\) and a possible value for \(b\), explaining how your values satisfy the above conditions.
    • Without carrying out any further calculations, explain why the standard deviation of all 29 times will be lower than your answer to part (d).
Edexcel Paper 3 2020 October Q4
10 marks Standard +0.3
  1. The discrete random variable \(D\) has the following probability distribution
\(d\)1020304050
\(\mathrm { P } ( D = d )\)\(\frac { k } { 10 }\)\(\frac { k } { 20 }\)\(\frac { k } { 30 }\)\(\frac { k } { 40 }\)\(\frac { k } { 50 }\)
where \(k\) is a constant.
  1. Show that the value of \(k\) is \(\frac { 600 } { 137 }\) The random variables \(D _ { 1 }\) and \(D _ { 2 }\) are independent and each have the same distribution as \(D\).
  2. Find \(\mathrm { P } \left( D _ { 1 } + D _ { 2 } = 80 \right)\) Give your answer to 3 significant figures. A single observation of \(D\) is made.
    The value obtained, \(d\), is the common difference of an arithmetic sequence.
    The first 4 terms of this arithmetic sequence are the angles, measured in degrees, of quadrilateral \(Q\)
  3. Find the exact probability that the smallest angle of \(Q\) is more than \(50 ^ { \circ }\)
Edexcel Paper 3 2020 October Q5
15 marks Standard +0.3
  1. A health centre claims that the time a doctor spends with a patient can be modelled by a normal distribution with a mean of 10 minutes and a standard deviation of 4 minutes.
    1. Using this model, find the probability that the time spent with a randomly selected patient is more than 15 minutes.
    Some patients complain that the mean time the doctor spends with a patient is more than 10 minutes. The receptionist takes a random sample of 20 patients and finds that the mean time the doctor spends with a patient is 11.5 minutes.
  2. Stating your hypotheses clearly and using a \(5 \%\) significance level, test whether or not there is evidence to support the patients' complaint. The health centre also claims that the time a dentist spends with a patient during a routine appointment, \(T\) minutes, can be modelled by the normal distribution where \(T \sim \mathrm {~N} \left( 5,3.5 ^ { 2 } \right)\)
  3. Using this model,
    1. find the probability that a routine appointment with the dentist takes less than 2 minutes
    2. find \(\mathrm { P } ( T < 2 \mid T > 0 )\)
    3. hence explain why this normal distribution may not be a good model for \(T\). The dentist believes that she cannot complete a routine appointment in less than 2 minutes.
      She suggests that the health centre should use a refined model only including values of \(T > 2\)
  4. Find the median time for a routine appointment using this new model, giving your answer correct to one decimal place.
Edexcel Paper 3 2021 October Q1
4 marks Moderate -0.8
  1. A particle \(P\) moves with constant acceleration \(( 2 \mathbf { i } - 3 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 2 }\)
At time \(t = 0 , P\) is moving with velocity \(4 \mathbf { i } \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
  1. Find the velocity of \(P\) at time \(t = 2\) seconds. At time \(t = 0\), the position vector of \(P\) relative to a fixed origin \(O\) is \(( \mathbf { i } + \mathbf { j } ) \mathrm { m }\).
  2. Find the position vector of \(P\) relative to \(O\) at time \(t = 3\) seconds.
Edexcel Paper 3 2021 October Q2
12 marks Standard +0.3
2. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{63363c3e-13fc-49a1-8cef-951e6e97e801-04_396_993_246_536} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} A small stone \(A\) of mass \(3 m\) is attached to one end of a string.
A small stone \(B\) of mass \(m\) is attached to the other end of the string.
Initially \(A\) is held at rest on a fixed rough plane.
The plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac { 3 } { 4 }\) The string passes over a pulley \(P\) that is 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.
Stone \(B\) hangs freely below \(P\), as shown in Figure 1.
The coefficient of friction between \(A\) and the plane is \(\frac { 1 } { 6 }\) Stone \(A\) is released from rest and begins to move down the plane.
The stones are modelled as particles.
The pulley is modelled as being small and smooth.
The string is modelled as being light and inextensible. Using the model for the motion of the system before \(B\) reaches the pulley,
  1. write down an equation of motion for \(A\)
  2. show that the acceleration of \(A\) is \(\frac { 1 } { 10 } \mathrm {~g}\)
  3. sketch a velocity-time graph for the motion of \(B\), from the instant when \(A\) is released from rest to the instant just before \(B\) reaches the pulley, explaining your answer. In reality, the string is not light.
  4. State how this would affect the working in part (b).
Edexcel Paper 3 2021 October Q3
10 marks Standard +0.3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{63363c3e-13fc-49a1-8cef-951e6e97e801-08_796_750_242_660} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} A beam \(A B\) has mass \(m\) and length \(2 a\).
