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OCR MEI S1 2006 January Q4
5 marks Easy -1.2
4 A company sells sugar in bags which are labelled as containing 450 grams.
Although the mean weight of sugar in a bag is more than 450 grams, there is concern that too many bags are underweight. The company can adjust the mean or the standard deviation of the weight of sugar in a bag.
  1. State two adjustments the company could make. The weights, \(x\) grams, of a random sample of 25 bags are now recorded.
  2. Given that \(\sum x = 11409\) and \(\sum x ^ { 2 } = 5206937\), calculate the sample mean and sample standard deviation of these weights.
OCR MEI S1 2006 January Q5
5 marks Moderate -0.8
5 A school athletics team has 10 members. The table shows which competitions each of the members can take part in.
Competiton
100 m200 m110 m hurdles400 mLong jump
\multirow{10}{*}{Athlete}Abel✓✓✓
Bernoulli✓✓
Cauchy✓✓✓
Descartes✓✓
Einstein✓✓
Fermat✓✓
Galois✓✓
Hardy✓✓✓
Iwasawa✓✓
Jacobi✓
An athlete is selected at random. Events \(A , B , C , D\) are defined as follows.
A: the athlete can take part in exactly 2 competitions. \(B\) : the athlete can take part in the 200 m . \(C\) : the athlete can take part in the 110 m hurdles. \(D\) : the athlete can take part in the long jump.
  1. Write down the value of \(\mathrm { P } ( A \cap B )\).
  2. Write down the value of \(\mathrm { P } ( C \cup D )\).
  3. Which two of the four events \(A , B , C , D\) are mutually exclusive?
  4. Show that events \(B\) and \(D\) are not independent.
OCR MEI S1 2006 January Q6
4 marks Easy -1.2
6 A band has a repertoire of 12 songs suitable for a live performance. From these songs, a selection of 7 has to be made.
  1. Calculate the number of different selections that can be made.
  2. Once the 7 songs have been selected, they have to be arranged in playing order. In how many ways can this be done?
OCR MEI S1 2006 January Q7
18 marks Moderate -0.8
7 At East Cornwall College, the mean GCSE score of each student is calculated. This is done by allocating a number of points to each GCSE grade in the following way.
GradeA*ABCDEFGU
Points876543210
  1. Calculate the mean GCSE score, \(X\), of a student who has the following GCSE grades: $$\mathrm { A } ^ { * } , \mathrm {~A} ^ { * } , \mathrm {~A} , \mathrm {~A} , \mathrm {~A} , \mathrm {~B} , \mathrm {~B} , \mathrm {~B} , \mathrm {~B} , \mathrm { C } , \mathrm { D } .$$ 60 students study AS Mathematics at the college. The mean GCSE scores of these students are summarised in the table below.
    Mean GCSE scoreNumber of students
    \(4.5 \leqslant X < 5.5\)8
    \(5.5 \leqslant X < 6.0\)14
    \(6.0 \leqslant X < 6.5\)19
    \(6.5 \leqslant X < 7.0\)13
    \(7.0 \leqslant X \leqslant 8.0\)6
  2. Draw a histogram to illustrate this information.
  3. Calculate estimates of the sample mean and the sample standard deviation. The scoring system for AS grades is shown in the table below.
    AS GradeABCDEU
    Score60504030200
    The Mathematics department at the college predicts each student's AS score, \(Y\), using the formula \(Y = 13 X - 46\), where \(X\) is the student's average GCSE score.
  4. What AS grade would the department predict for a student with an average GCSE score of 7.4 ?
  5. What do you think the prediction should be for a student with an average GCSE score of 5.5? Give a reason for your answer.
  6. Using your answers to part (iii), estimate the sample mean and sample standard deviation of the predicted AS scores of the 60 students in the department.
OCR MEI S1 2006 January Q8
18 marks Standard +0.3
8 Jane buys 5 jam doughnuts, 4 cream doughnuts and 3 plain doughnuts.
On arrival home, each of her three children eats one of the twelve doughnuts. The different kinds of doughnut are indistinguishable by sight and so selection of doughnuts is random. Calculate the probabilities of the following events.
  1. All 3 doughnuts eaten contain jam.
  2. All 3 doughnuts are of the same kind.
  3. The 3 doughnuts are all of a different kind.
  4. The 3 doughnuts contain jam, given that they are all of the same kind. On 5 successive Saturdays, Jane buys the same combination of 12 doughnuts and her three children eat one each. Find the probability that all 3 doughnuts eaten contain jam on
  5. exactly 2 Saturdays out of the 5 ,
  6. at least 1 Saturday out of the 5 .
