Questions — OCR MEI (4455 questions)

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OCR MEI FP3 2014 June Q2
24 marks Challenging +1.2
2 A surface \(S\) has equation \(\mathrm { g } ( x , y , z ) = 0\), where \(\mathrm { g } ( x , y , z ) = x ^ { 2 } + 3 y ^ { 2 } + 2 z ^ { 2 } + 2 y z + 6 x z - 4 x y - 24\). \(\mathrm { P } ( 2,6 , - 2 )\) is a point on the surface \(S\).
  1. Find \(\frac { \partial \mathrm { g } } { \partial x } , \frac { \partial \mathrm {~g} } { \partial y }\) and \(\frac { \partial \mathrm { g } } { \partial z }\).
  2. Find the equation of the normal line to the surface \(S\) at the point P .
  3. The point Q is on this normal line and close to P . At \(\mathrm { Q } , \mathrm { g } ( x , y , z ) = h\), where \(h\) is small. Find, in terms of \(h\), the approximate perpendicular distance from Q to the surface \(S\).
  4. Find the coordinates of the two points on the surface at which the normal line is parallel to the \(y\)-axis.
  5. Given that \(10 x - y + 2 z = 6\) is the equation of a tangent plane to the surface \(S\), find the coordinates of the point of contact.
OCR MEI FP3 2014 June Q3
24 marks Hard +2.3
3
  1. A curve has intrinsic equation \(s = 2 \ln \left( \frac { \pi } { \pi - 3 \psi } \right)\) for \(0 \leqslant \psi < \frac { 1 } { 3 } \pi\), where \(s\) is the arc length measured from a fixed point P and \(\tan \psi = \frac { \mathrm { d } y } { \mathrm {~d} x } . \mathrm { P }\) is in the third quadrant. The curve passes through the origin O , at which point \(\psi = \frac { 1 } { 6 } \pi . \mathrm { Q }\) is the point on the curve at which \(\psi = \frac { 3 } { 10 } \pi\).
    1. Express \(\psi\) in terms of \(s\), and sketch the curve, indicating the points \(\mathrm { O } , \mathrm { P }\) and Q .
    2. Find the arc length OQ .
    3. Find the radius of curvature at the point O .
    4. Find the coordinates of the centre of curvature corresponding to the point O .
    1. Find the surface area of revolution formed when the curve \(y = \frac { 1 } { 3 } \sqrt { x } ( x - 3 )\) for \(1 \leqslant x \leqslant 4\) is rotated through \(2 \pi\) radians about the \(y\)-axis.
    2. The curve in part (b)(i) is one member of the family \(y = \frac { 1 } { 9 } \lambda \sqrt { x } ( x - \lambda )\), where \(\lambda\) is a positive parameter. Find the equation of the envelope of this family of curves.
OCR MEI FP3 2014 June Q4
24 marks Challenging +1.8
4 The twelve distinct elements of an abelian multiplicative group \(G\) are $$e , a , a ^ { 2 } , a ^ { 3 } , a ^ { 4 } , a ^ { 5 } , b , a b , a ^ { 2 } b , a ^ { 3 } b , a ^ { 4 } b , a ^ { 5 } b$$ where \(e\) is the identity element, \(a ^ { 6 } = e\) and \(b ^ { 2 } = e\).
  1. Show that the element \(a ^ { 2 } b\) has order 6 .
  2. Show that \(\left\{ e , a ^ { 3 } , b , a ^ { 3 } b \right\}\) is a subgroup of \(G\).
  3. List all the cyclic subgroups of \(G\). You are given that the set $$H = \{ 1,7,11,13,17,19,23,29,31,37,41,43,47,49,53,59,61,67,71,73,77,79,83,89 \}$$ with binary operation multiplication modulo 90 is a group.
  4. Determine the order of each of the elements 11, 17 and 19 .
  5. Give a cyclic subgroup of \(H\) with order 4.
  6. By identifying possible values for the elements \(a\) and \(b\) above, or otherwise, give one example of each of the following:
    (A) a non-cyclic subgroup of \(H\) with order 12,
    (B) a non-cyclic subgroup of \(H\) with order 4.
