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Edexcel Paper 3 2021 October Q2
  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
  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
  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
  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
  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
    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
    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
    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
    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
    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
    6 Find the constant term in the expansion of \(\left( x ^ { 2 } + \frac { 1 } { x } \right) ^ { 15 }\).
    OCR MEI Paper 3 2018 June Q7
    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 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
    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 \(\mathrm { f } ^ { - 1 } ( x )\).
    OCR MEI Paper 3 2018 June Q10
    10 Point A has position vector \(\left( \begin{array} { l } a
    b
    0 \end{array} \right)\) where \(a\) and \(b\) can vary, point B has position vector \(\left( \begin{array} { l } 4
    2
    0 \end{array} \right)\) and point C has position vector \(\left( \begin{array} { l } 2
    4
    2 \end{array} \right)\). ABC is an isosceles triangle with \(\mathrm { AC } = \mathrm { AB }\).
    1. Show that \(a - b + 1 = 0\).
    2. Determine the position vector of A such that triangle ABC has minimum area. Answer all the questions.
      Section B (15 marks) The questions in this section refer to the article on the Insert. You should read the article before attempting the questions.
    OCR MEI Paper 3 2018 June Q11
    11 Line 8 states that \(\frac { a + b } { 2 } \geqslant \sqrt { a b }\) for \(a\), \(b \geqslant 0\). Explain why the result cannot be extended to apply in each of the following cases.
    1. One of the numbers \(a\) and \(b\) is positive and the other is negative.
    2. Both numbers \(a\) and \(b\) are negative.
    OCR MEI Paper 3 2018 June Q12
    12 Lines 5 and 6 outline the stages in a proof that \(\frac { a + b } { 2 } \geqslant \sqrt { a b }\). Starting from \(( a - b ) ^ { 2 } \geqslant 0\), give a detailed proof of the inequality of arithmetic and geometric means.
    OCR MEI Paper 3 2018 June Q13
    13 Consider a geometric sequence in which all the terms are positive real numbers. Show that, for any three consecutive terms of this sequence, the middle one is the geometric mean of the other two.
    OCR MEI Paper 3 2018 June Q14
    14
    1. In Fig. C1.3, angle CBD \(= \theta\). Show that angle CDA is also \(\theta\), as given in line 23 .
    2. Prove that \(h = \sqrt { a b }\), as given in line 24 .
    OCR MEI Paper 3 2018 June Q15
    15 It is given in lines \(31 - 32\) that the square has the smallest perimeter of all rectangles with the same area. Using this fact, prove by contradiction that among rectangles of a given perimeter, \(4 L\), the square with side \(L\) has the largest area. \section*{END OF QUESTION PAPER}
    OCR MEI Paper 3 2022 June Q1
    1 A curve for which \(y\) is inversely proportional to \(x\) is shown below.
    \includegraphics[max width=\textwidth, alt={}, center]{c30a926b-d832-46f5-aa65-0066ef482c3d-4_824_1125_561_242} Find the equation of the curve.
    OCR MEI Paper 3 2022 June Q2
    2 The function \(\mathrm { f } ( x ) = \sqrt { x }\) is defined on the domain \(x \geqslant 0\).
    The function \(\mathrm { g } ( x ) = 25 - x ^ { 2 }\) is defined on the domain \(\mathbb { R }\).
    1. Write down an expression for \(\mathrm { fg } ( x )\).
      1. Find the domain of \(\mathrm { fg } ( x )\).
      2. Find the range of \(\mathrm { fg } ( x )\).
    OCR MEI Paper 3 2022 June Q3
    3 An infinite sequence \(a _ { 1 } , a _ { 2 } , a _ { 3 } , \ldots\) is defined by \(a _ { \mathrm { n } } = \frac { \mathrm { n } } { \mathrm { n } + 1 }\), for all positive integers \(n\).
    1. Find the limit of the sequence.
    2. Prove that this is an increasing sequence.
    OCR MEI Paper 3 2022 June Q4
    4 In this question you must show detailed reasoning.
    Determine the exact solutions of the equation \(2 \cos ^ { 2 } x = 3 \sin x\) for \(0 \leqslant x \leqslant 2 \pi\).
    OCR MEI Paper 3 2022 June Q5
    5 A curve is defined implicitly by the equation \(2 x ^ { 2 } + 3 x y + y ^ { 2 } + 2 = 0\).
    1. Show that \(\frac { d y } { d x } = - \frac { 4 x + 3 y } { 3 x + 2 y }\).
    2. In this question you must show detailed reasoning. Find the coordinates of the stationary points of the curve.