OCR MEI S1 2007 January Q1
7 marks Moderate -0.8
1 The total annual emissions of carbon dioxide, \(x\) tonnes per person, for 13 European countries are given below. $$\begin{array} { c c c c c c c c c c c c c } 6.2 & 6.7 & 6.8 & 8.1 & 8.1 & 8.5 & 8.6 & 9.0 & 9.9 & 10.1 & 11.0 & 11.8 & 22.8 \end{array}$$
  1. Find the mean, median and midrange of these data.
  2. Comment on how useful each of these is as a measure of central tendency for these data, giving a brief reason for each of your answers.
OCR MEI S1 2007 January Q2
7 marks Easy -1.8
2 The numbers of absentees per day from Mrs Smith's reception class over a period of 50 days are summarised below.
Number of absentees0123456\(> 6\)
Frequency8151183410
  1. Illustrate these data by means of a vertical line chart.
  2. Calculate the mean and root mean square deviation of these data.
  3. There are 30 children in Mrs Smith's class altogether. Find the mean and root mean square deviation of the number of children who are present during the 50 days.
OCR MEI S1 2007 January Q3
6 marks Easy -1.8
3 The times taken for 480 university students to travel from their accommodation to lectures are summarised below.
Time \(( t\) minutes \()\)\(0 \leqslant t < 5\)\(5 \leqslant t < 10\)\(10 \leqslant t < 20\)\(20 \leqslant t < 30\)\(30 \leqslant t < 40\)\(40 \leqslant t < 60\)
Frequency3415318873275
  1. Illustrate these data by means of a histogram.
  2. Identify the type of skewness of the distribution.
OCR MEI S1 2007 January Q4
8 marks Moderate -0.3
4 A fair six-sided die is rolled twice. The random variable \(X\) represents the higher of the two scores. The probability distribution of \(X\) is given by the formula $$\mathrm { P } ( X = r ) = k ( 2 r - 1 ) \text { for } r = 1,2,3,4,5,6 .$$
  1. Copy and complete the following probability table and hence find the exact value of \(k\), giving your answer as a fraction in its simplest form.
    \(r\)123456
    \(\mathrm { P } ( X = r )\)\(k\)\(11 k\)
  2. Find the mean of \(X\). A fair six-sided die is rolled three times.
  3. Find the probability that the total score is 16 .
OCR MEI S1 2007 January Q5
8 marks Moderate -0.8
5 Each day the probability that Ashwin wears a tie is 0.2 . The probability that he wears a jacket is 0.4 . If he wears a jacket, the probability that he wears a tie is 0.3 .
  1. Find the probability that, on a randomly selected day, Ashwin wears a jacket and a tie.
  2. Draw a Venn diagram, using one circle for the event 'wears a jacket' and one circle for the event 'wears a tie'. Your diagram should include the probability for each region.
  3. Using your Venn diagram, or otherwise, find the probability that, on a randomly selected day, Ashwin
    (A) wears either a jacket or a tie (or both),
    (B) wears no tie or no jacket (or wears neither).
OCR MEI S1 2007 January Q6
18 marks Moderate -0.3
6 The birth weights in grams of a random sample of 1000 babies are displayed in the cumulative frequency diagram below. \includegraphics[max width=\textwidth, alt={}, center]{05b96db3-93c7-4921-a1c6-c7b2f8952a8f-4_1264_1553_486_296}
  1. Use the diagram to estimate the median and interquartile range of the data.
  2. Use your answers to part (i) to estimate the number of outliers in the sample.
  3. Should these outliers be excluded from any further analysis? Briefly explain your answer.
  4. Any baby whose weight is below the 10th percentile is selected for careful monitoring. Use the diagram to determine the range of weights of the babies who are selected. \(12 \%\) of new-born babies require some form of special care. A maternity unit has 17 new-born babies. You may assume that these 17 babies form an independent random sample.
  5. Find the probability that
    (A) exactly 2 of these 17 babies require special care,
    (B) more than 2 of the 17 babies require special care.
  6. On 100 independent occasions the unit has 17 babies. Find the expected number of occasions on which there would be more than 2 babies who require special care.