The beam rests in equilibrium with \(A\) on rough horizontal ground and with \(B\) against a smooth vertical wall. The beam is inclined to the horizontal at an angle \(\theta\), as shown in Figure 2.
The coefficient of friction between the beam and the ground is \(\mu\) The beam is modelled as a uniform rod resting in a vertical plane that is perpendicular to the wall. Using the model,
  1. show that \(\mu \geqslant \frac { 1 } { 2 } \cot \theta\) A horizontal force of magnitude \(k m g\), where \(k\) is a constant, is now applied to the beam at \(A\). This force acts in a direction that is perpendicular to the wall and towards the wall.
    Given that \(\tan \theta = \frac { 5 } { 4 } , \mu = \frac { 1 } { 2 }\) and the beam is now in limiting equilibrium,
  2. use the model to find the value of \(k\).
Edexcel Paper 3 2021 October Q4
10 marks Standard +0.3
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{63363c3e-13fc-49a1-8cef-951e6e97e801-12_453_990_244_539} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} A small stone is projected with speed \(65 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(O\) at the top of a vertical cliff.
Point \(O\) is 70 m vertically above the point \(N\).
Point \(N\) is on horizontal ground.
The stone is projected at an angle \(\alpha\) above the horizontal, where \(\tan \alpha = \frac { 5 } { 12 }\) The stone hits the ground at the point \(A\), as shown in Figure 3.
The stone is modelled as a particle moving freely under gravity.
The acceleration due to gravity is modelled as having magnitude \(\mathbf { 1 0 m ~ s } \mathbf { m ~ } ^ { \mathbf { - 2 } }\) Using the model,
  1. find the time taken for the stone to travel from \(O\) to \(A\),
  2. find the speed of the stone at the instant just before it hits the ground at \(A\). One limitation of the model is that it ignores air resistance.
  3. State one other limitation of the model that could affect the reliability of your answers.
Edexcel Paper 3 2021 October Q5
14 marks Standard +0.3
  1. At time \(t\) seconds, a particle \(P\) has velocity \(\mathbf { v } \mathrm { ms } ^ { - 1 }\), where
$$\mathbf { v } = 3 t ^ { \frac { 1 } { 2 } } \mathbf { i } - 2 t \mathbf { j } \quad t > 0$$
  1. Find the acceleration of \(P\) at time \(t\) seconds, where \(t > 0\)
  2. Find the value of \(t\) at the instant when \(P\) is moving in the direction of \(\mathbf { i } - \mathbf { j }\) At time \(t\) seconds, where \(t > 0\), the position vector of \(P\), relative to a fixed origin \(O\), is \(\mathbf { r }\) metres. When \(t = 1 , \mathbf { r } = - \mathbf { j }\)
  3. Find an expression for \(\mathbf { r }\) in terms of \(t\).
  4. Find the exact distance of \(P\) from \(O\) at the instant when \(P\) is moving with speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
Edexcel Paper 3 2021 October Q1
7 marks Moderate -0.8
  1. (a) State one disadvantage of using quota sampling compared with simple random sampling.
In a university 8\% of students are members of the university dance club.
A random sample of 36 students is taken from the university.
The random variable \(X\) represents the number of these students who are members of the dance club.
(b) Using a suitable model for \(X\), find
  1. \(\mathrm { P } ( X = 4 )\)
  2. \(\mathrm { P } ( X \geqslant 7 )\) Only 40\% of the university dance club members can dance the tango.
    (c) Find the probability that a student is a member of the university dance club and can dance the tango. A random sample of 50 students is taken from the university.
    (d) Find the probability that fewer than 3 of these students are members of the university dance club and can dance the tango. \section*{Question 1 continued.}
Edexcel Paper 3 2021 October Q2
6 marks Standard +0.3
  1. Marc took a random sample of 16 students from a school and for each student recorded
  • the number of letters, \(x\), in their last name
  • the number of letters, \(y\), in their first name
His results are shown in the scatter diagram on the next page.
  1. Describe the correlation between \(x\) and \(y\). Marc suggests that parents with long last names tend to give their children shorter first names.
  2. Using the scatter diagram comment on Marc's suggestion, giving a reason for your answer. The results from Marc's random sample of 16 observations are given in the table below.