OCR MEI FP3 2014 June Q5
24 marks Easy -2.5
5 In this question, give probabilities correct to 4 decimal places.
The speeds of vehicles are measured on a busy stretch of road and are categorised as A (not more than 30 mph ), B (more than 30 mph but not more than 40 mph ) or C (more than 40 mph ).
  • Following a vehicle in category A , the probabilities that the next vehicle is in categories \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) are \(0.9,0.07,0.03\) respectively.
  • Following a vehicle in category B , the probabilities that the next vehicle is in categories \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) are \(0.3,0.6,0.1\) respectively.
  • Following a vehicle in category C , the probabilities that the next vehicle is in categories \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) are \(0.1,0.7,0.2\) respectively.
This is modelled as a Markov chain with three states corresponding to the categories A, B, C. The speed of the first vehicle is measured as 28 mph .
  1. Write down the transition matrix \(\mathbf { P }\).
  2. Find the probabilities that the 10th vehicle is in each of the three categories.
  3. Find the probability that the 12th and 13th vehicles are in the same category.
  4. Find the smallest value of \(n\) for which the probability that the \(n\)th and \(( n + 1 )\) th vehicles are in the same category is less than 0.8, and give the value of this probability.
  5. Find the expected number of vehicles (including the first vehicle) in category A before a vehicle in a different category.
  6. Find the limit of \(\mathbf { P } ^ { n }\) as \(n\) tends to infinity, and hence write down the equilibrium probabilities for the three categories.
  7. Find the probability that, after many vehicles have passed by, the next three vehicles are all in category A. On a new stretch of road, the same categories are used but some of the transition probabilities are different.
    • Following a vehicle in category A , the probability that the next vehicle is in category B is equal to the probability that it is in category C .
    • Following a vehicle in category B , the probability that the next vehicle is in category A is equal to the probability that it is in category C .
    • Following a vehicle in category C , the probabilities that the next vehicle is in categories \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) are \(0.1,0.7,0.2\) respectively.
    In the long run, the proportions of vehicles in categories A, B, C are 50\%, 40\%, 10\% respectively.
  8. Find the transition matrix for the new stretch of road.
OCR MEI S1 Q1
17 marks Standard +0.3
1 A drug for treating a particular minor illness cures, on average, \(78 \%\) of patients. Twenty people with this minor illness are selected at random and treated with the drug.
  1. (A) Find the probability that exactly 19 patients are cured.
    (B) Find the probability that at most 18 patients are cured.
    (C) Find the expected number of patients who are cured.
  2. A pharmaceutical company is trialling a new drug to treat this illness. Researchers at the company hope that a higher percentage of patients will be cured when given this new drug. Twenty patients are selected at random, and given the new drug. Of these, 19 are cured. Carry out a hypothesis test at the \(1 \%\) significance level to investigate whether there is any evidence to suggest that the new drug is more effective than the old one.
  3. If the researchers had chosen to carry out the hypothesis test at the \(5 \%\) significance level, what would the result have been? Justify your answer.
OCR MEI S1 Q2
19 marks Standard +0.8
2 It is known that on average 85\% of seeds of a particular variety of tomato will germinate. Ramesh selects 15 of these seeds at random and sows them.
  1. (A) Find the probability that exactly 12 germinate.
    (B) Find the probability that fewer than 12 germinate The following year Ramesh finds that he still has many seeds left. Because the seeds are now one year old, he suspects that the germination rate will be lower. He conducts a trial by randomly selecting \(n\) of these seeds and sowing them. He then carries out a hypothesis test at the \(1 \%\) significance level to investigate whether he is correct.
  2. Write down suitable null and alternative hypotheses for the test. Give a reason for your choice of alternative hypothesis.
  3. In a trial with \(n = 20\), Ramesh finds that 13 seeds germinate. Carry out the test.
  4. Suppose instead that Ramesh conducts the trial with \(n = 50\), and finds that 33 seeds germinate. Given that the critical value for the test in this case is 35 , complete the test.
  5. If \(n\) is small, there is no point in carrying out the test at the \(1 \%\) significance level, as the null hypothesis cannot be rejected however many seeds germinate. Find the least value of \(n\) for which the null hypothesis can be rejected, quoting appropriate probabilities to justify your answer.
