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OCR MEI S1 2008 January Q6
18 marks Easy -1.2
6 The maximum temperatures \(x\) degrees Celsius recorded during each month of 2005 in Cambridge are given in the table below.
JanFebMarAprMayJunJulAugSepOctNovDec
9.27.110.714.216.621.822.022.621.117.410.17.8
These data are summarised by \(n = 12 , \Sigma x = 180.6 , \Sigma x ^ { 2 } = 3107.56\).
  1. Calculate the mean and standard deviation of the data.
  2. Determine whether there are any outliers.
  3. The formula \(y = 1.8 x + 32\) is used to convert degrees Celsius to degrees Fahrenheit. Find the mean and standard deviation of the 2005 maximum temperatures in degrees Fahrenheit.
  4. In New York, the monthly maximum temperatures are recorded in degrees Fahrenheit. In 2005 the mean was 63.7 and the standard deviation was 16.0 . Briefly compare the maximum monthly temperatures in Cambridge and New York in 2005. The total numbers of hours of sunshine recorded in Cambridge during the month of January for each of the last 48 years are summarised below.
    Hours \(h\)\(70 \leqslant h < 100\)\(100 \leqslant h < 110\)\(110 \leqslant h < 120\)\(120 \leqslant h < 150\)\(150 \leqslant h < 170\)\(170 \leqslant h < 190\)
    Number of years681011103
  5. Draw a cumulative frequency graph for these data.
  6. Use your graph to estimate the 90th percentile.
OCR MEI S1 2008 January Q7
18 marks Standard +0.3
7 A particular product is made from human blood given by donors. The product is stored in bags. The production process is such that, on average, \(5 \%\) of bags are faulty. Each bag is carefully tested before use.
  1. 12 bags are selected at random.
    (A) Find the probability that exactly one bag is faulty.
    (B) Find the probability that at least two bags are faulty.
    (C) Find the expected number of faulty bags in the sample.
  2. A random sample of \(n\) bags is selected. The production manager wishes there to be a probability of one third or less of finding any faulty bags in the sample. Find the maximum possible value of \(n\), showing your working clearly.
  3. A scientist believes that a new production process will reduce the proportion of faulty bags. A random sample of 60 bags made using the new process is checked and one bag is found to be faulty. Write down suitable hypotheses and carry out a hypothesis test at the \(10 \%\) level to determine whether there is evidence to suggest that the scientist is correct.
OCR MEI S1 2005 June Q1
5 marks Moderate -0.8
1 At a certain stage of a football league season, the numbers of goals scored by a sample of 20 teams in the league were as follows. \(\begin{array} { l l l l l l l l l l l l l l l l l l l l l } 22 & 23 & 23 & 23 & 26 & 28 & 28 & 30 & 31 & 33 & 33 & 34 & 35 & 35 & 36 & 36 & 37 & 46 & 49 & 49 \end{array}\)
  1. Calculate the sample mean and sample variance, \(s ^ { 2 }\), of these data.
  2. The three teams with the most goals appear to be well ahead of the other teams. Determine whether or not any of these three pieces of data may be considered outliers.
OCR MEI S1 2005 June Q2
8 marks Easy -1.3
2 Answer part (i) of this question on the insert provided.
A taxi driver operates from a taxi rank at a main railway station in London. During one particular week he makes 120 journeys, the lengths of which are summarised in the table.
Length
\(( x\) miles \()\)
\(0 < x \leqslant 1\)\(1 < x \leqslant 2\)\(2 < x \leqslant 3\)\(3 < x \leqslant 4\)\(4 < x \leqslant 6\)\(6 < x \leqslant 10\)
Number of
journeys
3830211498
  1. On the insert, draw a cumulative frequency diagram to illustrate the data.
  2. Use your graph to estimate the median length of journey and the quartiles. Hence find the interquartile range.
  3. State the type of skewness of the distribution of the data.
OCR MEI S1 2005 June Q3
8 marks Easy -1.2
3 Jeremy is a computing consultant who sometimes works at home. The number, \(X\), of days that Jeremy works at home in any given week is modelled by the probability distribution $$\mathrm { P } ( X = r ) = \frac { 1 } { 40 } r ( r + 1 ) \quad \text { for } r = 1,2,3,4 .$$
  1. Verify that \(\mathrm { P } ( X = 4 ) = \frac { 1 } { 2 }\).
  2. Calculate \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
  3. Jeremy works for 45 weeks each year. Find the expected number of weeks during which he works at home for exactly 2 days.