    \(x\)368753113454971066
    \(y\)7744685584745563
  3. Use your calculator to find the product moment correlation coefficient between \(x\) and \(y\) for these data.
  4. Test whether or not there is evidence of a negative correlation between the number of letters in the last name and the number of letters in the first name. You should
    • state your hypotheses clearly
    • use a \(5 \%\) level of significance
    \section*{Question 2 continued.}
    \includegraphics[max width=\textwidth, alt={}]{10736735-3050-43eb-9e76-011ca6fa48b8-05_1125_1337_294_372}
    \section*{Question 2 continued.} \section*{Question 2 continued.}
Edexcel Paper 3 2021 October Q3
8 marks Easy -1.2
  1. Stav is studying the large data set for September 2015
He codes the variable Daily Mean Pressure, \(x\), using the formula \(y = x - 1010\) The data for all 30 days from Hurn are summarised by $$\sum y = 214 \quad \sum y ^ { 2 } = 5912$$
  1. State the units of the variable \(x\)
  2. Find the mean Daily Mean Pressure for these 30 days.
  3. Find the standard deviation of Daily Mean Pressure for these 30 days. Stav knows that, in the UK, winds circulate
    • in a clockwise direction around a region of high pressure
    • in an anticlockwise direction around a region of low pressure
    The table gives the Daily Mean Pressure for 3 locations from the large data set on 26/09/2015
    LocationHeathrowHurnLeuchars
    Daily Mean Pressure102910281028
    Cardinal Wind Direction
    The Cardinal Wind Directions for these 3 locations on 26/09/2015 were, in random order, $$\begin{array} { l l l } W & N E & E \end{array}$$ You may assume that these 3 locations were under a single region of pressure.
  4. Using your knowledge of the large data set, place each of these Cardinal Wind Directions in the correct location in the table.
    Give a reason for your answer. \section*{Question 3 continued.}
Edexcel Paper 3 2021 October Q4
11 marks Standard +0.3
  1. A large college produces three magazines.
One magazine is about green issues, one is about equality and one is about sports.
A student at the college is selected at random and the events \(G , E\) and \(S\) are defined as follows \(G\) is the event that the student reads the magazine about green issues \(E\) is the event that the student reads the magazine about equality \(S\) is the event that the student reads the magazine about sports
The Venn diagram, where \(p , q , r\) and \(t\) are probabilities, gives the probability for each subset. \includegraphics[max width=\textwidth, alt={}, center]{10736735-3050-43eb-9e76-011ca6fa48b8-10_508_862_756_603}
  1. Find the proportion of students in the college who read exactly one of these magazines. No students read all three magazines and \(\mathrm { P } ( G ) = 0.25\)
  2. Find
    1. the value of \(p\)
    2. the value of \(q\) Given that \(\mathrm { P } ( S \mid E ) = \frac { 5 } { 12 }\)
  3. find
    1. the value of \(r\)
    2. the value of \(t\)
  4. Determine whether or not the events ( \(S \cap E ^ { \prime }\) ) and \(G\) are independent. Show your working clearly. \section*{Question 4 continued.} \section*{Question 4 continued.} \section*{Question 4 continued.}
Edexcel Paper 3 2021 October Q5
11 marks Standard +0.3
  1. The heights of females from a country are normally distributed with
  • a mean of 166.5 cm
  • a standard deviation of 6.1 cm
Given that \(1 \%\) of females from this country are shorter than \(k \mathrm {~cm}\),
  1. find the value of \(k\)
  2. Find the proportion of females from this country with heights between 150 cm and 175 cm A female, from this country, is chosen at random from those with heights between 150 cm and 175 cm
  3. Find the probability that her height is more than 160 cm The heights of females from a different country are normally distributed with a standard deviation of 7.4 cm Mia believes that the mean height of females from this country is less than 166.5 cm
    Mia takes a random sample of 50 females from this country and finds the mean of her sample is 164.6 cm
  4. Carry out a suitable test to assess Mia's belief. You should
    • state your hypotheses clearly
    • use a \(5 \%\) level of significance
    \section*{Question 5 continued.} \section*{Question 5 continued.} \section*{Question 5 continued.}
Edexcel Paper 3 2021 October Q6
7 marks Challenging +1.2
  1. The discrete random variable \(X\) has the following probability distribution
\(x\)\(a\)\(b\)\(c\)
\(\mathrm { P } ( X = x )\)\(\log _ { 36 } a\)\(\log _ { 36 } b\)\(\log _ { 36 } c\)
where
  • \(\quad a , b\) and \(c\) are distinct integers \(( a < b < c )\)
  • all the probabilities are greater than zero
    1. Find
      1. the value of a
      2. the value of \(b\)
      3. the value of \(c\)
Show your working clearly. The independent random variables \(X _ { 1 }\) and \(X _ { 2 }\) each have the same distribution as \(X\)
  • Find \(\mathrm { P } \left( X _ { 1 } = X _ { 2 } \right)\) \section*{Question 6 continued.} \section*{Question 6 continued.}
    •
    OCR MEI Paper 3 2018 June Q1
    3 marks Easy -1.2
    1 Triangle ABC is shown in Fig. 1. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{31bc8bde-8d37-4e97-94e2-e3e73aab55e9-4_451_565_520_744} \captionsetup{labelformat=empty} \caption{Fig. 1}
    \end{figure} Find the perimeter of triangle ABC .