OCR MEI S1 Q3
7 marks Moderate -0.3
3 At a dog show, three out of eleven dogs are to be selected for a national competition.
  1. Find the number of possible selections.
  2. Five of the eleven dogs are terriers. Assuming that the dogs are selected at random, find the probability that at least two of the three dogs selected for the national competition are terriers.
OCR MEI S1 Q1
16 marks Standard +0.3
1 A coffee shop provides free internet access for its customers. It is known that the probability that a randomly selected customer is accessing the internet is 0.35 , independently of all other customers.
  1. 10 customers are selected at random.
    (A) Find the probability that exactly 5 of them are accessing the internet.
    (B) Find the probability that at least 5 of them are accessing the internet.
    (C) Find the expected number of these customers who are accessing the internet. Another coffee shop also provides free internet access. It is suspected that the probability that a randomly selected customer at this coffee shop is accessing the internet may be different from 0.35 . A random sample of 20 customers at this coffee shop is selected. Of these, 10 are accessing the internet.
  2. Carry out a hypothesis test at the \(5 \%\) significance level to investigate whether the probability for this coffee shop is different from 0.35 . Give a reason for your choice of alternative hypothesis.
  3. To get a more reliable result, a much larger random sample of 200 customers is selected over a period of time, and another hypothesis test is carried out. You are given that 90 of the 200 customers were accessing the internet. You are also given that, if \(X\) has the binomial distribution with parameters \(n = 200\) and \(p = 0.35\), then \(\mathrm { P } ( X \geqslant 90 ) = 0.0022\). Using the same hypotheses and significance level which you used in part (ii), complete this test.
OCR MEI S1 Q2
8 marks Moderate -0.3
2 A manufacturer produces titanium bicycle frames. The bicycle frames are tested before use and on average \(5 \%\) of them are found to be faulty. A cheaper manufacturing process is introduced and the manufacturer wishes to check whether the proportion of faulty bicycle frames has increased. A random sample of 18 bicycle frames is selected and it is found that 4 of them are faulty. Carry out a hypothesis test at the \(5 \%\) significance level to investigate whether the proportion of faulty bicycle frames has increased.
OCR MEI S1 Q3
16 marks Standard +0.3
3 It is known that \(25 \%\) of students in a particular city are smokers. A random sample of 20 of the students is selected.
  1. (A) Find the probability that there are exactly 4 smokers in the sample.
    (B) Find the probability that there are at least 3 but no more than 6 smokers in the sample
    (C) Write down the expected number of smokers in the sample. A new health education programme is introduced. This programme aims to reduce the percentage of students in this city who are smokers. After the programme has been running for a year, it is decided to carry out a hypothesis test to assess the effectiveness of the programme. A random sample of 20 students is selected.
  2. (A) Write down suitable null and alternative hypotheses for the test.
    (B) Explain why the alternative hypothesis has the form that it does
  3. Find the critical region for the test at the \(5 \%\) level, showing all of your calculations.
  4. In fact there are 3 smokers in the sample. Complete the test, stating your conclusion clearly.
OCR MEI S1 Q1
18 marks Standard +0.3
1 Any patient who fails to turn up for an outpatient appointment at a hospital is described as a 'no-show'. At a particular hospital, on average \(15 \%\) of patients are no-shows. A random sample of 20 patients who have outpatient appointments is selected.
  1. Find the probability that
    (A) there is exactly 1 no-show in the sample,
    (B) there are at least 2 no-shows in the sample. The hospital management introduces a policy of telephoning patients before appointments. It is hoped that this will reduce the proportion of no-shows. In order to check this, a random sample of \(n\) patients is selected. The number of no-shows in the sample is recorded and a hypothesis test is carried out at the 5\% level.
  2. Write down suitable null and alternative hypotheses for the test. Give a reason for your choice of alternative hypothesis.
  3. In the case that \(n = 20\) and the number of no-shows in the sample is 1 , carry out the test.
  4. In another case, where \(n\) is large, the number of no-shows in the sample is 6 and the critical value for the test is 8 . Complete the test.
  5. In the case that \(n \leqslant 18\), explain why there is no point in carrying out the test at the \(5 \%\) level.
OCR MEI S1 Q2
17 marks Moderate -0.3
2 Mark is playing solitaire on his computer. The probability that he wins a game is 0.2 , independently of all other games that he plays.