OCR MEI S1 2005 June Q4
8 marks Moderate -0.8
4 An examination paper consists of three sections.
  • Section A contains 6 questions of which the candidate must answer 3
  • Section B contains 7 questions of which the candidate must answer 4
  • Section C contains 8 questions of which the candidate must answer 5
    1. In how many ways can a candidate choose 3 questions from Section A?
    2. In how many ways can a candidate choose 3 questions from Section A, 4 from Section B and 5 from Section C?
A candidate does not read the instructions and selects 12 questions at random.
  • Find the probability that they happen to be 3 from Section A, 4 from Section B and 5 from Section C.
    •
    OCR MEI S1 2005 June Q7
    16 marks Standard +0.3
    7 A game requires 15 identical ordinary dice to be thrown in each turn.
    Assuming the dice to be fair, find the following probabilities for any given turn.
    1. No sixes are thrown.
    2. Exactly four sixes are thrown.
    3. More than three sixes are thrown. David and Esme are two players who are not convinced that the dice are fair. David believes that the dice are biased against sixes, while Esme believes the dice to be biased in favour of sixes. In his next turn, David throws no sixes. In her next turn, Esme throws 5 sixes.
    4. Writing down your hypotheses carefully in each case, decide whether
      (A) David's turn provides sufficient evidence at the \(10 \%\) level that the dice are biased against sixes,
      (B) Esme's turn provides sufficient evidence at the \(10 \%\) level that the dice are biased in favour of sixes.
    5. Comment on your conclusions from part (iv).
    OCR MEI S1 2006 June Q1
    8 marks Easy -1.8
    1 Every day, George attempts the quiz in a national newspaper. The quiz always consists of 7 questions. In the first 25 days of January, the numbers of questions George answers correctly each day are summarised in the table below.
    Number correct1234567
    Frequency1233475
    1. Draw a vertical line chart to illustrate the data.
    2. State the type of skewness shown by your diagram.
    3. Calculate the mean and the mean squared deviation of the data.
    4. How many correct answers would George need to average over the next 6 days if he is to achieve an average of 5 correct answers for all 31 days of January?
    OCR MEI S1 2006 June Q2
    8 marks Moderate -0.8
    2 Isobel plays football for a local team. Sometimes her parents attend matches to watch her play.
    • \(A\) is the event that Isobel's parents watch a match.
    • \(B\) is the event that Isobel scores in a match.
    You are given that \(\mathrm { P } ( B \mid A ) = \frac { 3 } { 7 }\) and \(\mathrm { P } ( A ) = \frac { 7 } { 10 }\).
    1. Calculate \(\mathrm { P } ( A \cap B )\). The probability that Isobel does not score and her parents do not attend is 0.1 .
    2. Draw a Venn diagram showing the events \(A\) and \(B\), and mark in the probability corresponding to each of the regions of your diagram.
    3. Are events \(A\) and \(B\) independent? Give a reason for your answer.
    4. By comparing \(\mathrm { P } ( B \mid A )\) with \(\mathrm { P } ( B )\), explain why Isobel should ask her parents not to attend.
    OCR MEI S1 2006 June Q3
    7 marks Moderate -0.8
    3 The score, \(X\), obtained on a given throw of a biased, four-faced die is given by the probability distribution $$\mathrm { P } ( X = r ) = k r ( 8 - r ) \text { for } r = 1,2,3,4 .$$
    1. Show that \(k = \frac { 1 } { 50 }\).
    2. Calculate \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
    OCR MEI S1 2006 June Q4
    7 marks Moderate -0.8
    4 Peter and Esther visit a restaurant for a three-course meal. On the menu there are 4 starters, 5 main courses and 3 sweets. Peter and Esther each order a starter, a main course and a sweet.
    1. Calculate the number of ways in which Peter may choose his three-course meal.
    2. Suppose that Peter and Esther choose different dishes from each other.
      (A) Show that the number of possible combinations of starters is 6 .
      (B) Calculate the number of possible combinations of 6 dishes for both meals.
    3. Suppose instead that Peter and Esther choose their dishes independently.
      (A) Write down the probability that they choose the same main course.
      (B) Find the probability that they choose different dishes from each other for every course.
    OCR MEI S1 2006 June Q5
    6 marks Moderate -0.8
    5 Douglas plays darts, and the probability that he hits the number he is aiming at is 0.87 for any particular dart. Douglas aims a set of three darts at the number 20; the number of times he is successful can be modelled by \(\mathrm { B } ( 3,0.87 )\).