    OCR MEI Paper 3 2018 June Q2
    2 marks Easy -1.2
    2 The curve \(y = x ^ { 3 } - 2 x\) is translated by the vector \(\binom { 1 } { - 4 }\). Write down the equation of the translated curve. [2]
    OCR MEI Paper 3 2018 June Q3
    2 marks Challenging +1.2
    3 Fig. 3 shows a circle with centre O and radius 1 unit. Points A and B lie on the circle with angle \(\mathrm { AOB } = \theta\) radians. C lies on AO , and BC is perpendicular to AO . \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{31bc8bde-8d37-4e97-94e2-e3e73aab55e9-4_648_627_1507_717} \captionsetup{labelformat=empty} \caption{Fig. 3}
    \end{figure} Show that, when \(\theta\) is small, \(\mathrm { AC } \approx \frac { 1 } { 2 } \theta ^ { 2 }\).
    OCR MEI Paper 3 2018 June Q4
    10 marks Standard +0.3
    4 In this question you must show detailed reasoning.
    A curve has equation \(y = x - 5 + \frac { 1 } { x - 2 }\). The curve is shown in Fig. 4. \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{31bc8bde-8d37-4e97-94e2-e3e73aab55e9-5_723_844_424_612} \captionsetup{labelformat=empty} \caption{Fig. 4}
    \end{figure}
    1. Determine the coordinates of the stationary points on the curve.
    2. Determine the nature of each stationary point.
    3. Write down the equation of the vertical asymptote.
    4. Deduce the set of values of \(x\) for which the curve is concave upwards.
    OCR MEI Paper 3 2018 June Q5
    11 marks Moderate -0.3
    5 A social media website launched on 1 January 2017. The owners of the website report the number of users the site has at the start of each month. They believe that the relationship between the number of users, \(n\), and the number of months after launch, \(t\), can be modelled by \(n = a \times 2 ^ { k t }\) where \(a\) and \(k\) are constants.
    1. Show that, according to the model, the graph of \(\log _ { 10 } n\) against \(t\) is a straight line.
    2. Fig. 5 shows a plot of the values of \(t\) and \(\log _ { 10 } n\) for the first seven months. The point at \(t = 1\) is for 1 February 2017, and so on. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{31bc8bde-8d37-4e97-94e2-e3e73aab55e9-6_831_1442_609_388} \captionsetup{labelformat=empty} \caption{Fig. 5}
      \end{figure} Find estimates of the values of \(a\) and \(k\).
    3. The owners of the website wanted to know the date on which they would report that the website had half a million users. Use the model to estimate this date.
    4. Give a reason why the model may not be appropriate for large values of \(t\).
    OCR MEI Paper 3 2018 June Q6
    2 marks Moderate -0.3
    6 Find the constant term in the expansion of \(\left( x ^ { 2 } + \frac { 1 } { x } \right) ^ { 15 }\).
    OCR MEI Paper 3 2018 June Q7
    8 marks Standard +0.3
    7 In this question you must show detailed reasoning.
    Fig. 7 shows the curve \(y = 5 x - x ^ { 2 }\). \begin{figure}[h]
    \includegraphics[alt={},max width=\textwidth]{31bc8bde-8d37-4e97-94e2-e3e73aab55e9-7_511_684_383_694} \captionsetup{labelformat=empty} \caption{Fig. 7}
    \end{figure} The line \(y = 4 - k x\) crosses the curve \(y = 5 x - x ^ { 2 }\) on the \(x\)-axis and at one other point.
    Determine the coordinates of this other point.
    OCR MEI Paper 3 2018 June Q8
    8 marks Standard +0.8
    8 A curve has parametric equations \(x = \frac { t } { 1 + t ^ { 3 } } , y = \frac { t ^ { 2 } } { 1 + t ^ { 3 } }\), where \(t \neq - 1\).
    1. In this question you must show detailed reasoning. Determine the gradient of the curve at the point where \(t = 1\).
    2. Verify that the cartesian equation of the curve is \(x ^ { 3 } + y ^ { 3 } = x y\).
    OCR MEI Paper 3 2018 June Q9
    4 marks Standard +0.3
    9 The function \(\mathrm { f } ( x ) = \frac { \mathrm { e } ^ { x } } { 1 - \mathrm { e } ^ { x } }\) is defined on the domain \(x \in \mathbb { R } , x \neq 0\).
    1. Find \(\mathrm { f } ^ { - 1 } ( x )\).
    2. Write down the range of \(\mathrm { f } ^ { - 1 } ( x )\).