  1. Find the expected number of wins in 12 games.
  2. Find the probability that
    (A) he wins exactly 2 out of the next 12 games that he plays,
    (B) he wins at least 2 out of the next 12 games that he plays.
  3. Mark's friend Ali also plays solitaire. Ali claims that he is better at winning games than Mark. In a random sample of 20 games played by Ali, he wins 7 of them. Write down suitable hypotheses for a test at the \(5 \%\) level to investigate whether Ali is correct. Give a reason for your choice of alternative hypothesis. Carry out the test.
OCR MEI S1 Q3
18 marks Standard +0.3
3 A manufacturer produces tiles. On average 10\% of the tiles produced are faulty. Faulty tiles occur randomly and independently. A random sample of 18 tiles is selected.
  1. (A) Find the probability that there are exactly 2 faulty tiles in the sample.
    (B) Find the probability that there are more than 2 faulty tiles in the sample.
    (C) Find the expected number of faulty tiles in the sample. A cheaper way of producing the tiles is introduced. The manufacturer believes that this may increase the proportion of faulty tiles. In order to check this, a random sample of 18 tiles produced using the cheaper process is selected and a hypothesis test is carried out.
  2. (A) Write down suitable null and alternative hypotheses for the test.
    (B) Explain why the alternative hypothesis has the form that it does.
  3. Find the critical region for the test at the \(5 \%\) level, showing all of your calculations.
  4. In fact there are 4 faulty tiles in the sample. Complete the test, stating your conclusion clearly.
OCR MEI S1 Q1
5 marks Moderate -0.8
1 In a multiple-choice test there are 30 questions. For each question, there is a \(60 \%\) chance that a randomly selected student answers correctly, independently of all other questions.
  1. Find the probability that a randomly selected student gets a total of exactly 20 questions correct.
  2. If 100 randomly selected students take the test, find the expected number of students who get exactly 20 questions correct.
OCR MEI S1 Q2
19 marks Standard +0.3
2 An environmental health officer monitors the air pollution level in a city street. Each day the level of pollution is classified as low, medium or high. The probabilities of each level of pollution on a randomly chosen day are as given in the table.
Pollution levelLowMediumHigh
Probability0.50.350.15
  1. Three days are chosen at random. Find the probability that the pollution level is
    (A) low on all 3 days,
    (B) low on at least one day,
    (C) low on one day, medium on another day, and high on the other day.
  2. Ten days are chosen at random. Find the probability that
    (A) there are no days when the pollution level is high,
    (B) there is exactly one day when the pollution level is high. The environmental health officer believes that pollution levels will be low more frequently in a different street. On 20 randomly selected days she monitors the pollution level in this street and finds that it is low on 15 occasions.
  3. Carry out a test at the \(5 \%\) level to determine if there is evidence to suggest that she is correct. Use hypotheses \(\mathrm { H } _ { 0 } : p = 0.5 , \mathrm { H } _ { 1 } : p > 0.5\), where \(p\) represents the probability that the pollution level in this street is low. Explain why \(\mathrm { H } _ { 1 }\) has this form.
OCR MEI S1 Q3
18 marks Standard +0.3
3 The Department of Health 'eat five a day' advice recommends that people should eat at least five portions of fruit and vegetables per day. In a particular school, \(20 \%\) of pupils eat at least five a day.
  1. 15 children are selected at random.
    (A) Find the probability that exactly 3 of them eat at least five a day.
    (B) Find the probability that at least 3 of them eat at least five a day.
    (C) Find the expected number who eat at least five a day. A programme is introduced to encourage children to eat more portions of fruit and vegetables per day. At the end of this programme, the diets of a random sample of 15 children are analysed. A hypothesis test is carried out to examine whether the proportion of children in the school who eat at least five a day has increased.
  2. (A) Write down suitable null and alternative hypotheses for the test.
    (B) Give a reason for your choice of the alternative hypothesis.
  3. Find the critical region for the test at the \(10 \%\) significance level, showing all of your calculations. Hence complete the test, given that 7 of the 15 children eat at least five a day.