    1. Calculate the probability that Douglas hits 20 twice.
    2. Douglas aims fifty sets of 3 darts at the number 20. Find the expected number of sets for which Douglas hits 20 twice.
    3. Douglas aims four sets of 3 darts at the number 20. Calculate the probability that he hits 20 twice for two sets out of the four.
    OCR MEI S1 2006 June Q6
    18 marks Moderate -0.3
    6 It has been estimated that \(90 \%\) of paintings offered for sale at a particular auction house are genuine, and that the other \(10 \%\) are fakes. The auction house has a test to determine whether or not a given painting is genuine. If this test gives a positive result, it suggests that the painting is genuine. A negative result suggests that the painting is a fake. If a painting is genuine, the probability that the test result is positive is 0.95 .
    If a painting is a fake, the probability that the test result is positive is 0.2 .
    1. Copy and complete the probability tree diagram below, to illustrate the information above.
      [diagram]
      Calculate the probabilities of the following events.
    2. The test gives a positive result.
    3. The test gives a correct result.
    4. The painting is genuine, given a positive result.
    5. The painting is a fake, given a negative result. A second test is more accurate, but very expensive. The auction house has a policy of only using this second test on those paintings with a negative result on the original test.
    6. Using your answers to parts (iv) and (v), explain why the auction house has this policy. The probability that the second test gives a correct result is 0.96 whether the painting is genuine or a fake.
    7. Three paintings are independently offered for sale at the auction house. Calculate the probability that all three paintings are genuine, are judged to be fakes in the first test, but are judged to be genuine in the second test.
    OCR MEI S1 2006 June Q7
    18 marks Standard +0.3
    7 A geologist splits rocks to look for fossils. On average \(10 \%\) of the rocks selected from a particular area do in fact contain fossils. The geologist selects a random sample of 20 rocks from this area.
    1. Find the probability that
      (A) exactly one of the rocks contains fossils,
      (B) at least one of the rocks contains fossils.
    2. A random sample of \(n\) rocks is selected from this area. The geologist wants to have a probability of 0.8 or greater of finding fossils in at least one of the \(n\) rocks. Find the least possible value of \(n\).
    3. The geologist explores a new area in which it is claimed that less than \(10 \%\) of rocks contain fossils. In order to investigate the claim, a random sample of 30 rocks from this area is selected, and the number which contain fossils is recorded. A hypothesis test is carried out at the 5\% level.
      (A) Write down suitable hypotheses for the test.
      (B) Show that the critical region consists only of the value 0 .
      (C) In fact, 2 of the 30 rocks in the sample contain fossils. Complete the test, stating your conclusions clearly.
    OCR FP3 Specimen Q1
    5 marks Moderate -0.3
    1 Find the general solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } - \frac { y } { x } = x ,$$ giving \(y\) in terms of \(x\) in your answer.
    OCR FP3 Specimen Q2
    6 marks Standard +0.8
    2 The set \(S = \{ a , b , c , d \}\) under the binary operation * forms a group \(G\) of order 4 with the following operation table.
    \(*\)\(a\)\(b\)\(c\)\(d\)
    \(a\)\(d\)\(a\)\(b\)\(c\)
    \(b\)\(a\)\(b\)\(c\)\(d\)
    \(c\)\(b\)\(c\)\(d\)\(a\)
    \(d\)\(c\)\(d\)\(a\)\(b\)
    1. Find the order of each element of \(G\).
    2. Write down a proper subgroup of \(G\).
    3. Is the group \(G\) cyclic? Give a reason for your answer.
    4. State suitable values for each of \(a , b , c\) and \(d\) in the case where the operation \(*\) is multiplication of complex numbers.
    OCR FP3 Specimen Q3
    8 marks Standard +0.8
    3 The planes \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) have equations \(\mathbf { r } \cdot ( \mathbf { i } - 2 \mathbf { j } + 2 \mathbf { k } ) = 1\) and \(\mathbf { r } \cdot ( 2 \mathbf { i } + 2 \mathbf { j } - \mathbf { k } ) = 3\) respectively. Find
    1. the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\), correct to the nearest degree,
    2. the equation of the line of intersection of \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\), in the form \(\mathbf { r } = \mathbf { a } + t \mathbf { b }\).