OCR MEI S1 2007 January Q7
18 marks Standard +0.3
7 When onion seeds are sown outdoors, on average two-thirds of them germinate. A gardener sows seeds in pairs, in the hope that at least one will germinate.
  1. Assuming that germination of one of the seeds in a pair is independent of germination of the other seed, find the probability that, if a pair of seeds is selected at random,
    (A) both seeds germinate,
    (B) just one seed germinates,
    (C) neither seed germinates.
  2. Explain why the assumption of independence is necessary in order to calculate the above probabilities. Comment on whether the assumption is likely to be valid.
  3. A pair of seeds is sown. Find the expectation and variance of the number of seeds in the pair which germinate.
  4. The gardener plants 200 pairs of seeds. If both seeds in a pair germinate, the gardener destroys one of the two plants so that only one is left to grow. Of the plants that remain after this, only \(85 \%\) successfully grow to form an onion. Find the expected number of onions grown from the 200 pairs of seeds. If the seeds are sown in a greenhouse, the germination rate is higher. The seed manufacturing company claims that the germination rate is \(90 \%\). The gardener suspects that the rate will not be as high as this, and carries out a trial to investigate. 18 randomly selected seeds are sown in the greenhouse and it is found that 14 germinate.
  5. Write down suitable hypotheses and carry out a test at the \(5 \%\) level to determine whether there is any evidence to support the gardener's suspicions.
OCR FP3 2007 June Q1
3 marks Moderate -0.8
1
  1. By writing \(z\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), show that \(z z ^ { * } = | z | ^ { 2 }\).
  2. Given that \(z z ^ { * } = 9\), describe the locus of \(z\).
OCR FP3 2007 June Q2
5 marks Standard +0.3
2 A line \(l\) has equation \(\mathbf { r } = 3 \mathbf { i } + \mathbf { j } - 2 \mathbf { k } + t ( \mathbf { i } + 4 \mathbf { j } + 2 \mathbf { k } )\) and a plane \(\Pi\) has equation \(8 x - 7 y + 10 z = 7\). Determine whether \(l\) lies in \(\Pi\), is parallel to \(\Pi\) without intersecting it, or intersects \(\Pi\) at one point.
OCR FP3 2007 June Q3
6 marks Standard +0.8
3 Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } - 6 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 8 y = \mathrm { e } ^ { 3 x } .$$
OCR FP3 2007 June Q4
8 marks Standard +0.3
4 Elements of the set \(\{ p , q , r , s , t \}\) are combined according to the operation table shown below.
\(p\)\(q\)\(r\)\(s\)\(t\)
\(p\)\(t\)\(s\)\(p\)\(r\)\(q\)
\(q\)\(s\)\(p\)\(q\)\(t\)\(r\)
\(r\)\(p\)\(q\)\(r\)\(s\)\(t\)
\(s\)\(r\)\(t\)\(s\)\(q\)\(p\)
\(t\)\(q\)\(r\)\(t\)\(p\)\(s\)
  1. Verify that \(q ( s t ) = ( q s ) t\).
  2. Assuming that the associative property holds for all elements, prove that the set \(\{ p , q , r , s , t \}\), with the operation table shown, forms a group \(G\).
  3. A multiplicative group \(H\) is isomorphic to the group \(G\). The identity element of \(H\) is \(e\) and another element is \(d\). Write down the elements of \(H\) in terms of \(e\) and \(d\).
OCR FP3 2007 June Q5
8 marks Standard +0.8
5
  1. Use de Moivre's theorem to prove that $$\cos 6 \theta = 32 \cos ^ { 6 } \theta - 48 \cos ^ { 4 } \theta + 18 \cos ^ { 2 } \theta - 1 .$$
  2. Hence find the largest positive root of the equation $$64 x ^ { 6 } - 96 x ^ { 4 } + 36 x ^ { 2 } - 3 = 0 ,$$ giving your answer in trigonometrical form.
OCR FP3 2007 June Q6
10 marks Standard +0.8
6 Lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations $$\frac { x - 3 } { 2 } = \frac { y - 4 } { - 1 } = \frac { z + 1 } { 1 } \quad \text { and } \quad \frac { x - 5 } { 4 } = \frac { y - 1 } { 3 } = \frac { z - 1 } { 2 }$$ respectively.
  1. Find the equation of the plane \(\Pi _ { 1 }\) which contains \(l _ { 1 }\) and is parallel to \(l _ { 2 }\), giving your answer in the form r.n \(= p\).