OCR MEI S1 Q1
19 marks Standard +0.3
1 An online shopping company takes orders through its website. On average \(80 \%\) of orders from the website are delivered within 24 hours. The quality controller selects 10 orders at random to check when they are delivered.
  1. Find the probability that
    (A) exactly 8 of these orders are delivered within 24 hours,
    (B) at least 8 of these orders are delivered within 24 hours. The company changes its delivery method. The quality controller suspects that the changes will mean that fewer than \(80 \%\) of orders will be delivered within 24 hours. A random sample of 18 orders is checked and it is found that 12 of them arrive within 24 hours.
  2. Write down suitable hypotheses and carry out a test at the \(5 \%\) significance level to determine whether there is any evidence to support the quality controller's suspicion.
  3. A statistician argues that it is possible that the new method could result in either better or worse delivery times. Therefore it would be better to carry out a 2 -tail test at the \(5 \%\) significance level. State the alternative hypothesis for this test. Assuming that the sample size is still 18, find the critical region for this test, showing all of your calculations.
OCR MEI S1 Q2
8 marks Moderate -0.3
2 In a game of darts, a player throws three darts. Let \(X\) represent the number of darts which hit the bull's-eye. The probability distribution of \(X\) is shown in the table.
\(r\)0123
\(\mathrm { P } ( X = r )\)0.50.35\(p\)\(q\)
  1. (A) Show that \(p + q = 0.15\).
    (B) Given that the expectation of \(X\) is 0.67 , show that \(2 p + 3 q = 0.32\).
    (C) Find the values of \(p\) and \(q\).
  2. Find the variance of \(X\).
OCR MEI S1 Q3
8 marks Moderate -0.3
3 A psychology student is investigating memory. In an experiment, volunteers are given 30 seconds to try to memorise a number of items. The items are then removed and the volunteers have to try to name all of them. It has been found that the probability that a volunteer names all of the items is 0.35 . The student believes that this probability may be increased if the volunteers listen to the same piece of music while memorising the items and while trying to name them. The student selects 15 volunteers at random to do the experiment while listening to music. Of these volunteers, 8 name all of the items.
  1. Write down suitable hypotheses for a test to determine whether there is any evidence to support the student's belief, giving a reason for your choice of alternative hypothesis.
  2. Carry out the test at the \(5 \%\) significance level.
OCR MEI S1 Q4
18 marks Standard +0.3
4 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 Q1
18 marks Standard +0.3
1 A multinational accountancy firm receives a large number of job applications from graduates each year. On average \(20 \%\) of applicants are successful. A researcher in the human resources department of the firm selects a random sample of 17 graduate applicants.
  1. Find the probability that at least 4 of the 17 applicants are successful.
  2. Find the expected number of successful applicants in the sample.
  3. Find the most likely number of successful applicants in the sample, justifying your answer. It is suggested that mathematics graduates are more likely to be successful than those from other fields. In order to test this suggestion, the researcher decides to select a new random sample of 17 mathematics graduate applicants. The researcher then carries out a hypothesis test at the \(5 \%\) significance level.
  4. (A) Write down suitable null and alternative hypotheses for the test.
    (B) Give a reason for your choice of the alternative hypothesis.
  5. Find the critical region for the test at the \(5 \%\) level, showing all of your calculations.
  6. Explain why the critical region found in part (v) would be unaltered if a \(10 \%\) significance level were used.
OCR MEI S1 Q2
18 marks Standard +0.3
2 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 MEI S1 Q3
6 marks Moderate -0.8
3 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 Q4
15 marks Standard +0.3
4 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 MEI S1 Q1
8 marks Standard +0.3
1 Over a long period of time, \(20 \%\) of all bowls made by a particular manufacturer are imperfect and cannot be sold.
  1. Find the probability that fewer than 4 bowls from a random sample of 10 made by the manufacturer are imperfect. The manufacturer introduces a new process for producing bowls. To test whether there has been an improvement, each of a random sample of 20 bowls made by the new process is examined. From this sample, 2 bowls are found to be imperfect.
  2. Show that this does not provide evidence, at the \(5 \%\) level of significance, of a reduction in the proportion of imperfect bowls. You should show your hypotheses and calculations clearly.