    OCR FP3 Specimen Q4
    9 marks Standard +0.3
    4 In this question, give your answers exactly in polar form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(- \pi < \theta \leqslant \pi\).
    1. Express \(4 ( ( \sqrt { } 3 ) - \mathrm { i } )\) in polar form.
    2. Find the cube roots of \(4 ( ( \sqrt { } 3 ) - \mathrm { i } )\) in polar form.
    3. Sketch an Argand diagram showing the positions of the cube roots found in part (ii). Hence, or otherwise, prove that the sum of these cube roots is zero.
    OCR FP3 Specimen Q5
    9 marks Standard +0.8
    5 The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations $$\frac { x - 5 } { 1 } = \frac { y - 1 } { - 1 } = \frac { z - 5 } { - 2 } \quad \text { and } \quad \frac { x - 1 } { - 4 } = \frac { y - 11 } { - 14 } = \frac { z - 2 } { 2 } .$$
    1. Find the exact value of the shortest distance between \(l _ { 1 }\) and \(l _ { 2 }\).
    2. Find an equation for the plane containing \(l _ { 1 }\) and parallel to \(l _ { 2 }\) in the form \(a x + b y + c z = d\).
    OCR FP3 Specimen Q6
    10 marks Challenging +1.2
    6 The set \(S\) consists of all non-singular \(2 \times 2\) real matrices \(\mathbf { A }\) such that \(\mathbf { A Q } = \mathbf { Q A }\), where $$\mathbf { Q } = \left( \begin{array} { l l } 1 & 1 \\ 0 & 1 \end{array} \right)$$
    1. Prove that each matrix \(\mathbf { A }\) must be of the form \(\left( \begin{array} { l l } a & b \\ 0 & a \end{array} \right)\).
    2. State clearly the restriction on the value of \(a\) such that \(\left( \begin{array} { l l } a & b \\ 0 & a \end{array} \right)\) is in \(S\).
    3. Prove that \(S\) is a group under the operation of matrix multiplication. (You may assume that matrix multiplication is associative.)
    OCR FP3 Specimen Q7
    10 marks Challenging +1.2
    7
    1. Prove that if \(z = \mathrm { e } ^ { \mathrm { i } \theta }\), then \(z ^ { n } + \frac { 1 } { z ^ { n } } = 2 \cos n \theta\).
    2. Express \(\cos ^ { 6 } \theta\) in terms of cosines of multiples of \(\theta\), and hence find the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 3 } \pi } \cos ^ { 6 } \theta \mathrm {~d} \theta$$
    OCR FP3 Specimen Q8
    15 marks Challenging +1.2
    8
    1. Find the value of the constant \(k\) such that \(y = k x ^ { 2 } \mathrm { e } ^ { - 2 x }\) is a particular integral of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 4 y = 2 \mathrm { e } ^ { - 2 x }$$
    2. Find the solution of this differential equation for which \(y = 1\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\) when \(x = 0\).
    3. Use the differential equation to determine the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) when \(x = 0\). Hence prove that \(0 < y \leqslant 1\) for \(x \geqslant 0\).
    OCR MEI S1 2007 June Q1
    3 marks Easy -1.2
    1 A girl is choosing tracks from an album to play at her birthday party. The album has 8 tracks and she selects 4 of them.
    1. In how many ways can she select the 4 tracks?
    2. In how many different orders can she arrange the 4 tracks once she has chosen them?
    OCR MEI S1 2007 June Q2
    4 marks Moderate -0.8
    2 The histogram shows the amount of money, in pounds, spent by the customers at a supermarket on a particular day. \includegraphics[max width=\textwidth, alt={}, center]{5e4f3310-b96e-43db-9b6d-61da3270db06-2_977_1132_808_340}
    □ represents 20 customers
    1. Express the data in the form of a grouped frequency table.
    2. Use your table to estimate the total amount of money spent by customers on that day.
    OCR MEI S1 2007 June Q3
    8 marks Moderate -0.8
    3 The marks \(x\) scored by a sample of 56 students in an examination are summarised by $$n = 56 , \quad \Sigma x = 3026 , \quad \Sigma x ^ { 2 } = 178890 .$$
    1. Calculate the mean and standard deviation of the marks.
    2. The highest mark scored by any of the 56 students in the examination was 93 . Show that this result may be considered to be an outlier.
    3. The formula \(y = 1.2 x - 10\) is used to scale the marks. Find the mean and standard deviation of the scaled marks.