  2. Find the equation of the plane \(\Pi _ { 2 }\) which contains \(l _ { 2 }\) and is parallel to \(l _ { 1 }\), giving your answer in the form r.n \(= p\).
  3. Find the distance between the planes \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\).
  4. State the relationship between the answer to part (iii) and the lines \(l _ { 1 }\) and \(l _ { 2 }\).
OCR FP3 2007 June Q8
10 marks Standard +0.3
8
  1. Find the general solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + y \tan x = \cos ^ { 3 } x$$ expressing \(y\) in terms of \(x\) in your answer.
  2. Find the particular solution for which \(y = 2\) when \(x = \pi\).
OCR FP3 2007 June Q9
12 marks Standard +0.3
9 The set \(S\) consists of the numbers \(3 ^ { n }\), where \(n \in \mathbb { Z }\). ( \(\mathbb { Z }\) denotes the set of integers \(\{ 0 , \pm 1 , \pm 2 , \ldots \}\).)
  1. Prove that the elements of \(S\), under multiplication, form a commutative group \(G\). (You may assume that addition of integers is associative and commutative.)
  2. Determine whether or not each of the following subsets of \(S\), under multiplication, forms a subgroup of \(G\), justifying your answers.
    1. The numbers \(3 ^ { 2 n }\), where \(n \in \mathbb { Z }\).
    2. The numbers \(3 ^ { n }\), where \(n \in \mathbb { Z }\) and \(n \geqslant 0\).
    3. The numbers \(3 ^ { \left( \pm n ^ { 2 } \right) }\), where \(n \in \mathbb { Z }\). 4
OCR MEI S1 2008 January Q1
7 marks Easy -1.8
1 Alice carries out a survey of the 28 students in her class to find how many text messages each sent on the previous day. Her results are shown in the stem and leaf diagram.
000113577788
1012334469
201337
357
4
58
Key: 2 | 3 represents 23
  1. Find the mode and median of the number of text messages.
  2. Identify the type of skewness of the distribution.
  3. Alice is considering whether to use the mean or the median as a measure of central tendency for these data.
    (A) In view of the skewness of the distribution, state whether Alice should choose the mean or the median.
    (B) What other feature of the distribution confirms Alice's choice?
  4. The mean number of text messages is 14.75 . If each message costs 10 pence, find the total cost of all of these messages.
OCR MEI S1 2008 January Q2
5 marks Easy -1.8
2 Codes of three letters are made up using only the letters A, C, T, G. Find how many different codes are possible
  1. if all three letters used must be different,
  2. if letters may be repeated.
OCR MEI S1 2008 January Q3
8 marks Moderate -0.8
3 Steve is going on holiday. The probability that he is delayed on his outward flight is 0.3 . The probability that he is delayed on his return flight is 0.2 , independently of whether or not he is delayed on the outward flight.
  1. Find the probability that Steve is delayed on his outward flight but not on his return flight.
  2. Find the probability that he is delayed on at least one of the two flights.
  3. Given that he is delayed on at least one flight, find the probability that he is delayed on both flights.
OCR MEI S1 2008 January Q4
8 marks Moderate -0.8
4 A company is searching for oil reserves. The company has purchased the rights to make test drillings at four sites. It investigates these sites one at a time but, if oil is found, it does not proceed to any further sites. At each site, there is probability 0.2 of finding oil, independently of all other sites. The random variable \(X\) represents the number of sites investigated. The probability distribution of \(X\) is shown below.
\(r\)1234
\(\mathrm { P } ( X = r )\)0.20.160.1280.512
  1. Find the expectation and variance of \(X\).
  2. It costs \(\pounds 45000\) to investigate each site. Find the expected total cost of the investigation.
  3. Draw a suitable diagram to illustrate the distribution of \(X\).
OCR MEI S1 2008 January Q5
8 marks Moderate -0.3
5 Sophie and James are having a tennis competition. The winner of the competition is the first to win 2 matches in a row. If the competition has not been decided after 5 matches, then the player who has won more matches is declared the winner of the competition. For example, the following sequences are two ways in which Sophie could win the competition. (S represents a match won by Sophie; \(\mathbf { J }\) represents a match won by James.) \section*{SJSS SJSJS}
  1. Explain why the sequence \(\mathbf { S S J }\) is not possible.
  2. Write down the other three possible sequences in which Sophie wins the competition.
  3. The probability that Sophie wins a match is 0.7 . Find the probability that she wins the competition in no more than 4 matches.